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Volume 10,Number 1 January 2008 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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ON INVARIANT APPROXIMATION FORNONCOMMUTATIVE MAPPINGS IN LOCALLY CONVEX

SPACES

M. S. KhanDepartment of Mathematics and Statistics

College of ScienceSultan Qaboos University, P. O. Box 36, PCode 123, Al -Khod,

Muscat, Sultanate of Oman.e-mail: [email protected]

Hemant Kumar NashineDepartment of Mathematics, Raipur Institute of Technology

Chhatauna, Mandir Hasaud, Raipur-492101(Chhattisgarh), INDIA.e-mail: [email protected]

nashine [email protected]

ABSTRACTThe aim of this paper is to generalize the results related to invariant approx-

imation by weakening commutativity hypothesis and by increasing the numberof mappings involved.

2000 Mathematics Subject Classification. 41A50, 46A03, 47H10.

Key Words and Phrases: Best approximant, common fixed points, com-mutating mappings, compatible mapping, demiclosed mapping, locally convexspace.

1. INTRODUCTION

During the last four decades several interesting and valuable results as anapplication of fixed point theorems were studied extensively in the field of aninvariant approximation theory. An excellent reference can be seen in [16].

Meinardus [9] was the first who employed a fixed-point theorem to establishthe existence of an invariant approximation. Further, Brosowski [1] obtained acelebrated result and generalized the Meinardus’s result. Afterwards, a numberof results exist has been proved in the direction of Brosowski [1] (see in [3, 12,13, 17]). In a paper, Jungck and Sessa [5] further weakened the hypothesisof Sahab, Khan and Sessa [12] by replacing the weak and strong topology forrelatively nonexpansive commutative maps.

Recently, Nashine [10] obtained invariant approximation results for a classof contraction commutative three mappings in locally convex space and he ex-tended all the previous related results. Some works on best approximation forweakly commutative mappings in locally convex spaces are done by Khan andHussain [7, 11].

1

7JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.1,7-15,2008, COPYRIGHT 2008 EUDOXUS PRESS, LLC

2 M. S. KHAN AND HEMANT KUMAR NASHINE

In [4] the concept of compatible mappings was introduced as a generaliza-tion of commuting mappings. The purpose of this paper is to further emulatethe compatible mapping concept. We extend the result of Nashine [10] by em-ploying compatible mappings in lieu of commuting mappings, and by using fourmappings as opposed to three in the setup of in locally convex space. To achieveour goal, we use the concept given by Kothe [8], Tarafdar [18] and the result ofJungck [5]. In this way, we give new direction to the line of investigation givenby Brosowski [1]. Finally, we derive some consequences from our main result.

2. PRELIMINARIES

Before we prove our main result, let us recall following definitions:

Definition 2.1. [8, 7, 11]. In the sequel (E , τ) will be a Hausdorff locally convextopological vector space. A family pα : α ∈ I of seminorms defined on E issaid to be an associated family of seminorms for τ if the family γU : γ > 0,where U =

⋂ni=1 Uαi

and Uαi= x : pαi

(x) < 1, forms a base of neighbourhoodof zero for τ . A family pα : α ∈ I of seminorms defined on E is called anaugmented associated family for τ if pα : α ∈ I is an associated family withthe property that the seminorm maxpα, pβ ∈ pα : α ∈ I for any α, β ∈ I.The associated and augmented families of seminorms will be denoted by A(τ)and A∗(τ), respectively. It is well known that if given a locally convex space(E , τ), there always exists a family pα : α ∈ I of seminorms defined of E suchthat pα : α ∈ I = A∗(τ). A subset M of E is τ -bounded if and only if eachpα is bounded on M.

The following construction will be crucial. Suppose that M is a τ -boundedsubset of E . For this set M, we can select a number λα > 0 for each α ∈ Isuch that M ⊂ λαUα where Uα = x : pα(x) ≤ 1. Clearly, B =

⋂α λαUα

is τ - bounded, τ -closed, absolutely convex and contains M. The linear spanEB of B in E is

⋃∞n=1 nB. The Minkowski functional of B is a norm ‖.‖B on

EB. Thus, (EB, ‖.‖B) is a normed space with B as its closed unit ball andsupα pα(x/λα) = ‖x‖B for each x ∈ EB.

Definition 2.2. Let I and T be selfmaps on M. The map T is called(i) A∗(τ)-nonexpansive if for all x, y ∈M

pα(T x− T y) ≤ pα(x− y),

for each pα ∈ A∗(τ).(ii)A∗(τ)-I−nonexpansive if for all x, y ∈M

pα(T x− T y) ≤ pα(Ix− Iy),

for each pα ∈ A∗(τ).

For simplicity, we shall call A∗(τ)-nonexpansive (A∗(τ) − I- nonexpansive)maps to be nonexpansive (I−nonexpansive).

Following the concept of compatible due to Jungck [4], we have

8

ON INVARIANT APPROXIMATION FOR NONCOMMUTATIVE MAPPINGS..... 3

Definition 2.3. [4]. A pair of self-mappings (T , I) of a locally convex space(E , τ) is said to be compatible, if pα(T Ixn − IT xn) → 0, whenever xn is asequence in E such that T xn, Ixn → t ∈ E .

Every commuting pair of mappings is compatible but the converse is not truein general.

Definition 2.4. [10]. Let x0 ∈M. We denote by PM(x0) the set of best M−approximant to x0, i.e., if PM(x0) = y ∈M : pα(y − x0) = dpα

(x0,M) for allpα ∈ A∗(τ), where

dpα(x0,M) = infpα(x0 − z) : z ∈M.

Definition 2.5. [10]. The map T : M→ E is said to be demiclosed at 0 if forevery net xn in M converging weakly to x and T xn converging strongly to0, we have T x = 0.

Throughout this paper F(T )(resp.F(I)) denotes the set of fixed point ofmapping T (resp.I).

The following result of Jungck [5] is needed in the sequel:

Theorem 2.6. [5]. Let A,S, I and J be continuous self mappings of a compactmetric space (X , d) with A(X ) ⊆ J (X ) and S(X ) ⊆ I(X ). If (A, I) and (S,J )are compatible pairs and satisfying

d(Ax,Sy) < maxd(Ix,J y), d(Ix,Ax), d(J y,Sy),12[d(Ix,Sy)+d(J y,Ax)] > 0,

then A,S, I and J have a unique common fixed point in X .

3. MAIN RESULT

Lemma 3.1. Let A and I be compatible self-maps of a τ -bounded subset M ofa Hausdorff locally convex space (E , τ). Then A and I be two compatible on Mwith respect to ‖.‖B.

Proof. By hypothesis for each pα ∈ A∗(τ),

(3.1) pα(AIxn − IAxn) → 0,

whenever xn is a sequence in M such that Axn, Ixn → t ∈M.Taking supremum on both sides, we get

supα

pα(AIxn − IAxn

λα) → 0

‖AIxn − IAxn‖B → 0whenever xn is a sequence in M such that Axn, Ixn → t ∈M.

We use a technique of Tarafdar [18] to obtain the following common fixedpoint theorems which generalize Theorem 2.6.

9

Theorem 3.2. Let M be a nonempty τ -bounded, τ -sequentially compact subsetof a Hausdorff locally convex space (E , τ). Let A,S, I and J be self mappingsof M with A(M) ⊆ J (M) and S(M) ⊆ I(M). If (A, I) and (S,J ) arecompatible pairs, A and S are continuous, I and J are nonexpansive, andsatisfying

(3.2) pα(Ax− Sy) < N(x, y)

whereN(x, y) = maxpα(Ix− J y), pα(Ix−Ax), pα(J y − Sy),

12 [pα(Ix− Sy) + pα(J y −Ax)]

for all x, y ∈ M and pα ∈ A∗(τ), then A,S, I and J have a unique commonfixed point in M.

Proof. Since the norm topology on EB has a base of neighbourhood of zeroconsisting of τ -closed sets and M is τ -sequentially compact, therefore, M is a‖.‖B-sequentially compact subset of (EB, ‖.‖B) (Theorem 1.2, [18]). By Lemma3.1, A and I are ‖.‖B-compatible maps of M. Similarly, by the Lemma 3.1, Sand J are ‖.‖B-compatible maps of M. From (3.2) we obtain for x, y ∈M,

supα pα(Ax−Syλα

) < maxsupα pα(Ix−J yλα

), supα pα(Ix−Axλα

), supα pα(J y−Syλα

),

12 [supα pα(Ix−Sy

λα) + supα pα(J y−Ax

λα)].

Thus

(3.3)‖Ax− Sy‖B < max‖Ix− J y‖B, ‖Ix−Ax‖B, ‖J y − Sy‖B,

12 [ ‖Ix− Sy‖B + ‖J y −Ax‖B].

Note that, if I and J are nonexpansive on a τ -bounded, τ -sequentiallycompact subset M of E , then I and J are also nonexpansive with respectto ‖.‖B and hence ‖.‖B-continuous ([8]). A comparison of our hypothesis withthat of Theorem 2.6 tells that we can apply Theorem 2.6 to M as a sub-set of (EB, ‖.‖B) to conclude that there exists a unique v ∈ M such thatv = Av = Sv = Iv = J v.

Theorem 3.3. Let M be a nonempty τ -bounded, τ -sequentially complete andp−starshaped subset of a Hausdorff locally convex space (E , τ). Let A,S, I andJ be self mappings of M with A(M) ⊆ J (M) and S(M) ⊆ I(M). Suppose(A, I) and (S,J ) are compatible pairs, A and S are continuous, I and J arenonexpansive, and affine, I(M) = M = J (M), p ∈ F(I) ∩ F(J ). If A,S, Iand J satisfy the following:

(3.4) pα(Ax− Sy) < N(x, y)



10

for all x, y ∈ M and pα ∈ A∗(τ), then A,S, I and J have a common fixedpoint in M provided one of the following conditions hold:

(i) M is τ -sequentially compact;

(ii) M is weakly compact in (E , τ), I and J are weakly continuous andI − A and J − S are demiclosed at 0.

Proof. Choose a monotonically nondecreasing sequence kn of real numberssuch that 0 < kn < 1 and lim sup kn = 1. For each n ∈ N, define An,Sn : M→M as follows:

(3.5) Anx = knAx + (1− kn)p, Snx = knSx + (1− kn)p.

Obviously, for each n, An and Sn mapM into itself sinceM is p−starshaped.As I is affine, (A, I) is compatible and p ∈ F(I), so

AnIx = knAIx + (1− kn)p

IAnxn = I(knAx + (1− kn)p = knIAx + (1− kn)Ip.

Since (A, I) is compatible, therefore

0 ≤ limn pα(AmIxn − IAmxn)

≤ limn pα(AIxn − IAxn) + limn(1− km)pα(p− Ip)

= 0,

whenever limn Inxn = limnAxn = t ∈ M for all n, for each x ∈ M. HenceAn and I are compatible for each n and An(M) ⊆ M = J (M). Similarly,we can prove Sn and J are compatible for each n and Sn(M) ⊆M = I(M).

For all x, y ∈ M, pα ∈ A∗(τ) and for all j ≥ n, (n fixed), we obtain from(3.4) and (3.10) that

pα(Anx− Sny) = knpα(Ax− Sy) ≤ kjpα(Ax− Sy)

≤ pα(Ax− Sy)

< maxpα(Ix− J y), pα(Ix−Ax), pα(J y − Sy),


< maxpα(Ix− J y), pα(Ix−Anx) + pα(Anx−Ax),

pα(J y − Sny) + pα(Sny − Sy),

12 [pα(Ix− Sny) + pα(Sny − Sy)+

pα(J y −Anx) + pα(Anx−Ax)]

11

< maxpα(Ix− J y), pα(Ix−Anx)

+(1− kn)pα(p−Ax), pα(J y − Sny)

+(1− kn)pα(p− J y), 12 [pα(Ix− Sny)

+(1− kn)pα(p− Sy) + pα(J y −Anx)

+(1− kn)pα(p−Ax)].

Hence for all j ≥ n, we have

(3.6)pα(Anx− Sny) < maxpα(Ix− J y), pα(Ix−Anx)

+(1− kj)pα(p−Ax), pα(J y − Sny)

+(1− kj)pα(p− Sy), 12 [pα(Ix− Sny)

+(1− kj)pα(p− Sy) + pα(J y −Anx)

+(1− kj)pα(p−Ax)].

As lim kj = 1, from (3.6), for every n ∈ N, we have

(3.7)pα(Anx− Sny) = limj pα(Anx− Sny)

< limjmaxpα(Ix− J y), pα(Ix−Anx)

+(1− kj)pα(p−Ax), pα(J y − Sny)

+(1− kj)pα(p− Sy), 12 [pα(Ix− Sny)

+(1− kj)pα(p− Sy) + pα(J y −Anx)

+(1− kj)pα(p−Ax)].

This implies that for every n ∈ N,(3.8)

pα(Anx− Sny) < maxpα(Ix− J y), pα(Ix−Anx), pα(J y − Sny),

12 [pα(Ix− Sny) + pα(J y −Anx)],

for all x, y ∈M and for all pα ∈ A∗(τ).Moreover, I and J being nonexpansive on M, implies that I and J are

‖.‖B-nonexpansive and, hence, ‖.‖B-continuous. Since the norm topology onEB has a base of neighbourhood of zero consisting of τ -closed sets and M isτ -sequentially complete, therefore, M is a ‖.‖B-sequentially complete subset of

12

(EB, ‖.‖B) (see proof of Theorem 1.2 in [18]). Thus from Theorem 3.2 with thecondition (a) or (b), for every n ∈ N, An,Sn, I and J have unique commonfixed point xn in M, i.e.,

(3.9) xn = Anxn = Snxn = Ixn = J xn,

for each n ∈ N.

(i) As M is τ -sequentially compact and xn is a sequence in M, so xnhas a convergent subsequence xm such that xm → y ∈M. As I andS, T are continuous and

xm = Ixm = Amxm = kmAxm + (1− km)p,

xm = J xm = Smxm = kmSxm + (1− km)p,

so it follows that y = T y = Sy = Iy = J y.

(ii) The sequence xn has a subsequence xm converges to u ∈M. SinceI is weakly continuous and so as in (i), we have Iu = u. Now,

xm = Ixm = Amxm = kmAxm + (1− km)pimplies that

Ixm −Axm = (1− km)[p−Axm] → 0as m →∞. The demiclosedness of I − A at 0 implies that (I − A)u =

0. Hence Iu = u = Au. Similarly, we can show Su = u = J u,whenJ − S is demiclosed at 0. This completes the proof.

An immediate consequence of the Theorem 3.3 is as follows:

Corollary 3.4. Let M be a nonempty τ -bounded, τ -sequentially complete andp−starshaped subset of a Hausdorff locally convex space (E , τ). Let A,S, I andJ be self mappings of M with A(M) ⊆ J (M) and S(M) ⊆ I(M). Suppose(A, I) and (S,J ) are compatible pairs, A and S are continuous, I and J arenonexpansive, and affine, I(M) = M = J (M), p ∈ F(I) ∩ F(J ). If A,S, Iand J satisfy the following:

(3.10) pα(Ax− Sy) < N(x, y)


12pα(Ix− Sy), 1

2pα(J y −Ax)for all x, y ∈ M and pα ∈ A∗(τ), then A,S, I and J have a common fixedpoint in M under each of the conditions (i)− (ii) of Theorem 3.3.

An immediate consequence of the Theorem 3.3 and Corollary 3.4 is as follows:

13


Corollary 3.5. Let M be a nonempty τ -bounded, τ -sequentially complete andp−starshaped subset of a Hausdorff locally convex space (E , τ). Let A,S, I andJ be self mappings of M with A(M) ⊆ J (M) and S(M) ⊆ I(M). SupposeA, I,S and J are commutative, A and S are continuous, I and J are non-expansive, and affine, I(M) = M = J (M), p ∈ F(I) ∩ F(J ). If A,S, I andJ satisfy (3.4) or (3.10) for all x, y ∈ M and pα ∈ A∗(τ), then A,S, I andJ have a common fixed point in M under each of the conditions (i) − (ii) ofTheorem 3.3.

An application of Theorem 3.3, we prove the following more general resultin invariant approximation theory:

Theorem 3.6. Let A,S, I and J be selfmaps of a Hausdorff locally convexspace (E , τ) and M a subset of E such that A,S(∂M) ⊆M, where ∂M standsfor the boundary of M and x0 ∈ F(A) ∩ F(S) ∩ F(I) ∩ F(J ). Suppose thatA and S are continuous, (A, I) and (S,J ) are compatible pairs, I and J arenonexpansive and affine on D = PM(x0). Further, suppose A,S, I and Jsatisfy (3.4) for each x, y ∈ D, pα ∈ A∗(τ). If D is nonempty p−starshapedwith p ∈ F(I) ∩ F(J ) and I(D) = D = J (D), then A,S, I and J have acommon fixed point in D provided one of the following conditions hold:

(i) D is τ -sequentially compact;

(ii) D is weakly compact in (E , τ), I and J are weakly continuous andI − A and J − S are demiclosed at 0.

Proof. First, we show that A and S are self map on D, i.e., A,S : D → D.Let y ∈ D, then Iy,J y ∈ D, since I(D) = D = J (D). Also, if y ∈ ∂M, thenAy ∈ M, since A(∂M) ⊆ M. Now since Ax0 = Sx0 = x0 = Ix0 = J x0, sofor each pα ∈ A∗(τ), we have from (3.4)

pα(Ay − x0) = pα(Ay − Sx0) ≤ N(y, x0).

Now, Ay ∈ M and Iy ∈ D, this imply that Ay is also closest to x0, so Ay ∈D. Similarly Sy ∈ D. Consequently A,S, I and J are selfmaps on D. Theconditions of Theorem 3.3 ((i) − (ii)) are satisfied and, hence, there exists aw ∈ D such that Aw = Sw = w = Iw = Jw. This completes the proof.

An immediate consequence of the Theorem 3.6 is as follows:

Corollary 3.7. Let A,S, I and J be selfmaps of a Hausdorff locally convexspace (E , τ) and M a subset of E such that A,S(∂M) ⊆M, where ∂M standsfor the boundary of M and x0 ∈ F(A) ∩ F(S) ∩ F(I) ∩ F(I). Suppose thatA,S are continuous, (A, I) and (S,J ) are compatible pairs, I and J are non-expansive and affine on D = PM(x0). Further, suppose A,S, I and J satisfy(3.4) for each x, y ∈ D, pα ∈ A∗(τ). If D is nonempty p−starshaped withp ∈ F(I) ∩ F(J ) and I(D) = D = J (D), then A,S, I and J have a commonfixed point in D under each of the conditions (i)− (ii) of Theorem 3.6.

14


An immediate consequence of the Theorem 3.6 and Corollary 3.7 is as follows:

Corollary 3.8. Let A,S, I and J be selfmaps of a Hausdorff locally convexspace (E , τ) and M a subset of E such that A,S(∂M) ⊆M, where ∂M standsfor the boundary of M and x0 ∈ F(A) ∩ F(S) ∩ F(I) ∩ F(I). Suppose thatA,S are continuous, A, I,S and J are commutative, I and J are nonexpansiveand affine on D = PM(x0). Further, suppose A,S, I and J satisfy (3.4) or(3.10) for each x, y ∈ D, pα ∈ A∗(τ). If D is nonempty p−starshaped withp ∈ F(I) ∩ F(J ) and I(D) = D = J (D), then A,S, I and J have a commonfixed point in D under each of the conditions (i)− (ii) of Theorem 3.6.

Remark 3.9. With the remark given by Jungck [4] that every commuting pair ofmappings is compatible but the converse is not true in general, and by using fourmappings as opposed to three, our results generalize the results of Nashine [10]and consequently other related results (see in [1, 2, 3, 6, 9, 12, 13, 14, 15, 17]).

References

[1] B. Brosowski, Fixpunktsatze in der Approximationstheorie, Mathematica(Cluj) 11

(1969), 165 - 220.[2] A. Carbone. Some results on invariant approximation. Internat. J. Math. Math. Soc. (3)

17 (1994), 483 - 488.

[3] T. L. Hicks and M. D. Humpheries. A note on fixed point theorems. J. Approx. Theory34 (1982), 221-225.

[4] G. Jungck. Compatible mappings and common fixed points. Internat. J. Math. Math.

Sci. (4) 9(1986), 771-779.[5] G. Jungck. Common fixed points for commuting and compactible maps on compacta.

Proc. Amer. Math. Soc. 103(1988), 977-993.[6] G. Jungck and S. Sessa. Fixed point theorems in best approximation theory. Math. Japon-

ica. (2) 42(1995), 249-252.

[7] A. R. Khan and N. Hussain. An extension of a theorem of Sahab, Khan and Sessa.Internat. J. Math. Math. Soc. 27 (11) (2001), 701-706.

[8] G. Kothe. Topological vector spaces I. Die Grundlehren der mathematischen Wis-

senschaften, Vol. 159, Springer-Verlag, New York, 1969.[9] G. Meinardus. Invarianze bei Linearen Approximationen. Arch. Rational Mech. Anal.

14(1963), 301 - 303.

[10] Hemant Kumar Nashine. Existence of best approximation result in locally convex space.Kungpook Math. J. 46(2006), 389-397.

[11] N. Hussain and A. R. Khan. Common fixed point results in best approximation theory.

Applied Math. Letters. 16 (2003), 575-580.[12] S. A. Sahab, M. S. Khan and S. Sessa. A result in best approximation theory. J. Approx.

Theory 55 (1988), 349-351.[13] S. P. Singh. An application of a fixed point theorem to approximation theory. J. Approx.

Theory 25(1979), 89 - 90.

[14] S. P. Singh. Application of fixed point theorems to approximation theory in: V. Laksh-mikantam (Ed.), Applied nonlinear Analysis, Academic Press, New York, 1979.

[15] S. P. Singh. Some results on best approximation in locally convex spaces. J. Approx.Theory 28 (1980), 329 - 332.

[16] S. P. Singh, B. Watson and P. Srivastava. Fixed point theory and best approximation:The KKM-Map Principle Vol. 424, Kluwer Academic Publishers, 1997.

[17] P. V. Subrahmanyam. An application of a fixed point theorem to best approximations.J. Approx. Theory 20(1977), 165- 172.

[18] E. Tarafdar. Some fixed point theorems on locally convex linear topological spaces. Bull.Austral. Math. Soc. 13(1975), 241-254.

15

A new application of almost increasing sequences

Huseyin Bor

Department of Mathematics, Erciyes University, 38039 Kayseri, TURKEY

E-mail:[email protected], URL:http://fef.erciyes.edu.tr/math/hbor.htm

Abstract

In the present paper, a theorem on | N , pn |k summability factors of infinite series has

been proved under weaker conditions. Also we have obtained a new result concerning the

| C, 1 |k summability factors.

1 Introduction

Let∑

an be a given infinite series with partial sums (sn). We denote by uαn and tαn the

n-th Cesaro means of order α, with α > −1, of the sequence (sn) and (nan), respectively,

i.e.,

uαn =

1Aα

n

n∑

v=0

Aα−1n−vsv, (1)

tαn =1

Aαn

n∑

v=1

Aα−1n−vvav, (2)

where

Aαn = O(nα), α > −1, Aα

0 = 1 and Aα−n = 0 for n > 0. (3)

The series∑

an is said to be summable | C,α |k, k ≥ 1, if (see [5])

∞∑

n=1

nk−1 | uαn − uα

n−1 |k=∞∑

v=1

| tαn |kn

< ∞, (4)

2000 AMS Subject Classification: 40D15, 40F05, 40G99.

Key Words: Absolute summability, almost increasing sequence.

1

where tαn = n(uαn − uα

n−1) (see [7]).

Let (pn) be a sequence of positive numbers such that

Pn =n∑

v=0

pv →∞ as n →∞, (P−i = p−i = 0, i ≥ 1). (5)

The sequence-to-sequence transformation

σn =1Pn

n∑

v=0

pvsv (6)

defines the sequence (σn) of the Riesz mean or simply the (N , pn) mean of the sequence

(sn), generated by the sequence of coefficients (pn) (see [6]). The series∑

an is said to be

summable | N , pn |k, k ≥ 1, if (see [2], [3])

∞∑

n=1

(Pn/pn)k−1 | ∆σn−1 |k< ∞, (7)

where

∆σn−1 = − pn

PnPn−1

n∑

v=1

Pv−1av, n ≥ 1. (8)

In the special case pn = 1 for all values of n, | N , pn |k summability is the same as | C, 1 |ksummability.

2. Known Results. Mishra and Srivastava [9] have proved the following theorem for

| N , pn | summability.

Theorem A. Let (Xn) be a positive non-decreasing sequence and let there be sequences

(βn) and (λn) such that

| ∆λn |≤ βn, (9)

βn → 0 as n →∞, (10)∞∑

n=1

n | ∆βn | Xn < ∞, (11)

| λn | Xn = O(1). (12)

Ifn∑

v=1

| sv |v

= O(Xn) as n →∞ (13)

2

BOR18

and (pn) is a sequence such that

Pn = O(npn), (14)

Pn∆pn = O(pnpn+1), (15)

then the series∑∞

n=1 anPnλnnpn

is summable | N , pn |.Later on Bor [4] has proved Theorem A for | N , pn |k summability in the following form.

Theorem B. Let (Xn) be a positive non-decreasing sequence and the sequences (βn) and

(λn) are such that conditions (9)-(15) of Theorem A are satisfied with the condition (13)

replaced by:n∑

v=1

| sv |kv

= O(Xn) as n →∞. (16)

Then the series∑∞

n=1 anPnλnnpn

is summable | N , pn |k, k ≥ 1.

It may be noted that if we take k = 1 in Theorem B, then we get Theorem A.

3. Main result. The aim of this paper is to prove Theorem B under weaker conditions.

For this we need the concept of almost increasing sequence. A positive sequence (bn)

is said to be almost increasing if there exists a positive increasing sequence (cn) and two

positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). Obviously every increasing

sequence is almost increasing. However, the converse need not be true as can be seen by

taking the example, say bn = ne(−1)n.

Now, we shall prove the following theorem.

Theorem. Let (Xn) be an almost increasing sequence. If the conditions (9)-(12) and

(14)-(16) are satisfied, then the series∑∞

n=1 anPnλnnpn

is summable | N , pn |k, k ≥ 1.

Remark. It should be noted that, from the hypotheses of the Theorem, (λn) is bounded

and ∆λn = O(1/n) (see [4]).

We require the following lemma for the proof of the theorem.

Lemma ([8]). If (Xn) be an almost increasing sequence, then under the conditions

(10)-(11) we have that

nXnβn = O(1), (17)∞∑

n=1

βnXn < ∞. (18)

3

ALMOST INCREASING SEQUENCES 19

4. Proof of the Theorem. Let (Tn) be the sequence of (N , pn) mean of the series∑∞

n=1anPnλn

npn. Then, by definition, we have

Tn =1Pn

n∑

v=1

pv

v∑

r=1

arPrλr

rpr=

1Pn

n∑

v=1

(Pn − Pv−1)avPvλv

vpv. (19)

Then

Tn − Tn−1 =pn

PnPn−1

n∑

v=1

Pv−1Pvavλv

vpv, n ≥ 1. (20)

Using Abel’s transformation, we get

Tn − Tn−1 =pn

PnPn−1

n∑

v=1

sv∆(

Pv−1Pvλv

vpv

)+

λnsn

n

=snλn

n+

pn

PnPn−1

n−1∑

v=1

svPv+1Pv∆λv

(v + 1)pv+1

+pn

PnPn−1

n−1∑

v=1

Pvsvλv∆(

Pv

vpv

)− pn

PnPn−1

n−1∑

v=1

svPvλv1v

= Tn,1 + Tn,2 + Tn,3 + Tn,4, say.

To prove the theorem, by Minkowski’s inequality, it is sufficient to show that

∞∑

n=1

(Pn

pn

)k−1

| Tn,r |k< ∞, for r = 1, 2, 3, 4. (21)

Firstly by using Abel’s transformation, we have that

m∑

n=1

(Pn

pn

)k−1

| Tn,1 |k =m∑

n=1

(Pn

npn

)k−1

| λn |k−1| λn | | sn |kn

= O(1)m∑

n=1

| λn | | sn |kn

= O(1)m−1∑

n=1

∆ | λn |n∑

v=1

| sv |kv

+ O(1) | λm |m∑

n=1

| sn |kn

= O(1)m−1∑

n=1

| ∆λn | Xn + O(1) | λm | Xm

= O(1)m−1∑

n=1

βnXn + O(1) | λm | Xm = O(1) as m →∞,

4

BOR20

by virtue of the hypotheses of the Theorem and Lemma.

Now, using the fact that Pv+1 = O((v + 1)pv+1) by (14), we have

m+1∑

n=2

(Pn

pn

)k−1

| Tn,2 |k = O(1)m+1∑

n=2

pn

PnP kn−1

|n−1∑

v=1

Pvsv∆λv |k

= O(1)m+1∑

n=2

pn

PnP kn−1

n−1∑

v=1

Pv

pv| sv | pv | ∆λv |

k

Now applying Holder’s inequality, we have that

m+1∑

n=2

(Pn

pn

)k−1

| Tn,2 |k = O(1)m+1∑

n=2

pn

PnPn−1

n−1∑

v=1

(Pv

pv

)k

| sv |k pv | ∆λv |k

×(

1Pn−1

n−1∑

v=1

pv

)k−1

= O(1)m∑

v=1

(Pv

pv

)k

| sv |k pv | ∆λv |km+1∑

n=v+1

pn

PnPn−1

= O(1)m∑

v=1

(Pv | ∆λv |

pv

)k−1

| sv |k| ∆λv |

= O(1)m∑

v=1

| sv |k| ∆λv |(

Pv

vpv

)k−1

= O(1)m∑

v=1

vβv| sv |k

v

= O(1)m−1∑

v=1

∆(vβv)v∑

r=1

| sr |kr

+ O(1)mβm

m∑

v=1

| sv |kv

= O(1)m−1∑

v=1

| ∆(vβv) | Xv + O(1)mβmXm

= O(1)m−1∑

v=1

v | ∆βv | Xv + O(1)m−1∑

v=1

| βv | Xv + O(1)mβmXm = O(1)

as m →∞, in view of the hypotheses of the Theorem and Lemma.

Again, since ∆( Pvvpv

) = O( 1v ), by (14) and (15) (see [9]), as in Tn,1 we have that

m+1∑

n=2

(Pn

pn

)k−1

| Tn,3 |k = O(1)m+1∑

n=2

pn

PnP kn−1

n−1∑

v=1

Pv | sv || λv | 1v

k

= O(1)m+1∑

n=2

pn

PnP kn−1

n−1∑

v=1

(Pv

pv

)pv | sv || λv | 1

v

k

5


= O(1)m+1∑

n=2

pn

PnPn−1

n−1∑

v=1

(Pv

vpv

)k

pv | sv |k| λv |k

×

1Pn−1

n−1∑

v=1

pv

k−1

= O(1)m∑

v=1

(Pv

vpv

)k

| sv |k pv | λv |km+1∑

n=v+1

pn

PnPn−1

= O(1)m∑

v=1

(Pv

vpv

)k

pv | sv |k| λv |k 1Pv

.v

v

= O(1)m∑

v=1

(Pv

vpv

)k−1

| λv |k−1| λv | | sv |kv

= O(1)m∑

v=1

| λv | | sv |kv

= O(1)m−1∑

v=1

Xvβv + O(1)Xm | λm |= O(1) as m →∞.

Finally, using Holder’s inequality, as in Tn,3 we have

m+1∑

n=2

(Pn

pn

)k−1

| Tn,4 |k =m+1∑

n=2

pn

PnP kn−1

|n−1∑

v=1

svPv

vλv |k

=m+1∑

n=2

pn

PnP kn−1

|n−1∑

v=1

svPv

vpvpvλv |k

≤m+1∑

n=2

pn

PnPn−1

n−1∑

v=1

| sv |k(

Pv

vpv

)k

pv | λv |k

×(

1Pn−1

n−1∑

v=1

pv

)k−1

= O(1)m∑

v=1

(Pv

vpv

)k

| sv |k pv | λv |k 1Pv

.v

v

= O(1)m∑

v=1

| λv | | sv |kv

= O(1)m−1∑

v=1

Xvβv + O(1)Xm | λm |= O(1) as m →∞.

Therefore we getm∑

n=1

(Pn

pn

)k−1

| Tn,r |k= O(1) as m →∞, for r = 1, 2, 3, 4.

6

BOR22

This completes the proof of the Theorem.

Finally if we take pn = 1 for all values of n in this theorem, then we get a new result

concerning the | C, 1 |k summability factors.

References

[1] S.Aljancic and D.Arandelovic, O-regularly varying functions, Publ. Inst. Math. 22

(1977), 5-22.

[2] H.Bor, On two summability methods, Math. Proc. Camb. Philos Soc. 97 (1985),

147-149.

[3] H.Bor, A note on two summability methods, Proc. Amer. Math. Soc. 98 (1986), 81-84.

[4] H.Bor, A note on | N , pn |k summability factors of infinite series, Indian J. Pure Appl.

Math. 18 (1987), 330-336.

[5] T.M. Flett, On an extension of absolute summability and some theorems of Littlewood

and Paley, Proc. London Math. Soc. 7 (1957), 113-141.

[6] G.H.Hardy, Divergent Series, Oxford Univ. Press. Oxford (1949).

[7] E. Kogbetliantz, Sur les series absolument sommables par la methode des moyennes

arithmetiques, Bull. Sci. Math. 49 (1925), 234-256.

[8] S. M. Mazhar, A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica

25 (1997), 233-242.

[9] K. N. Mishra and R. S. L. Srivastava, On | N , pn | summability factors of infinite

series, Indian J. Pure Appl. Math. 15 (1984), 651-656.

7


On ∗-Homomorphisms between JC∗-Algebras

Choonkil Park, Won-Gil Park and Hee-Jeong Wee

Choonkil Park: Department of Mathematics, Hanyang University, Seoul 133–791, Republicof Korea, [email protected] Park: National Institute for Mathematical Sciences, Daejeon 305–340, Republic ofKorea, [email protected]

Hee-Jeong Wee: Department of Mathematics, Chungnam National University, Daejeon 305–

764, Republic of Korea, [email protected]

Abstract. It is shown that every almost unital almost linear mapping f : A → B ofJC∗-algebra A to a JC∗-algebra B is a homomorphism when f(2nu y) = f(2nu) f(y)holds for all unitaries u ∈ A, all y ∈ A, and all n = 0, 1, 2, · · ·, and that every almost unitalalmost linear continuous mapping f : A → B of a JC∗-algebra A of real rank zero to aJC∗-algebra B is a homomorphism when f(2nuy) = f(2nu)f(y) holds for all u ∈ v ∈ A |v = v∗, ‖v‖ = 1, v is invertible, all y ∈ A, and all n = 0, 1, 2, · · ·.

Furthermore, we are going to prove the generalized Hyers–Ulam–Rassias stability of∗-homomorphisms between JC∗-algebras, and C-linear ∗-derivations on JC∗-algebras.

2000 Mathematics Subject Classification. Primary 47E10, 39B52, 17Cxx, 46L05.

Key words and Phrases. Hyers–Ulam–Rassias stability, homomorphism in JC∗-algebra, real

rank 0, linear derivation

1. Introduction

Our knowledge concerning the continuity properties of epimorphisms on Banach al-

gebras, Jordan–Banach algebras, and, more generally, nonassociative complete normed

algebras, is now fairly complete and satisfactory (see [9] and [10]). A basic continuity

problem consists in determining algebraic conditions on a Banach algebra A which ensure

———————–

This work was supported by Korea Research Foundation Grant KRF-2005-041-C00027.

1


that derivations on A are continuous. In 1996, Villena [10] proved that derivations on

semisimple Jordan–Banach algebras are continuous.

Let E1 and E2 be Banach spaces with norms || · || and ‖ · ‖, respectively. Consider

f : E1 → E2 to be a mapping such that f(tx) is continuous in t ∈ R for each fixed x ∈ E1.

Assume that there exist constants θ ≥ 0 and p ∈ [0, 1) such that

‖f(x + y)− f(x)− f(y)‖ ≤ θ(||x||p + ||y||p)

for all x, y ∈ E1. Rassias [8] showed that there exists a unique R-linear mapping T : E1 →E2 such that

‖f(x)− T (x)‖ ≤ 2θ

2− 2p||x||p

for all x ∈ E1. Gavruta [2] generalized the Rassias’ result.

Jun, Kim and Shin [4] proved the following: Let X and Y be Banach spaces. Denote

by ϕ : X ×X → [0,∞) a function such that

ε(x) :=∞∑

j=1

2−j(ϕ(2j−1x, 0) + ϕ(0, 2j−1x) + ϕ(2j−1x, 2j−1x)) < ∞

for all x ∈ X. Suppose that f, g, h : X → Y are mappings satisfying

‖2f(x + y

2)− g(x)− h(y)‖ ≤ ϕ(x, y)

for all x, y ∈ X. Then there exists a unique additive mapping T : X → Y such that

‖2f(x

2)− T (x)‖ ≤ ‖g(0)‖+ ‖h(0)‖+ ε(x),

‖g(x)− T (x)‖ ≤ ‖g(0)‖+ 2‖h(0)‖+ ϕ(x, 0) + ε(x),

‖h(x)− T (x)‖ ≤ 2‖g(0)‖+ ‖h(0)‖+ ϕ(0, x) + ε(x)

for all x ∈ X.

B.E. Johnson [3], Theorem 7.2 also investigated almost algebra ∗-homomorphisms

between Banach ∗-algebras : Suppose that U and B are Banach ∗-algebras which satisfy

the conditions of [3], Theorem 3.1. Then for each positive ε and K there is a positive δ

such that if T ∈ L(U ,B) with ‖T‖ < K, ‖T∨‖ < δ and ‖T (x∗)∗−T (x)‖ ≤ δ‖x‖ (x ∈ U)

then there is a ∗-homomorphism T ′ : U → B with ‖T −T ′‖ < ε. Here L(U ,B) is the space

of bounded linear maps from U into B, and T∨(x, y) = T (xy) − T (x)T (y) (x, y ∈ U).

See [3] for details.

PARK ET AL26

The original motivation to introduce the class of nonassociative algebras known as

Jordan algebras came from quantum mechanics (see [9]). Let H be a complex Hilbert

space, regarded as the “state space” of a quantum mechanical system. Let L(H) be the

real vector space of all bounded self-adjoint linear operators on H, interpreted as the

(bounded) observables of the system. In 1932, Jordan observed that L(H) is a (nonasso-

ciative) algebra via the anticommutator product x y := xy+yx2

. A commutative algebra

X with product x y is called a Jordan algebra if x2 (x y) = x (x2 y) holds.

A complex Jordan algebra C with product x y and involution x 7→ x∗ is called a

JB∗-algebra if C carries a Banach space norm ‖ · ‖ satisfying ‖x y‖ ≤ ‖x‖ · ‖y‖ and

‖xx∗x|| = ‖x‖3. Here xy∗z := x (y∗ z)−y∗ (z x)+z (xy∗) denotes the Jordan

triple product of x, y, z ∈ C. A unital Jordan C∗-subalgebra of a C∗-algebra, endowed

with the anticommutator product, is called a JC∗-algebra.

Throughout this paper, let A be a JC∗-algebra with norm || · || and unit e, and Ba JC∗-algebra with norm ‖ · ‖ and unit e′. Let U(A) = u ∈ A | u∗u = uu∗ = e,Asa = x ∈ A | x = x∗, and I1(Asa) = v ∈ Asa | ‖v‖ = 1, v is invertible.

In this paper, we prove that every almost unital almost linear mapping h : A → Bis a homomorphism when h(3nu y) = h(3nu) h(y) holds for all u ∈ U(A), all y ∈ A,

and all n = 0, 1, 2, · · ·, and that for a JC∗-algebra A of real rank zero (see [1]), every

almost unital almost linear continuous mapping h : A → B is a homomorphism when

h(3nu y) = h(3nu) h(y) holds for all u ∈ I1(Asa), all y ∈ A, and all n = 0, 1, 2, · · ·.Furthermore, we are going to prove the generalized Hyers–Ulam–Rassias stability of

∗-homomorphisms between JC∗-algebras, and C-linear ∗-derivations on JC∗-algebras.

2. ∗-homomorphisms between JC∗-algebras

We are going to investigate ∗-homomorphisms between JC∗-algebras.

Theorem 2.1. Let f, g, h : A → B be mappings satisfying f(0) = 0, g(0) = 0 and

h(0) = 0, and let f(2nu y) = f(2nu) f(y), g(2nu y) = g(2nu) g(y) and h(2nu y) =

h(2nu) h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, · · ·, for which there exists a

function ϕ : A \ 0 × A \ 0 → [0,∞) such that

ϕ(x, y) :=∞∑

j=0

2−jϕ(2j−1x, 2j−1y) < ∞, (1)

JC*-ALGEBRAS 27

‖2f(µx + µy

2)− µg(x)− µh(y)‖ ≤ ϕ(x, y), (2)

‖f(2nu∗)− f(2nu)∗‖ ≤ ϕ(2nu, 2nu) (3)

for all µ ∈ T1 := λ ∈ C | |λ| = 1, all u ∈ U(A), all x, y ∈ A, and all n = 0, 1, 2, · · ·.Assume that

limn→∞

f(2ne)

2n= e′. (4)

Then the mappings f, g, h : A → B are ∗-homomorphisms.

Proof. Put µ = 1 ∈ T1. It follows from Corollary 2.5 of [4] that there exists a unique

additive mapping H : A → B such that

‖2f(x

2)−H(x)‖ ≤ ε(x),

‖g(x)−H(x)‖ ≤ ϕ(x, 0) + ε(x),

‖h(x)−H(x)‖ ≤ ϕ(0, x) + ε(x) (5)

for all x ∈ A \ 0, where

ε(x) :=∞∑

j=1

2−j(ϕ(2j−1x, 0) + ϕ(0, 2j−1x) + ϕ(2j−1x, 2j−1x)) < ∞

for all x ∈ A \ 0. The additive mapping H : A → B is given by

H(x) = limn→∞

1

2nf(2nx)

for all x ∈ A, and

limn→∞ 2−nf(2nx) = lim

n→∞ 2−ng(2nx) = limn→∞ 2−nh(2nx)

for all x ∈ A. Let f(x) = 2f(x2) for all x ∈ A, then

limn→∞

1

2nf(2nx) = lim

n→∞1

2nf(2nx)

for all x ∈ A.

By the assumption,

‖f(2nµx)− µf(2nx)‖= ‖f(2nµx)− 1

2µg(2nx)− 1

2µh(2nx) + 1

2µg(2nx) + 1

2µh(2nx)− µf(2nx)‖

≤ 12ϕ(2nx, 2nx) + 1

2|µ|ϕ(2nx, 2nx) = ϕ(2nx, 2nx)

PARK ET AL28

for all µ ∈ T1 and all x ∈ A \ 0. Thus 2−n‖f(2nµx)− µf(2nx)‖ → 0 as n →∞ for all

µ ∈ T1 and all x ∈ A \ 0. Hence

H(µx) = limn→∞

f(2nµx)

2n= lim

n→∞µf(2nx)

2n= µH(x) (6)

for all µ ∈ T1 and all x ∈ A \ 0.Now let λ ∈ C (λ 6= 0) and M an integer greater than 2|λ|. Then | λ

M| < 1

2=

1 − 24. By Theorem 1 of [5], there exist four elements µ1, µ2, µ3, µ4 ∈ T1 such that

4 λM

= µ1+µ2+µ3+µ4. And H(x) = H(2· 12x) = 2H(1

2x) for all x ∈ A. So H(1

2x) = 1

2H(x)

for all x ∈ A. Thus by (6)

H(λx) = H(M

4· 4 λ

Mx) = M ·H(

1

4· 4 λ

Mx)

=M

4H(4

λ

Mx) =

M

4H(µ1x + µ2x + µ3x + µ4x)

=M

4(H(µ1x) + H(µ2x) + H(µ3x) + H(µ4x))

=M

4(µ1 + µ2 + µ3 + µ4)H(x) =

M

4· 4 λ

MH(x)

= λH(x)

for all x ∈ A. Hence

H(ζx + ηy) = H(ζx) + H(ηy) = ζH(x) + ηH(y)

for all ζ, η ∈ C \ 0 and all x, y ∈ A. And H(0x) = 0 = 0H(x) for all x ∈ A. So the

unique additive mapping H : A → B is a C-linear mapping.

By (1) and (3), we get

H(u∗) = limn→∞

f(2nu∗)2n

= limn→∞

f(2nu)∗

2n= ( lim

n→∞f(2nu)

2n)∗ = H(u)∗

for all u ∈ U(A). Since H is C-linear and each x ∈ A is a finite linear combination of

unitary elements (cf. [6]), say, x =∑m

j=1 λjuj (λj ∈ C, uj ∈ U(A)),

H(x∗) = H(m∑

j=1

λju∗j) =

m∑

j=1

λjH(u∗j) =m∑

j=1

λjH(uj)∗

= (m∑

j=1

λjH(uj))∗ = H(

m∑

j=1

λjuj)∗

= H(x)∗

JC*-ALGEBRAS 29

for all x ∈ A.

Since f(2nu y) = f(2nu) f(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, · · ·,

H(u y) = limn→∞

1

2nf(2nu y) = lim

n→∞1

2nf(2nu) f(y) = H(u) f(y) (7)

for all u ∈ U(A) and all y ∈ A. By the additivity of H and (7),

2nH(u y) = H(2nu y) = H(u (2ny)) = H(u) f(2ny)

for all u ∈ U(A) and all y ∈ A. Hence

H(u y) =1

2nH(u) f(2ny) = H(u) 1

2nf(2ny) (8)

for all u ∈ U(A) and all y ∈ A. Taking the limit in (8) as n →∞, we obtain

H(u y) = H(u) H(y) (9)

for all u ∈ U(A) and all y ∈ A. Since H is C-linear and each x ∈ A is a finite linear

combination of unitary elements, i.e., x =∑m

j=1 λjuj (λj ∈ C, uj ∈ U(A)), it follows

from (9) that

H(x y) = H(m∑

j=1

λjuj y) =m∑

j=1

λjH(uj y)

=m∑

j=1

λjH(uj) H(y) = H(m∑

j=1

λjuj) H(y)

= H(x) H(y)

for all x, y ∈ A. By (7) and (9),

H(e) H(y) = H(e y) = H(e) f(y)

for all y ∈ A. Since limn→∞f(2ne)

2n = H(e) = e′,

H(y) = f(y)

for all y ∈ A. Similarly, H(y) = g(y) = h(y) for all y ∈ A. Therefore, the mapping

f, g, h : A → B are ∗-homomorphisms, as desired. 2

Corollary 2.2. Let f, g, h : A → B be mappings satisfying f(0) = 0, g(0) = 0 and

h(0) = 0 and let f(2nu y) = f(2nu) f(y), g(2nu y) = g(2nu) g(y) and h(2nu y) =

PARK ET AL30

h(2nu) h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, · · ·, for which there exist

constants θ ≥ 0 and p ∈ [0, 1) such that

‖2f(µx + µy

2)− µg(x)− µh(y)‖ ≤ θ(||x||p + ||y||p),‖f(2nu∗)− f(2nu)∗‖ ≤ 2np+1θ

for all µ ∈ T1, all u ∈ U(A), n = 0, 1, · · ·, and all x, y ∈ A \ 0. Assume that

limn→∞f(2ne)

2n = e′. Then the mappings f, g and h are ∗-homomorphisms.

Proof. Define ϕ(x, y) = θ(||x||p + ||y||p) for all x, y ∈ A\0, and then apply Theorem

2.1. 2

Theorem 2.3. Let f, g, h : A → B be mappings satisfying f(0) = 0, g(0) = 0 and

h(0) = 0 and let f(2nu y) = f(2nu) f(y), g(2nu y) = g(2nu) g(y) and h(2nu y) =

h(2nu) h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, · · ·, for which there exists a

function ϕ : A×A → [0,∞) satisfying (1), (3), and (4) such that

‖2f(µx + µy

2)− µg(x)− µh(y)‖ ≤ ϕ(x, y) (10)

for µ = 1, i, and all x, y ∈ A \ 0. If f(tx) is continuous in t ∈ R for each fixed x ∈ A,

then the mappings f, g, h : A → B are ∗-homomorphisms.

Proof. Put µ = 1 in (10). By the same reasoning as the proof of Theorem 2.1, there

exists a unique additive mapping H : A → B satisfying the inequality (5). By the same

reasoning as the proof of [8], Theorem, the additive mapping H : A → B is R-linear.

Put µ = i in (10). By the same method as the proof of Theorem 2.1, one can obtain

that

H(ix) = limn→∞

f(2nix)

2n= lim

n→∞if(2nx)

2n= iH(x)

for all x ∈ A.

For each element λ ∈ C, λ = s + it, where s, t ∈ R. So

H(λx) = H(sx + itx) = sH(x) + tH(ix) = sH(x) + itH(x) = (s + it)H(x)

= λH(x)

for all λ ∈ C and all x ∈ A. So

H(ζx + ηy) = H(ζx) + H(ηy) = ζH(x) + ηH(y)

JC*-ALGEBRAS 31

for all ζ, η ∈ C, and all x, y ∈ A. Hence the additive mapping H : A → B is C-linear.

The rest of the proof is the same as the proof of Theorem 2.1. 2

From now on, assume that A is a JC∗-algebra of real rank zero, where “real rank

zero” means that the set of invertible self-adjoint elements is dense in the set of self-

adjoint elements (see [1]).

Now we are going to investigate continuous ∗-homomorphisms between JC∗-algebras.

Theorem 2.4. Let f, g, h : A → B be continuous mappings satisfying f(0) = 0,

g(0) = 0 and h(0) = 0 and let f(2nu y) = f(2nu) f(y), g(2nu y) = g(2nu) g(y)

and h(2nu y) = h(2nu) h(y) for all u ∈ I1(Asa), all y ∈ A, and all n = 0, 1, 2, · · ·, for

which there exists a function ϕ : A×A → [0,∞) satisfying (1), (2), (3), and (4). Then

the mappings f, g, h : A → B are ∗-homomorphisms.

Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique

C-linear involutive mapping H : A → B satisfying the inequality (5).

Since f(2nu y) = f(2nu) f(y) for all u ∈ I1(Asa), all y ∈ A, and all n = 0, 1, 2, · · ·,

H(u y) = limn→∞

1

2nf(2nu y) = lim

n→∞1

2nf(2nu) f(y) = H(u) f(y) (11)

for all u ∈ I1(Asa) and all y ∈ A. By the additivity of H and (11),

2nH(u y) = H(2nu y) = H(u (2ny)) = H(u) f(2ny)

for all u ∈ I1(Asa) and all y ∈ A. Hence

H(u y) =1

2nH(u) f(2ny) = H(u) 1

2nf(2ny) (12)

for all u ∈ I1(Asa) and all y ∈ A. Taking the limit in (12) as n →∞, we obtain

H(u y) = H(u) H(y) (13)

for all u ∈ I1(Asa) and all y ∈ A.

By (11) and (13),

H(e) H(y) = H(e y) = H(e) f(y)

for all y ∈ A. Since limn→∞f(2ne)

2n = H(e) = e′,

H(y) = f(y)

PARK ET AL32

for all y ∈ A. Similarly, H(y) = g(y) = h(y) for all y ∈ A. So H : A → B is continuous.

But by the assumption that A has real rank zero, it is easy to show that I1(Asa) is dense

in x ∈ Asa | ||x|| = 1. So for each w ∈ z ∈ Asa | ||z|| = 1, there is a sequence κjsuch that κj → w as j →∞ and κj ∈ I1(Asa). Since H : A → B is continuous, it follows

from (13) that

H(w y) = H( limj→∞

κj y) = limj→∞

H(κj y)

= limj→∞

H(κj) H(y) = H( limj→∞

κj) H(y) (14)

= H(w) H(y)

for all w ∈ z ∈ Asa | ||z|| = 1 and all y ∈ A.

For each x ∈ A, x = x+x∗2

+ ix−x∗2i

, where x1 := x+x∗2

and x2 := x−x∗2i

are self-adjoint.

First, consider the case that x1 6= 0, x2 6= 0. Since H : A → B is C-linear, it follows

from (14) that

H(x y) = H(x1 y + ix2 y) = H(||x1|| x1

||x1|| y + i||x2|| x2

||x2|| y)

= ||x1||H(x1

||x1|| y) + i||x2||H(x2

||x2|| y)

= ||x1||H(x1

||x1||) H(y) + i||x2||H(x2

||x2||) H(y)

= H(||x1|| x1

||x1||) + iH(||x2|| x2

||x2||) H(y) = H(x1 + ix2) H(y)

= H(x) H(y)

for all y ∈ A.

Next, consider the case that x1 6= 0, x2 = 0. Since H : A → B is C-linear, it follows

from (14) that

H(x y) = H(x1 y) = H(||x1|| x1

||x1|| y) = ||x1||H(x1

||x1|| y)

= ||x1||H(x1

||x1||) H(y) = H(||x1|| x1

||x1||) H(y) = H(x1) H(y)

= H(x) H(y)

for all y ∈ A.

Finally, consider the case that x1 = 0, x2 6= 0. Since H : A → B is C-linear, it follows

from (14) that

H(x y) = H(ix2 y) = H(i||x2|| x2

||x2|| y) = i||x2||H(x2

||x2|| y)

JC*-ALGEBRAS 33

= i||x2||H(x2

||x2||) H(y) = H(i||x2|| x2

||x2||) H(y) = H(ix2) H(y)

= H(x) H(y)

for all y ∈ A. Hence

H(x y) = H(x) H(y)

for all x, y ∈ A.

Therefore, the mappings f, g, h : A → B are ∗-homomorphisms, as desired. 2

Corollary 2.5. Let f, g, h : A → B be continuous mappings satisfying f(0) = 0,

g(0) = 0 and h(0) = 0, and let f(2nu y) = f(2nu) f(y), g(2nu y) = g(2nu) g(y) and

h(2nu y) = h(2nu) h(y) for all u ∈ I1(Asa), all y ∈ A, and all n = 0, 1, 2, · · ·, for which

there exist constants θ ≥ 0 and p ∈ [0, 1) such that

‖2f(µx + µy

2)− µg(x)− µh(y)‖ ≤ θ(||x||p + ||y||p),‖f(2nu∗)− f(2nu)∗‖ ≤ 2np+1θ

for all µ ∈ T1, all u ∈ I1(Asa), all x, y ∈ A\0, and all n = 0, 1, 2, · · ·. If limn→∞f(2ne)

2n =

e′, the mappings f, g, h : A → B are ∗-homomorphisms.

Proof. Define ϕ(x, y) = θ(||x||p + ||y||p) for all x, y ∈ A\0, and then apply Theorem

2.4. 2

Theorem 2.6. Let f, g, h : A → B be continuous mappings satisfying f(0) = 0,

g(0) = 0 and h(0) = 0, and let f(2nu y) = f(2nu) f(y), g(2nu y) = g(2nu) g(y) and

h(2nu y) = h(2nu) h(y) for all u ∈ I1(Asa), all y ∈ A, and all n = 0, 1, 2, · · ·, for which

there exists a function ϕ : A × A → [0,∞) satisfying (1), (3), (4), and (10). Then the

mappings f, g, h : A → B are ∗-homomorphisms.


C-linear mapping H : A → B satisfying the system of the inequalities (5).

The rest of the proof is the same as the proofs of Theorems 2.1 and 2.4. 2

3. Stability of ∗-homomorphisms in JC∗-algebras

We prove the generalized Hyers–Ulam–Rassias stability of ∗-homomorphisms in JC∗-

algebras.

PARK ET AL34

Theorem 3.1. Let f, g, h : A → B be mappings with f(0) = 0, g(0) = 0 and h(0) = 0

for which there exists a function ϕ : A4 → [0,∞) such that

ϕ(x, y, z, w) =∞∑

j=0

2−jϕ(2jx, 2jy, 2jz, 2jw) < ∞, (15)

‖2f(µx + µy + z w

2)− µg(x)− µh(y)− f(z) f(w)‖ (16)

≤ ϕ(x, y, z, w),

‖f(2nu∗)− f(2nu)∗‖ ≤ ϕ(2nu, 2nu, 0, 0) (17)

for all µ ∈ T1, all u ∈ U(A), all x, y, z, w ∈ A \ 0, and all n = 0, 1, 2, · · ·. Then there

exists a unique ∗-homomorphism H : A → B such that

‖2f(x2)−H(x)‖ ≤ ε(x),

‖g(x)−H(x)‖ ≤ ϕ(x, 0, 0, 0) + ε(x),

‖h(x)−H(x)‖ ≤ ϕ(0, x, 0, 0) + ε(x) (18)

for all x ∈ A \ 0, where

ε(x) :=∞∑

j=1

2−j(ϕ(2j−1x, 0, 0, 0) + ϕ(0, 2j−1x, 0, 0) + ϕ(2j−1x, 2j−1x, 0, 0)) < ∞.

Proof. Put z = w = 0 and µ = 1 ∈ T1 in (16). By the same reasoning as the proof of

Theorem 2.1, there exists a unique C-linear involutive mapping H : A → B satisfying the

inequality (18). The C-linear mapping H : A → B is given by

H(x) = limn→∞

1

2nf(2nx) (19)

for all x ∈ A.

It follows from (19) that

H = limn→∞

f(22nx)

22n(20)

for all x ∈ A. Let x = y = 0 in (16). Then we get

‖2f(z w

2)− f(z) f(w)‖ ≤ ϕ(0, 0, z, w)

for all z, w ∈ A. Since

1

22nϕ(0, 0, 2nz, 2nw) ≤ 1

2nϕ(0, 0, 2nz, 2nw),

JC*-ALGEBRAS 35

1

22n‖2f(

1

22nz 2nw)− f(2nz) f(2nw)‖ ≤ 1

22nϕ(0, 0, 2nz, 2nw)

≤ 1

2nϕ(0, 0, 2nz, 2nw) (21)

for all z, w ∈ A. By (19), (20), and (21),

2H(z w

2) = lim

n→∞2f(1

222nz w)

22n= lim

n→∞2f(1

22nz 2nw)

2n · 2n

= limn→∞(

f(2nz)

2n f(2nw)

2n) = lim

n→∞f(2nz)

2n lim

n→∞f(2nw)

2n

= H(z) H(w)

for all z, w ∈ A. But since H is C-linear,

H(z w) = 2H(z w

2) = H(z) H(w)

for all z, w ∈ A. Hence the C-linear mapping H : A → B is a ∗-homomorphism satisfying

the inequality (18) as desired. 2

Corollary 3.2. Let f, g, h : A → B be a mapping with f(0) = 0, g(0) = 0 and

h(0) = 0 for which there exist constants θ ≥ 0 and p ∈ [0, 1) such that


2)− µg(x)− µh(y)− f(z) f(w)‖≤ θ(‖x‖p + ‖y‖p + ‖z‖p + ‖w‖p),

‖f(2nu∗)− f(2nu)∗‖ ≤ 2np+1θ

for all µ ∈ T1, all u ∈ U(A), all x, y, z, w ∈ A \ 0, and all n = 0, 1, 2, · · ·. Then there

exists a unique ∗-homomorphism H : A → B such that

‖2f(x2)−H(x)‖ ≤ 1

2−2p θ‖x‖p

‖g(x)−H(x)‖ ≤ 3−2p

2−2p θ‖x‖p

‖h(x)−H(x)‖ ≤ 3−2p

2−2p θ‖x‖p

for all x ∈ A \ 0.

Proof. Define ϕ(x, y, z, w) = θ(‖x‖p + ‖y‖p + ‖z‖p + ‖w‖p), and then apply Theorem

3.1. 2

PARK ET AL36

Theorem 3.3. Let f, g, h : A → B be a mapping with f(0) = 0, g(0) = 0 and

h(0) = 0 for which there exists a function ϕ : A4 → [0,∞) satisfying (15) and (16) such

that


2)− µg(x)− µh(y)− f(z) f(w)‖ ≤ ϕ(x, y, z, w)

for µ = 1, i, and all x, y, z, w ∈ A \ 0. If f(tx) is continuous in t ∈ R for each fixed

x ∈ A, then there exists a unique ∗-homomorphism H : A → B satisfying the inequality

(18).


C-linear mapping H : A → B satisfying the inequality (18).

The rest of the proof is the same as the proofs of Theorems 2.1 and 3.1. 2

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6. R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, ElementaryTheory, Academic Press, New York (1983).

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8. Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc.72 (1978), 297–300.

9. H. Upmeier, Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics, RegionalConference Series in Mathematics No. 67, Amer. Math. Soc., Providence (1987).

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JC*-ALGEBRAS 37

Hyers–Ulam–Rassias Stability of Isometric

Homomorphisms in Quasi-Banach Algebras

Sang-Hyuk Lee† and Choonkil Park∗

†Sang-Hyuk Lee: School of Mechatronics, Changwon National University, Gyangwon 641–773, Republic of Korea, [email protected]∗Choonkil Park: Department of Mathematics, Hanyang University, Seoul 133–791, Republic

of Korea, [email protected]

Abstract. In this paper, we prove the Hyers–Ulam–Rassias stability of isometric homo-morphisms in quasi-Banach algebras. This is applied to investigate isometric isomorphismsbetween quasi-Banach algebras.

2000 Mathematics Subject Classification. Primary 39B52, 46B03, 46B04, 47B48, 47Jxx.

Key words and Phrases. Hyers–Ulam–Rassias stability, additive mapping, quasi-Banach

algebra, isometric isomorphism, p-Banach algebra

1 Introduction and preliminaries

The stability problem of functional equations originated from a question of S.M. Ulam [39]

concerning the stability of group homomorphisms: Let (G1, ∗) be a group and let (G2, ¦, d)

be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ(ε) > 0 such

that if a mapping h : G1 → G2 satisfies the inequality d(h(x ∗ y), h(x) ¦ h(y)) < δ for all

x, y ∈ G1, then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all

x ∈ G1?

———————–∗The corresponding author was supported by the research fund of Hanyang University (HY-2006-N).

1

If the answer is affirmative, we would say that the equation of homomorphism H(x ∗y) = H(x) ¦H(y) is stable. The concept of stability for a functional equation arises when

we replace the functional equation by an inequality which acts as a perturbation of the

equation. Thus the stability question of functional equations is that how do the solutions

of the inequality differ from those of the given functional equation?

D.H. Hyers [13] gave a first affirmative answer to the question of Ulam for Banach

spaces. Let X and Y be Banach spaces. Assume that f : X → Y satisfies

‖f(x + y)− f(x)− f(y)‖ ≤ ε

for all x, y ∈ X and some ε ≥ 0. Then there exists a unique additive mapping T : X → Y

such that

‖f(x)− T (x)‖ ≤ ε

for all x ∈ X.

Let X and Y be Banach spaces with norms || · || and ‖ · ‖, respectively. Consider

f : X → Y to be a mapping such that f(tx) is continuous in t ∈ R for each fixed x ∈ X.

Th.M. Rassias [28] introduced the following inequality: Assume that there exist constants

θ ≥ 0 and p ∈ [0, 1) such that

‖f(x + y)− f(x)− f(y)‖ ≤ θ(||x||p + ||y||p)

for all x, y ∈ X. Th.M. Rassias [28] showed that there exists a unique R-linear mapping

T : X → Y such that

‖f(x)− T (x)‖ ≤ 2θ

2− 2p||x||p

for all x ∈ X. The above inequality has provided a lot of influence in the development

of what is known as Hyers–Ulam–Rassias stability of functional equations. Beginning

around the year 1980 the topic of approximate homomorphisms, or the stability of the

equation of homomorphism, was studied by a number of mathematicians. Gavruta [11]

generalized the Rassias’ result. The stability problems of several functional equations

have been extensively investigated by a number of authors and there are many interesting

results concerning this problem (see [2], [5]–[6], [10], [11], [14]–[18], [20]–[26], [29]–[33],

[36], [37]).

We recall some basic facts concerning quasi-Banach spaces and some preliminary re-

sults.

LEE-PARK40

Definition 1.1 ([4, 35]) Let X be a real linear space. A quasi-norm is a real-valued

function on X satisfying the following:

(1) ‖x‖ ≥ 0 for all x ∈ X and ‖x‖ = 0 if and only if x = 0.

(2) ‖λx‖ = |λ| · ‖x‖ for all λ ∈ R and all x ∈ X.

(3) There is a constant K ≥ 1 such that ‖x + y‖ ≤ K(‖x‖+ ‖y‖) for all x, y ∈ X.

The pair (X, ‖ · ‖) is called a quasi-normed space if ‖ · ‖ is a quasi-norm on X. The

smallest possible K is called the modulus of concavity of ‖ · ‖.A quasi-Banach space is a complete quasi-normed space.

A quasi-norm ‖ · ‖ is called a p-norm (0 < p ≤ 1) if

‖x + y‖p ≤ ‖x‖p + ‖y‖p

for all x, y ∈ X. In this case, a quasi-Banach space is called a p-Banach space.

Given a p-norm, the formula d(x, y) := ‖x−y‖p gives us a translation invariant metric

on X. By the Aoki–Rolewicz theorem [35] (see also [4]), each quasi-norm is equivalent to

some p-norm. Since it is much easier to work with p-norms than quasi-norms, henceforth

we restrict our attention mainly to p-norms.

Definition 1.2 ([1]) Let (A, ‖ · ‖) be a quasi-normed space. The quasi-normed space

(A, ‖ · ‖) is called a quasi-normed algebra if A is an algebra and there is a constant C > 0

such that ‖xy‖ ≤ C‖x‖ · ‖y‖ for all x, y ∈ A.

A quasi-Banach algebra is a complete quasi-normed algebra.

If the quasi-norm ‖ · ‖ is a p-norm then the quasi-Banach algebra is called a p-Banach

algebra.

Definition 1.3 Let A and B be quasi-Banach algebras with norms ‖ · ‖A and ‖ · ‖B. An

algebra homomorphism H : A → B is called an isometric homomorphism if the algebra

homomorphism H : A → B satisfies

‖H(x)−H(y)‖B = ‖x− y‖A

for all x, y ∈ A. If, in addition, the algebra homomorphism H : A → B is bijective, then

the algebra homomorphism H : A → B is called an isometric isomorphism.

The stability of isometries in normed spaces and Banach algebras have been investi-

gated in several papers (see [3, 8, 9, 12, 19]).

ON QUASI-BANACH ALGEBRAS 41

The paper is organized as follows: In Section 2, we prove the Hyers–Ulam–Rassias

stability of isometric homomorphisms in quasi-Banach algebras, associated to the Cauchy

functional equation and the Jensen functional equation.

In Section 3, we investigate isometric isomorphisms between quasi-Banach algebras.

2 Stability of isometric homomorphisms

in quasi-Banach algebras

Throughout this section, assume that A is a quasi-normed algebra with quasi-norm ‖ · ‖A

and that B is a p-Banach algebra with p-norm ‖ · ‖B. Let K be the modulus of concavity

of ‖ · ‖B.

We prove the Hyers–Ulam–Rassias stability of isometric homomorphisms in quasi-

Banach algebras, associated to the Cauchy functional equation.

Theorem 2.1 Let r > 2 and θ be positive real numbers, and let f : A → B be a mapping

such that

‖f(x + y)− f(x)− f(y)‖B ≤ θ(‖x‖rA + ‖y‖r

A), (1)

‖f(xy)− f(x)f(y)‖B ≤ θ(‖x‖rA + ‖y‖r

A), (2)

| ‖f(x)‖B − ‖x‖A | ≤ θ‖x‖rA (3)

for all x, y ∈ A. If f(tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a

unique isometric homomorphism H : A → B such that

‖f(x)−H(x)‖B ≤ 2θ

(2pr − 2p)1p

‖x‖rA (4)

for all x ∈ A.

Proof. Letting y = x in (1), we get

‖f(2x)− 2f(x)‖B ≤ 2θ‖x‖rA (5)

for all x ∈ A. So

‖f(x)− 2f(x

2)‖B ≤ 2θ

2r‖x‖r

A

LEE-PARK42

for all x ∈ A. Since B is a p-Banach algebra,

‖2lf(x

2l)− 2mf(

x

2m)‖p

B ≤m−1∑

j=l

‖2jf(x

2j)− 2j+1f(

x

2j+1)‖p

B ≤2pθp

2pr

m−1∑

j=l

2pj

2prj‖x‖pr

A (6)

for all nonnegative integers m and l with m > l and all x ∈ A. It follows from (6) that

the sequence 2nf( x2n ) is a Cauchy sequence for all x ∈ A. Since B is complete, the

sequence 2nf( x2n ) converges. So one can define the mapping H : A → B by

H(x) := limn→∞ 2nf(

x

2n)

for all x ∈ A.


‖H(x + y)−H(x)−H(y)‖B = limn→∞ 2n‖f(

x + y

2n)− f(

x

2n)− f(

y

2n)‖B

≤ limn→∞

2nθ

2nr(‖x‖r

A + ‖y‖rA) = 0

for all x, y ∈ A. So

H(x + y) = H(x) + H(y)

for all x, y ∈ A. Moreover, letting l = 0 and passing the limit m →∞ in (6), we get (4).

By the same reasoning as in the proof of Theorem of [28], the mapping H : A → B is

R-linear.


‖H(xy)−H(x)H(y)‖B = limn→∞ 4n‖f(

xy

2n · 2n)− f(

x

2n)f(

y

2n)‖B

≤ limn→∞

4nθ

2nr(‖x‖r

A + ‖y‖rA) = 0


H(xy) = H(x)H(y)

for all x, y ∈ A.

Now, let T : A → B be another Cauchy additive mapping satisfying (4). Then we

have

‖H(x)− T (x)‖B = 2n‖H(x

2n)− T (

x

2n)‖B

≤ 2nK(‖H(x

2n)− f(

x

2n)‖B + ‖T (

x

2n)− f(

x

2n)‖B)

≤ 2n+2Kθ

(2pr − 2p)1p 2nr

‖x‖rA,


which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = T (x) for

all x ∈ A. This proves the uniqueness of H.


| ‖2nf(x

2n)‖B − ‖x‖A | = 2nθ

2nr‖x‖r

A,

which tends to zero as n →∞ for all x ∈ A. So

‖H(x)‖B = limn→∞ ‖2

nf(x

2n)‖B = ‖x‖A

for all x ∈ A. Hence

‖H(x)−H(y)‖B = ‖H(x− y)‖B = ‖x− y‖A

for all x, y ∈ A. So the mapping H : A → B is an isometry. Thus the mapping H : A → B

is a unique isometric homomorphism satisfying (4). 2

Theorem 2.2 Let r < 1 and θ be positive real numbers, and let f : A → B be a mapping

satisfying (1), (2) and (3). If f(tx) is continuous in t ∈ R for each fixed x ∈ A, then there

exists a unique isometric homomorphism H : A → B such that

‖f(x)−H(x)‖B ≤ 2θ

(2p − 2pr)1p

‖x‖rA (7)

for all x ∈ A.

Proof. It follows from (5) that

‖f(x)− 1

2f(2x)‖B ≤ θ‖x‖r

A


‖ 1

2lf(2lx)− 1

2mf(2mx)‖p

B ≤m−1∑

j=l

‖ 1

2jf(2jx)− 1

2j+1f(2j+1x)‖p

B ≤ θpm−1∑

j=l

2prj

2pj‖x‖pr

A (8)


the sequence 12n f(2nx) is a Cauchy sequence for all x ∈ A. Since B is complete, the

sequence 12n f(2nx) converges. So one can define the mapping H : A → B by

H(x) := limn→∞

1

2nf(2nx)

for all x ∈ A.

The rest of the proof is similar to the proof of Theorem 2.1. 2

We prove the Hyers–Ulam–Rassias stability of isometric homomorphisms in quasi-

Banach algebras, associated to the Jensen functional equation.

LEE-PARK44

Theorem 2.3 Let r < 1 and θ be positive real numbers, and let f : A → B be a mapping

with f(0) = 0 satisfying (2) and (3) such that

‖2f(x + y

2)− f(x)− f(y)‖B ≤ θ(‖x‖r

A + ‖y‖rA) (9)

for all x, y ∈ A. If f(tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a

unique isometric homomorphism H : A → B such that

‖f(x)−H(x)‖B ≤ K(3 + 3r)θ

(3p − 3pr)1p

‖x‖rA (10)

for all x ∈ A.

Proof. Letting y = −x in (9), we get

‖ − f(x)− f(−x)‖B ≤ 2θ‖x‖rA

for all x ∈ A. Letting y = 3x and replacing x by −x in (9), we get

‖2f(x)− f(−x)− f(3x)‖B ≤ (3r + 1)θ‖x‖rA

for all x ∈ A. Thus

‖3f(x)− f(3x)‖B ≤ K(3r + 3)θ‖x‖rA (11)

for all x ∈ A. So

‖f(x)− 1

3f(3x)‖B ≤ K(3r + 3)θ

3‖x‖r

A


‖ 1

3lf(3lx)− 1

3mf(3mx)‖p

B ≤m−1∑

j=l

‖ 1

3jf(3jx)− 1

3j+1f(3j+1x)‖p

B

≤ Kp(3r + 3)pθp

3p

m−1∑

j=l

3prj

3pj‖x‖pr

A (12)


the sequence 13n f(3nx) is a Cauchy sequence for all x ∈ A. Since B is complete, the

sequence 13n f(3nx) converges. So one can define the mapping H : A → B by

H(x) := limn→∞

1

3nf(3nx)

for all x ∈ A.

By (9),

‖2H(x + y

2)−H(x)−H(y)‖B = lim

n→∞1

3n‖2f(3n · x + y

2)− f(3nx)− f(3ny)‖B

≤ limn→∞

3rn

3nθ(‖x‖r

A + ‖y‖rA) = 0


2H(x + y

2) = H(x) + H(y)

for all x, y ∈ A. Moreover, letting l = 0 and passing the limit m → ∞ in (12), we get

(10).


‖H(xy)−H(x)H(y)‖B = limn→∞

1

9n‖f(9nxy)− f(3nx)f(3ny)‖B

≤ limn→∞

3nrθ

9n(‖x‖r

A + ‖y‖rA) = 0


H(xy) = H(x)H(y)

for all x, y ∈ A.

Now, let T : A → B be another Jensen additive mapping satisfying (10). Then we

have

‖H(x)− T (x)‖pB =

1

3pn‖H(3nx)− T (3nx)‖p

B

≤ 1

3pn(‖H(3nx)− f(3nx)‖p

B + ‖T (3nx)− f(3nx)‖pB)

≤ 2 · 3prn

3pn· Kp(3 + 3r)pθp

3p − 3pr‖x‖pr

A ,

which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = T (x) for

all x ∈ A. This proves the uniqueness of H.


Theorem 2.4 Let r > 2 and θ be positive real numbers, and let f : A → B be a mapping

with f(0) = 0 satisfying (2), (3) and (9). If f(tx) is continuous in t ∈ R for each fixed

x ∈ A, then there exists a unique isometric homomorphism H : A → B such that

‖f(x)−H(x)‖B ≤ K(3r + 3)θ

(3pr − 3p)1p

‖x‖rA (13)

for all x ∈ A.

LEE-PARK46

Proof. It follows from (11) that

‖f(x)− 3f(x

3)‖B ≤ K(3r + 3)θ

3r‖x‖r

A


‖3lf(x

3l)− 3mf(

x

3m)‖p

B ≤m−1∑

j=l

‖3jf(x

3j)− 3j+1f(

x

3j+1)‖p

B

≤ Kp(3r + 3)pθp

3pr

m−1∑

j=l

3pj

3prj‖x‖pr

A (14)


the sequence 3nf( x3n ) is a Cauchy sequence for all x ∈ A. Since B is complete, the

sequence 3nf( x3n ) converges. So one can define the mapping H : A → B by

H(x) := limn→∞ 3nf(

x

3n)

for all x ∈ A.

The rest of the proof is similar to the proofs of Theorems 2.1 and 2.3. 2

3 Isometric isomorphisms between quasi-Banach

algebras

Throughout this section, assume that A is a quasi-Banach algebra with quasi-norm ‖ · ‖A

and unit e and that B is a p-Banach algebra with p-norm ‖ · ‖B and unit e′. Let K be

the modulus of concavity of ‖ · ‖B.

We investigate isometric isomorphisms between quasi-Banach algebras, associated to

the Cauchy functional equation.

Theorem 3.1 Let r > 2 and θ be positive real numbers, and let f : A → B be a bijective

mapping satisfying (1) and (3) such that

f(xy) = f(x)f(y) (15)

for all x, y ∈ A. If limn→∞2nf( e2n ) = e′ and f(tx) is continuous in t ∈ R for each fixed

x ∈ A, then the mapping f : A → B is an isometric isomorphism.


Proof. Since f(xy) − f(x)f(y) = 0 for all x, y ∈ A, the mapping f : A → B satisfies

(2). By Theorem 2.1, there exists an isometric homomorphism H : A → B satisfying (4).

The mapping H : A → B is defined by

H(x) = limn→∞ 2nf(

x

2n)

for all x ∈ A.


H(x) = H(ex) = limn→∞ 2nf(

ex

2n) = lim

n→∞ 2nf(e

2n· x) = lim

n→∞ 2nf(e

2n)f(x)

= e′f(x) = f(x)

for all x ∈ A. So the bijective mapping f : A → B is an isometric isomorphism. 2

Theorem 3.2 Let r < 1 and θ be positive real numbers, and let f : A → B be a bijective

mapping satisfying (1), (3) and (15). If f(tx) is continuous in t ∈ R for each fixed x ∈ A

and limn→∞ 12n f(2ne) = e′, then the mapping f : A → B is an isometric isomorphism.


(2). By Theorem 2.2, there exists an isometric homomorphism H : A → B satisfying (7).

The mapping H : A → B is defined by

H(x) = limn→∞

1

2nf(2nx)

for all x ∈ A.


We investigate isometric isomorphisms between quasi-Banach algebras, associated to

the Jensen functional equation.

Theorem 3.3 Let r < 1 and θ be positive real numbers, and let f : A → B be a bijective

mapping with f(0) = 0 satisfying (3), (9) and (15). If f(tx) is continuous in t ∈ R for

each fixed x ∈ A and limn→∞ 13n f(3ne) = e′, then the mapping f : A → B is an isometric

isomorphism.


(2). By Theorem 2.3, there exists an isometric homomorphism H : A → B satisfying

(10). The mapping H : A → B is defined by

H(x) = limn→∞

1

3nf(3nx)

LEE-PARK48

for all x ∈ A.


Theorem 3.4 Let r > 2 and θ be positive real numbers, and let f : A → B be a bijective

mapping with f(0) = 0 satisfying (3), (9) and (15). If f(tx) is continuous in t ∈ R for

each fixed x ∈ A and limn→∞ 3nf( e3n ) = e′, then the mapping f : A → B is an isometric

isomorphism.


(2). By Theorem 2.4, there exists an isometric homomorphism H : A → B satisfying

(13). The mapping H : A → B is defined by

H(x) = limn→∞ 3nf(

x

3n)

for all x ∈ A.


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BETTER ERROR ESTIMATION FORSZASZ-MIRAKJAN-BETA OPERATORS

OKTAY DUMAN, MEHMET ALI OZARSLAN, AND HUSEYIN AKTUGLU

Abstract. In this paper, we present a modification of a sequence ofmixed summation-integral type operators having Szasz and Beta basisfunctions in summation and integration, the so-called Szasz-Mirakjan-Beta operators. Then we show that our modified operators have a bettererror estimation on the interval [0, 2]. Furthermore, we give an r-thorder generalization of the modified Szasz-Mirakjan-Beta operators andinvestigate their approximation properties.

1. Introduction

The classical Szasz-Mirakjan operators are defined by

Sn(f ; x) := e−nx∞∑

k=0

(nx)k

k!f

(k

n

),

where f ∈ C[0,∞), x ≥ 0 and n ∈ N.Some approximation properties of the Szasz-Mirakjan operators and their

modifications were studied by Agrawal and Kasana [1], Duman and Ozarslan[3], Finta [4], Finta, Govil and Gupta [5], Gupta [6], Gupta and Noor [7],Gupta, Noor and Beniwal [8], Gupta and Pant [9], Srivastava and Gupta[11], Totik [12], Zeng and Piriou [13]. Further properties and general ap-proximation results on these operators may be found in the monograph byAltomore and Campiti [2].

Recently Gupta and Noor [7] have proposed a sequence of mixed summation-integral type operators, the so-called Szasz-Mirakjan-Beta operators, as fol-lows:(1.1)

Un(f ; x) = e−nx∞∑

k=1

(nx)k

k!B(n + 1, k)

∫ ∞

0f(t)

tk−1

(1 + t)n+k+1dt + e−nxf(0),

where f ∈ C[0,∞) such that |f(t)| ≤ M(1 + t)γ for some M > 0, γ > 0.Now, for the operators Un given by (1.1), the following lemma follows

from [7] immediately.

Key words and phrases. Szasz-Mirakjan-Beta operators; positive linear operators; theKorovkin-type approximation theorem; modulus of continuity; Lipschitz class functionals.

2000 Mathematics Subject Classification. 41A25, 41A36.

1


2 OKTAY DUMAN, MEHMET ALI OZARSLAN, AND HUSEYIN AKTUGLU

Lemma A [7]. Let ei(x) = xi, i = 0, 1, 2. Then, for each x ≥ 0 and n > 1,we have

(a) Un(e0; x) = 1,(b) Un(e1; x) = x,

(c) Un(e2; x) =1

n− 1(nx2 + 2x

).

Lemma A shows that the operators Un preserve the test functions e0(x) =1 and e1(x) = x. Actually, many well-known approximating operators pre-serve these test functions, such as Bernstein polynomials, Meyer-Konig andZeller operators, Szasz-Mirakjan operators, Baskakov operators etc. Ob-serve that these operators do not preserve the test function e2(x) = x2.However, by modifying the Bernstein polynomials, King [10] presented anon-trivial sequence of positive linear operators which approximate eachcontinuous function on [0, 1] while preserving the functions e0 and e2. Thenit is proved that these modified operators have a better rate of convergencethan the classical Bernstein polynomials on the interval [0, 1/3]. Thus a nat-ural question arises: can we construct a sequence of positive linear operatorspreserving the test functions e0 and e2 so that our modified operators have abetter error estimation than Szasz-Mirakjan-Beta operators. In the presentpaper we mainly focus on this problem.

2. Construction of the operators

We first consider the Banach lattice

Cγ [0,∞) := f ∈ C[0, +∞) : |f(x)| ≤ M(1 + x)γ for some M > 0, γ > 0endowed with the norm

‖f‖γ := supx∈[0,+∞)

|f(x)|(1 + x)γ

.

Then, the set e0, e1, e2 is a K+−subset of Cγ [0,∞); also the space Cγ [0,∞)is isomorphic to C[0, 1] (see, for details, [2]).

Let rn(x) be sequence of real-valued continuous functions defined on[0,∞) with 0 ≤ rn(x) < ∞. Then we have

Un(f ; rn(x)) = e−nrn(x)∞∑

k=1

(nrn(x))k

k!B(n + 1, k)

∫ ∞

0f(t)

tk−1

(1 + t)n+k+1dt +e−nrn(x)f(0),

where x ∈ [0,∞), f ∈ Cγ [0,∞), γ > 0 and n ∈ N. Now, if we replace rn(x)by r∗n(x) defined as

(2.1) r∗n(x) :=1n

(−1 +

√1 + n(n− 1)x2

), x ≥ 0 and n ∈ N,

then we get the following positive linear operators(2.2)

U∗n(f ; x) := e−nr∗n(x)

∞∑

k=1

(nr∗n(x))k

k!B(n + 1, k)

∫ ∞

0f(t)

tk−1

(1 + t)n+k+1dt +e−nr∗n(x)f(0),

54

SZASZ-MIRAKJAN-BETA OPERATORS 3

where f ∈ Cγ [0,∞), γ > 0 and x ≥ 0.Then, observe that every U∗

n maps CB[0, +∞), the space of all boundedand continuous functions on [0, +∞), into itself.

On the other hand, from Lemma A we obtain the following result at once.

Lemma 2.1. For each x ≥ 0, we have(a) U∗

n(e0; x) = 1,

(b) U∗n(e1; x) =

1n

(−1 +

√1 + n(n− 1)x2

),

(c) U∗n(e2; x) = x2.

Now, fix b > 0 and consider the lattice homomorphism Tb : C[0, +∞) →C[0, b] defined by Tb(f) := f |[0,b] for every f ∈ C[0, +∞). In this case, wesee that, for each i = 0, 1, 2,

(2.3) limn→∞Tb (U∗

n(ei)) = Tb(ei) uniformly on [0, b].

Thus, with the universal Korovkin-type property with respect to mono-tone operators (see Theorem 4.1.4 (vi) of [2, p. 199]) we have the follow-ing: “Let X be a compact set and H be a cofinal subspace of C(X). IfE is a Banach lattice, S : C(X) → E is a lattice homomorphism and ifLn is a sequence of positive linear operators from C(X) into E such thatlimn→∞ Ln(h) = S(h) for all h ∈ H, then limn→∞ Ln(f) = f provided thatf belongs to the Korovkin closure of H”.

Hence, by using (2.3) and the above property we obtain the followingKorovkin-type approximation result.

Theorem 2.2. limn→∞ U∗n(f ; x) = f(x) uniformly with respect to x ∈ [0, b]

provided f ∈ Cγ [0,∞), γ > 0 and b > 0.

3. Better error estimation

In this section we compute the rate of convergence of the operators U∗n

defined by (2.2). Then, we will show that our operators has better errorestimation on the interval [0, 2] than that of the Szasz-Mirakjan-Beta oper-ators Un given by (1.1). To achieve this we use the modulus of continuityand the elements of Lipschitz class functionals.

If we define the function ψx, (x ≥ 0), by ψx(y) = y − x, then by Lemma2.1 one can get the following result, immediately.

Lemma 3.1. For every x ≥ 0, we have

(a) U∗n(ψx; x) = −x +

1n

(−1 +

√1 + n(n− 1)x2

),

(b) U∗n(ψ2

x; x) = 2x

(x +

1n−

√1 + n(n− 1)x2

n

).

Let f ∈ CB[0, +∞) and x ≥ 0. Then, the modulus of continuity of fdenoted by ω(f, δ), is defined to be

ω(f, δ) = sup|y−x|≤δ; x,y∈[0,+∞)

|f(y)− f(x)| .

55


Then we have the following

Theorem 3.2. For every f ∈ CB[0,+∞), x ≥ 0 and n > 1, we have

|U∗n(f ;x)− f(x)| ≤ 2ω(f, δn,x),

where δn,x :=√

2x (x− r∗n(x)) and r∗n(x) is given by (2.1).

Proof. Now, let f ∈ CB[0, +∞) and x ≥ 0. Using linearity and monotonicityof U∗

n we easily get, for every δ > 0 and n ∈ N, that

|U∗n(f ; x)− f(x)| ≤ ω(f, δ)

1 +

1δ

√U∗

n (ψ2x; x)

.

Now applying Lemma 3.1 (b) and choosing δ = δn,x the proof is completed.¤

Remark. For the Szasz-Mirakjan-Beta operators given by (1.1) we maywrite that, for every f ∈ CB[0,+∞), x ≥ 0 and n > 1,

(3.1) |Un(f ; x)− f(x)| ≤ 2ω(f, αn,x),

where αn,x :=√

x(2+x)n−1 (see [7]).

Now we claim that the error estimation in Theorem 3.2 is better thanthat of (3.1) provided f ∈ CB[0, +∞) and x ∈ [0, 2]. Indeed, for 0 ≤ x ≤ 2,we have x2

4 ≤ 1. Also since (n− 12)2 − n(n− 1) = 1

4 , we can write that

x2

[(n− 1

2)2 − n(n− 1)

]≤ 1,

or1 + n(n− 1)x2 ≥ (n− 1

2)2x2

which gives√

1 + n(n− 1)x2 ≥(

2n− 12

)x.

Then we obtain

− 1n

+1n

√1 + n(n− 1)x2 ≥ − 1

n+

(2n− 1

2n

)x.

Using the above inequality we have

x− r∗n(x) ≤ 2 + x

2nor

(3.2) 2x (x− r∗n(x)) ≤ x(2 + x)n

≤ x(2 + x)n− 1

for x ∈ [0, 2] and n > 1. This guarantees that δn,x ≤ αn,x for x ∈ [0, 2] andn > 1, which corrects our claim.

Now we can also compute the rate of convergence of the operators U∗n

by means of the elements of the Lipschitz class LipM (α), (α ∈ (0, 1]). To

56


get this, we recall that a function f ∈ CB[0,∞) belongs to LipM (α) if theinequality

(3.3) |f(y)− f(x)| ≤ M |y − x|α (x, y ∈ [0,∞))

holds.

Theorem 3.3. For every f ∈ LipM (α), x ≥ 0 and n > 1, we have

|U∗n(f ;x)− f(x)| ≤ M 2x (x− r∗n(x))α

2 ,

where r∗n(x) is given by (2.1).

Proof. Since f ∈ LipM (α) and x ≥ 0, using inequality (3.3) and then apply-ing the Holder inequality with p = 2

α , q = 22−α we get

|U∗n(f ; x)− f(x)| ≤ U∗

n (|f(y)− f(x)| ;x)≤ M U∗

n (|y − x|α ; x)≤ M

U∗

n

(ψ2

x; x)α

2

≤ M 2x (x− r∗n(x))α2 ,

whence the result. ¤

Notice that as in the proof of Theorem 3.2, since Un(ψ2x; x) = x(2+x)

n−1 , theSzasz-Mirakjan-Beta operators defined by (1.1) satisfy

(3.4) |Un(f ; x)− f(x)| ≤ M

x(2 + x)n− 1

α2

for every f ∈ LipM (α), x ≥ 0 and n > 1. So, it follows from (3.2) that theabove claim also holds for Theorem 3.2, i.e., the rate of convergence of theoperators U∗

n by means of the elements of the Lipschitz class functionals isbetter than the ordinary error estimation given by (3.4) whenever x ∈ [0, 2].

4. r − th order generalization of the operators U∗n

Let C(r)γ [0,∞), r = 0, 1, 2, ..., denote the space of all functions f ∈

Cγ [0,∞) such that the r-th derivative f (r) ∈ Cγ [0,∞) (for some γ > 0)with f (0)(x) := f(x). In the case of r = 0, the space C

(0)γ [0,∞) coincides

with Cγ [0,∞). Now we consider the following r−th order generalization ofthe positive linear operators U∗

n defined by (2.2):

(4.1)

U∗n,r(f ; x) = e−nr∗n(x)

∞∑k=0

r∑i=0

(nr∗n(x))k

k!B(n + 1, k)

∞∫0

f (i)(t)(t− x)i

i!dt,

+e−nr∗n(x)r∑

i=0

(−1)ixif i(0)i!

where f ∈ C(r)γ [0,∞), γ > 0, r = 0, 1, 2, ..., n ∈ N and r∗n(x) is given by

(2.1). Observe that U∗n,0 = U∗

n.

57


Now using the definition of the operators U∗n,r we may write that

(4.2) U∗n,r(f ; x) =

∫ ∞

0

r∑

i=0

Wn(x, t)f (i)(t)(t− x)i

i!dt,

where

Wn(x, t) = e−nr∗n(x)∞∑

k=1

(nr∗n(x))k

k!B(n + 1, k)tk−1

(1 + t)n+k+1+ e−nr∗n(x)δ(t)

and δ(t) is the Dirac delta function.Thus we have the following

Theorem 4.1. For all f ∈ C(r)γ [0,∞), γ > 0, such that f (r) ∈ LipM (α),

and for every x ≥ 0 we have∣∣U∗

n,r(f ; x)− f(x)∣∣ ≤ M

(r − 1)!α

α + rB(α, r)

∣∣U∗n

(|t− x|r+α ;x)∣∣ ,

where r = 1, 2, ... and B(α, r) is the beta function.

Proof. By (4.2) and Lemma 2.1 (a) one can write that

(4.3) f(x)− U∗n,r(f ; x) =

∫ ∞

0Wn(x, t)

f(x)−

r∑

i=0

f (i)(t)(t− x)i

i!

dt

Then we known from Taylor’s formula that

(4.4)f(x)−

r∑i=0

f (i)(t)(x− t)i

i!=

(x− t)r

(r − 1)!

×1∫0

(1− s)r−1f (r) (t + s(x− t))− f (r)(t)

ds.

Since f (r) ∈ LipM (α),

(4.5)∣∣∣f (r) (t + s(x− t))− f (r)(t)

∣∣∣ ≤ M sα |t− x|α .

Using (4.5) and the usual definition of the beta integral in (4.4) we concludethat

(4.6)

∣∣∣∣∣f(x)−r∑

i=0

f (i)(t)(x− t)i

i!

∣∣∣∣∣ ≤M

(r − 1)!α

α + rB(α, r) |t− x|r+α .

Thus, the proof is completed by considering (4.3) and (4.6). ¤

Finally, for the uniform convergence of the operators U∗n,r given by (4.2)

we obtain the next result.

Theorem 4.2. For every f ∈ C(r)γ [0,∞), γ > 0, r = 1, 2, ..., such that

f (r) ∈ LipM (α), we have

limn→∞U∗

n,r(f ; x) = f(x) uniformly with respect to x ∈ [0, b], b > 0.

58


Proof. Let x ∈ [0, b] and define the function g by g(t) = |t− x|r+α . Then,from Theorem 2.2, it is clear that

limn→∞U∗

n(g; x) = g(x) = 0 uniformly with respect to x ∈ [0, b].

So the proof follows from Theorem 4.1 immediately. ¤

References

[1] P.N. Agrawal and H.S. Kasana, On simultaneous approximation by Szasz-Mirakjanoperators, Bull. Inst. Math. Acad. Sinica, 22, 181-188 (1994).

[2] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applica-tion, Walter de Gruyter Studies in Math. 17, de Gruyter & Co., Berlin, 1994.

[3] O. Duman and M.A. Ozarslan, Szasz-Mirakjan type operators providing a bettererror estimation, Appl. Math. Lett., (2006); in press (doi:10.1016/j.aml.2006.10.007).

[4] Z. Finta, On converse approximation theorems, J. Math. Anal. Appl., 312, 159-180(2005).

[5] Z. Finta, N.K. Govil and V. Gupta, Some results on modified Szasz-Mirakjan opera-tors, J. Math. Anal. Appl., 327, 1284-1296 (2007).

[6] V. Gupta, Simultaneous approximation for Bezier variant of Szasz-Mirakyan-Durrmeyer operators, J. Math. Anal. Appl., (in press).

[7] V. Gupta and M.A. Noor, Convergence of derivatives for certain mixed Szasz-Betaoperators, J. Math. Anal. Appl., 321, 1-9 (2006).

[8] V. Gupta, M.A. Noor and Beniwal, Rate of convergence in simultaneous approxima-tion for Szasz-Mirakyan-Durrmeyer operators, J. Math. Anal. Appl., 322, 964-970(2006).

[9] V. Gupta and R.P. Pant, Rate of convergence for the modified Szasz-Mirakyan oper-ators on functions of bounded variation, J. Math. Anal. Appl., 223, 476-483 (1999).

[10] J.P. King, Positive linear operators which preserve x2, Acta. Math. Hungar., 99,203-208 (2003).

[11] H.M. Srivastava and V. Gupta, A certain family of summation integral type operators,Math. Comput. Modelling, 37, 1307-1315 (2003).

[12] V. Totik, Uniform approximation by Szasz-Mirakjan type operators, Acta Math. Hun-gar., 41, 291-307 (1983).

[13] X.-M. Zeng and A. Piriou, Rate of pointwise approximation for locally boundedfunctions by Szasz operators, J. Math. Anal. Appl., 307, 433-443 (2005).

TOBB Economics and Technology University, Faculty of Arts and Sci-ences, Department of Mathematics, Sogutozu 06530, Ankara, Turkey

E-mail address: [email protected]

Eastern Mediterranean University, Faculty of Arts and Sciences, Depart-ment of Mathematics, Gazimagusa, Mersin 10, Turkey


Eastern Mediterranean University, Faculty of Arts and Sciences, Depart-ment of Mathematics, Gazimagusa, Mersin 10, Turkey


59

SOME THEOREMS ON IF -COMPACT LINEAR OPERATORS

HAKAN EFE

Abstract. The purpose of this paper to introduce IF -compact operators inIF -normed linear spaces in the sense of Lael and Nourouzi [7]. Also classicalcompact operator by means of IF -concept have investigated.

1. Introduction

The notion of fuzzy norm on a linear space was introduced by Katsaras [8] in 1984rstly. In 1992, Felbin [4] gave a denition of a fuzzy norm on a linear space whoseassociated metric is Kaleva type [5]. In 1994, Chang and Mordeson [3] introducedanother idea of a fuzzy norm on a linear space whose associated metric is Kramosiland Michalek type [9]. Xiao and Zhu [11] redened the idea of Felbins [4] denitionof fuzzy norm of a linear operator from a fuzzy normed linear space to another fuzzynormed linear space. In 2003, Bag and Samanta [1] introduced a denition of a fuzzynorm and proved a decomposition theorem of a fuzzy norm into a family of crispnorms. In 2005, Bag and Samanta [2] gave an idea of fuzzy norm of a linear operatorfrom a fuzzy normed linear space to another fuzzy normed linear space. They alsodened various notion of continuities operators and boundedness of linear operatorsover fuzzy normed linear spaces. In 2004, Park [10] using the idea of intuitionisticfuzzy sets, dened the notion of intuitionistic fuzzy metric spaces with the helpof continuous t-norm and continuous t-conorm as a generalization of fuzzy metricspace. In 2006, Lael and Nourouzi [6] introduced fuzzy compact operators betweenfuzzy normed spaces. Very recently Lael and Nourouzi [7] gave a new denition forIF -normed linear space and proved some theorems: open mapping, closed graphand uniform boundedness in IF -normed linear spaces.In this paper, we introduce IF -compact operators in IF -normed linear spaces

in the sense of Lael and Nourouzi. We have investigated classical compact operatorby means of IF -concept.

2. Preliminaries

Denition 1 ([7]). The 3-tuple (X;; ) is said to be an IF -normed linear spaceif X is a real vector space, and , are F -sets of X R satisfying the followingconditions for every x; y 2 X and t; s 2 R,

(i) (x; t) = 0, for all non-positive real number t,(ii) (x; t) = 1 for all t 2 R+ if and only if x = 0,(iii) (cx; t) = (x; tjcj ), for all t 2 R

+ and c 6= 0,(iv) (x+ y; s+ t) minf(x; t); (y; s)g,(v) limt!1 (x; t) = 1 and limt!0 (x; t) = 0,

2000 Mathematics Subject Classication. 46S40, 47A30, 47B07.Key words and phrases. IF -normed linear space, IF -compact operator.

1


2 HAKAN EFE

(vi) (x; t) = 1, for all non-positive real number t,(vii) (x; t) = 0 for all t 2 R+ if and only if x = 0,(viii) (cx; t) = (x; tjcj ), for all t 2 R

+ and c 6= 0,(ix) (x+ y; s+ t) maxf(x; t); (y; s)g,(x) limt!1 (x; t) = 0 and limt!0 (x; t) = 1.

In this case, we will call (; ) an IF -norm on X. In addition, (X;) is calledan F -normed space.It is easy to see that for every x 2 X, the functions (x; ) and (x; ) are

nondecreasing and nonincreasing on R, respectively.

Lemma 1 ([7]). Let (X;) be an F -normed linear space and (x; t) = 1 (x; t)for all x 2 X and t 2 R. Then (X;; ) is an IF -normed linear space.

Example 1. Let (X; jj jj) be a normed space and that 0; 0; 1 and 1 be F -setson X R dened by

0(x; t) =

tt+jjxjj if t > 0

0 if t 0 and 0(x; t) =

(jjxjjt+jjxjj if t > 0

1 if t 0,

1(x; t) =

(exp

jjxjj

t

if t > 0

0 if t 0and 1(x; t) =

(1 exp

jjxjj

t

if t > 0

1 if t 0:

Then (0; 0) and (1; 1) ae two IF -norms on X.

Denition 2 ([7]). Let (X;; ) be an IF -normed linear space and (xn)n be asequence in X. Then (xn)n is said to be convergent to x 2 X if limn!1 (xn x; t) = 1 and limn!1 (xn x; t) = 0, for all t > 0. We denote it by xn ! x.

Denition 3. Let (X;; ) be an IF -normed linear space and (xn)n be a sequencein X. Then (xn)n is said to be a Cauchy sequence, if limn!1 (xn+p xn; t) = 1and limn!1 (xn+p xn; t) = 0 for all t > 0, p 2 N.

Theorem 1 ([7]). Let (X;; ) be an IF -normed linear space. Assume furtherthat

(F) (x; t) = 0 for all t > 0 implies x = 0.

Dene

jjxjj = ^ft > 0 : (x; t) , (x; t) 1 g,

where 2 (0; 1). Then fjj jj : 2 (0; 1)g is an ascending family of norms on X,and they are called -norms on X corresponding to (or induced by) the IF -norm(; ) on X.

Remark 1. Let (X;; ) be an IF -normed linear space. Assume further that, forx 6= 0, (x; ) and (x; ) are continuous functions of R and is strictly increasingon ft > 0 : 0 < (x; t) < 1g and is strictly decreasing on ft > 0 : 0 < (x; t) < 1g.Let we show this (FF).

Lemma 2 ([7]). Let (X;; ) be an IF -normed linear space satisfying (F) and(xn)n be a sequence in X. Then xn ! x i¤ limn!1 jjxnxjj = 0 for all 2 (0; 1).

62

IF -NORMED LINEAR SPACES 3

3. Main Results

Denition 4. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces andT : X ! Y be a linear operator. The operator T is called IF -continuous at z 2 Xif for any " > 0, and 2 (0; 1) there exist > 0 and 2 (0; 1) such that for allx 2 X, if 1(x z; ) > and 1(x z; ) < 1 then 2(T (x) T (z); ") > and 2(T (x) T (z); ") < 1 .

If T is IF -continuous at each point of X, then T is said to be IF -continuous onX.

Denition 5. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces andT : X ! Y be a linear operator. The operator T is called strongly IF -continuousat z 2 X if for any " > 0, there exist > 0 such that for all x 2 X, 2(T (x) T (z); ") 1(x z; ) and 2(T (x) T (z); ") 1(x z; ).

If T is strongly IF -continuous at each point of X, then T is said to be stronglyIF -continuous on X.

Denition 6. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces andT : X ! Y be a linear operator. The operator T is called weakly IF -continuous atz 2 X if for any " > 0, and 2 (0; 1) there exist > 0 such that for all x 2 X,if 1(x z; ) and 1(x z; ) 1 then 2(T (x) T (z); ") and2(T (x) T (z); ") 1 .

If T is weakly IF -continuous at each point of X, then T is said to be weaklyIF -continuous on X.

Denition 7. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces andT : X ! Y be a linear operator. The operator T is called sequentially IF -continuous at x 2 X if for any sequence (xn)n, xn 2 X for all n, with xn ! ximplies T (xn) ! T (x). I.e., for all t > 0, if limn!1 1(xn x; t) = 1 andlimn!1 1(xnx; t) = 0 then limn!1 2(T (xn)T (x); t) = 1 and limn!1 2(T (xn)T (x); t) = 0.

If T is sequentially IF -continuous at each point of X, then T is said to besequentially IF -continuous on X.

Remark 2. If a linear operator is strongly IF -continuous then it is weakly IF -continuous.

Theorem 2. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces and T :X ! Y be a linear operator. If T is strongly IF -continuous then it is sequentiallyIF -continuous but not conversely.

Proof. We assume that T is strongly IF -continuous at z 2 X. Then for any " > 0,there exist > 0 such that for all x 2 X,

((i)) 2(T (x) T (z); ") 1(x z; ) and 2(T (x) T (z); ") 1(x z; ).

Let (xn)n be a sequence in X such that xn ! z, i.e.,

((ii)) limn!1

1(xn z; t) = 1 and limn!1

1(xn z; t) = 0

63

4 HAKAN EFE

for all t > 0. Now from (i), 2(T (xn) T (z); ") 1(xn z; ) and 2(T (xn) T (z); ") 1(xn z; ) for n = 1; 2; :::. This implies,

limn!1

2(T (xn) T (z); ") limn!1

1(xn z; ) and

limn!1

2(T (xn) T (z); ") limn!1

1(xn z; ).

Hence, limn!1 2(T (xn) T (z); ") = 1 and limn!1 2(T (xn) T (z); ") = 0 by(ii). Since " is a small arbitrary number, it follows that T (xn)! T (z).

To show that sequential IF -continuity of T does not imply strong IF -continuityof T , consider the following example:

Example 2. Let (X = R; jj jj) be a normed linear space where jjxjj = jxj for allx 2 R. Dene 1, 2, 1 and 2 : X R! [0; 1] by

1(x; t) =

tt+jxj if t > 0

0 if t 0 and 1(x; t) =

(jxjt+jxj if t > 0

1 if t 0,

2(x; t) =

tt+kjxj if t > 0

0 if t 0 and 2(x; t) =

(jxj

t+kjxj if t > 0

1 if t 0,

where k > 0 is a constant. Then (X;1; 1) and (Y; 2; 2) are IF -normed lin-ear spaces. If we consider the function T (x) = x4

1+x2 , then T is sequentially IF -continuous but not strongly IF -continuous.

Theorem 3. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces andT : X ! Y be a linear operator. Then T is IF -continuous i¤ it is sequentiallyIF -continuous.

Proof. Suppose T is IF -continuous at z 2 X. Let (xn)n be a sequence in Xsuch that xn ! z. Let " > 0 be given and choose 2 (0; 1). Since T is IF -continuous at z 2 X, then there exist > 0 and 2 (0; 1) such that for all x 2 X,if 1(x z; ) > and 1(x z; ) < 1 then 2(T (x) T (z); ") > and2(T (x) T (z); ") < 1 . Since xn ! z in X, there exists pozitive integer n0such that 1(xn z; ) > and 1(xn z; ) < 1 for all n n0. Then2(T (xn) T (z); ") > and 2(T (xn) T (z); ") < 1 for all n n0. Sofor a given " > 0 and for any 2 (0; 1), there exists pozitive integer n0 such that2(T (xn)T (z); ") > and 2(T (xn)T (z); ") < 1 for all n n0. This implieslimn!1 2(T (xn) T (z); ") = 1 and limn!1 2(T (xn) T (z); ") = 0. Since " > 0is arbitrary, thus T (xn)! T (z) in (Y; 2; 2).Next we suppose that T is sequentially IF -continuous at z 2 X. If possible

assume that T is not IF -continuous at z 2 X. Thus there exists " > 0 and 2 (0; 1) such that for any > 0 and 2 (0; 1), there exists y(depending on , )such that

1(z y; ) > and 1(z y; ) < 1 but (i)

2(T (z) T (y); ") and 2(T (z) T (y); ") 1 .Thus for = 1 1

n+1 , =1

n+1 , n = 1; 2; :::, there exists yn such that 1(z yn;

1n+1 ) > 1 1

n+1 and 1(z yn;1

n+1 ) <1

n+1 but 2(T (z) T (yn); ") and2(T (z) T (yn); ") 1 . Taking > 0, there exists n0 such that 1

n+1 < forall n n0. Then 1(z yn; ) 1(z yn; 1

n+1 ) > 1 1n+1 and 1(z yn; )

64

1(z yn; 1n+1 ) <

1n+1 for all n n0. This implies limn!1 1(z yn; ) 1 and

limn!1 1(zyn; ) 0. Hence yn ! z. But from (i), 2(T (z)T (yn); ") and2(T (z)T (yn); ") 1 so 2(T (z)T (yn); ")9 1 and 2(T (z)T (yn); ")9 0as n ! 1. Thus T (yn) does not convergence to T (z) whereas yn ! z (w.r.t.(1; 1)), which is a contradiction to our assuption. Hence T is IF -continuous atz.

Denition 8 ([7]). Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spacesand T : X ! Y be a linear operator. The operator T is called weakly IF -bounded iffor any 2 (0; 1), there exists constant h > 0 such that for every x 2 X and " > 0,1(hx; ") and 1(hx; ") 1) 2(T (x); ") and 2(T (x); ") 1.

Denition 9 ([7]). Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spacesand T : X ! Y be a linear operator. The operator T is called strongly IF -boundedif there exists constant h > 0 such that for every x 2 X and " > 0, 2(T (x); ") 1(hx; ") and 2(T (x); ") 1(hx; ").

Theorem 4 ([7]). Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spacessatisfying (F), where 2(x; ) and 2(x; ) are continuous function on R for allx 2 X. If the linear operator T : (X;1; 1) ! (Y; 2; 2) is bounded w.r.t. -norms of (1; 1) and (2; 2), then T is weakly IF -bounded.

Theorem 5. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces andT : X ! Y be a linear operator. Then

(i) T is strongly IF -continuous on X i¤ T is strongly IF -continuous at a pointx0 2 X.

(ii) T is strongly IF -continuous i¤ T is strongly IF -bounded.

Proof. (i) Since T is strongly IF -continuous at x0 2 X, thus for each " > 0,there exist > 0 such that for all x 2 X, 2(T (x) T (x0); ") 1(x x0; ) and2(T (x)T (x0); ") 1(xx0; ). Taking any y 2 X and replacing x by x+x0ywe get

2(T (x+ x0 y) T (x0); ") 1(x+ x0 y x0; )) 2(T (x) + T (x0) T (y) T (x0); ") 1(x y; )) 2(T (x) T (y); ") 1(x y; ),

and

2(T (x+ x0 y) T (x0); ") 1(x+ x0 y x0; )) 2(T (x) + T (x0) T (y) T (x0); ") 1(x y; )) 2(T (x) T (y); ") 1(x y; ).

Since y 2 X is arbitrary, it follows that T is strongly IF -continuous on X.(ii) We suppose that T is strongly IF -continuous. Using continuity of T at

x = 0, for " = 1, there exists > 0 such that 2(T (x) T (0); 1) 1(x 0; ) and2(T (x) T (0); ") 1(x 0; ), for all x 2 X.Suppose that x 6= 0 and t > 0. Putting u = x=t then

2(T (x); t) = 2(tT (u); t) = 2(T (u); 1) 1(u; ) = 1xt; = 1

x;t

h

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6 HAKAN EFE

and

2(T (x); t) = 2(tT (u); t) = 2(T (u); 1) 1(u; ) = 1xt; = 1

x;t

h

where h = 1=. So 2(T (x); t) 1 (hx; t) and 2(T (x); t) 1 (hx; t).If x 6= 0 and t 0 then 2(T (x); t) = 0 = 1 (hx; t) and 2(T (x); t) = 0 =

1 (hx; t).If x = 0 and t 2 R then T (0) = 0 and

2(0; t) = 1

0;t

h

= 1 and 2(0; t) = 1

0;t

h

= 0 if t > 0,

2(0; t) = 1

0;t

h

= 0 and 2(0; t) = 1

0;t

h

= 1 if t 0.

Hence T is strongly IF -bounded.Conversely suppose that T is strongly IF -bounded. Then there exists h > 0

such that 2(T (x); ") 1(hx; ") and 2(T (x); ") 1(hx; ") for all x 2 X and forall " > 0. This implies

2(T (x) T (0); ") 1(x 0;"

h) and 2(T (x) T (0); ") 1(x 0;

"

h)

for all x 2 X and for all " > 0. Hence

2(T (x) T (0); ") 1(x 0; ) and 2(T (x) T (0); ") 1(x 0; )

where = "h . This implies that T is strongly IF -continuous at 0 and hence it is

strongly IF -continuous on X.

Theorem 6. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces andT : X ! Y be a linear operator. Then

(i) T is weakly IF -continuous on X i¤ T is weakly IF -continuous at a pointx0 2 X.

(ii) T is weakly IF -continuous i¤ T is weakly IF -bounded.

Proof. (i) Since T is weakly IF -continuous at x0 2 X, thus for each " > 0, thereexist > 0 such that for all x 2 X, 1(x x0; ) and 1(x x0; ) 1 )2(T (x) T (x0); ") and 2(T (x) T (x0); ") 1 . Taking any y 2 X andreplacing x by x+ x0 y we get

1(x+ x0 y x0; ) and 1(x+ x0 y x0; ) 1

) 2(T (x+ x0 y) T (x0); ") and 2(T (x+ x0 y) T (x0); ") 1 .I.e.,

1(x y; ) and 1(x y; ) 1 ) 2(T (x) T (y); ") and 2(T (x) T (y); ") 1 .

Since y 2 X is arbitrary, it follows that T is weakly IF -continuous on X.(ii) We suppose that T is weakly IF -continuous. Using continuity of T at x = 0,

for " = 1, there exists > 0 such that 1(x 0; ) and 1(x 0; ) 1 implies 2(T (x) T (0); 1) and 2(T (x) T (0); 1) 1 , for all x 2 X. I.e.,1(x; ) and 1(x; ) 1 implies 2(T (x); 1) and 2(T (x); 1) 1 ,for all x 2 X.

66


Suppose that x 6= 0 and t > 0. Putting x = u=t then1(u=t; ) and 1(u=t; ) 1

) 2(T (u=t); 1) and 2(T (u=t); 1) 1 , i.e.,1(u; t) and 1(u; t) 1

) 2(T (u); t) and 2(T (u); t) 1 , i.e.,1(hu; t) and 1(hu; t) 1

) 2(T (u); t) and 2(T (u); t) 1 where h = 1

. This implies T is weakly IF -bounded.If x 6= 0 and t 0 then 1 (hx; t) = 2(T (x); t) = 0 and 1 (hx; t) =

2(T (x); t) = 1 for any h > 0.If x = 0 and t 2 R then for h > 0,1 (h 0; t) = 2(T (0); t) = 1 and 1 (h 0; t) = 2(T (0); t) = 0 if t > 0,1 (h 0; t) = 2(T (0); t) = 0 and 1 (h 0; t) = 2(T (0); t) = 1 if t 0.

Hence T is weakly IF -bounded.Conversely suppose that T is weakly IF -bounded. Then there exists h > 0 such

that 1(hx; t) and 1(hx; t) 1 ) 2(T (x); t) and 2(T (x); t) 1 for all x 2 X and for all t 2 R. This implies

1

x 0; t

h

and 1

x 0; t

h

1

) 2(T (x) T (0); t) and 2(T (x) T (0); t) 1 .Then,

1

x 0; "

h

and 1

x 0; "

h

1

) 2(T (x) T (0); ") and 2(T (x) T (0); ") 1 for " > 0. Hence

1 (x 0; ) and 1 (x 0; ) 1 ) 2(T (x) T (0); ") and 2(T (x) T (0); ") 1

where = "h. This implies that T is weakly IF -continuous at x = 0 and hence it

is weakly IF -continuous on X.

Lemma 3. Let (X;; ) be an IF -normed linear spaces satisfying (F) and (FF)and fjj jj : 2 (0; 1)g be the family of corresponding -norms of (; ) on X.Then for x0 2 X, x0 6= 0, (x0; jjx0jj) and (x0; jjx0jj) 1 for all 2 (0; 1).

Proof. Let jjx0jj = , then > 0. There exists a sequence (tn)n, tn > 0, n 2 Nsuch that (x0; tn) , (x0; tn) 1 and tn # . Therefore,limn!1

(x0; tn) and limn!1

(x0; tn) 1

) (x0; limn!1

tn) and (x0; limn!1

tn) 1 by (FF)

) (x0; jjx0jj) and (x0; jjx0jj) 1 forall 2 (0; 1).

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8 HAKAN EFE

Lemma 4. Let (X;; ) be an IF -normed linear spaces satisfying (F) and (FF)and fjjjj : 2 (0; 1)g be the family of corresponding -norms of (; ) on X. Thenfor x0 6= 0, 2 (0; 1) and t0 > 0, jjx0jj = t0 i¤ (x0; t0) = and (x0; t0) = 1.

Proof. Let 2 (0; 1), x0 6= 0 and t0 = jjx0jj = ^fs : (x0; s) , (x0; s) 1 g. Since (x; ) and (x; ) are continuous (by (FF)), from Lemma 3 wehave (x0; t0) and (x0; t0) 1 . Also (x0; t0) (x0; s) if (x0; s) and (x0; t0) (x0; s) if (x0; s) 1 . If possible, let (x0; t0) > and(x0; t

0) < 1 , then by the continuity of (x0; ) and (x0; ) at t0, there existst00 < t0 such that (x0; t00) > and (x0; t00) < 1 which is impossible, sincet0 = ^fs : (x0; s) , (x0; s) 1g. Thus (x0; t0) and (x0; t0) 1.Hence we get (x0; t0) = and (x0; t0) = 1 .Next if (x0; t0) = and (x0; t0) = 1 , 2 (0; 1), then from the denition

jjx0jj = ^ft : (x0; t) , (x0; t) 1g = t0 (Since (x0; ) is strictly increasingon ft > 0 : 0 < (x; t) < 1g and is strictly decreasing on ft > 0 : 0 < (x; t) < 1g).This completes the proof.

Theorem 7. Let (X;; ) be an IF -normed linear spaces satisfying (F) and (FF)and fjj jj : 2 (0; 1)g be the family of corresponding -norms of (; ) on X.Then for any increasing (decreasing) sequence (n)n in (0; 1), n ! impliesjjxjjn ! jjxjj for all x 2 X.

Proof. For x = 0, clearly n ! implies jjxjjn ! jjxjj.Suppose x 6= 0. From Lemma 4, for x 6= 0, 2 (0; 1) and t0 > 0 we have jjxjj =

t0 i¤ (x; t0) = and (x; t0) = 1. Let (n)n be an increasing sequence in (0; 1)such that n ! 2 (0; 1). Let jjxjjn ! tn and (x; t) = and (x; t) = 1 .Since fjj jj : 2 (0; 1)g is an increasing family of norms, (tn)n is an increas-ing sequence of real numbers and it is bounded above by t (since jjxjjn jjxjjfor all n 2 N). Hence (tn)n is convergent. Thus limn!1 (x; tn) = limn!1 nand limn!1 (x; tn) = limn!1(1 n) which implies (x; limn!1 tn) = and(x; limn!1; tn) = 1. Thus we have (x; limn!1 tn) = (x; t) and (x; limn!1; tn) =(x; t), which implies limn!1 tn = t by (FF). Therefore limn!1 jjxjjn = jjxjj.Similarly, if (n)n is a decreasing sequence in (0; 1) and n ! 2 (0; 1), then itcan be shown that jjxjjn ! jjxjj for all x 2 X.

Theorem 8. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces sat-isfying (F) and (FF) and T : X ! Y be a linear operator . Then T is weaklyIF -bounded i¤ T is bounded w.r.t. -norms of (1; 1) and (2; 2), 2 (0; 1).

Proof. First we suppose that T is weakly fuzzy bounded. Thus for all 2 (0; 1),there exists h > 0 such that for all x 2 X, for all t 2 R we have

1 (hx; t) and 1 (hx; t) 1 ) 2(T (x); t) and 2(T (x); t) 1 .

I.e.

_f 2 (0; 1) : jjhxjj1 tg ) (i)

_f 2 (0; 1) : jjT (x)jj2 tg .

Now we show that,

_f 2 (0; 1) : jjhxjj1 tg , jjhxjj1 t.

68

If x = 0 then the relation is obvious. Suppose x 6= 0. Now if(ii) _f 2 (0; 1) : jjhxjj1 tg > then jjhxjj1 t.

If _f 2 (0; 1) : jjhxjj1 tg = , then there exists an increasing sequence (n)nin (0; 1) such that n " and jjhxjj1n t. Then by Theorem 7 we have

(iii) jjhxjj1 t.Thus from (ii) and (iii) we get,

(iv) _f 2 (0; 1) : jjhxjj1 tg ) jjhxjj1 t.

Next we suppose that jjhxjj1 t.If jjhxjj1 < t then 1 (hx; t) and 1 (hx; t) 1 . I.e.

(v) _f 2 (0; 1) : jjhxjj1 tg .

If jjhxjj1 = t i.e. ^fs : 1 (hx; s) and 1 (hx; s) 1 g = t, then thereexists a decreasing sequence (sn)n in R such that sn # t and 1 (hx; sn) and1 (hx; sn) 1 ) limn!1 1 (hx; sn) and limn!1 1 (hx; sn) 1 ) 1 (hx; limn!1 sn) and 1 (hx; limn!1 sn) 1 by (FF).) 1 (hx; t) and 1 (hx; t) 1 .

(vi) ) _f 2 (0; 1) : jjhxjj1 tg .From (v) and (vi) it follows that,

(vii) jjhxjj1 t) _f 2 (0; 1) : jjhxjj1 tg .Hence from (iv) and (vii) we have,

(viii) _f 2 (0; 1) : jjhxjj1 tg , jjhxjj1 t.In a similar way we can show that,

(ix) _f 2 (0; 1) : jjT (x)jj2 tg , jjT (x)jj2 t.Therefore from (viii) and (ix) we have

1 (hx; t) and 1 (hx; t) 1) 2 (T (x); t) and 2 (T (x); t) 1then jjhxjj1 t ) jjT (x)jj2 t. This implies that jjT (x)jj2 hjjxjj1 for all 2 (0; 1).Conversely suppose that for all 2 (0; 1), there exists h > 0 such that

jjT (x)jj2 hjjxjj1 for all x 2 X.Then for x 6= 0, jjhxjj1 t) jjT (x)jj2 t, for all t > 0, i.e.,

^fs : 1 (hx; s) and 1 (hx; s) 1 g t)^fs : 2 (T (x); s) and 2 (T (x); s) 1 g t.

In a similar way as above we can show that

^fs : 1 (hx; s) and 1 (hx; s) 1 g t, 1 (hx; s) and 1 (hx; s) 1

and

^fs : 2 (T (x); s) and 2 (T (x); s) 1 g t, 2 (T (x); t) and 2 (T (x); t) 1 .

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10 HAKAN EFE

Thus we have

1 (hx; t) and 1 (hx; t) 1) 2 (T (x); t) and 2 (T (x); t) 1for all x 2 X.If x 6= 0, t 0 and x = 0, t > 0 then the above relation is obvious. Hence the

theorem follows.

4. IF -Compact Operators

Denition 10. A subset A of an IF -normed linear space (X;; ) is said to beIF -bounded i¤ there exist t > 0 and 0 < r < 1 such that (x; t) > 1 r and(x; r) < r for all x 2 A.

Denition 11. A subset A of an IF -normed linear space (X;; ) is said to beIF -compact if any sequence (xn)n in A has a subsequence converging to an elementof A.

Denition 12. The IF -closure of a subset B of an IF -normed linear space (X;; )is denoted by B and dened by the set of all x 2 X such that there is a sequence(xn)n of elements of B with xn ! x. We say that B is IF -closed if B = B.

Denition 13. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces. Alinear operator T : X ! Y is called IF -compact operator if for every IF -boundedsubset M of X the subset of T (M) of Y is relatively compact, that is the IF -closureof T (M) is a IF -compact set.

Example 3. Let (X; jj jj1) and (Y; jj jj2) be two ordinary normed linear spaces,and T : X ! Y be a compact operator. Then it is easy to see that T : (X;1; 1)!(Y; 2; 2) is a IF -compact operator, where (1; 1) and (2; 2) are the standardIF -norms induced by ordinary norms jj jj1 and jj jj2, respectively, i.e.,

i(x; t) =

tt+jjxjji if t > 0, t 2 R,0 if t 0 ,

and

i(x; t) =

(jjxjjit+jjxjji if t > 0, t 2 R,0 if t 0

for i = 1; 2.

Example 4. Let C[0; 1] be the set of all real valued continuous functions on [0; 1]with the IF -norm

('(x); t) =t

t+ supx2[0;1] j'(x)jand ('(x); t) =

supx2[0;1] j'(x)jt+ supx2[0;1] j'(x)j

,

where '(x) 2 C[0; 1] and t > 0. If k(x; y) with x; y 2 [0; 1] is a real valued continu-ous function, then the operator T : C[0; 1]! C[0; 1] dened by

(T')(x) =

Z 1

0

k(x; y)'(y)dy,

where ' 2 C[0; 1] is an IF -compact operator.

Theorem 9. Let T : (X;1; 1) ! (Y; 2; 2) be a linear operator. Then T isIF -compact i¤ it maps every IF -bounded sequence (xn)n in X onto a sequence(T (xn))n in Y which has an IF -convergent subsequence.

70

Proof. Suppose that T be a IF -compact operator and (xn)n be an IF -boundedsequence in (X;1; 1). The IF -closure of fT (xn) : n 2 Ng is an IF -compact set.So (T (xn))n has an IF -convergent subsequence by denition. Conversely, let Abe a IF -bounded subset of (X;1; 1). We show that the IF -closure of T (A) isIF -compact. Let (xn)n be a sequence in the closure of T (A). For given " > 0,n 2 N and t > 0, there exists (yn)n in T (A) such that 2(xn yn; t2 ) > 1 " and2(xn yn; t2 ) < ". Let yn = T (zn), where zn 2 A. Since A is IF -bounded set, sois fzn : n 2 Ng. On the other hand, because T is IF -compact operator, T (zn) hasan IF -convergent subsequence (ynk)k = (T (znk))k. Let ynk ! y for some y 2 Y .Hence 2

ynk y; t2

> 1 " and 2

ynk y; t2

< " for all nk > n0. We have

2 (xnk y; t) min

2

xnk ynk ;

t

2

; 2

ynk y;

t

2

> 1 ",

2 (xnk y; t) max

2

xnk ynk ;

t

2

; 2

ynk y;

t

2

< "

for all nk > n0. Hence (xnk)k is an IF -convergent subsequence of (xn)n. Thus theIF -closure of T (A) is an IF -compact set.

Denition 14. Let (X;; ) be an IF -normed linear space. We dene the followingsubset of X:

B[x; r] = fy 2 X : (x y; t) and (x y; t) 1 g,where x 2 X, 2 (0; 1) and r > 0.

Theorem 10. Let (X;; ) be an IF -normed linear space satisfying (F)and, (x; )and (x; ) are continuous functions on R. Then X is nite dimensional i¤ B[x; r]is an IF -compact set in X, for each 2 (0; 1) and r > 0.

Proof. Let A[x; r] = fy 2 X : jjx yjj rg, where 2 (0; 1) and r > 0.We rst show that B[x; r] = A[x; r]. If y 2 B[x; r], then (x y; t) and (x y; t) 1 . Since jjx yjj r, then y 2 A[x; r]. Now if y 2 A[x; r],then jjx yjj r, or ^ft > 0 : (x y; t) and (x y; t) 1 g r.If ^ft > 0 : (x y; t) and (x y; t) 1 g < r, then (x y; t) and (x y; t) 1 . Thus y 2 B[x; r]. If ^ft > 0 : (x y; t) and (x y; t) 1 g = r, there is (tn)n such that tn ! r, and (x y; t) and (x y; t) 1 . By continuity of (x; ) and (x; ) we obtain (x y; r) =limn!1 (x y; tn) and (x y; r) = limn!1 (x y; tn) 1 . Hencey 2 B[x; r]. Consequently B[x; r] = A[x; r].Suppose now that dimX <1, x 2 X, and r > 0. Choose the sequence (xn)n in

B[x; r]. It is clear that A[x; r] is a compact subset of (X; jj jj). Hence there is asubsequence (xnk)k of (xn)n and v 2 A[x; r] such that xnk

jjjj! v. Because in nite

dimensional spaces all norms are equivalent, xnkjjjj! v, for all 2 (0; 1). Thus by

Lemma 2 we obtain xnk(;)! v. Since B[x; r] = A[x; r], we have v 2 B[x; r].

Conversely, let B[x; r] be IF -compact. To show that X is nite dimensional,it su¢ ces to prove that A[x; r] is compact with respect to -norm. Choose asequence (xn)n of A[x; r]. Since B[x; r] is IF -compact, it has a IF -convergentsubsequence (xnk)k. Lemma 2 implies that (xnk)k is convergent under jj jj. ThusA[x; r] is compact in normed linear space (X; jj jj). This shows that X is nitedimensional.

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12 HAKAN EFE

Lemma 5. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces satisfying(F) and T : X ! Y be an IF -fuzzy compact operator. Then T : (X; jj jj1) !(Y; jj jj2) is an ordinary compact operator for all 2 (0; 1).Proof. We show that for each bounded sequence (xn)n in (X; jj jj1), the sequence(T (xn))n has a convergent subsequence in (Y; jj jj2). Let (xn)n be a boundedsequence in (X; jj jj1). There exists M > 0 such that jjxnjj1 < M for all n 2 N.Hence 1 (xn;M) and 1 (xn;M) 1 , for all n, that is (xn)n is IF -bounded. Thus (T (xn))n has an IF -convergent subsequence (T (xnk))k. By Lemma2, (T (xnk))k is convergent under jj jj2. Theorem 11. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces sat-isfying (F) and (FF). Then

(a) Every IF -compact linear operator T : X ! Y is weakly IF -continuous.(b) If dimX =1 then the identity operator I : X ! X is not an IF -compact

operator.

Proof. (a) Choose 2 (0; 1). Let jj jj1 and jj jj2 are -norms on X and Ycorresponding to the IF -norms (1; 1) and (2; 2), respectively. By Lemma 5, T :(X; jj jj1)! (Y; jj jj2) is a compact operator. Since compact operator is bounded,there existsM > 0 such that jjT (x)jj2 Mjjxjj1. Hence T is weakly IF -boundedby Theorem 8. Now Theorem 6 implies that T is weakly IF -continuous.(b) The identity operator I maps B[0; 1] to itself. Suppose on the contrary that

I is a IF -compact operator. Then B[0; 1] is IF -compact for all 2 (0; 1). NowB[0; 1] A[0; 1] = B[0; 1] implies that B[0; 1] is closed and so IF -compact.Thus X is nite dimensional by Theorem 10, which is a contradiction. Theorem 12. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces. Thenthe set of all IF -compact linear operators from X into Y is a linear subspace ofIF 0(X;Y ).

Proof. Suppose that T1 and T2 are IF -compact linear operators from X intoY . Let (xn)n be any IF -bounded sequence in X. The sequence (T1(xn))n hasa IF -convergent subsequence (T1(xnk))k. The sequence (T2(xnk))k also has anIF -convergent subsequence (T2(zn))n. Hence (T1(zn))n and (T2(zn))n are IF -convergent sequences. Let T1(zn)! u, and T2(zn)! v. If t > 0, we have

limn!1

2((T1+T2)(zn)uv; t) limn!1

min

2

T1(zn) u;

t

2

; 2

T2(zn) v;

t

2

,

and

limn!1

2((T1+T2)(zn)uv; t) limn!1

max

2

T1(zn) u;

t

2

; 2

T2(zn) v;

t

2

.

Thus, limn!1 2((T1 + T2)(zn) u v; t) = 1 and limn!1 2((T1 + T2)(zn) u v; t) = 0, for all t > 0. This implies T1 + T2 is an IF -compact operator. Nowif T1(xnk)! y, then

limn!1

2 (T1(xnk) y; t) = limn!1

2

T1(xnk) y;

t

jj

= 1 and

limn!1

2 (T1(xnk) y; t) = limn!1

2

T1(xnk) y;

t

jj

= 0

for all 2 Rnf0g, and t > 0. Hence T1 is also an IF -compact operator.

72

Theorem 13. Let (X;; ) be an IF -normed linear space, T : X ! X be anIF -compact linear operator, and S : X ! X be a strongly IF -continuous linearoperator. Then ST and TS are IF -compact operators.

Proof. Let (xn)n be any IF -bounded sequence in X. Then (T (xn))n has an IF -convegent subsequence (T (xnk))k. Let limn!1 T (xnk) = y. Since S is stronglyIF -continuous, by Theorem 3 we have S(T (xnk)) ! S(y). Hence ST (xn) has anIF -convergent subsequence. This proves ST is IF -compact. Now to show that TSis IF -compact, choose any IF -bounded sequence (xn)n. There exist t0 > 0 and r0 2(0; 1) such that 1 (xn; t0) > 1 r0 and 1 (xn; t0) < r0 for all n 1. By Theorem6 we conclude that the operator S is a strongly IF -bounded linear operator. Thusthere exists M > 0 such that 1 (S(xn); t0M) > 1 r0 and 1 (S(xn); t0M) < r0,for all n. It follows that (S(xn))n is IF -bounded sequence in S(X). Because T isfuzzy compact, (TS(xn))n has an IF -convergent subsequence. This completes theproof.

Lemma 6. Let (X;; ) be an IF -normed linear space satisfying (F), (x; ) and (x; ) are continuous functions on R and dimX < 1. Then each IF -boundedsequence (xn)n in (X;; ) has an IF -convergent subsequence.

Proof. Let (xn)n be an IF -bounded sequence in (X;; ). There are t0 > 0 andr0 2 (0; 1) such that (xn; t0) > 1 r0 and (xn; t0) < r0, for all n 2 N. Hence(xn)n 2 B1r0 [0; t0]. By Theorem 10, B1r0 [0; t0] is an IF -compact set, so (xn)nhas an IF -convergent subsequence.

Theorem 14. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces sat-isfying (F) and (FF). If T : X ! Y is a linear operator where dimX <1, thenT is weakly IF -continuous.

Proof. Since (X;1; 1) and (Y; 2; 2) satisfy (F), we may suppose that jj jj1 andjjjj2 are -norms on X and Y corresponding to the IF -norms (1; 1) and (2; 2),respectively. Since T is of nite dimension, thus T : (X; jj jj1) ! (X; jj jj2) is abounded linear operator for each 2 (0; 1). Thus by Theorem 8, it follows that Tis weakly IF -bounded.

Theorem 15. Let (X;1; 1) and (Y; 2; 2) be two IF -normed linear spaces sat-isfying (F) and 2(x; ) and 2(x; ) are continuous function on R, and T : X ! Ya linear operator. Then the following hold:

(a) If T is weakly IF -bounded and dimT (X) < 1, then T is an IF -compactoperator.

(b) In addition if (X;1; 1) and (Y; 2; 2) satisfying (FF) and dimT (X) <1, then T is an IF -compact operator.

Proof. (a) Let (xn)n be an IF -bounded sequence of X. There are t0 > 0 andr0 2 (0; 1) such that (xn; t0) > 1 r0 and (xn; t0) < r0, for all n 2 N. Since Tis weakly IF -bounded, there is M1r0 > 0 such that for all n,

1(xn; t0) 1 r0 and 1(xn; t0) r0 )

2

T (xn);

t0M1r0

1 r0 and 2

T (xn);

t0M1r0

r0.

73

14 HAKAN EFE

It follows that (T (xn))n is an IF -bounded sequence in T (X). Since dimT (X) <1,the sequence (T (xn))n has a convergent subsequence by Lemma 6. Hence T is IF -compact.(b) Theorem 14 implies that T is weakly IF -continuous. We also imply by

Theorem 6 that T is weakly IF -bounded. Since dimT (X) < 1, by the (a) weconclude that T is an IF -compact operator.

References

[1] T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11(2003), 687-705.

[2] T. Bag, S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst., 151 (2005), 513-547.[3] S.C. Cheng, J.N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Cal.

Math. Soc., 86 (1994), 429-436.[4] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst., 48 (1992), 239-248.[5] O. Kaleva, S. Seikala, On fuzzy metric spaces, Fuzzy Sets Syst., 12 (1984), 215-229.[6] F. Lael, K. Nourouzi, Fuzzy compact linear operators, Chaos Solitons and Fractals in press

doi: 10.1016/j.chaos.2006.04.055.[7] F. Lael, K. Nourouzi, Some results on the IF -normed spaces, Chaos Solitons and Fractals in

press doi: 10.1016/j.chaos.2006.10.019.[8] A.K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets Syst., 12 (1984), 143-154.[9] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975),

326-334.[10] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons and Fractals, 22 (2004), 1039-

1046.[11] J.-z. Xiao, X.-h. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and

Syst., 133 (2003), 389-399.

(Hakan Efe), Department of Mathematics, Faculty of Science and Arts, Gazi Univer-sity, Teknikokullar, 06500 Ankara, TurkeyE-mail address : [email protected]

74

I-CONVERGENCE OF POSITIVE LINEAR OPERATORS ON LpWEIGHTED SPACES

F. Dirika, A. Aralb, K. Demircic;

a;cDepartment of Mathematics, Faculty of Sciences and Arts, Sinop University, 57000Sinop, Turkey

bK¬r¬kkale University Department of Mathematics, 71450 Yahsihan, K¬r¬kkale, Turkey

Abstract. In this paper, using the concept of I-convergence we prove aKorovkin type approximation by means of positive linear operators dened on theweighted space Lp;! (R). Also we state its n-dimensional analogue for the weightedspace Lp;(Rn). Also we display an example such that our method of convergence isstronger than the usual convergence in the weighted spaces Lp;! (R) and Lp;(Rn).

1. IntroductionThe concept of convergence of a sequence of real numbers has been extended to statisticalconvergence by Steinhaus [7] and Fast [2]. If K N, the set of natural numbers, thenKn denotes the set fk 2 K : k ng and jKnj denotes the cardinality of Kn. The naturaldensity of K is given by d (K) := limn 1

n jKnj, if limit exists. A sequence x = (xk) of realnumbers is statistically convergent to L if for every " > 0 the set fk 2 N : jxk Lj "ghas the natural density zero.A generalization of statistical convergence which is based on the structure of the ideal

I of subsets of N is given by Kostyrko, Maµcaj, alát and Sleziak [6].A non-void class I P (N) is called the ideal if I is additive ( i.e., A, B 2 I )

A [B 2 I) and hereditary (i.e., A 2 I, B A) B 2 I).Throughout in this paper we consider admissible ideals, i.e. those which are di¤erent

from P (N) and contain all singletons. It is easy to check that I = fK N : d (K) = 0gforms an admissible ideal. A sequence x = (xk) of real numbers is I-convergent to L iffk : jxk Lj "g 2 I for every " > 0. In this case we write I limx = L. It is knownthat any convergent sequence is I-convergent, but not conversely. Some examples andproperties of I-convergence may be found in [6]. 12

2. The Main ResultsFor a sequence (Ln) of positive linear operators on C (a; b), which is the space of continuousfunctions on [a; b] and bounded on real axis R, Korovkin [5] solved a problem which basedon the existence of the limit limn Ln (f ;x) = f (x). Also, Curtis [1] has extended thistheorem for the functions in Lp (; ). Gadjiev [3] stated and proved weighted Korovkintype theorems in the space of locally integrable functions in R. Recently, Gadjiev andAral [4] have investigated Korovkin type approximation theorems in the weighted spacesLp; ! (R) and Lp; (Rn).The purpose of the present paper is to study a Korovkin type approximation theorem

via I-convergence in the weighted space Lp; ! (R) and Lp; (Rn).12000 AMS Subject Classications: Primary 41A10, 41A25, 41A36; Secondary 40A05, 40A30.Key words and phrases. I-convergence, positive linear operator, Korovkin theorem, weighted space.2 Corresponding authorEmail Address : [email protected] (F. Dirik), [email protected] (A. Aral), [email protected] (K.

Demirci)

1


I-CONVERGENCE OF POSITIVE LINEAR OPERATORS ON LpWEIGHTED SPACES2

For a xed p 2 [1;1); let ! be a positive continuous function on R satisfying thecondition Z

R

t2p! (t) dt <1. (2.1)

By Lp; ! (R) (1 p <1) we denote the linear space of all functions f which are measur-able, pabsolutely integrable on R with respect to the weight function !, that is,

Lp; ! (R) =

8><>:f : R! R : kfkp; ! :=

0@ZR

jf (t)jp ! (t) dt

1A 1p

<1

9>=>; .The minimum and maximum values of the function ! on nite intervals will be denoted

by !min and !max respectively.

Theorem 1. Let I be an admissible ideal in N. Let (Ln)n2N be the sequence of positivelinear operators Ln : Lp; ! (R) ! Lp; ! (R) and let the sequence fkLnkg be a uniformlybounded. If

I lim Ln ti; x xi p; ! = 0; i = 0; 1; 2, (2.2)

then for any function f 2 Lp; ! (R) ; we have

I lim kLnf fkp; ! = 0:

Proof. We follow the proof of Theorem 1 in [4] up to a certain stage.We can choose a such large number A such that for every " > 0 fA2 p; ! < ", (2.3)

where A1 be characteristic function of the interval [A; A] and A2 (t) = 1 A1 (t).Since Ln is linear operator, we have

kLnf fkp; ! = Ln A1 + A2 f A1 + A2 f p; ! (2.4)

Ln A1 f A1 f p; ! + Ln A2 f A2 f p; !

= I0

n + I00

n .

Firstly, we compute I00

n : Since fLng is a uniformly bounded sequence, there exists a con-stant K > 0 such that

kLnkp; ! K. (2.5)

Hence, from (2:3), we have

I00

n Ln A2 f p; ! + A2 f p; ! (2.6)

(K + 1) A2 f p; !

< (K + 1) ":

Furthermore, for every function f 2 Lp;!(R) the inequality

kA1 fkp !1=pmin kfkp;!

DIRIK ET AL76

implies Lp;!(R) Lp(A;A). Since the space of continuous functions is dense in Lp(A;A),given f 2 Lp;!(R), for each "0 > 0; there exists a continuous function ' on [A; A] sat-isfying the condition '(x) = 0 for jxj > A such that

(f ')A1 p < "0

(K + 1)!1=pmax

.

We can write from these inequalities and (2:5)

I0

n = Ln A1 f A1 f p; ! (2.7)

Ln (f ')A1 p; ! + Ln 'A1 'A1 p; ! + (f ')A1 p; !

Ln 'A1 'A1 p; ! + "0.

Since A12

A1 ' = 0 for some A1 > A, we obtain the following inequality for the rst

term of (2:7) Ln 'A1 'A1 p; ! = A1

1 + A12

Ln'A1

A11 + A1

2

'A1

p; !

(2.8)

Ln 'A1 'A1 A1

1

p; !

+ A1

2 Ln'A1

p; !

.

Now, by denoting M' = maxt2R

j' (t)jA1 (t), we get

A12 Ln

'A1

p; !

=

0B@ Zjtj>A1

Ln 'A1 ; tp ! (t) dt1CA

1p

M'

0B@ Zjtj>A1

jLn (1; t) 1jp ! (t) dt

1CA1p

+M'

0@ZR

A12 (t)! (t) dt

1A 1p

.

According to the hypotheses of the theorem, since ! 2 L1(R) we can choose the numberA1 such that 0@Z

R

A12 (t)!(t)dt

1A 1p

<"0

M':

So we have A12 Ln

'A1

p; !

M' kLn (1; x) 1kp; ! + "0. (2.9)

Substituting these inequalities into (2:7), we have

I 0n 2"0 +M' kLn (1; x) 1kp; ! + Ln 'A1 'A1 A1 p; ! . (2.10)

77

By the continuity of 'A1 on [A; A], for given any "0 > 0 there is a > 0 such that' (t)A1 (t) ' (x)A1 (x) < "0 + 2M'(t x)2

2.

Thus, Ln 'A1 'A1 A11

p; !

Ln ' (t)A1 (t) ' (x)A1 (x) ; xA1

1 (x) p; !(2.11)

+ ' (x)A1

1 (x) (Ln (1; x) 1) p; !

"0 +

2M'

2A2 +M'

kLn(1; x) 1kp;!

+4M'

2A kLn (t; x) xkp; ! +

2M'

2 Ln t2; x x2 .

Substituting (2:11) into (2:10), we get

I 0n 2"0 +

"0 +

2M'

2A2 +M'

kLn(1; x) 1kp;! +

4M'

2AkLn (t; x) xkp; !

+2M'

2 Ln t2; x x2 p; ! .

Then, the inequality (2:4) is obtained the following,

kLnf fkp; ! 2"0 + (K + 1) "+

"0 +

2M'

2A2 +M'

kLn(1; x) 1kp;!

+4M'

2A kLn (t; x) xkp; ! +

2M'

2 Ln t2; x x2 p; ! .

We conclude that

kLnf fkp; ! 2"0 + (K + 1) "+BnkLn(1; x) 1kp;! + kLn (t; x) xkp; !

+ Ln t2; x x2 p; !o

where B := maxn"0 +

2M'

2A2 +M';

4M'

2;2M'

2

o. The last inequality shows that, for any

"00> 0,n

n : kLnf fkp; ! "00o

fn : 2"0 + (K + 1) "+B fkLn(1; x) 1kp;! (2.12)

+ kLn (t; x) xkp; !+ Ln t2; x x2 p; ! "00oo .

DIRIK ET AL78

Now we write

D : =nn : kLnf fkp; ! "

00o,

D1 : =

(n : kLn (1; x) 1kp; !

"00 (2"0 + (K + 1) ")

3B

),

D2 : =

(n : kLn(t;x) xkp;!

"00 (2"0 + (K + 1) ")

3B

),

D3 : =

(n : Ln(t2;x) x2 p;! "

00 (2"0 + (K + 1) ")

3B

).

Then, it follows from (2:12) that D D1 [D2 [D3. By (2:2), Di 2 I for each i = 1; 2; 3.So, by the denition of ideal, D1 [D2 [D3 2 I, which yields D 2 I. So we haven

n : kLnf fkp; ! "00o2 I

whence the result.Now we give an example of a sequence of positive linear operators such that this

operator satises the conditions of Theorem 1 but does not satisfy the conditions ofclassical Korovkin theorem in weighted space Lp; ! (R).

Example 2. We choose ! (x) =

11+x6m

p, p 1. By Lp;m (R) we denote the space of

Lp; ! (R). Note that this selection of ! (x) satises the condition (2:1). Also note that for1 p <1

Lp;m (R) =nf : R! R :

1 + x6m

1f (x) 2 Lp (R)

o,

where m is a positive integer.The Kantorovich variant of the Szasz-Mirakyan operators [8] by replacing f

kbnn

with

with an integral mean of f (x) over the interval [(k + 1) bn=n; kbn=n] as follows:

Kn (f ;x) :=n

bn

1Xk=0

Pn;k (x)

Z (k+1)bn=n

kbn=n

f (t) dt; n 2 N; x 2 [0; bn) (2.13)

where (bn) is a sequence of positive real numbers satisfying the condition

limn!1

bnn= 0 and lim

n!1bn =1:

and

Pn;k (x) := enx=bn (nx)

k

k!bkn, k = 0; 1; 2; :::.

This operator satises the conditions of the classical Korovkin theorem in [4]. Now deneTn : Lp;m (R+) ! Lp;m (R+) by Tn (f ;x) = (1 + n)Kn (f ;x) where (Kn) is dened by(2:13) and (n) is an I-convergent null sequence but not convergent. So it is possible toconstruct such an (n). Without of generality we may assume that (n) is a non-negative.It is known that

Kn (1;x) = 1, Kn (t;x) = x+bnnand Kn

t2;x

= x2 +

2bnnx+

b2n3n2

.

79

Hence, (Tn) satises all conditions of Theorem 1. So, for any function f 2 Lp; ! (R), wehave

I lim kLnf fkp; ! = 0,

but (Tn) does not satisfy the classical Korovkin theorem.

Now we establish an analogue of Theorem 1 for the weight space Lp;(Rn).Let be a positive continuous function in Rn, satisfying the conditionZ

Rn

jtj2p (t) dt <1

and for 1 p <1

Lp; (Rn) =

8><>:f : Rn ! R; kfkp; =

0@ZRn

jf (t)jp (t) dt

1A1=p

<1

9>=>; .Theorem 3. Let I be an admissible ideal in N. Let (Ln)n2N be the sequence of positivelinear operators Ln : Lp; (Rn)! Lp; (Rn) and let the sequence fkLnkg be a uniformlybounded. If

I limn!1

kLn (1; x) 1kp; = 0,

I limn!1

kLn (ti; x) xikp; = 0, i = 1; 2; ::; n,

I limn!1

Ln jtj2 ; x jxj2 p;

= 0,

then for any function f 2 Lp; (Rn),we have

I limn!1

kLnf fkp; = 0.

Proof. Using a similar idea, the wanted result is obtained.

Conclusion 4. If we dene If = fK N : jKj <1g, Id = fK N : d (K) = 0g, IdA= fK N : dA (K) = 0g and I = fK N : (K) = 0g, then we get the denitions ofusual convergence, statistical convergence, A-statistical convergence and -statistical con-vergence, respectively. Details may be found in [5]. So, Theorem 1 and Theorem 3 arevalid in this cases.

References[1] P. C., Jr Curtis, The degree of approximation by positive convolution operators,Michi-

gan Math. J., No:2, 12, 153-160 (1965)

[2] H. Fast, Sur la convergence statistique, Colloq. Math., 2, 241-244, (1951).

[3] A. D. Gadjiev, R. O. Efendiyev, E. Ibikli, On Korovkin type theorem in the space oflocally integrable functions, Czechoslovak Mat. Journal 53, No:1, (128), 45-53, (2003).

[4] A. D. Gadjiev and A. Aral, Weighted Lp-approximation with positive linear operatorson unbounded sets, Appl. Math. Lett., doi:10.1016/j.aml.2006.12.007, (2007).

DIRIK ET AL80


[5] P. P. Korovkin, Linear operators and the theory of approximation, India, Delhi, (1960).

[6] P. Kostryko, M. Maµcaj, T. alát, M. Sleziak, I-convergence and Extremal I-limitpoints, Math. Slovaca, 55, No. 4, 443-464, (2005).

[7] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq.math. 2, 73-74, (1951).

[8] O. Szász, Generalization of S. Bernsteins polynomials to innite interval, J. ResearchNat. Bur. Standards, 45, 239-245, (1959).

81

Identication of planar screens at lowfrequencies in thermoelasticity

Drossos Gintides and Kiriakie KiriakiDepartment of Mathematics

National Technical University of AthensZografou Campus, 15780 Athens, Greece

Abstract

In this paper the problem of determining a screen in an isotropicand homogeneous thermoelastic medium at low frequencies is consid-ered. We formulate the direct problem for the planar screen in thethermoelastic medium and present an equivalent model for the prob-lem under consideration at low-frequencies based on an non - homoge-neous formulation via appropriate Dirac measures. We prove that thecorresponding inverse problem of reconstructing the planar screen fortwo important cases: the thermal stress dislocation and the thermaldisplacement dislocation from boundary measurements has a uniquesolution. Finally, we present a reconstruction method for the abovecases based on a proper use of certain vector test functions and theapplication of the two-sided Laplace transform.

1 Introduction

Scattering theory has played a central role in the scientic area of math-ematical physics. The inverse scattering problem is a basic one in manyapplications, so, it concentrates the most interest and is in the foreground ofthe mathematical research. An excellent presentation of theoretical resultsand methods exploiting the inverse problem can be found in [10]. Problemsconnected with the scattering of waves by a very thin obstacle became veryimportant, nding applications especially in non-destructive tests. For the

1


identication of cracks by boundary measurements signicant results amongothers have been obtained by Friedman and Vogelius [14], Alessandrini etal [1] and Kress et al [9]. Amari et al in a series of papers, between them[2, 3], have proposed a new method for solving inverse problems using low-frequency waves. More precisely in [2] a computational attractive methodis introduced for the identication of planar cracks located deep inside aheterogeneous conducting body based on low-frequency asymptotic analysisof Maxwells equations. In [4] complete asymptotic expansions of solutionsof the system of elastostatics in the presence of an inclusion are presentedwhile in [15] identication of elastic inclusions and elastic moment tensors byboundary measurements is investigated.In this work we examine an inverse scattering problem for a screen in

thermoelasticity. Thermoelasticity combines the theory of elastodynamicswith the theory of heat conduction. The governing equations form a coupledsystem of three hyperbolic and one parabolic equation. The hyperbolic equa-tions have a source term which is proportional to the temperature gradient,while the parabolic equation has a source term which is proportional to thedivergence of the velocity. In [16] can be found an excellent introduction ofthe theory of thermoelasticity. In [11] the scattering process, the scatteringamplitudes and cross sections in thermoelasticity are presented. The the-ory of thermoelastic waves in the low-frequency region has been developedin [11]. Results for specic applications at low frequencies can be found in[8, 12]. In [7] a complete analysis of the three-dimensional thermoelasticscattering problems from screens is presented. In a recent work [5] the reci-procity gap principle is exploited in order to identify planar cracks in whichare all located in the same plane in a homogeneous and isotropic thermoelas-tic medium. Their method estimates the plane by an explicit formula.In Section 2 we formulate the direct scattering problem for the planar

screen in the thermoelastic medium. Two types of screens are considered,thermal stress dislocations and displacements dislocations. The governingequations, the boundary conditions for each case and the radiation condi-tions for the well-posedness of the problem are presented. In Section 3 anequivalent model for the problem under consideration at low-frequencies ispresented. Based on results for the elastic case given in [6] we prove that thedirect problem for the screen can be replaced by a model prescribed by a non-homogeneous equation having a force term corresponding to a thermoelasticsources distribution on the planar screen. In Section 4 we face the inverseproblem to determine the shape of the screen and the estimation of the cor-

2

GINTIDES-KIRIAKI84

responding Burgers vectors which describe mainly the physical properties ofthe dislocation [6]. The reconstruction method for the thermal stress disloca-tion as well as the thermal displacement dislocation is presented. Firstly, themethod gives the plane of the screen and secondly provides a reconstructionof the screen by an inversion of a two-sided Laplace transform applied ondata produced by Greens formula.

2 Formulation of the direct scattering prob-lem

We consider the scattering process in a homogeneous and isotropic ther-moelastic medium of Biot type in R3. The Biot medium is characterized bythe real Lamé constants ; where > 0, + 2 > 0 and mass density ,the coe¢ cient of thermal di¤usity and the coupling constants and .We assume that inside the thermoelastic medium we have a planar dis-

location which is a bounded, simply connected, orientable smooth surfacewith a smooth non-self intersecting boundary. We denote the two sides of as + and and we will use the superscripts + and - to indicate that thecorresponding quantities are measured at neighboring points on + and respectively. The unit normal vector is denoted as b and, due to the factthat we consider a planar dislocation, it is the same on each point of . Weassume that b = b = b+.If we consider only time harmonic elds then the Biot system assumes

the following spectral form + !2

u (x) = r (x) ; x 2R3n (1)

+i !

(x) = i ! r u (x) ; x 2R3n (2)

where u (x) is the elastic displacement, (x) denotes the thermal variationeld, is the Lamé operator and ! is the frequency. We dene as q = i !

the spectral thermal constant. Note that whenever ! 0+ , ! 0+ theBiot system decouples into the Navier equation of dynamic elasticity and theheat conduction equation. Introducing a four dimensional vector notation asin [11], we have

U (x) :=

u (x) (x)

(3)

3

LOW FREQUENCES IN THERMOELASTICITY 85

and the Biot system (1, 2) is simply written as

LU (x) = 0; ;x 2R3n (4)

where L is the di¤erential operator L :=

( + !2) I3 rq r + q

. The

thermoelastic operator L is a 4x4 matrix elliptic di¤erential operator. Itis not self-adjoint and the adjoint operator L may be obtained by L byreplacing with i ! and vice versa.A unied eldU (x) which satises (4) admits a decomposition into three

vector elds

U (x) = U1 (x) +U2 (x) +Us (x) (5)

with

U1 (x) =u1 (x) ; 1 (x)

; U2 (x) =

u2 (x) ; 2 (x)

;Us (x) = (us (x) ; 0)

(6)The displacement elds u1 (x) ;u2 (x) and us (x) satisfy the following vecto-rial Helmholtz equations

+ k21u1 (x) = 0;

+ k22

u2 (x) = 0;

+ k2s

us (x) = 0 (7)

The temperature elds satisfy the scalar Helmholtz equations+ k21

1 (x) = 0;

+ k22

2 (x) = 0 (8)

The dispersion relations characterizing equation (4) are written as

k21 + k22 = q (1 + ) + k

2p; k

21k22 = q k

2p; k

2s = !

2 (9)

where k1 and k2 are the complex wavenumbers of the elastothermal andthermoelastic waves respectively and are given by

kj = !=vj + idj; vj > 0; dj > 0; j = 1; 2

vj are the phase velocities and dj determine the corresponding dissipation co-e¢ cients, ks = !

p= is the wavenumber of the uncoupled transverse wave;

kp = !p= (+ 2) is the wavenumber of the longitudinal wave in the ab-

sence of thermal interactions and = = (+ 2) is the dimensionlessthermoelastic coupling constant. From the dispersion relations it comes out

4

GINTIDES-KIRIAKI86

that the transverse eld is not a¤ected by the existence of the thermal eld.The longitudinal eld gives birth to two curl free elds, u1 (x) and u2 (x),which are characterized as the elastothermal and thermoelastic eld respec-tively and in view of (9) are involved in the thermoelastic propagation ofwaves.On the surface of the dislocation we assume that one of the following

boundary conditions holds

Bj (@x; b)U (x) = 0; x 2; j = 1; 2; 3; 4 (10)

where the unied thermoelastic total eld U (x) is given by

U (x) = Uin (x) +Usc (x) (11)

Uin (x) is the incident eld, which is an entire solution of (4), and Usc (x) isthe scattered eld. If the incident eld is a plane which propagates in the bddirection then it admits the following form

Uin (x) = A1

bd1

ei k1

bdx+A22bd1

ei k2

bdx+A3 bb0

ei ks

bdx (12)where 1 =

i k1qk21q

; 1 =ik2

(k21k22):ThefourboundarydifferentialoperatorsBj

are given by

B1 (@x; b) = I3 00 1

(13)

B2 (@x; b) = R (@x; b) = T (@x; b) b0 @v

(14)

B3 (@x; b) = I3 00 @v

(15)

B4 (@x; b) = T (@x; b) b0 1

(16)

I3 is 3x3 unit matrix and the operator T (@x; b) is the surface traction oper-ator of elasticity and is given by

T (@x; b) = 2 b r+ br + b r (17)

5


The rst thermoelastic problem corresponds to a rigid screen at constanttemperature, it is of Dirichlet type, the second to a screen with a Neumanntype condition in thermal insulation, the third to a Dirichlet condition inthermal insulation and the fourth to a Neumann type problem at a constanttemperature.For the well posedness of the exterior boundary value problems, the scat-

tered eld Usc (x) must satisfy the Kupradze radiation conditions [16] asr = jxj ! 1 for i = 1; 2; 3 and j = 1; 2

uj (x) = o

1

r

; @xiu

j (x) = O

1

r2

;

j (x) = o

1

r

; @xi

j (x) = O

1

r2

us (x) = O

1

r

; r (@xiu

s (x) i ksus (x)) = O1

r

(18)

Uniqueness and existence theorems for thermoelastic screens in R3 have beenproved by Cakoni [7].

3 The equivalent model at low frequencies

In what follows, we will consider the rst two boundary value problems corre-sponding to the boundary operators (13) and (14). For the remaining bound-ary conditions, corresponding to the boundary operators (15) and (16), theconsideration is similar and all the remaining analysis is straightforward.Let us assume that the unied thermoelastic vector U (x) and the sur-

face traction - ux vector R (@x; b)U (x) be discontinuous across . Wedene as usually with the bracket notation the discontinuities of the thermaldisplacement eld

[U (x)] := U+ (x)U (x) (19)

and the thermal stress eld

[R (@x; b)U (x)] := R (@x; b)U+ (x)R (@x; b)U (x) (20)

We consider a su¢ ciently large domain containing the dislocation . Using

6

GINTIDES-KIRIAKI88

the thermoelastic Greens integral formula [16] in , we haveZn

(WL UUL W) dx = Z+

W [R U] ds

Z+

[U] RWds+

Z@

(WR UURW) ds (21)

where R is the adjoint di¤erential operator to R and can be applied byinterchanging with i ! . The unied thermoelastic fundamental solution~E (x;x0) satises the equation

L~E (x;x0) = 4 (x x0)~I4 (22)

and its explicit form can be found in [7]. Applying Greens formula (21) forthe fundamental solution ~E (x;x0) and the eld U and taking into account(22) we conclude to

U (x) = Z+

~E (x;x0) [R U (x0)] ds (x0)

Z+

[U (x0)] R ~ET (x;x0) ds (x0)

+

Z@

~E (x;x0) RU (x0)U (x0) R ~ET (x;x0)

ds (x0)

(23)

From the properties of the delta function, equation (23) yields

U (x) =

Z

~E (x;x0) Z+

(x0y) [R U (y)] ds (y)

Z+

[U (y)] (R (x0 y)) ds (y)dx0

+

Z@

~E (x;x0) RU (x0)U (x0) R ~E

T(x;x0)

ds (x0)

(24)

From equation (24) we infer that any boundary value problem concern-ing a dislocation problem on ; can be considered as an equivalent non-homogeneous problem

LU (x) = [R U (x0)] [U (x0)] (R ) ;x 2R3 (25)

7


where is the Dirac measure on :Now, if we have the rst boundary value problem, the eld U (x0) van-

ishes on which means that [U (x0)] = 0 on . So, for this case only thediscontinuity of the unied thermal stress eld remains in (25). Similarly,for the second boundary value problem, only the second term remains in theright hand side of (25) .In what follows we will examine the problem in the frame of low frequen-

cies. We can consider problems in the low frequency regime in cases wherethe characteristic dimension of the scatterer is much less than the wavelengthof the incident eld [12].Let us now consider the thermal stress dislocation. In the right hand side

of (25) only the term concerning the discontinuity of the stress eld appearsbecause the other term vanishes due to the continuity of the displacementeld on .The low frequency properties of the Biot system of thermoelasticity is

presented in [12]. In the low frequency limit we have ! ! 0 and this meansthat all wavenumbers tend to 0. We dene the constants

k = ks; p =kpks; q =

pq

ks; m =

kmks; m = 1; 2

Using this notation, it is shown in [12] that the scattered eld U (x) can beexpanded in the form

U (x) =1Xn=0

(i k)n

n!Un (x)

for certain vector functions Un (x).The incident plane wave of the form (12) assumes the following expansion:

Uin (x) =1Xn=0

(i k)n

n!

24A1n10@ bdbd bxnn1

bd bxn11A+ A2n2

0@ 2bdbd bxnnbd bxn1

1A+As

bdbd bxn0

!(26)

We observe that the rst term in the expansion is only of elastic character,that is Uin (x) = U0 (x) +O (k) where

Uin;0 (x) = A1 01

bd0

+ A2 02

2bd0

+ As

bd0

8

GINTIDES-KIRIAKI90

Since we are interested to consider scattering problems in the low frequencyregion, we assume that the incident eld is approximately constant. Thesame result holds also for the stress eld, since the stress tensor has rstorder derivatives, the rst order terms are deduced from Uin;1 (x) where the

application of the derivatives onbd bx produce only constant terms. From

this argument we can assume that the non - homogeneous term in (25) forthe thermal stress dislocation is approximately the same at all points on thescreen. So, supposing that

[R U (x0)] = B (27)

where B =bb

is a constant vector, equation (25) is written as

LU (x) = B ; x 2R3 (28)

For the case of the thermal displacement dislocation, assuming again that atlow frequencies the displacement eld is constant at all points on the screen,equation (25) yields the form

LU (x) = A (R ) ; x 2R3 (29)

where A =

aa

is an unknown constant vector.

We next derive an identity which is crucial to the reconstruction method.For any C2 vectorW using the corresponding Greens formula [16] and equa-tion (25), we conclude that

1

jj

Z

W (x0) [R U (x0)] ds (x0) +Zn

ULWdx

=

Z@

(U (x0) RW (x0)W (x0) R U (x0)) ds (x0) (30)

where jj is the area of . Assuming thatW :=

ww

equation (30) can

be written as1

jj

Z

(w b+ w b) ds+Zn

ULWdx

= Z@

(w (Tu qk b ) + w b r) ds+

Z@

(u (Tw b w ) + b rw ) ds (31)

9


4 Uniqueness of the inverse thermoelastic prob-lem for a screen

This uniqueness of the inverse scattering problem is strongly related withthe unique continuation principle. It is well known that for elliptic equationsthe unique continuation principle is equivalent with the uniqueness of theCauchy problem. In our case, the uniqueness for the Cauchy problem issatised since we have homogeneous and isotropic thermoelastic medium,the coe¢ cients are analytic and Holmgrens theorem can be applied [17].The inverse problem we face is the determination of the shape of the

planar screen and the vector B if we have a thermal stress dislocation orthe vector A if we have displacement dislocation. These vectors characterizethe physical properties of the dislocation. We assume that the availableinformation is the knowledge of the eld U on the surface of a sphere totally enclosing the planar screen.We assume that we know a priori the type of the unknown screen that is

we know that we have a thermal stress dislocation or a displacement dislo-cation. We prove the following uniqueness theorem for this inverse thermalstress dislocation problem:

Theorem 1 Let 1 and 2 be two planar screens and B1; B2 be two constantvectors. Assume that Uj; j = 1; 2 are solutions of the scattering problem:

LUj (x) = j Bj; j = 1; 2; x 2R3 (32)

and the elds are satisfying the radiation conditions (18). If U1 = U2 on @then 1 = 2 and B1 = B2.

Proof. Suppose 1 6= 2 and we have two elds U1 and U2 such thatU1 = U2 on @. Then there exists a point x0 2 1 such that x0 =2 1. Itis well known that solutions of the thermoelasticity for a homogeneous andisotropic material admit the unique continuation property [17]. So, fromequality of the vector elds U1 = U2 on @ and unique continuation weinfer that there exists a constant 0 such that

U1 (x0 + b) = U2 (x0 + b) ; 0 < < 0 (33)

From the singular behavior of the Greens dyadic as x approaches x0 we knowthat its measure is of order

O

1

jx x0j

10

GINTIDES-KIRIAKI92

and consequently the same behavior has the eld U1 in the vicinity of thepoint x0. So,U2 remains bounded as ! 0 but jU1j !

!01which contradicts

equality in the vicinity of x0 stated in (33). Therefore, 1 = 2:The equality of the vectorsB1 andB2 follows directly from the application

of the denition of the Dirac measure: We take the inner product in (32)with any vector distribution and integrate over , that is

RLUj (x

0) (x0) dv = jj j Bj (x) ; j = 1; 2. But we have proved that U1 = U2

on , 1 = 2 and obviously j1j = j2j so, B1 (x) = B2 (x) for anydistribution on = 1 = 2. which means that B1 = B2.With some slight modications we prove that uniqueness holds for the

inverse problem from a thermal displacement dislocation:

Theorem 2 Let 1 and 2 be two planar screens andA1; A2 be two constantvectors. Assume that Uj; j = 1; 2 are solutions of the scattering problem:

LUj (x) = AjR j

; j = 1; 2;x 2R3 (34)

and the elds are satisfying the radiation conditions (18). If U1 = U2 on @then 1 = 2 and A1 = A2.

Proof. Using the same argument as before we prove that there exists apoint x0 2 1 such that x0 =2 1 and

U1 (x0 + b) = U2 (x0 + b) ; 0 < < 0 (35)

and consequently the same behavior has the eld U1 in the vicinity of thepoint x0. So, U2 remains bounded as ! 0 but jU1j !

!01 [6] which

contradicts equality in the vicinity of x0 stated in (33). Therefore, 1 = 2:The equality of the vectors A1 and A2 follows from the application of thedenition of the Dirac measure: We take the inner product in (32) with anyvector distribution and integrate over , that isZ

LUj (x0) (x0) dv = jj j Aj R (x) ; j = 1; 2

But we have proved that U1 = U2 on , 1 = 2 and obviously j1j = j2jso, A1 R (x) = A2 R (x) for any distribution on = 1 = 2. whichmeans that A1 = A2.

11

5 Reconstruction method for thermal stressdislocation and thermal displacement dis-location

The purpose of our work is to extract geometrical and physical informationabout a screen in thermoelasticity at low frequencies. The reconstruction ofthe shape of the screen and the estimation of its physical characteristicsas the vector b and the b is of great importance in applications. Followingthe ideas presented in [2] we examine rst the case of the thermal stressdislocation. We assume that the elds u; ; Tu and r are known on @:Suppose now that we want to simplify the form of (31) in low frequencies

in such a way that we can use it for the inverse problem. If U is a solution ofthe basic di¤erential operator then the term which is not possible to computeisRnUL

Wdx since in the inverse problem is unknown. The integralover @ for arbitrary test function is computable because measurements areperformed there. The operator L can be written as L = L0 + L

1 where

L0 =

0 r

and L1 =

!2I3 0i ! r i !=

: We observe that if we

choose a functionW =

ww

which is a solution of the static adjoint ver-

sion of the basic equation, then it holds that the application of L onW isapproximately of order O (!), that is LW = O (!2). Static solutions ofthe adjoint operator have to satisfy the system of equations

w = 0w = r w

In all cases we will use only elastic displacement elds satisfying r w =0and simple harmonic functions w.In the sequel, we will make simple choices of functions w and w in (31)

in order to extract information about the screen. We rst take asww

the constant test functions

ej0

, j = 1; 2; 3 where ej are the unit vectors

of the coordinate system. Then equation (31) gives the stress dislocationvector b =b bb = b (b1; b2; b3). We have that

12

GINTIDES-KIRIAKI94

ejb =b bj =Z@

fej (Tu qk b )g ds+O (!) ; j = 1; 2; 3 (36)

and summing

b =

vuut 3Xj=1

Z@

jej (Tu qk b )j2 ds+O p!If we dene as b(1) the approximate magnitude of the vectors b, that is

b(1) =qP3

j=1

R@jej (Tu qk b )j2 ds then the components of the vector

b, can be approximated from (36) by

b(1)j =

1

b(1)

Z@

fej (Tu qk b )g dsin the sense that bj = b

(1)j +O (

p!).

The thermal term b can be derived similarly using the test vector01

,

with this choice equation (31) is written as

b =

Z@

f b r + qk u bg ds+O (!) (37)

We will face, now, the most important problem, which is the determinationof the shape of the screen. Suppose that the equation of the plane where thescreen is located is given by x3 = A x1 + B x2 + C. We will compute theconstants A, B and C which a priori are not known. Dene as

f (w;w) =

Z@

(u (Tw b w ) + b rw ) dsZ@

(w (Tu qk b ) + w b r) ds (38)

Then using the test functions of the formxiej0

; i; j = 1; 2; 3 we have

1

jj

Z

xids bj = f (xiej; 0)

13


and consequently we derive the linear system

Af (x1ej; 0)

b1+B

f (x2ej; 0)

b2+ C =

f (x3ej; 0)

b3; j = 1; 2; 3 (39)

from which the equation of the plane is always determined. Of course, sincewe have in a preceding step computed approximately the constants bj; de-noted by b(1)j the computation of the constants A,B, C is again up to anerror.The next important problem is to estimate the shape of the planar dis-

location. For simplicity we assume that x3 = 0 and in this case the unitnormal is e3.We consider the harmonic function (x) = ei x, where = 0. We

construct the function w : = r (x) = i ei x and set W =

r (x)0

.

Then from (31) we infer that

1

jj

Z

iei x

b ds

= Z@

i ei x (Tu qk b ) uT i ei x ds (40)

In this case we want to estimate approximately the shape of the planardislocation (1) writing (40) in the form

1

j(1)j

Z(1)

ei x ds = g ( ) (41)

where

g ( ) =1 b

Z@

ei x (Tu qk b ) uT ei x ds

The integral 1

j(1)jR(1)

ei x ds can be written as 1

j(1)jRR3 e

i x0h(1) (x0) dx0

where h(1) (x0) is the characteristic function of (1). Writing x0 = (x1; x2; x3)

and =0 ;p0 0

; 0=( 1; 2) then equation (41) yields

1

j(1)j

ZR3ei

0x0h(1) (x0) dx0 = ei

p0 0 x3g ( ) ; x3 = 0 (42)

14

GINTIDES-KIRIAKI96

From the above equation we see that eip0 0 x3h ( ) is the Laplace

transform of 1

j(1)jh(1) (x0) with compact support. Therefore

eip0 0 x3h ( )

can be uniquely determined by the inverse Laplace transform since this func-tion can be uniquely extended as an analytic function of the complex variable0: The inverse Laplace is

(1) = sup p

Z ip+1

ip1ei

p0 0 x3g

0 ;p0 0

d0 (43)

For the reconstruction of the displacement dislocation we start with theequation (21)

1

jj

Z

RW (x0) [U (x0)] ds (x0) +Zn

ULWdx

=

Z@

(U (x0) RW (x0)W (x0) R U (x0)) ds (x0) (44)

and setting approximately [U (x0)] = A =

a baa

where A is a constant

vector, we derive the relation

1

jj

Z

[a ba (Tw b w ) + ab rw ] ds+ Zn

ULWdx

= Z@

(w (Tu qk b ) + w b r) dsZ@

(u (Tw b w ) + b rw ) ds (45)

This case is more complicated although the idea is the same: compute the

unknown parameters using test functions which have constant terms in Tw b w and b rw. For simplicity we assume that ba = b which simulates aslitsituation. Then using only the thermal constant test function

01

we have

ajj

Z

ds = Z@

(b r + b ) ds+O (!)15


from which we compute

a =1

Z@

(b r + b ) ds+O (!)Again for simplicity that we know that the plane where the screen is locatedis the plane x3 = c . Then v1 = v2 = 0 and v23 = 1. Using the test

functionsw4

0

; where w4 = (2x21;x22;x23), we infer that .

2a1

jj

Z

2v21x1 v22x2 v23x3

ds = g (w4) (46)

or2a c = g (w4)() c =

1

2a g (w4) (47)

If we do not have a plane parallel to the x3 = 0 then we have to use morefunctions like (x21; 2x22;x23) and (x21;x22; 2x23).To estimate the shape of the planar dislocation we can use the same

procedure. Assume that we have the plane x3 = 0. We use the samefunction (x) = ei x, where = 0 and construct the test function

W =

r (x)0

. Then from (44) we infer that

2jj

Z

iei x

( bn) ( ba) ds

=

Z@

2 i (u ) (b ) ei x i ei x (Tu qk b ) ds

(48)

In this case we want to estimate approximately the shape of the planardislocation (1) writing (40) in the form

1

j(1)j

Z(1)

ei x ds = g ( ) (49)

where

g ( ) =1

2 ( bn) ( ba)Z@

2 i (u ) (b ) ei x

i ei x (Tu qk b ) ds (50)

16

GINTIDES-KIRIAKI98

Using the same inversion procedure we can reconstruct the shape (1) of thescreen.We mention here that this work can be directly applied to a problem in

elasticity by vanishing all quantities describing thermal e¤ects.

References

[1] G. Alessandrini, E. Beretta and S. Vessela, Determining linear cracksby boundary measurements: Lipschitz stability, SIAM J. Math. Anal.,27, 361-75 (1996).

[2] H. Ammari and G. Bao, Identication of cracks by boundary measure-ments at low frequencies, Inverse Problems, 16, 133-143 (2000).

[3] H. Ammari, G. Bao and G. A. Fleming, An inverse source problem forMaxwells equations in magnetoengefalography, SIAM J. Appl. Math.,62, 1360-1382 (2002).

[4] H. Ammari, H. Kang, G. Nakamura and K. Tanuma, Complete as-ymptotic expansions of the solutions of the system of elastostatics inthe presence of an inclusion of small diameter and detection of an inclu-sion, J. Elasticity, 67, 97-129 (2002).

[5] S. Andrieux and H. D. Bui, Écart à la réciprocité et identication dessures en thermoélasticité isotrope transitoire", C. R. Mecanique 334(2006).

[6] A. Ben-Menahem and S. Jit Singh, Seismic Waves and Sources,Springer Verlag, (1981).

[7] F. Cakoni, Boundary Integral Method for Thermoelastic Screen in Scat-tering Problem in R3, Mathematical Methods in the Applied Sciences,23, 441-466, (2000).

[8] F. Cakoni and G. Dassios, The coated thermoelastic body within a low-frequency elastodynamic eld, Int. J. Enging. Sci. Vol.36, 1815-1838,(1998).

17


[9] R. Chapko, R. Kress and L. Monch, On the numerical solution of a hy-persingular integral equation for elastic scattering from a planar crack,IMA J. Numer. Anal. 20, No 4, 601-619 (2000).

[10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scatter-ing Theory, Springer, Berlin, (1992).

[11] G. Dassios and V. Kostopoulos, The scattering amplitudes and crosssections in the theory of thermoelasticity, SIAM J. Appl. Math., Vol.48, No 1, 79-98, (1988).

[12] G. Dassios and V. Kostopoulos, On Rayleigh expansions in thermoeas-tic scattering, SIAM J. Appl. Math., Vol. 50, No 5, 1300-1324, (1990).

[13] G. Dassios and V. Kostopoulos, Scattering of elastic waves by a smallthermoelastic body, Int. J. Enging. Sci. Vol.32, 1593-1603, (1994).

[14] A. Friedman and M. Vogelius, Determining cracks by boundary mea-surements, Indiana Iniv. Math. J. 38, 527-556 (1989).

[15] H. Kang, E. Kim and J-Y Lee, Identication of elastic inclusions andelastic moment tensors by boundary measurements, Inverse Problems,19, 703-724 (2003).

[16] V. D. Kupradze,Three-Dimensional problems of the mathematical the-ory of elasticity and thermoelasticity, North-Holland, (1979).

[17] G. Nakamura, G. Uhlmann and J-N Wang, Unique continuation prop-erty for elliptic systems and crack determination in anisotropic elas-ticity, Contemporary Mathematics, 362, 321338, Amer. Math. Soc.,(2004).

18

GINTIDES-KIRIAKI100

(λ, σ) - DOUBLE SEQUENCE SPACES VIA ORLICZ FUNCTION

EKREM SAVAS & RICHARD F. PATTERSON

Abstract. In this paper we define and study two concepts which arise fromthe notions of invariant means and de la Valle-Poussin mean namely: stronglydouble (λ, σ)- convergence defined by Orlicz function and uniform (λ, σ)-statisticalconvergence and establish natural characterization for the underline sequencespaces.

1. Introduction and Background

Let l∞ be the Banach space of bounded x = (xk) with the usual norm ||x|| =supn|xn|. A sequence x ∈ l∞ is said to be almost convergent if all of its Banachlimits coincide. Let c denote the space of all almost convergent sequences. Lorentz[5] proved that

c = x ∈ l∞ : limm

tm,n(x) exists uniformly in nwhere

tm,n(x) =xn + xn+1 + · · ·+ xm+n

m + 1.

The following space of strongly almost convergent sequence was introduced by Mad-dox in [7]

[c] = x ∈ l∞ : limm

tm,n(|x− Le|) exists uniformly in n for some L ∈ cwhere e = (1, 1, . . .).

Let σ be a one-to-one mapping from the set of natural numbers into itself. Acontinuous linear functional φ on l∞ is said to be an invariant mean or a σ-meanprovided that

i φ(x) ≥ 0 when the sequence x = (xk) is such that xk ≥ 0 for all k,ii φ(e) = 1 where e = (1, 1, 1, . . .), andiii φ(x) = φ(xσ(k)) for all x ∈ l∞.

For certain class of mapping σ every invariant mean ϕ extends the limit functionalon space c, in the sense that ϕ(x) = lim x for all x ∈ c. Consequently, c ⊂ Vσ

where Vσ is the bounded sequence of all whose σ-mean are equal. If x = (xk), setTx = (Txk) = (xσ(n)), it can be shown that

Vσ =

x ∈ l∞ : limm

tm,n(x) = L uniformly in n, L = σ − limx

wheretm,n(x) =

xn + xσ(n) + · · ·+ xσm(n)

m + 1, t−1,n(x) = 0,

Date: February 3, 2006.2000 Mathematics Subject Classification. Primary 40A99; Secondary 40A05.Key words and phrases. Double sequence spaces, Orlicz function, de la Valle-Poussin means,

double statistical convergent .1


2 EKREM SAVAS & RICHARD F. PATTERSON

(see, Schaefer [21]) . We say that a bounded sequence x = (xk) is σ-convergent ifand only if x ∈ Vσ such that σk(n) 6= n for all n ≥ 0 , k ≥ 1. The space [Vσ] isof strongly σ-convergent sequence was introduced by Mursaleen [13] as follows: Asequence x = (xk) is said to be strongly σ-convergent if there exists a number Lsuch that

(1.1)1k

k∑

i=1

|xσi(m) − L| → 0

as k →∞ uniformly in m. We will denote [Vσ] as the set of all strongly σ-convergentsequences. When (1.1) holds we write [Vσ] − lim x = L. If we let σ(m) = m + 1,then [Vσ] = [c]. Thus strong σ-convergence generalizes the concept of strong almostconvergence sequence space.

Recall in [3] that an Orlicz function M : [0,∞) → [0,∞) is continuous, convex,non-decreasing function such that M(0) = 0 and M(x) > 0 for x > 0, and M(x) →∞ as x →∞.

Subsequently the notion of Orlicz function was used to define sequence spacesby Parashar and B.Choudhary [18] and other authors. An Orlicz function M canbe represented in the following integral form: M(x) =

∫ x

0p(t)dt where p is the

known kernel of M , right differential for t ≥ 0 , p(0) = 0, p(t) > 0 for t > 0, p isnon-decreasing and p(t) →∞ as t →∞.

If the convexity property of Orlicz function M is replaced with M(x + y) ≤M(x) + M(y) then this function is called Modulus function, which was presentedand discussed by Ruckle [15] and Maddox [6]. Let ω

′′denote the set of all double

sequences of real numbers. In 1900 Pringsheim presented the following definitionfor the convergence of double sequences.

Definition 1.1 (Pringsheim, [14]). A double sequence x = [xk,l] has Pringsheimlimit L (denoted by P-limx = L) provided that given ε > 0 there exists N ∈ N suchthat |xk,l − L| < ε whenever k, l > N . We shall describe such an x more briefly as“P-convergent”.

We shall denote the space of all P-convergent sequences by c′′. By a bounded

double sequence we shall mean there exists a positive number M such that |xk,l| <M for all (k, l), and denote such bounded by ||x||(∞,2) = supk,l |xk,l| < ∞. Weshall also denote the set of all bounded double sequences by l

′′∞. We also note in

contrast to the case for single sequence, a P-convergent double sequence need not bebounded. Let λ = (λi) be a non-decreasing sequence of positive numbers tendingto ∞ such that

λi+1 ≤ λi + 1, λ1 = 0.

The generalized de la Valee-Poussin mean is defined by

ti(x) =1λi

∑

k∈Ii

xk

where Ii = [i − λi + 1, i]. A sequence x = (xn) is said to be (V, λ)-summable to anumber L, if ti(x) → L as i →∞ (see [4]) .

Definition 1.2. Let λ = (λi) and µ = (µj) be two non-decreasing sequences ofpositive real numbers both of which tends to ∞ as i and j approach ∞, respectively.

102

DOUBLE SEQUENCE SPACES 3

Also let λi+1 ≤ λi + 1, λ1 = 0 and µj+1 ≤ µj + 1, µ1 = 0. We write the generalizeddouble de la Valee-Poussin mean by

ti,j(x) =1

λiµj

∑

(k,l)∈Ii,j

xk,l,

where Ij = [j − µj + 1, j].

A sequence x = (xk,l) is said to be (V′′, λ, µ)-summable to a number L, if

tij(x) → L as i, j → ∞ in the Pringsheim sense. Throughout this paper we shalldenote λi,j by λiµj , (σk(m), σl(n)) by σk,l(m, n), and (k ∈ Ii, l ∈ Ij) by (k, l) ∈ Ii,j .

We shall now generalize Definition 1.2 via double σ-convergence as follows:

Definition 1.3. A bounded double sequence x = (xk,l) of real number is said to be(λ, σ)-convergent to L provided that

P − limi,j

T i,jm,n = L uniformly in (m,n),

where

T i,jm,n =

1λi,j

∑

(k,l)∈Ii,j

xσk,l(m,n).

In this case we write (λ, σ) − limx = L. We shall also denote the set of all(λ, σ)-convergent sequences by V

′′(λ,σ)

. Clearly V′′(λ,σ)

⊂ l′′∞. One can see that in

contrast to the case for single sequences, a P-convergent sequence need not be(λ, σ)-convergent. But, it is easy to see that every bounded P-convergent doublesequence is (λ, σ)-convergent. In addition, if we let σ(m) = m + 1, σ(n) = n + 1,and λi,j = ij in the above definition then (λ, σ)-convergence reduces to the almostP-convergence which was defined by Moricz and Rhoades in [8]. It is quite naturalto expect that the sets of sequences that are strongly double (λ, σ)-summable tozero, strongly double (λ, σ)-summable and strongly double (λ, σ)-bounded by dela Valee-Poussin mean method can be defined by combining the concepts of Orliczfunction, λ-method, and σ-mean. Such a combination would be a multidimensionalanalogues of the definition presented by E. Savas and R. Savas in [20]. We nowready to present the multidimensional sequence spaces.

Definition 1.4. Let M be an Orlicz function and p = (pk,l) be any factorabledouble sequence of strictly positive real numbers. Let λ = (λi) and µ = (µj) be thesame as in above.

[V′′σ , λ,M, p] = x ∈ w

′′: P − lim

i,j

1λi,j

∑

(k,l)∈Ii,j

[M

(xσk,l(m,n) − l

ρ

)]pk,l

= 0

uniformly in (m, n), for some ρ > 0 and some l > 0,

[V′′σ , λ,M, p]0 = x ∈ w

′′: P − lim

i,j

1λi,j

∑

(k,l)∈Ii,j

[M

(xσk,l(m,n)

ρ

)]pk,l

= 0

uniformly in (m,n), for some ρ > 0,

103

and

[V′′σ , λ,M, p]∞ = x ∈ w

′′: sup

i,j,m,n

1λi,j

∑

(k,l)∈Ii,j

[M

(xσk,l(m,n)

ρ

)]pk,l

< ∞

for some ρ > 0.If we consider various assignment of M , λ, and p in the above sequence spaces

we are granted the following:

(1) If M(x) = x, λi,j = ij, and pk,l = 1 for all (k, l) then [V′′σ ,M, λ, p] = [V

′′σ ],

[V′′σ M, λ, p]0 = [V

′′σ ]0, and [V

′′σ M, λ, p]∞ = [V

′′σ ]∞.

(2) If pk,l = 1 for all (k, l) then [V′′σ ,M, λ, p] = [V

′′σ ,M, λ], [V

′′σ ,M, λ, p]0 =

[V′′σ , M, λ]0, and [V

′′σ ,M, λ, p]∞ = [V

′′σ ,M, λ]∞.

(3) If pk,l = 1 for all (k, l) and λi,j = ij then [V′′σ ,M, λ, p] = [V

′′σ ,M ], [V

′′σ ,M, λ, p]0 =

[V′′σ , M ]0, and [V

′′σ , M, λ, p]∞ = [V

′′σ ,M ]∞.

(4) If M(x) = x and pk,l = p for all (k, l) then [V′′σ ,M, λ, p] = [V

′′σ , λ]p,

[V′′σ ,M, λ, p]0 = [V

′′σ , λ]p0, and [V

′′σ ,M, λ, p]∞ = [V

′′σ , λ]p∞.

(5) If λi,j = ij then [V′′σ ,M, λ, p] = [V

′′σ ,M, p], [V

′′σ ,M, λ, p]0 = [V

′′σ ,M, p]0, and

[V′′σ , M, λ, p]∞ = [V

′′σ ,M, p]∞ which was studied by Savas and Patterson in

[19].(6) If M(x) = x, λi,j = ij , σ(m) = m + 1, and σ(n) = n + 1, then

[V′′σ , M, λ, p] = [c

′′, p], [V

′′σ , M, λ, p]0 = [c

′′, p]0, and [V

′′σ ,M, λ, p]∞ = [c

′′, p]∞.

(7) If σ(m) = m+1 and σ(n) = n+1 then [V′′σ ,M, λ, p] = [c

′′,M, λ], [V

′′σ ,M, λ, p]0 =

[c′′,M, λ]0, and [V

′′σ , M, λ, p]∞ = [c

′′,M, λ]∞.

(8) If M(x) = x, λi,j = ij, σ(m) = m+1, σ(n) = n+1, and pk,l = 1 for all (k, l)[V

′′σ , M, λ, p] = [c

′′], [V

′′σ ,M, λ, p]0 = [c

′′]0, and [V

′′σ ,M, λ, p]∞ = [c

′′]∞, [c

′′]

was studied in [8].

The following inequalities will be used throughout this paper. Let p = (pk,l)be a double sequence of positive real numbers with 0 < pk,l ≤ supk,l pk,l = H

and D = max1, 2H−1. Then for the factorable sequences ak and bk in thecomplex plane, we have as in Maddox [7]

|ak,l + bk,l|pk,l ≤ D(|ak,l|pk,l + |bk,l|pk,l).

2. Main Results

We begin the characterization of the above sequence spaces by presenting thefollowing theorems:

Theorem 2.1. Let the sequence (pk,l) be bounded then [V′′σ ,M, λ, p], [V

′′σ ,M, λ, p]0,

and [V′′σ ,M, λ, p]∞ are linear spaces over the set of complex numbers.

Proof. We shall only prove that [V′′σ ,M, λ, p]0 is linear. The proof of the others

follow in a similar manner. If α, β ∈ C then there exist two positive numbers ρ1

and ρ2 such that

P − limi,j→∞

1λi,j

∑

(k,l)∈Ii,j

[M

(xσk,l(m,n)

ρ1

)]pk,l

= 0

104

uniformly in (m,n) and

P − limi,j→∞

1λi,j

∑

(k,l)∈Ii,j

[M

(yσk,l(m,n)

ρ2

)]pk,l

= 0

uniformly in (m,n). Let ρ3 = max2|α|ρ1, 2|β|ρ2. Since M is non-decreasing andconvex, we have

1λi,j

∑

(k,l)∈Ii,j

[M

( |αxσk,l(m,n) + βyσk,l(m,n)|ρ3

)]pk,l

=1

λi,j

∑

(k,l)∈Ii,j

[M

( |αxσk,l(m,n)|ρ3

+|βyσk,l(m,n)|

ρ3

)]pk,l

≤ 1λi,j

∑

(k,l)∈Ii,j

12pk,l

[M

( |xσk,l(m,n)|ρ1

)+ M

( |yσk,l(m,n)|ρ2

)]pk,l

≤ 1λi,j

∑

(k,l)∈Ii,j

[M

( |xσk,l(m,n)|ρ1

)+ M

( |yσk,l(m,n)|ρ2

)]pk,l

≤ D1

λi,j

∑

(k,l)∈Ii,j

[M

( |xσk,l(m,n)|ρ1

)]pk,l

+ D1

λi,j

∑

(k,l)∈Ii,j

[M

( |yσk,l(m,n)|ρ2

)]pk,l

.

Now since each statement of the last inequality tends to zero as (i,j) approaches inthe Pringsheim sense, uniformly in (m,n), [V

′′σ ,M, λ, p]0 is linear. ¤

Definition 2.1. An Orlicz function M is said to satisfy ∆2-condition for all valuesof u, if there exists a constant K > 0 such that M(2u) ≤ KM(u) for all u ≥ 0.The ∆2-condition is equivalent to the inequality M(lu) ≤ K(l)M(u) for all valuesof u and for l ≥ 1 being satisfied.

Theorem 2.2. Let M be an Orlicz function. If β = limt→∞M(t\ρ)

t ≥ 1, then[V

′′σ ,M, λ] = [V

′′σ , λ].

Proof. Let x ∈ [V′′σ , λ], then

Si,jm,n = P − lim

i,j

1λi,j

∑

(k,l)∈Ii,j

|xσk,l(m,n) − L| = 0, uniformly in (m, n).

Let ε > 0 be given and choose 0 < δ < 1 such that M(u) < ε for every 0 ≤ u ≤ δ.We can write for each (m,n)

∑

(k,l)∈Ii,j

M

( |xσk,l(m,n) − L|ρ

)=

∑

(k,l)∈Ii,j&|xσk(m),σl(n)−L|≤δ

M


)

+∑

(k,l)∈Ii,j&|xσk,l(m,n)−L|>δ

M


).

105

It is clear that:∑

(k,l)∈Ii,j&|xσk,l(m,n)−L|≤δ

M


)< ε(λi,j).

On the other hand, we use the fact that

|xσk,l(m,n) − L| < 1 +[ |xσk,l(m,n) − L|

ρ

]

where [h] denotes the integer part of h.Since M is an Orlicz function we have:

M


)≥ M(1).

Now, let us consider the second part where the sum is taken over |xσk,l(m,n)−L| > δ.Thus

∑

(k,l)∈Ii,j&|xσk,l(m,n)−L|>δ

M


)≤

∑

(k,l)∈Ii,j

M

(1 +

[ |xσk,l(m,n) − L|ρ

])

≤ 2M(1)1δ(λi,j)T i,j

m,n.

Therefore∑

(k,l)∈Ii,j

M(|xσk,l(m,n) − L|) ≤ ε(λi,j) + 2M(1)

1δ(λi,j)T i,j

m,n

for every (m,n). Hence x ∈ [V′′σ , M, λ]. Observe that in this part of the proof we did

not need β ≥ 1. Let β ≥ 1 and x ∈ [V′′σ ,M, λ]. Since β ≥ 1 we have M(t) ≥ β(t)

for all t ≥ 0. It follows that xk,l → L[V′′σ ,M, λ] implies xk,l → L[V

′′σ , λ]. This

implies [V′′σ , M, λ] = [V

′′σ , λ]. ¤

3. Double statistical convergence

A real number sequence x is said to be statistically convergent to the number Lif for every ε > 0

limn

1n|k < n : |xk − L| ≥ ε| = 0,

where by k < n we mean that k = 0, 1, 2, ..., n and the vertical bars indicate thenumber of elements in the enclosed set. In this case we write st1−limx = L or xk →L(st1). Statistical convergence is a generalization of the usual notion of convergencefor real valued sequences that parallels the usual theory of convergence. The ideaof statistical convergence was first introduce by Fast [1], but the rapid developmentwere started after the papers of Fridy [2] and Salat, [16]. Now statistical convergencehas become one of the most active area of research in the field of Summabilitytheory.

Before we present the new definitions and the main theorems, we shall state afew known results. The following definition was presented by Mursaleen in [12].A sequence x is said to be λ-statistically convergent or Sλ-convergent to L, if forevery ε > 0

limn

1λn|k ∈ In : |xk − L ≥ ε|| = 0,

106

where the vertical bars indicate the numbers of elements in the enclosed set. Inthis case we write Sλ− limx = L or xk → L(Sλ). Savas [17] presented and studiedthe concepts of uniformly λ-statistical convergence as follows: A sequence x is saidto be uniformly λ-statistically convergent or Sλ-convergent to L, if for every ε > 0

limn

1λn

maxm

|k ∈ In : |xk+m − L ≥ ε|| = 0.

In this case we write Sλ − limx = L or xk → L(Sλ).Recently Mursaleen and Edely [10] presented the notion statistical convergence

for double sequence x = (xk,l) as follows: A real double sequence x = (xk,l) is saidto be statistically convergent to L, provided that for each ε > 0

P − limm,n

1mn

|(k, l) : k ≤ m and l ≤ n, |xk,l − L| ≥ ε| = 0

. In this section of this paper our goal is to define and study (λ, σ)-statisticalanalogue of convergence for double sequence. We now present (λ, σ)- statisticalanalogues for double sequence x = (xk,l) as follows:

Definition 3.1. A double sequence x = (xk,l) is said to be uniformly S′′(λ,σ)

-convergent or uniformly (λ, σ)-statistical convergent to L, provided that for everyε > 0

P − limi,j

1λ i,j

maxm,n

|(k, l) ∈ Ii,j : |xσk,l(m,n) − L| > ε| = 0.

In this case we write S′′(λ,σ)

− limx = L or xk,l → L(S′′(λ,σ)

) and S′′(λ,σ)

= x :

∃L ∈ R, S′′(λ,σ)

− limx = L. If we take λi,j = ij the above definition reduce to thefollowing which was defined in [19]:

Definition 3.2. A double sequence x = (xk,l) of real numbers is said to be uni-formly double (σ

′′)-statistically convergent to 0, if

P − limi,j

1ij

maxm,n≥0

|k < i and l < j : |xσk,l(m,n)| ≥ ε| = 0.

We denote by (S′′σ ), the set of sequences x = (xk,l) which are uniformly double σ-

statistically convergent to L. In particular, if σ is the translation in both dimension(S

′′σ )0 reduces to the set of sequences x = (xk,l) which are uniformly double almost

statistically convergent to 0, which was defined in [9] as follows: A double sequencex = (xk,l) of real numbers is said to be uniformly almost statistically convergent to0 if

P − limi,j

1ij

maxm,n≥0

|k < iandl < j : |xk+m,l+n| ≥ ε| = 0.

The set of double almost statistically convergent sequences shall be denoted by S′′.

Theorem 3.1. Let λ = (λi,j) be the same as in above, and let 0 < p < ∞, then

(1) xk,l → L[V′′σ , λ]p implies xk,l → L(S

′′(λ,σ)

);

(2) if x ∈ l′′∞ and xk,l → L(S

′′(λ,σ)

) then xk,l → L[V′′σ , λ]p;

(3) S′′(λ,σ)

∩ l′′∞ = [V

′′σ , λ]p ∩ l

′′∞.

Proof.

107


Let ε > 0 and xk,l → L[V′′σ , λ]p and since

∑

(k,l)∈Ii,j

|xσk,l(m,n) − L|p ≥∑

(k,l)∈Ii,j

|xσk,l(m,n) − L|p for |xσk,l(m,n)| ≥ ε

≥ εp|(k, l) ∈ Ii,j : |xσk,l(m,n) − L|| ≥ ε.

We have xk,l → L(S′′(λ,σ)

).

Suppose that xk,l → L(S′′(λ,σ)

) and x ∈ l′′∞ such that |xσk,l(m,n)| ≤ K for all (k, l)

and (m,n). Let ε > 0 be given and Nε such that

1λi,j

|(k, l) ∈ Ii,j : |xσk,l(m,n) − L| ≥( ε

2

) 1p | ≤ ε

2Kp

for all i, j ≥ Nε. Also let

Li,j = (k, l) ∈ Ii,j : |xσk,l(m,n)| ≥( ε

2

) 1p .

Now for all i, j > Nε we are granted1

λi,j

∑

(k,l)∈Ii,j

|xσk,l(m,n) − L|p =1

λi,j

∑

(k,l)∈Li,j

|xσk,l(m,n) − L|p

+1

λi,j

∑

(k,l)6∈Li,j

|xσk,l(m,n) − L|p

≤ 1λi,j

λi,jε

2KpKp

+1

λi,jλi,j

ε

2= ε.

Hence xk,l → L[V′′σ , λ]p. The third part follows immediately from (1) and (2). ¤

If we let λi,j = ij and p = 1 in Theorem 3.1. then we have the following corollary:

Corollary 3.1. (1) xk,l → L[V′′σ ] implies xk,l → L(S

′′σ )

(2) If x ∈ l′′∞ and xk,l → L(S

′′σ ) then xk,l → L[V

′′σ ]

(3) S′′σ ∩ l

′′∞ = [V

′′σ ] ∩ l

′′∞.

Theorem 3.2. S′′σ − lim x = L implies S

′′(λ,σ)

− limx = L if and only if

P − lim infi,j

λi,j

ij> 0.(3.1)

Proof. For given ε > 0 we have (k, l) ∈ Ii,j : |xσk,l(m,n)−L| ≥ ε ⊂ k ≤ i&l ≤ j :|xσk,l(m,n) − L| ≥ ε. Therefore

1ij|k ≤ i, l ≤ j : |xσk,l(m,n) − L| ≥ ε| ≥ 1

ij|(k, l) ∈ Ii,j : |xσk,l(m,n) − L| ≥ ε|

≥ λi,j

ij

1λi,j

|(k, l) ∈ Ii,j : |xσk,l(m,n) − L| ≥ ε|.

Taking the limit as (i, j) →∞ in the Pringsheim sense and the hypothesis, we havethe following: xk,l → L[S

′′σ ] implies xk,l → L[S

′′(λ,σ)

].

108

Conversely, suppose that P − lim infi,jλi,j

ij = 0 and x ∈ S′′σ . Then as in [12] we

can choose subsequences i(p)∞p=1 and j(q)∞q=1 such that λi(p)i(p) < 1

p and µj(q)j(q) < 1

q .Let us consider the following double sequence:

xk,l :=

1, if k ∈ Ii(p) and l ∈ Ij(q), (p, q = 1, 2, 3, . . .)0, otherwise .

Then x ∈ [V′′σ ] and hence by Corollary 3.1 x ∈ S

′′σ . But on the other hand x 6∈

[V′′(λ,σ)

]p and thus Theorem 3.1 implies x 6∈ (S′′(λσ)

); a contradiction and hence (3.1)holds. This completes the proof. ¤

Theorem 3.3. If M be an Orlicz function and 0 < h = inf pk ≤ pk ≤ supk pk =H < ∞ then [V

′′σ ,M, λ, p] ⊂ S

′′(λ,σ)

Proof. Let x ∈ [V′′σ ,M, λ, p]. Then there exists ρ > 0 such that

1λi,j

∑

(k,l)∈Ii,j

[M

(xσk,l(m,n)−L

ρ

)]pk,l

→ 0

as (i, j) →∞ in the Pringsheim sense uniformly in (m,n). If ε > 0 and let ε1 = ερ ,

then we obtain the following:

1λi,j

∑

(k,l)∈Ii,j

[M

(xσk,l(m,n) − L

ρ

)]pk,l

=1

λi,j

∑

(k,l)∈Ii,j&|xσk,l(m,n)|≥ε

[M

(xσk,l(m,n) − L

ρ

)]pk,l

+1

λi,j

∑

(k,l)∈Ii,j&|xσk,l(m,n)|<ε

[M

(xσk,l(m,n) − L

ρ

)]pk,l

≥ 1λi,j

∑


[M

(xσk,l(m,n) − L

ρ

)]pk,l

≥ 1λi,j

∑


[M(ε1)]pk,l

≥ 1λi,j

∑


min[M(ε1)]inf pk,l , [M(ε1)]H

≥ 1λi,j

|(k, l) ∈ Ii,j : |xσk,l(m,n) − L| ≥ ε|min[M(ε1)]h

, [M(ε1)]H.

Hence x ∈ S′′(λ,σ)

.¤

Theorem 3.4. Let M be a bounded Orlicz function and 0 < h = inf pk ≤ pk ≤supk pk = H < ∞. Then S

′′(λ,σ)

⊂ [V′′σ ,M, λ, p].

109

Proof. Suppose that M is bounded. Then there exists an integer K such thatM(x) < K for x > 0. Thus

1λi,j

∑

(k,l)∈Ii,j

[M


)]pk,l

=1

λi,j

∑

(k,l)∈Ii,j&|xσk,l(m,n)−L|≥ε

[M


)]pk,l

+1

λi,j

∑

(k,l)∈Ii,j&|xσk,l(m,n)−L|<ε

[M


)]pk,l

≤ 1λi,j

∑

(k,l)∈Ii,j&|xσk,l(m,n)−L|≥ε

max Kh,KH

+1

λi,j

∑

(k,l)∈Ii,j&|xσk,l(m,n)−L|<ε

[M

(ε

ρ

)]pk,l

≤ max Kh, KH 1λi,j

|(k, l) ∈ Ii,j : |xσk,l(m,n) − L| ≥ ε|

+ max

[M

(ε

ρ

)]h

,

[M

(ε

ρ

)]H

.

Hence x ∈ [V′′σ , M, λ, p]. This completes the proof.

¤

References

[1] Fast, H. Sur la convergence statistique, Collog. Math. 2 (1951), 241-244.[2] Fridy, J. A. On Statistical Convergence, Analysis, 5 (1985), 301-313.[3] Krasnoselskii, M. A. and Rutisky, Y. B. Convex function and Orlicz spaces, Groningen, Nether-lands, 1961.

[4] Leindler, L. ber die verallgemeinerte de la Valle-Poussinsche Summierbarkeit allgemeiner Or-thogonalreihen, Acta Math. Acad. Sci. Hungar. 16 (1965) 375–387.

[5] Lorentz, G.G. A contribution to the theory of divergent sequences, Acta. Math., 80 (1948),167-190.

[6] Maddox, I.J. Sequence spaces defined by a modulus, Math. Proc. Camb. Philos. Soc., 100(1986), 161-166.

[7] Maddox, I. J. Spaces of strongly summable sequences, Quart. J. Math. Oxford Ser. 18(2) (1967)345-355.

[8] Moricz, F. and Rhoades, B. E. Almost convergence of double sequences and strong regularityof summability matrices, Math. Proc. Comb. Phil. Soc., 104 (1988), 283-294.

[9] Mursaleen, M. and Edely, O. H. Generalized statistical convergence, Inform. Sci. 162(3-4)(2004), 287-294.

[10] Mursaleen and Edely, O. H. Statistical convergence of double sequences, J. Math. Anal. Appl.,288(1) (2003), 223-231.

[11] Mursaleen, Khan, Q.A., and Chishti, T. A. Some new Convergent sequences spaces definedby Orlicz Functions and Statistical convergence, Ital. J. Pure Appl. Math., 9 (2001), 25-32.

[12] Mursaleen, M. λ-statistical convergence, Math. Slovaca 50(1) (2000), 111-115.[13] Mursaleen, M. Matrix transformations between some new sequence spaces, Houston J. Math.9(4) (1983), 505-509.

[14] Pringsheim, A. Zur theorie der zweifach unendlichen Zahlenfolgen, Mathematische Annalen,53 (1900), 289 - 321.

110


[15] Ruckle, W. H. FK Spaces in which the sequence of coordinate vectors in bounded, Cand, J.Math. 25 (1973) 973-978.

[16] Salat, T. On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980),139-150.

[17] Savas, E. Strong almost convergence and almost λ-statistical convergence, Hokkaido Math.J. 29(3) (2000) 531-536.

[18] Parashar, S.D. and Choudhary, B. Sequence spaces defined by Orlicz functions, Indian J.Pure appl. Math., 25(4) (1994), 419-428.

[19] Savas, E. and Patterson, R. F. Some σ-Double Sequence Spaces Defined by Orlicz Function,(in press).

[20] Savas, E. and Savas, R. Some λ-sequence spaces defined by Orlicz functions, Indian J. PureAppl. Math. 34(12) (2003), 1673-1680.

[21] Schaefer, P. Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972),104-110.

Istanbul Ticaret University , Department of Mathematics, Uskudar/Istanbul- TURKEYE-mail address: [email protected]

Department of Mathematics and Statistics, University of North Florida Jacksonville,Florida, 32224


111

Journal of Computational Analysis and Applications Vol. 0 No 0. 000-000, 2007,

COPYRIGHT 2007, EUDOXUS PRESS, LLC.

STRONG CONVERGENCE OF AN ITERATIVE

ALGORITHM FOR ACCRETIVE OPERATORS

IN BANACH SPACES

Yeol Je Cho, Yonghong Yao∗ and Haiyun Zhou

Department of Mathematics Education and the RINS

Gyeongsang National University

Chinju 660-701, Korea

E-mail: [email protected]

Department of Mathematics

Tianjin Polytechnic University

Tianjin 300160, P. R. China


Department of Mathematics

Shijiazhuang Mechanical Engineering College

Shijiazhuang 050003, P. R. China


Abstract. In this paper, motivated by the result of Aoyama et al., we firstintroduce a new iterative scheme for an accretive operator in a Banach spaceand obtain some strong convergence theorems in a Banach space under appro-priate conditions on parameters.

∗ Corresponding author.Keywords: Strong convergence; accretive operators; fixed point; uniformly smooth and

uniformly convex Banach spaces.2000 AMS Mathematics Subject Classifications: 47H05, 47H10.


2 Yeol Je Cho, Yonghong Yao and Haiyun Zhou

1. Introduction

Let H be a real Hilbert space with norm ‖ · ‖ and inner product 〈·, ·〉, Cbe a nonempty closed convex subset of H and A be a monotone operator ofC into H.

The variational inequality problem is formulated as finding a point x∗ ∈ Csuch that

〈x− x∗, Ax∗〉 ≥ 0 (1)

for all x ∈ C. Such a point x∗ ∈ C is called a solution of the problem (1). Theset of solutions of the variational inequality problem is denoted by VI(C, A).In the case when C = H, VI(H, A) = A−10 holds, where

A−10 = x∗ ∈ H : Ax∗ = 0.

An element of A−10 is called a zero point of A. An operator A of C into His said to be α-inverse strongly monotone if there exists a positive real numberα such that

〈x− y,Ax−Ay〉 ≥ α‖Ax−Ay‖2 (2)

for all x, y ∈ C (see [1]-[3]). It is known that, if T is a nonexpansive mappingof C into itself, then A = (I −T ) is 1

2 -inverse strongly monotone and F (T ) =V I(C, A), where F (T ) is the set of fixed points of T .

Let P : H → C be a mapping and x ∈ H. Then there exists a unique pointPx ∈ C such that

‖x− Px‖ = d(x,C).

Such a mapping P of H onto C is called the metric projection onto C. It iswell-known that ‖x − y‖ = d(x,C) if and only if 〈x − y, y − z〉 ≥ 0 for allz ∈ C.

In the case of C = H = RN , to find a zero point of an inverse stronglymonotone operator, in 1979, Golshtein and Tretyakov [4] proved the followingtheorem:

Theorem GT. Let RN be the N -dimensional Euclidean space and A be anα-inverse strongly monotone operator of R into itself with A−10 6= ∅. Letxn be a sequence defined as follows:

x1 = x ∈ RN ,

xn+1 = xn − λnAxn

(3)

CHO ET AL114

An Iterative Algorithm for Accretive Operators 3

for all n = 1, 2, · · · , where λn is a sequence in [0, 2α]. If λn is chosen sothat λn ∈ [a, b] for some a, b with 0 < a < b < 2α, then xn defined by (3)converges to some element of A−10.

To find a solution of the variational inequality for an inverse strongly mono-tone operator, Iiduka et al. [5] proved the following weak convergence theorem:

Theorem I. Let C be a nonempty closed convex subset of a real Hilbertspace H and A be an α-inverse strongly monotone operator of C into H withV I(C, A) 6= ∅. Let xn be a sequence defined as follows:

x1 = x ∈ C,

xn+1 = PC(αnxn + (1− αn)PC(xn − λnAxn))(4)

for all n = 1, 2, · · · , where PC is the metric projection from H onto C, αn isa sequence in [−1, 1] and λn is a sequence in [0, 2α]. If αn and λn arechosen so that αn ∈ [a, b] for some a, b with −1 < a < b < 1 and λn ∈ [c, d]for some c, d with 0 < c < d < 2(1 + a)α, then xn defined by (4) convergesweakly to some element of V I(C, A).

Let X be a Banach space, X∗ be the dual space of X and 〈·, ·〉 denotethe pairing between X and X∗. For q > 1, the generalized duality mappingJq : X → 2X∗

is defined by

Jq(x) = f ∈ X∗ : 〈x, f〉 = ‖x‖q, ‖f‖ = ‖x‖q−1for all x ∈ X. In particular, if q = 2, the mapping J2 is called the normalizedduality mapping and, usually, write J2 = J . Further, we have the followingproperties of the generalized duality mapping Jq:

(1) Jq(x) = ‖x‖q−2J2(x) for all x ∈ X with x 6= 0.(2) Jq(tx) = tq−1Jq(x) for all x ∈ X and t ∈ [0,∞).(3) Jq(−x) = −Jq(x) for all x ∈ X.

Let U = x ∈ X : ‖x‖ = 1. A Banach space X is said to uniformly convexif, for any ε ∈ (0, 2], there exists δ > 0 such that, for any x, y ∈ U ,

‖x− y‖ ≥ ε implies∥∥∥x + y

2

∥∥∥ ≤ 1− δ.

It is known that a uniformly convex Banach space is reflexive and strictlyconvex. A Banach space X is said to be smooth if the limit

limt→0

‖x + ty‖ − ‖x‖t

(5)

ACCRETIVE OPERATORS IN BANACH SPACES 115

exists for all x, y ∈ U . It is also said to be uniformly smooth if the limit(5) is attained uniformly for x, y ∈ U . The norm of X is said to be Frechetdifferentiable if, for any x ∈ U , the limit (5) is attained uniformly for ally ∈ U . The modulus of smoothness of X is defined by

ρ(τ) = sup12(‖x + y‖+ ‖x− y‖)− 1 : x, y ∈ X, ‖x‖ = 1, ‖y‖ = τ,

where ρ : [0,∞) → [0,∞) is a function. It is known that X is uniformlysmooth if and only if limτ→0

ρ(τ)τ = 0. Let q be a fixed real number with

1 < q ≤ 2. A Banach space X is said to be q-uniformly smooth if there existsa constant c > 0 such that

ρ(τ) ≤ cτ q

for all τ > 0.

Note that(1) X is a uniformly smooth Banach space if and only if Jq is single-valued

and uniformly continuous on any bounded subset of X.(2) All Hilbert spaces, Lp (or lp) spaces (p ≥ 2) and the Sobolev spaces W p

m

(p ≥ 2) are 2-uniformly smooth, while Lp (or lp) and W pm spaces (1 < p ≤ 2)

are p-uniformly smooth.

Let C be a nonempty closed convex subset of a Banach space X. Anoperator A of C into X is said to be accretive if there exists j(x−y) ∈ J(x−y)such that

〈Ax−Ay, j(x− y)〉 ≥ 0

for all x, y ∈ C. And an operator A of C into X is said to be α-inverse stronglyaccretive if, for any α > 0,

〈Ax−Ay, J(x− y)〉 ≥ α‖Ax−Ay‖2

for all x, y ∈ C. Evidently, the definition of the inverse strongly accretiveoperator is based on that of the inverse strongly monotone operator.

Let D be a subset of C and Q be a mapping of C into D. Then Q is saidto be sunny if

Q(Qx + t(x−Qx)) = Qx,

whenever Qx + t(x −Qx) ∈ C for x ∈ C and t ≥ 0. A mapping Q of C intoitself is called a retraction if Q2 = Q. If a mapping Q of C into itself is aretraction, then Qz = z for all z ∈ R(Q), where R(Q) is the range of Q. Asubset D of C is called a sunny nonexpansive retract of C if there exists asunny nonexpansive retraction from C onto D. We know the following lemmaconcerning sunny nonexpansive retraction:

CHO ET AL116


Lemma 1.1. ([10]) Let C be a closed convex subset of a smooth Banach spaceX, D be a nonempty subset of C and Q be a retraction from C onto D. ThenQ is sunny and nonexpansive if and only if

〈u− Pu, j(y − Pu)〉 ≤ 0

for all u ∈ C and y ∈ D.

Remark 1.1. (1) If X is a Hilbert space, then a sunny nonexpansive retrac-tion QC is coincident with the metric projection from X onto C.

(2) If C is a nonempty closed convex subset of a uniformly convex anduniformly smooth Banach space X and T is a nonexpansive mapping of Cinto itself with F (T ) 6= ∅, then the set F (T ) is a sunny nonexpansive retractof C.

Recently, Aoyama et al. [6] first considered the following generalized vari-ational inequality problem in a Banach space:

Problem A. Let X be a smooth Banach space and C be a nonempty closedconvex subset of X. Let A be an accretive operator of C into X. Find a pointx∗ ∈ C such that, for some j(x− x∗) ∈ J(x− y),

〈Ax∗, j(x− x∗)〉 ≥ 0 (6)

for all x ∈ C.

Problem A is connected with the fixed point problem for nonlinear map-pings, the problem of finding a zero point of an accretive operator and soon. For the problem of finding a zero point of an accretive operator by theproximal point algorithm, see Kamimura and Takahashi [7].

In order to find a solution of Problem A, Aoyama et al. [6] introduced thefollowing iterative scheme for an accretive operator A in a Banach space X:

x1 = x ∈ C,

xn+1 = αnxn + (1− αn)QC(xn − λnAxn)(7)

for all n = 1, 2, · · · , where QC is a sunny nonexpansive retraction from X ontoC. Then they proved a weak convergence theorem in a Banach space whichis generalized simultaneously theorems of [1] and [4] as follows:


Theorem A. Let X be a uniformly convex and 2-uniformly smooth Banachspace and C be a nonempty closed convex subset of X. Let QC be a sunnynonexpansive retraction from X onto C, α > 0 and A be an α-inverse stronglyaccretive operator of C into X with S(C,A) 6= ∅, where

S(C, A) = x∗ ∈ C : 〈Ax∗, J(x− x∗)〉 ≥ 0, x ∈ C.If λn and αn are chosen so that λn ∈ [a, α

K2 ] for some a > 0 andαn ∈ [b, c] for some b, c with 0 < b < c < 1, then the sequence xn defined by(6) converges weakly to some element z of S(C, A), where K is the 2-uniformlysmoothness constant of X.

In this paper, motivated by Theorem A, we first introduce the followingiterative algorithm xn for an accretive operator A in a Banach space X: forany fixed u ∈ C,

x0 ∈ C,

xn+1 = αnu + βnxn + γnQC(xn − λnAxn)(8)

for all n = 1, 2, · · · , where QC is a sunny nonexpansive retraction from Xonto C, αn, βn, γn are three sequences in (0, 1) and λn is a sequenceof real numbers, and obtain some strong convergence theorems in a Banachspace under appropriate conditions on parameters.

2. Preliminaries

We need the following lemmas for proof of our main results:

Lemma 2.1. ([9]) Let q be a given real number with 1 < q ≤ 2 and X be aq-uniformly smooth Banach space. Then

‖x + y‖q ≤ ‖x‖q + q〈y, Jq(x)〉+ 2‖Ky‖q

for all x, y ∈ X, where K is the q-uniformly smoothness constant of X.

The following lemma is characterized the set of solutions of Problem A byusing sunny nonexpansive retractions.

Lemma 2.2. ([6]) Let C be a nonempty closed convex subset of a smoothBanach space X. Let QC be a sunny nonexpansive retraction from X onto Cand let A be an accretive operator of C into X. Then, for all λ > 0,

S(C,A) = F (Q(I − λA)).

CHO ET AL118

Lemma 2.3. ([11]) Let C be a nonempty bounded closed convex subset of auniformly convex Banach space X and T be nonexpansive mapping of C intoitself. If xn is a sequence of C such that xn → x weakly and xn−Txn → 0,then x is a fixed point of T .

Lemma 2.4. ([12]) Let xn, zn be bounded sequences in a Banach spaceX and αn be a sequence in [0, 1] which satisfies the following condition:

0 < lim infn→∞

αn ≤ lim supn→∞

αn < 1.

Suppose thatxn+1 = αnxn + (1− αn)zn

for all n = 0, 1, 3, · · · and

lim supn→∞

(‖zn+1 − zn‖ − ‖xn+1 − xn‖) ≤ 0.

Then limn→∞ ‖zn − xn‖ = 0.

Lemma 2.5. ([10]) Assume that an is a sequence of nonnegative real num-bers such that

an+1 ≤ (1− γn)an + δn

for all n = 0, 1, 3, · · · , where γn is a sequence in (0, 1) and δn is a sequencein R such that

(i)∑∞

n=0 γn = ∞;(ii) lim supn→∞

δn

γn≤ 0 or

∑∞n=0 |δn| < ∞.

Then limn→∞ an = 0.

3. The Main Results

In this section, we obtain some strong convergence theorems for finding asolution of Problem A for an inverse strongly accretive operator in a uniformlyconvex and 2-uniformly smooth Banach space.

Theorem 3.1. Let X be a uniformly convex and 2-uniformly smooth Banachspace and C be a nonempty closed convex subset of X. Let QC be a sunnynonexpansive retraction from X onto C, α > 0 and A be an α-inverse stronglyaccretive operator of C into X with S(C,A) 6= ∅. Let αn, βn, γn are

three sequences in (0, 1) and λn is a real number sequence in [a, αK2 ] for

some a > 0 satisfying the following conditions:(i) αn + βn + γn = 1 for all n = 0, 1, 3, · · · ;(ii) αn → 0 and

∑∞n=0 αn = ∞;

(iii) 0 < lim infn→∞ βn ≤ lim supn→∞ βn ≤ 1;(iv) limn→∞(λn+1 − λn) = 0.

Then the sequence xn defined by (7) converges strongly to Q′u, where Q′ isa sunny nonexpansive retraction of C onto S(C, A).

Proof. First, we observe that I −λnA is nonexpansive. Indeed, for all x, y ∈C and λn ∈

(0, α

K2

], from Lemma 2.1, we have

‖(I − λnA)x− (I − λnA)y‖2= ‖(x− y)− λn(Ax−Ay)‖2≤ ‖x− y‖2 − 2λn〈Ax−Ay, J(x− y)〉

+ 2K2λ2n‖Ax−Ay‖2

≤ ‖x− y‖2 − 2λnα‖Ax−Ay‖2+ 2K2λ2

n‖Ax−Ay‖2= ‖x− y‖2 + 2λn(K2λn − α)‖Ax−Ay‖2.

(9)

So, if 0 < λn ≤ αK2 , then I − λnA is a nonexpansive mapping.

Letting p ∈ S(C,A), it follows from Lemma 2.2 that p = QC(p − λnAp).Putting yn = QC(xn − λnAxn), it follows from (9) that

‖yn − p‖ = ‖QC(xn − λnAxn)−QC(p− λnAp)‖≤ ‖(xn − λnAxn)− (p− λnAp)‖≤ ‖xn − p‖.

(10)

Thus we have, from (7) and (10),

‖xn+1 − p‖ = ‖αn(u− p) + βn(xn − p) + γn(yn − p)‖≤ αn‖u− p‖+ βn‖xn − p‖+ γn‖yn − p‖≤ αn‖u− p‖+ (1− αn)‖xn − p‖≤ max‖u− p‖, ‖x0 − p‖.

Therefore, xn is bounded and so is yn.

CHO ET AL120


Define a sequence xn in C by

xn+1 = βnxn + (1− βn)zn

for all n = 0, 1, 2, · · · . Then it follows that

zn+1 − zn =xn+2 − βn+1xn+1

1− βn+1− xn+1 − βnxn

1− βn

=αn+1u + γn+1yn+1

1− βn+1− αnu + γnyn

1− βn

=( αn+1

1− βn+1− αn

1− βn

)u +

γn+1

1− βn+1(yn+1 − yn)

+( γn+1

1− βn+1− γn

1− βn

)yn.

(11)

Note that

‖yn+1 − yn‖= ‖QC(xn+1 − λn+1Axn+1)−QC(xn − λnAxn)‖≤ ‖(xn+1 − λn+1Axn+1)− (xn − λnAxn)‖= ‖(xn+1 − λn+1Axn+1)− (xn − λn+1Axn) + (λn − λn+1)Axn‖≤ ‖xn+1 − xn‖+ |λn+1 − λn|‖Axn‖.

(12)

From (11) and (12), we have

‖zn+1 − zn‖ ≤∣∣∣ αn+1

1− βn+1− αn

1− βn

∣∣∣‖u‖+γn+1

1− βn+1‖yn+1 − yn‖

+∣∣∣ γn+1

1− βn+1− γn

1− βn

∣∣∣‖yn‖

≤∣∣∣ αn+1

1− βn+1− αn

1− βn

∣∣∣(‖u‖+ ‖yn‖)+ ‖xn+1 − xn‖+ |λn+1 − λn|‖Axn‖,

which implies that

lim supn→∞

(‖zn+1 − zn‖ − ‖xn+1 − xn‖) ≤ 0.

Hence, from Lemma 2.4, we obtain limn→∞ ‖zn − xn‖ = 0.



Next, we show that

lim supn→∞

〈u−Q′u, j(xn −Q′u)〉 ≤ 0. (13)

To show (13), we can choose a sequence xni of xn converges weakly toz such that

lim supn→∞

〈u−Q′u, j(xn −Q′u)〉 = lim supi→∞

〈u−Q′u, j(xni −Q′u)〉. (14)

We first prove z ∈ S(C,A). Since λn is in [a, αK2 ] for some a > 0, it follows

that λni is bounded and so there exists a subsequence λnij of λni which

converges to λ0 ∈ [a, αK2 ]. We may assume, without loss of generality, that

λni → λ0. Since QC is nonexpansive, it follows from yni = QC(xni−λniAxni)that

‖QC(xni − λ0Axni)− xni‖≤ ‖QC(xni − λ0Axni)− yni‖+ ‖yni − xni‖≤ ‖(xni − λ0Axni)− (xni − λniAxni)‖+ ‖yni − xni‖≤ |λni − λ0|‖Axni‖+ ‖yni − xni‖,

which implies that

limi→∞

‖QC(I − λ0A)xni − xni‖ = 0. (15)

By Lemma 2.3 and (15), since we have z ∈ F (QC(I − λ0A)), it follows fromLemma 2.2 that z ∈ S(C, A).

Now, from (14) and Lemma 1.1, we have

lim supn→∞

〈u−Q′u, j(xn −Q′u)〉 = lim supi→∞

〈u−Q′u, j(xni −Q′u)〉

= 〈u−Q′u, j(z −Q′u)〉≤ 0.

CHO ET AL122


Therefore, we have (noting that (10))

‖xn+1 −Q′u‖2= 〈αnu + βnxn + γnyn −Q′u, j(xn+1 −Q′u)〉= αn〈u−Q′u, j(xn+1 −Q′u)〉+ βn〈xn −Q′u, j(xn+1

−Q′u)〉+ γn〈yn −Q′u, j(xn+1 −Q′u)〉≤ αn〈u−Q′u, j(xn+1 −Q′u)〉+

12βn(‖xn −Q′u‖2 + ‖xn+1 −Q′u‖2)

+12γn(‖yn −Q′u‖2 + ‖xn+1 −Q′u‖2)

≤ 12(1− αn)(‖xn −Q′u‖2 + ‖xn+1 −Q′u‖2)

+ αn〈u−Q′u, j(xn+1 −Q′u)〉,which implies that

‖xn+1 −Q′u‖2≤ (1− αn)‖xn −Q′u‖2 + 2αn〈u−Q′u, j(xn+1 −Q′u)〉. (16)

Finally, by Lemma 2.5 and (16), we conclude that xn converges stronglyto Q′u. This completes the proof.

Remark 3.1. (1) From (9), we know that Q(I − λnA) is nonexpansive.(2) If S(C, A) 6= ∅, it follows from Remark 1.1 and Lemma 2.2 that there

exists a sunny nonexpansive retraction Q′ of C onto F (Q(I−λnA)) = S(C, A).

Let C be a subset of a smooth Banach space X and α > 0. An operator Aof C into X is said to be α-strongly accretive if there exists j(x−y) ∈ J(x−y)such that

〈Ax−Ay, j(x− y)〉 ≥ α‖x− y‖2for all x, y ∈ C. Let L > 0. An operator A of C into X is said to be L-Lipschitzcontinuous if

‖Ax−Ay‖ ≤ L‖x− y‖for all x, y ∈ C.

Let C be a nonempty closed convex subset of a Hilbert space H. Onemethod of finding a point x∗ ∈ V I(C, A) is to use the projection algorithmxn defined by

x1 = x ∈ C,

xn+1 = PC(xn − λAxn)


for all n = 1, 2, · · · , where PC is the metric projection from H onto C, A is amonotone (accretive) operator of C into H and λ is a positive real number. Itis well known that if A is an α-strongly accretive and L-Lipschitz continuousoperator of C into H and λ ∈ (0, 2α

L2 ), then the operator PC(I − λA) is acontraction of C into itself.

Now, we prove a strong convergence theorem for strongly accretive opera-tor.

Theorem 3.3. Let X be a uniformly convex and 2-uniformly smooth Banachspace and C be a nonempty closed convex subset of X. Let QC be a sunnynonexpansive retraction from X onto C, α > 0 and A be an α-strongly ac-cretive and L-Lipschitz continuous operator of C into X with S(C,A) 6= ∅.Let αn, βn, γn are three sequences in (0, 1) and λn is a real numbersequence in [a, α

K2L2 ] for some a > 0 satisfying the following conditions:(i) αn + βn + γn = 1 for all n = 0, 1, 2, · · · ;(ii) αn → 0 and

∑∞n=0 αn = ∞;

(iii) 0 < lim infn→∞ βn ≤ lim supn→∞ βn ≤ 1;(iv) limn→∞(λn+1 − λn) = 0.

For fixed u ∈ C, define a sequence xn as follows:

x0 ∈ C,

xn+1 = αnu + βnxn + γnQC(xn − λnAxn)

for all n = 0, 1, 2, · · · . Then the sequence xn converges strongly to Q′u,where Q′ is a sunny nonexpansive retraction of C onto S(C, A).

Proof. Since A is an α-strongly accretive and L-Lipschitz continuous opera-tor of C into X, we have

〈Ax−Ay, J(x− y)〉 ≥ α‖x− y‖2 ≤ α

L2‖Ax−Ay‖2

for all x, y ∈ C. Therefore, A is αL2 -inverse strongly accretive. Using Theorem

3.1, we can obtain that xn converges strongly to Q′u. This completes theproof.

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2. F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalitiesand convergence rates, Set-Valued Anal. 6 (1998), 313–344.

CHO ET AL124


3. W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappingsand monotone mappings, J. Optim. Theory Appl. 118 (2003), 417–428.

4. E. G. Golshtein and N. V. Tretyakov, Modified Lagrangians in convex programmingand their generalizations, Math. Programming Study 10 (1979), 86–97.

5. H. Iiduka, W. Takahashi and M. Toyoda, Approximation of solutions of variationalinequalities for monotone mappings, PanAmer. Math. J. 14 (2004), 49–61.

6. K. Aoyama, H. Iiduka and W. Takahashi, Weak convergence of an iterative sequencefor accretive operators in Banach spaces, Fixed Point Theory and Applications (2006),1–13.

7. S. Kamimura and W. Takahashi, Weak and strong convergence of solutions to accretiveoperator inclusions and applications, Set-Valued Anal. 8 (2000), 361–374.

8. Y. Takahashi, K. Hashimoto and M. Kato, On sharp uniform convexity, smoothness,and strong type, co-type inequalities, J. Nonlinear and Convex Anal. 3 (2002), 267–281.

9. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991),1127–1138.

10. H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal.Appl. 298 (2004), 279–291.

11. F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banachspaces, Proc. Symp. Pure. Math. 18 (1976), 78–81.

12. T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl.305 (2005), 227–239.


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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.10,NO.1,2008 ON INVARIANT APPROXIMATION FOR NONCOMMUTATIVE MAPPINGS IN LOCALLY CONVEX SPACES,M.KHAN,H.K.NASHINE,…………………………7 A NEW APPLICATION OF ALMOST INCREASING SEQUENCES,H.BOR,…….17 ON *-HOMOMORPHISMS BETWEEN JC*-ALGEBRAS,C.PARK,W-G.PARK,H-J.WEE,…………………………………………………………………………………25 HYERS-ULAM-RASSIAS STABILITY OF ISOMETRIC HOMOMORPHISMS IN QUASI-BANACH ALGEBRAS,S-H.LEE,C.PARK,………………………………...39 BETTER ERROR ESTIMATION FOR SZASZ-MIRAKJAN-BETA OPERATORS, O.DUMAN,M.OZARSLAN,H.AKTUGLU,………………………………………….53 SOME THEOREMS ON IF-COMPACT LINEAR OPERATORS,H.EFE,…………..61 I-CONVERGENCE OF POSITIVE LINEAR OPERATORS ON Lp WEIGHTED SPACES,F.DIRIK,A.ARAL,K.DEMIRCI,……………………………………………75 IDENTIFICATION OF PLANAR SCREENS AT LOW FREQUENCIES IN THERMOELASTICITY,D.GINTIDES,K.KIRIAKI,…………………………………83 (l,s)-DOUBLE SEQUENCE SPACES VIA ORLICZ FUNCTION,E.SAVAS, R.PATTERSON,………………………………………………………………………101 STRONG CONVERGENCE OF AN ITERATIVE ALGORITHM FOR ACCRETIVE OPERATORS IN BANACH SPACES,Y.CHO,Y.YAO,H.ZHOU,…………………..113

Volume 10,Number 2 April 2008 ISSN:1521-1398 PRINT,1572-9206 ONLINE







10) Ding-Xuan Zhou





28) T. E. Simos











33) Xin-long Zhou

A Sequential Probability Ratio Test

Iuliana Florentina Iatan,

Department of Mathematics and Informatics,Technical University of Civil Engineering, Bucharest, Romania,

e-mail [email protected]

Abstract

Sequential procedures are concerned with statistical analysis ofdata when the number of observations is not predetermined.

If the method of weight functions of Wald is not feasible and ifFraser sufficiency and invariance do not apply, then one should lookfor a procedure in which sample estimates replace the true values ofthe nuisance parameters.

The maximum likelihood SPRT’s (SPRT= Sequential ProbabilityRatio Test) was proposed by Bartlett and by Cox.

The aim of this paper is to proof the theorem of Cox from [2],after we modified Lemma 1 of Cramer and LeCam (see [2]).

AMS Subject Classification: 62-xx, 62Lxx, 62L12.Keywords: maximum likelihood SPRT’s, sequential likelihood

ratio, sequential testing, statistics hypothesis, sequential procedures.

1 Introduction

In this problems we have to use the sequential testing of the statisticshypothesis.

Let f(X; γ, δ) be the probability density of a sequence of independentand identically distributed random variables Xii≥1, where γ ∈ Γ andδ ∈ ∆ and let Θ = Γ×∆ be the parameter space.

We shall present Lemma 1 which is from [3] and which it was correctedof me in order to use it in proof of Theorem 1.

Lemma 1 ([3]) For each (γ, δ) ∈ Θ, assume that

1


(i) ∂ ln f/∂γ and ∂ ln f/∂δ exist and are absolutely bounded by somefunctions that are integrable with respect to the density and

(ii) ∂2 ln f/∂γ2, ∂2 ln f/∂γ∂δ and ∂2 ln f/∂δ2 exist, are absolutely boundedby some functions of X and their expectation exist.

Then, we have

n−1 ∂2 ln f(Xn; γ, δ)/∂γ2 → −Iγγ in probability as n →∞n−1 ∂2 ln f(Xn; γ, δ)/∂γ∂δ → −Iγδ in probability as n →∞n−1 ∂2 ln f(Xn; γ, δ)/∂δ2 → −Iδδ in probability as n →∞

(1)

where f(Xn; γ, δ) =∏n

i=1 f(Xi; γ, δ),

E(∂ ln f/∂γ) = 0 = E(∂ ln f/∂δ),Var(∂ ln f/∂γ) = −E(∂2 ln f/∂γ2) = nIγγ ,Cov(∂ ln f/∂γ, ∂ ln f/∂δ) = −E(∂2 ln f/∂γ∂δ) = nIγδ,Var(∂ ln f/∂δ) = −E(∂2 ln f/∂δ2) = nIδδ.

(2)

2 Maximum Likelihood SPRT’s

We are interested in testing

H0 : γ = γ0 cu alternativa H1 : γ = γ1,

where δ is the nuisance parameter, no prior information about which isavailable.

One assume that

|γ1 − γ| and |γ0 − γ| are of the order n−1/2,

where γ denotes the true value.Let γn and δn denote the maximum likelihood estimators of γ and δ

based on (X1, . . . , Xn).The consider the sequential procedure based on

Z′n = ln

f(Xn; γ1, δn)

f(Xn; γ0, δn), (3)

which is called the sequential likelihood ratio.

2

IATAN140

We shall give Cox’s procedure which is a slight modification of Bartlett’sprocedure.

The Taylor’s expansions for ln f(Xn; γi, δn), for i = 0, 1 about thetrue value (γ, δ) give

ln f(Xn; γ0, δn) ≈ ln f(Xn; γ, δ) + (γ0 − γ)∂ ln f(Xn; γ, δ)

∂γ+

+12(γ0 − γ)2

∂2 ln f(Xn; γ, δ)∂γ2

+ (γ0 − γ)(δn − δ)∂2 ln f(Xn; γ, δ)

∂γ∂δ(4)

and

ln f(Xn; γ1, δn) ≈ ln f(Xn; γ, δ) + (γ1 − γ)∂ ln f(Xn; γ, δ)

∂γ+

+12(γ1 − γ)2


+ (γ1 − γ)(δn − δ)∂2 ln f(Xn; γ, δ)

∂γ∂δ(5)

Substracting (4) from (5) we obtain

Z′n = (γ1−γ0)

∂ ln f(Xn; γ, δ)∂γ

+12(γ1−γ0)(γ1+γ0−2γ)


+

+(γ1 − γ0)(δn − δ)∂2 ln f(Xn; γ, δ)

∂γ∂δ+ Rn(Xn), (6)

where Rn contains the second order derivatives and it converges to zeroin probability when γi − γ, i = 0, 1 are sufficiently small and the secondderivatives are uniform continue.

Expanding the functions ∂ ln f(Xn; γ, δ)∂γ and ∂ ln f(Xn; γ, δ)

∂δ about γn and

δn, which are the solutions of the equations

∂ ln f(Xn; γn, δn)∂γn

= 0 =∂ ln f(Xn; γn, δn)

∂δn

(7)

we shall obtain


= −(γn−γ)∂2 ln f(Xn; γ, δ)

∂γ2−(δn−δ)

∂2 ln f(Xn; γ, δ)∂γ∂δ

+R′n(Xn)

(8)and

3

SEQUENTIAL PROBABILITY TEST 141

∂ ln f(Xn; γ, δ)∂δ

= −(γn−γ)∂2 ln f(Xn; γ, δ)

∂γ∂δ−(δn−δ)

∂2 ln f(Xn; γ, δ)∂δ2

+R”n(Xn),

(9)where R

′n and R”

n converge to zero in probability and therefore we canneglect them for sufficiently large n.

Theorem 1 ([3]) Under suitable regularity conditions, we have

Z′n ≈ (γ1 − γ0)Iγγn

(γn −

12(γ0 + γ1)

), (10)

where nγn is asymptotically normal with mean nγ and variance

nIδδ

(IγγIδδ − I2γδ)

2.

Proof [I. Iatan]Substituting (8) in (6) we shall have

Z′n = (γ1−γ0)

[−(γn − γ)


− (δn − δ)∂2 ln f(Xn; γ, δ)

∂γ∂δ

]+

+12(γ1−γ0)(γ1+γ0−2γ)


+(γ1−γ0)(δn−δ)∂2 ln f(Xn; γ, δ)

∂γ∂δ;

and further,

Z′n ≈ (γ1 − γ0)

[(γn − γ)nIγγ + (δn − δ)nIγδ

]−

−12(γ1 − γ0)(γ1 + γ0 − 2γ)nIγγ − (γ1 − γ0)(δn − δ)nIγδ.

Finally, ignoring terms of order O(1) we shall obtain

Z′n ≈ (γ1 − γ0)nIγγ

(γn − γ − 1

2γ1 −

12γ0 + γ

);

therefore

Z′n ≈ (γ1 − γ0)Iγγn

(γn −

12(γ0 + γ1)

).

Using (1) in (8) and (9) it results

4

IATAN142

n(γn − γ)Iγγ + n(δn − δ)Iγδ = ∂ ln f(Xn; γ, δ)

∂γ

n(γn − γ)Iγδ + n(δn − δ)Iδδ = ∂ ln f(Xn; γ, δ)∂δ .

(11)

If we multiply the first equation from (11) by Iδδ and the second by−Iγδ and we sum the two equations, it obtains

n(γn − γ)[IγγIδδ − (Iγδ)2

]= Iδδ


− Iγδ∂ ln f(Xn; γ, δ)

∂δ.

Therefore,

nγn =[IγγIδδ − I2

γδ

]−1[Iδδ



∂δ

]+ nγ

(12)We have

E(nγn) =[IγγIδδ − I2

γδ

]−1[IδδE

(∂ ln f(Xn; γ, δ)

∂γ

)− IγδE


∂δ

)]+nγ

(13)Using Lema 1, from (13) it results

E(nγn) = nγ.

We shall calculate

Var(nγn) = E((nγn − E(nγn))2

)= E

((nγn − nγ)2

). (14)

Taking into account (12), the relation (14) becomes

Var(nγn) = E

([IγγIδδ − I2

γδ

]−2[Iδδ



∂δ

]2)

=

=[IγγIδδ − I2

γδ

]−2E

([Iδδ



∂δ

]2)

;

finally,

Var(nγn) =[IγγIδδ − I2

γδ

]−2 · [T − 2U + W ], (15)

where

5


T = I2δδE

((∂ ln f(Xn; γ, δ)

∂γ

)2)

,

U = IδδIγδE


∂γ· ∂ ln f(Xn; γ, δ)

∂δ

),

W = I2γδE

((∂ ln f(Xn; γ, δ)

∂δ

)2)

.

We know that:

Var(


)= E

((∂ ln f(Xn; γ, δ)

∂γ

)2)−[E(


)]2

,

Cov(


,∂ ln f(Xn; γ, δ)

∂δ

)=

= E(


· ∂ ln f(Xn; γ, δ)∂δ

)−

−E(


)· E(


),

Var(


)= E

((∂ ln f(Xn; γ, δ)

∂δ

)2)−[E(


)]2

.

Using Lema 1 we shall obtain

Var(


)= E

((∂ ln f(Xn; γ, δ)

∂γ

)2)

,

Cov(


,∂ ln f(Xn; γ, δ)

∂δ

)=

= E(


· ∂ ln f(Xn; γ, δ)∂δ

),

Var(


)= E

((∂ ln f(Xn; γ, δ)

∂δ

)2)

,

6

IATAN144

namely,

E

((∂ ln f(Xn; γ, δ)

∂γ

)2)

= Var(


)= nIγγ , (16)

E(

∂2 ln f(Xn; γ, δ)∂γ∂δ

)= Cov


∂γ,


)= nIγδ,

(17)

E

((∂ ln f(Xn; γ, δ)

∂δ

)2)

= Var(


)= nIδδ. (18)

Substituting (16)- (18) into (15) it results

Var(nγn) =[IγγIδδ − I2

γδ

]−2 · [I2δδ · nIγγ − 2IδδIγδ · nIγδ + I2

γδ · nIδδ] =

=[IγγIδδ − I2

γδ

]−2·[I2δδ·nIγγ−IδδIγδ·nIγδ] =

[IγγIδδ − I2

γδ

]−2·nIδδ(IδδIγγ−I2γδ);

therefore

Var(nγn) =nIδδ

(IγγIδδ − I2γδ)

2.

From (12) we note that for large n, nγn is the sum of independent andidentically distributed random variables Yii=1, n, where

Yi − γ =[IγγIδδ − I2

γδ

]−1[Iδδ

∂ ln f(Xn; γ; δ)∂γ

− Iγδ∂ ln f(Xn; γ; δ)

∂δ

].

(19)We can use Wald’s ([3]) approximations for the boundary values in

terms of error probabilities.Thus,

B =(1− I2

γδ/IγγIδδ

)−1ln [β/(1− α)] ≈ cn ln [β/(1− α)] ,

A =(1− I2

γδ/IγγIδδ

)−1ln [(1− β)/α] ≈ cn ln [(1− β)/α] ,

(20)

where

cn =(1− I2

γnδn/IγnγnIδnδn

)−1.

7


3 Conclusions

Sequential procedures are concerned with statistical analysis of data whenthe number of observations is not predetermined.

Our goal is to use the sequential testing of the statistics hypothesis.If the method of weight functions of Wald is not feasible and if Fraser

sufficiency and invariance do not apply, then one should look for a pro-cedure in which sample estimates replace the true values of the nuisanceparameters. We confine ourselves to maximum likelihood estimators, whichhave desirable large sample properties.

The maximum likelihood SPRT’s was proposed by Bartlett and by Cox.In this paper we proof the theorem of Cox from [3], after we modified

Lemma 1.

References

[1] Bishop, C.,M., 2006. Pattern Recognition and Machine Learning.Springer, Heidelberg.

[2] Duda, D.,O., Hart P., E., Stork, D., G., 2001. Pattern Classification.John Wiley, second edition.

[3] Govindarajulu, Z., 1975. Sequential Statistical Procedures. AcademicPress, New York.

[4] Jain, A., Dubes, R., 1988. Algorithms for Clustering Data. Prentice-Hall, Englewood Cliffs, New Jersey 07632.

[5] MacKay, D., J., C., 2003. Information Theory, Inference, and LearningAlgorithms. Cambridge University Press.

[6] Mihoc, Gh., Craiu, V., 1979. Sequential analysis. Vol. III, Ed.Academiei, Bucharest.

[7] Siegmund, D., 1985. Sequential Analysis. Springer.

[8] Webb, A., 2002. Statistical Pattern Recognition. John Wiley and Sons,N.York, second edition.

8

IATAN146

CONVERGENCE THEOREMS OF ITERATIVE SEQUENCES

FOR GENERALIZED P -QUASICONTRACTIVE

MAPPINGS IN P -CONVEX METRIC SPACES

Jong Kyu Kim

Department of Mathematics Education, Kyungnam University,Masan, Kyungnam, 631-701, KoreaE-mail: [email protected]

Sung-Ae Chun

Department of Mathematics, Kyungnam UniversityMasan, Kyungnam 631-701, KoreaE-mail: [email protected]

Young Man Nam

Department of Mathematics, Education, Kyungnam UniversityMasan, Kyungnam 631-701, KoreaE-mail: [email protected]

Abstract. We first introduce the concept of a p-convex metric space and ageneralized p-quasicontractive mapping(or generalized quasicontractive map-ping with respect to a weak distance p.) And also, we prove some convergencetheorems of iterative sequences with errors for generalized p-quasicontractivemapping in p-convex metric spaces.

AMS Mathematics Subject Classification: 47H05, 47H10, 47H15, 54H25.

Key words and phrases : p-convex metric space, fixed point, Ishikawa typeiterative sequence with errors, generalized p-quasicontractive mapping.

1. Introduction and preliminaries

Let K be a nonempty closed convex subset of a Banach space E and letT : K → K be a nonlinear pseudo-contractive mapping or accretive mapping.Recently concerning the problem of the Ishikawa iterative sequence xn [9]


defined by

x0 ∈ K,

yn = βnxn + (1− βn)Txn,

xn+1 = αnxn + (1− αn)Tyn, n ≥ 0(1.1)

converging strongly to a fixed point of T or to a solution of the equationTx = f has been considered by many authors ([1,3-8,11-18,21,24,26), whereαn and βn are sequences in [0, 1] satisfying the certain conditions.

Rhoades [23] and Naimpally-Singh [22] suggested the following open ques-tion:

Can the Ishikawa iterative procedure be extended to nonlinear quasi-contra-ctive mapping in a metric space?

This question is in fact solved in the affirmative (see Liu [19,20] and Xu[26]) for the Hilbert or Banach space setting.

Definition 1.1. [2] Let (E, d) be a metric space and I = [0, 1]. For any posi-tive integer n ≥ 2, denote by En = E × E × · · · × E︸︷︷︸

n

, In = I × I × · · · × I︸︷︷︸n

. A

mapping W : En× In → E is said to be a convex structure on E if it satisfiesthe following conditions: for any u, x1, x2, · · · , xn ∈ E and for any α1, α2,· · · , αn ∈ I with

∑ni=1 αi = 1

(1) W (x1, x2, · · · , xn; 0, 0, · · · , αi, 0, · · · , 0) = αixi = xi, i = 1, 2, · · · , n;

(2) d(u,W (x1, x2, · · · , xn; α1, α2, · · · , αn)) ≤n∑

i=1

αid(u, xi).

E together with a metric d and a convex structure W is called a convex metricspace, and denote it by (E, d, W ).

In 1988, Ding [8] proved the following fixed point theorem using Ishikawatype iterative scheme: Let K be a nonempty closed convex subset of a com-plete convex metric space X with convex structure W and let T : K → Kbe a quasi-contractive mapping [7], i.e., there exists a constant q ∈ [0, 1) suchthat for all x, y ∈ K,

d(Tx, Ty) ≤ q ·maxd(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx). (1.2)

Suppose that xn is a sequence defined by

x0 ∈ K,

xn+1 = W (Tyn, xn, αn),yn = W (Txn, xn, βn), n ≥ 0,

(1.3)

KIM ET AL148

where αn and βn satisfy 0 ≤ αn, βn ≤ 1 and∞∑

n=0

αn diverges. Then xnconverges to a unique fixed point of T in K.

In 2003, Chang-Kim [2] introduced the concept of new type iterative se-quence xn with errors in a metric space. Let (E, d) be a convex metric spacewith a convex structure W : E3 × I3 → E and T : E → E be a generalizedquasi contractive mapping. Define a sequence xn as follows:

x0 ∈ E,

yn = W (xn, Txn, vn; ξn, ηn, δn),xn+1 = W (xn, T yn, un; αn, βn, γn), n ≥ 0

(1.4)

where αn, βn, γn, ξn, ηn and δn are sequences in [0, 1] withαn + βn + γn = 1, ξn + ηn + δn = 1, n = 0, 1, 2, · · · and un, vn aresequences in E, then xn is called the Ishikawa type iterative sequence witherrors of T.

Especially, if ηn = 0 and δn = 0 for all n ≥ 0, it follows from the definitionof the convex structure W that yn = xn. Hence from (1.4) we have

xn+1 = W (xn, Txn, un; αn, βn, γn). (1.5)

The sequence defined by (1.5) is called the Mann type iterative sequence witherrors of T. It should be pointed that if E is a linear normed space, then E is aconvex metric space with a convex structure W (x, y; 1−λ, λ) = (1−λ)x+λy,∀x, y ∈ E, λ ∈ I. Therefore, the Ishikawa iterative sequence (1.1) is a specialcases of (1.4) with γn = 0, δn = 0 and un = vn = 0, for all n ≥ 0 and also,(1.4) with δn = γn = un = vn = 0 for all n ≥ 0 reduces to (1.3).

And also Chang-Kim [2] proved the following theorem for the Ishikawa typeiterative sequences with errors (1.4): Let (E, d,W ) be a complete convex met-ric space, W : E3 × I3 → E be the convex structure of E, T be a generalizedquasi-contractive mapping defined by

d(Tx, Ty)

≤ Φ(

maxd(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx))

,(1.6)

for all x, y ∈ E, and xn be the Ishikawa type iterative sequence with errorsof T defined by (1.4). Then the sequence xn converges to a unique fixedpoint z of T in E.

On the other hand, Kada-Suzuki-Takahashi [10] introduced the concept ofω-distance on a metric space as follows:

CONVERGENCE THEOREMS...P-CONVEX METRIC SPACES 149

Let E be a metric space with a metric d. Then a function p : E×E → [0,∞)is called a ω-distance on E if the following satisfied:

(1) p(x, z) ≤ p(x, y) + p(y, z) for any x, y, z ∈ E;(2) for any x ∈ E, p(x, ·) : E → [0,∞) is lower semi-continuous;(3) for any ε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ

imply d(x, y) ≤ ε.

Then we know that a metric d is a w-distance but the w-distance p cannot bea metric (see the examples in [10]).

Now, we introduce the concepts of a p-convex metric space and a p-convexstructure Wp for the w-distance p.

Definition 1.2. Let (E, d) be a metric space with a ω-distance p and I =[0, 1]. For any positive integer n ≥ 2, denote by En = E × E × · · · × E︸︷︷︸

n

, In =

I × I × · · · × I︸︷︷︸n

. A mapping Wp : En× In → E is said to a p-convex structure

of E if it satisfies the following conditions: for any u, x1, x2, · · · , xn ∈ E and

for any α1, α2, · · · , αn ∈ I withn∑

i=1

αi = 1,

(1) Wp(x1, x2, · · · , xn; 0, 0, · · · , αi, 0, · · · , 0) = αixi = xi, i = 1, 2, · · · , n;

(2) p(u,Wp(x1, x2, · · · , xn;α1, α2, · · · , αn)) ≤n∑

i=1

αip(u, xi); (1.7)

(3) p(Wp(x1, x2, · · · , xn;α1, α2, · · · , αn), u) ≤n∑

i=1

αip(xi, u).

E together with a p-convex structure Wp is called a p-convex metric space,and denote it by (E, d, p, Wp).

A nonempty subset K of a p-convex metric space E with a p-convex struc-ture Wp is said to be p-convex (cf. [25]) if

Wp(x1, x2, · · · , xi, · · · , xn; α1, α2, · · · , αi, · · · , αn) ∈ K,

for all (x1, x2, · · · , xi, · · · , xn;α1, α2, · · · , αi, · · · , αn) ∈ Kn × In.

Since a metric d is w-distance, if we put p = d, then a convex metric space(E, d, W ) is a p-convex metric space with a w-distance d.

The purpose of this paper is to prove some new convergence theorems forIshikawa type iterative sequence with errors in p-convex metric spaces with aw-distance p and a p-convex structure Wp. The results of this paper not onlyextend and improve the well known results in Chang-Kim [2], Ciric [7], Ding

KIM ET AL150

[8], Kim et al. [11-18], Liu [19,20], Naimpally-Singh [22], Rhoades [23,24] andXu [26] but also give an affirmative answer to the open question of Naimpally-Singh [22] in p-convex metric spaces.

Definition 1.3. (cf. [2]) A function Φ : [0,∞) → [0,∞) is said to satisfy thecondition (CΦ), if it is nondecreasing, continuous from right, Φ(t) < t, for allt > 0 and Φ(0) = 0, and the graph of Φ intersects with the graph parallelmoved toward the positive direction with respect to t-axis of the identityfunction I at only one point, that is,

(t, Φ(t)) : t ≥ 0 ∩ (t + a, I(t)) : t ≥ 0, a ≥ 0 is a singleton.

Definition 1.4. Let (E, d) be a metric space with a ω-distance p and T :E → E be a mapping. If there exists a function Φ : [0,∞) → [0,∞) satisfyingthe condition (CΦ) such that

p(Tx, Ty)

≤ Φ(

maxp(x, x), p(x, y), p(x, Tx), p(y, Ty), p(x, Ty), p(y, Tx),

p(y, y), p(y, x), p(Tx, x), p(Ty, y), p(Ty, x), p(Tx, y))

,

for all x, y ∈ E then T is said to be a generalized p-quasicontractive mapping(or generalized quasicontractive mapping with respect to a weak distance p).

If we put p = d, then a generalized p-quasicontractive mapping is general-ized quasi-contractive mapping (1.6) and define Φ(t) = kt, for all t ∈ [0, 1) thenthe generalized quasi-contractive mapping (1.6) is quasi-contractive mapping(1.2).

2. Main results

We introduce the following lemma which plays a crucial role in the proofsof our main theorems.

Lemma 2.1. [10] Let (E, d) be a metric space with a ω-distance p. Let xnbe a sequence in E, let an be a sequence in [0,∞) converging to zero, andlet x, y, z ∈ E. Then the following hold:

(i) if p(x, y) = 0 and p(x, z) = 0, then y = z,(ii) if p(xn, xm) ≤ an for any n,m ∈ N with m > n, then xn is a Cauchy

sequence.

Using the concepts of w-distance p introduced by Kada-Suzuki-Takahashi[10] and a p-convex metric space, we have the following theorem which is ageneralization of the result in [2].

Theorem 2.1. Let K be a nonempty closed p-convex subset of a completep-convex metric space E with a ω-distance p and a p-convex structure Wp.Let T : K → K be a generalized p-quasicontractive mapping

p(Tx, Ty) (2.1)

≤ Φ(

maxp(x, x), p(x, y), p(x, Tx), p(y, Ty), p(x, Ty), p(y, Tx),


,

for all x, y ∈ K and some Φ : [0,∞) → [0,∞) satisfying the condition (CΦ)and such that

infp(x, y) + p(x, Tx) | x ∈ K > 0 (2.2)

for every y ∈ K with y 6= Ty. Suppose that xn is Ishikawa type iterativesequence with errors of T defined by

x0 ∈ K,

yn = Wp(xn, Txn, vn; ξn, ηn, δn),xn+1 = Wp(xn, T yn, un; αn, βn, γn), n ≥ 0

(2.3)

where αn, βn, γn, ξn, ηn and δn are sequences in [0, 1] with

αn +βn +γn = 1, ξn +ηn + δn = 1, n = 0, 1, 2, · · · and∞∑

n=0

βn = ∞, and un,vn are sequences in E, and the following condition is satisfied:For any nonnegative integers n,m with 0 ≤ n < m, if δ(An,m) > 0, then

maxn≤i,j≤m

p(x, y), p(y, x) | x ∈ ui, vi, y ∈ xj , yj , Txj , T yj , uj , vj< δ(An,m),

(2.4)

whereAn,m = xi, yi, Txi, T yi, ui, vi : n ≤ i ≤ m

andδ(An,m) = sup

x,y∈An,m

p(x, y).

Then there is a unique fixed point z of T in K and the sequence xn convergesto z.

Proof. Let N be the set of all nonnegative integers. Then, for any n,m ∈ N,0 ≤ n < m, we have

δ(An,m) = maxD1, D′1, D2, D3, D

′3, D4, D

′4, D5, D6, D

′6, (2.5)

KIM ET AL152

where

D1 = maxp(xn, Txi), p(xn, T yi) | n ≤ i ≤ m,D′

1 = maxp(Txi, xn), p(Tyi, xn) | n ≤ i ≤ m,D2 = maxp(Txi, Txj), p(Txi, T yj), p(Tyi, T yj), p(Tyj , Txi) | n ≤ i, j ≤ m,D3 = maxp(xi, Txj), p(xi, T yj) | n < i ≤ m, n ≤ j ≤ m,D′

3 = maxp(Txj , xi), p(Tyj , xi) | n < i ≤ m, n ≤ j ≤ m,

D4 = maxp(yi, Txj), p(yi, T yj) | n ≤ i, j ≤ m,D′

4 = maxp(Txj , yi), p(Tyj , yi) | n ≤ i, j ≤ m,D5 = maxp(xi, xj), p(xi, yj), p(yi, yj), p(yj , xi) | n ≤ i, j ≤ m,D6 = maxp(x, y) | x ∈ ui, vi, y ∈ xj , yj , Txj , T yj , uj , vj : n ≤ i, j ≤ m,D′

6 = maxp(y, x) | x ∈ ui, vi, y ∈ xj , yj , Txj , T yj , uj , vj : n ≤ i, j ≤ m.Now we prove that

δ(An,m) = maxD1, D′1. (2.6)

For the proof, we consider the following several steps.(I) Since T : K → K is a generalized p-quasicontractive mappings,

D2 ≤ Φ(δ(An,m)). (2.7)

(II) It follows from (2.3) and (1.7) that if n n, then by the same way, we have

p(xi−1, Txj) ≤ maxp(xi−2, Txj), Φ(δ(An,m)), D6.

By induction, for n < i ≤ m, n ≤ j ≤ m, we can obtain

p(xi, Txj) ≤ maxp(xi−1, Txj),Φ(δ(An,m)), D6≤ maxp(xi−2, Txj),Φ(δ(An,m)), D6≤ · · ·≤ maxp(xn, Txj),Φ(δ(An,m)), D6.

Similarly, for n < i ≤ m, n ≤ j ≤ m, we have

p(xi, T yj) ≤ maxp(xn, T yj), Φ(δ(An,m)), D6.

This implies that

D3 = maxp(xi, Txj), p(xi, T yj) | n < i ≤ m,n ≤ j ≤ m≤ maxp(xn, Txj), p(xn, T yj), Φ(δ(An,m)), D6 | n ≤ j ≤ m= maxD1, Φ(δ(An,m)), D6.

(2.8)

(III) Using the same way of the step (II), we can obtain

D′3 = maxp(Txj , xi), p(Tyj , xi) | n < i ≤ m,n ≤ j ≤ m≤ maxp(Txj , xn), p(Tyj , xn), Φ(δ(An,m)), D′

6 | n ≤ j ≤ m= maxD′

1, Φ(δ(An,m)), D′6.

(2.9)

(IV) For n ≤ i, j ≤ m, by (2.7) and (2.8) we have

p(yi, Txj) = p(Wp(xi, Txi, vi; ξi, ηi, δi), Txj)

≤ ξip(xi, Txj) + ηip(Txi, Txj) + δip(vi, Txj)

≤ maxp(xi, Txj),Φ(δ(An,m)), D6≤ maxD1,Φ(δ(An,m)), D6.

Similarly, we have

p(yi, T yj) ≤ maxD1, Φ(δ(An,m)), D6.

Hence we have

D4 = maxp(yi, Txj), p(yi, Tyj) | n ≤ i, j ≤ m≤ maxD1, Φ(δ(An,m)), D6.

(2.10)

(V) Similarly, from (2.7) and (2.9) we can obtain

D′4 = maxp(Txj , yi), p(Tyj , yi) | n ≤ i, j ≤ m≤ maxD′

1, Φ(δ(An,m)), D′6.

(2.11)

(VI) Since

D5 = maxp(xi, xj), p(xi, yj), p(yi, yj), p(yj , xi) | n ≤ i, j ≤ m,

KIM ET AL154

we can consider the following:(a) We first make an estimation for maxp(xi, xj) | n ≤ i, j ≤ m. Let

A1 = maxp(xi, xj) | n ≤ i, j ≤ m.

Then there exists k, ` with n ≤ k, ` ≤ m such that A1 = p(xk, x`) and

if n ≤ k ≤ ` ≤ m =⇒ p(xk, x`−1) < p(xk, x`) = A1, (2.12)

if n ≤ ` ≤ k ≤ m =⇒ p(xk−1, x`) < p(xk, x`) = A1. (2.13)

If n ≤ k ≤ ` ≤ m, then

A1 = p(xk,Wp(x`−1, T y`−1, u`−1; α`−1, β`−1, γ`−1))

≤ α`−1p(xk, x`−1) + β`−1p(xk, Ty`−1) + γ`−1p(xk, u`−1)

≤ α`−1p(xk, x`−1) + β`−1D1 + γ`−1D′6.

(2.14)

If α`−1 = 0, then from (2.14), A1 ≤ maxD1, D′6.

If α`−1 6= 0, then from (2.12) and (2.14),

A1 < α`−1p(xk, x`) + β`−1D1 + γ`−1D′6

≤ maxA1, D1, D′6.

Therefore,A1 ≤ maxD1, D

′6 for n ≤ k ≤ ` ≤ m. (2.15)

If n ≤ ` ≤ k ≤ m, then

A1 = p(Wp(xk−1, T yk−1, uk−1; αk−1, βk−1, γk−1), x`)

≤ αk−1p(xk−1, x`) + βk−1p(Tyk−1, x`) + γk−1p(uk−1, x`)

≤ αk−1p(xk−1, x`) + βk−1D′1 + γk−1D6.

(2.16)

If αk−1 = 0, then from (2.16), A1 ≤ maxD′1, D6.

If αk−1 6= 0, then from (2.13) and (2.16),

A1 < αk−1p(xk, x`) + βk−1D′1 + γk−1D6

≤ maxA1, D′1, D6.

Therefore,A1 ≤ maxD′

1, D6, for n ≤ ` ≤ k ≤ m. (2.17)

Hence, from (2.15) and (2.17),

A1 ≤ maxD1, D′1, D6, D

′6. (2.18)

(b) Secondly, we make an estimation for maxp(xi, yj) | n ≤ i, j ≤ m. Let

A2 = maxp(xi, yj) | n ≤ i, j ≤ m.

Since yj = Wp(xj , Txj , vj ; ξj , ηj , δj), using (2.18) and (2.8),

A2 = maxp(xi, Wp(xj , Txj , vj ; ξj , ηj , δj)) | n ≤ i, j ≤ m≤ ξjp(xi, xj) + ηjp(xi, Txj) + δjp(xi, vj) | n ≤ i, j ≤ m≤ maxD1, D

′1, D6, D

′6,Φ(δ(An,m)).

(2.19)

(c) And also, we can do it for maxp(yj , xi) | n ≤ i, j ≤ m and then weobtain same estimation (2.19). Let

A3 = maxp(yj , xi) | n ≤ i, j ≤ m.

Then from (2.18) and (2.9),

A3 = maxp(Wp(xj , Txj , vj ; ξj , ηj , δj), xi) | n ≤ i, j ≤ m≤ ξjp(xj , xi) + ηjp(Txj , xi) + δjp(vj , xi) | n ≤ i, j ≤ m≤ maxD1, D

′1, D6, D

′6,Φ(δ(An,m)).

(2.20)

(d) Finally, we make an estimation for maxp(yi, yj) | n ≤ i, j ≤ m. Let

A4 = maxp(yi, yj) | n ≤ i, j ≤ m.

Then by using (2.20) and (2.10), we have

A4 = maxp(yi,Wp(xj , Txj , vj ; ξj , ηj , δj)) | n ≤ i, j ≤ m≤ maxξjp(yi, xj) + ηjp(yi, Txj) + δjp(yi, vj) | n ≤ i, j ≤ m≤ maxp(yi, xj), p(yi, Txj), p(yi, vj) | n ≤ i, j ≤ m≤ maxD1, D

′1, D6, D

′6, Φ(δ(An,m)).

(2.21)

Similarly, using (2.19) and (2.11) we obtain the same result (2.21)

A4 = maxp(Wp(xi, Txi, vi; ξi, ηi, δi), yj) | n ≤ i, j ≤ m≤ maxξjp(xi, yj) + ηjp(Txi, yj) + δjp(vi, yj) | n ≤ i, j ≤ m≤ maxp(xi, yj), p(Txi, yj), p(vi, yj) | n ≤ i, j ≤ m≤ maxD1, D

′1, D6, D

′6, Φ(δ(An,m)).

KIM ET AL156

Consequently, we have

A4 ≤ maxD1, D′1, D6, D

′6,Φ(δ(An,m)). (2.22)

It follows from (2.18)∼(2.20) and (2.22),

D5 = maxp(xi, xj), p(xi, yj), p(yj , xi), p(yi, yj) | n ≤ i, j ≤ m≤ maxD1, D

′1, D6, D

′6,Φ(δ(An,m)). (2.23)

Combining (2.7)∼(2.11) and (2.23), it follows from (2.5),

δ(An,m) = maxD1, D2, D3, D4, D5, D6, D′1, D

′3, D

′4, D

′6

≤ maxD1, D′1, D6, D

′6, Φ(δ(An,m)). (2.24)

If maxD1, D′1 < Φ(δ(An,m)), maxD1, D

′1 < D6 or maxD1, D

′1 < D′

6,then δ(An,m) > 0. In fact, maxD1, D

′1, D6, D

′6,Φ(δ(An,m)) is Φ(δ(An,m)),

D6 or D′6. Since Φ satisfying the condition (CΦ), by using (2.4) we have

maxD1, D′1 < Φ(δ(An,m)) < δ(An,m),

maxD1, D′1 < D6 < δ(An,m)

ormaxD1, D

′1 < D′

6 < δ(An,m).

Therefore, from (2.24) we have δ(An,m) < δ(An,m), which is a contradic-tion. So, we have δ(An,m) ≤ maxD1, D

′1. However, it is obvious that

maxD1, D′1 ≤ δ(An,m). Therefore we have

δ(An,m) = maxD1, D′1.

This completes the proof of conclusion (2.6).Taking n = 0 in (2.6), we have

δ(A0,m)

= maxp(x0, Txi), p(x0, Tyi), p(Txi, x0), p(Tyi, x0) | 0 ≤ i ≤ m≤ p(x0, Tx0) + p(Tx0, x0)

+ maxp(Tx0, Txi), p(Tx0, T yi), p(Txi, Tx0), p(Tyi, Tx0) | 0 ≤ i ≤ m≤ p(x0, Tx0) + p(Tx0, x0) + Φ(δ(A0,m)).

From the condition (CΦ) (see Definition 1.3), there exists the (I − Φ)−1.Therefore, we have

δ(A0,m) ≤ (I − Φ)−1 (p(x0, Tx0) + p(Tx0, x0)) , ∀m ≥ 0. (2.25)

This implies that the sequence δ(A0,m) is bounded.On the other hand, for any positive integers n, m, 1 ≤ n < m, it follows

from (2.6)δ(An,m)

= maxD1, D′1

= maxp(xn, Txi), p(xn, T yi), p(Txi, xn), p(Tyi, xn) | n ≤ i ≤ m= max

n≤i≤mp(Wp(xn−1, T yn−1, un−1;αn−1, βn−1, γn−1), Txi),

p(Wp(xn−1, T yn−1, un−1;αn−1, βn−1, γn−1), T yi),

p(Txi,Wp(xn−1, T yn−1, un−1;αn−1, βn−1, γn−1)),

p(Tyi,Wp(xn−1, T yn−1, un−1;αn−1, βn−1, γn−1))≤ max

n≤i≤mαn−1p(xn−1, Txi) + βn−1p(Tyn−1, Txi) + γn−1p(un−1, Txi),

αn−1p(xn−1, T yi) + βn−1p(Tyn−1, T yi) + γn−1p(un−1, T yi),

αn−1p(Txi, xn−1) + βn−1p(Txi, T yn−1) + γn−1p(Txi, un−1),

αn−1p(Tyi, xn−1) + βn−1p(Tyi, T yn−1) + γn−1p(Tyi, un−1)≤ αn−1δ(An−1,m) + βn−1Φ(δ(An−1,m)) + γn−1δ(An−1,m)

= (1− βn−1)δ(An−1,m) + βn−1Φ(δ(An−1,m))

= (I − βn−1(I − Φ))δ(An−1,m).

By induction and using (2.25),δ(An,m) ≤ (I − βn−1(I − Φ))δ(An−1,m)

≤ (I − βn−1(I − Φ))(I − βn−2(I − Φ))δ(An−2,m)≤ · · ·

≤k−1∏

j=0

(I − βj(I − Φ))δ(Aj,m)

≤ · · ·

≤n−1∏

j=0

(I − βj(I − Φ))δ(A0,m)

≤n−1∏

j=0

(I − βj(I − Φ))(t0),

where t0 = (I − Φ)−1(p(x0, Tx0) + p(Tx0, x0)). Let

an =n−1∏

j=0

(I − βj(I − Φ))(t0), ∀n ∈ N.

KIM ET AL158

Then for any n,m ∈ N,δ(An,m) ≤ an.

And, since Φ : [0,∞) → [0,∞) satisfies the condition (CΦ), for any t > 0,

Φ(t) < t, i.e., (I −Φ)(t) > 0, for all t > 0, and∞∑

j=0

βj = ∞, the sequence an

converges to zero as n → ∞. Therefore, by Lemma 2.1 (ii), this implies thatxn is a Cauchy sequence in K.Let lim

n→∞xn = z (∈ K). Then, since p(xn, ·) is lower semi-continuous,

p(xn, z) ≤ lim infm→∞

p(xn, xm)

≤ lim infm→∞

δ(An,m)

≤ an.

(2.26)

Note that for any n,p(xn, Txn) ≤ an. (2.27)

Assume that z 6= Tz. Then by (2.2), (2.26) and (2.27), we have

0 < infp(xn, z) + p(xn, Txn) | n ∈ N= 2 inf

n∈Nan = 0.

This is a contradiction. Therefore, we have z = Tz.If z, v ∈ F (T ), then, since T is a generalized p-quasicontractive mapping,

p(z, z) ≤ Φ(p(z, z)),

p(v, v) ≤ Φ(p(v, v)),

p(z, v) ≤ Φ(maxp(z, v), p(v, z), 0)(2.28)

andp(v, z) ≤ Φ(maxp(z, v), p(v, z), 0). (2.29)

Without loss of generality, if 0 < p(z, v) ≤ p(v, z), then p(z, v) ≤ Φ(p(v, z))and p(v, z) ≤ Φ(p(v, z)). This implies

p(z, z) = p(v, v) = p(z, v) = p(v, z) = 0.

Then by Lemma 2.1 (i), z = v. Therefore, xn converges to a unique fixedpoint z of T in K. This completes the proof. ¤

We also, easily can get the following theorem for a p-quasicontractive map-ping from Theorem 2.1.

Theorem 2.2. Let E and K be the same as in Theorem 2.1, T : K → K bea p-quasicontractive mapping defined by

p(Tx, Ty)

≤ k

(maxp(x, x), p(x, y), p(x, Tx), p(y, Ty), p(x, Ty), p(y, Tx),


,

for all x, y ∈ K, k ∈ [0, 1), and (2.2) is satisfied. Let xn be the Ishikawatype iterative sequence with errors of T defined by (2.3) and (2.4) is satisfied.Then there is a unique fixed point z of T in K and the sequence xn convergesto z.

For a Mann type iterative sequence with errors, we have the following result.

Theorem 2.3. Let E and K be the same as in Theorem 2.1, T : K → K bea generalized p-quasicontractive mapping (2.1) satisfying condition (2.2). Letxn be a Mann type iterative sequence with errors of T defined by

x0 ∈ K,

xn+1 = Wp(xn, Txn, un; αn, βn, γn), n ≥ 0.(2.30)

where αn, βn, γn, and un are satisfied the conditions of in Theorem2.1 and (2.4) is satisfied. Then there is a unique fixed point z of T in K andthe sequence xn converges to z.

Remark 2.1. Since a convex structure W is a p-convex structure Wp witha w-distance p = d, the Ishikawa type iterative sequence with errors (2.3)becomes to (1.4) and especially, if ηn = 0 and δn = 0 ∀n ≥ 0, it follows fromthe definition of p-convex structure Wp that yn = xn and hence (2.3) reducesto the Mann type iterative sequence with errors (2.30).

Acknowledgements. This work was supported by the Korea Research Foun-dation Grant funded by the Korean Government (MOEHRD, Basic ResearchPromotion Fund)(KRF-2006-311-C00201).

References

1. S. S. Chang, Some results for asymptotically pseudo-contractive mappings and asymp-totically nonexpansive mappings, Proc. Amer. Math. Soc. 129 (3) (2001), 845–853.

2. S. S. Chang and J. K. Kim, Convergence theorems of the Ishikawa type iterative se-quences with errors for generalized quasi-contractive mappings in convex metric spaces,Applied Mathematics Letters 16 (2003), 535-542.

KIM ET AL160

3. S. S. Chang, J. K. Kim and D. S. Jin, Iterative sequences with errors for asymptoticallyquasi nonexpansive mappings in convex metric spaces, Archives Inequ. and Appl. 2(4)(2004), 365-374.

4. S. S. Chang, J. K. Kim and S. M. Kang, Approximating fixed points of asymptoti-cally quasi-nonexpansive type mappings by the Ishikawa iterative sequences with mixederrors, Dynamic Systems and Appl. 13 (2004), 179–186.

5. C. E. Chidume, Convergence theorems for strongly pseudo-contractive and stronglyaccretive mappings, J. Math. Anal. Appl. 228 (1998), 254–264.

6. C. E. Chidume, Global iterative schemes for strongly pseudo-contractive maps, Proc.Amer. Math. Soc. 126 (1998), 2641–2649.

7. L. J. Ciric, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc.45 (1974), 267–273.

8. X. P. Ding, Iteration process for nonlinear mappings in convex metric spaces, J. Math.Anal. Appl. 132 (1988), 114–122.

9. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974),147–150.

10. O. Kada, T. Suzuki and W. Takahashi, Nonconvex minimization theorems and fixedpoint theorems in complete metric spaces, Math. Japon. 44 (1996), 381–391.

11. J. K. Kim, Z. Liu, Y. M. Nam and S.-A. Chun, Strong convergence theorems and sta-bility problems for Mann and Ishikawa iterative sequences for strictly hemi-contractivemappings, J. of Nonlinear and Convex Analysis 5 (2) (2004), 285-294.

12. J. K. Kim, Z. Liu, and S. M. Kang, Almost stability of Ishikawa iterative schemeswith errors for φ-strongly quasi-accretive and φ-hemicontractive operators, Commun.Korean Math. Soc. 19(2) (2004), 267-281.

13. J. K. Kim, S. M. Jang and Z. Liu, Convergence theorems and stability problemsof Ishikawa iterative sequences for nonlinear operator equations of the accretive andstrongly accretive operators, Commun. on Applied Nonlinear Analysis 10(3) (2003),85-98.

14. J. K. Kim, K. H. Kim and K. S. Kim, Convergence theorems of modified three-stepiterative sequences with mixed errors for asymptotically quasi-nonexpansive mappingsin Banach spaces, PanAmerican Math. Jour. 14 (2004), 45–54.

15. J. K. Kim, K. H. Kim and K. S. Kim, Three-step iterative sequences with errors forasymptotically quasi-nonexpansive mappings in convex metric spaces, Nonlinear Anal-ysis and Convex Analysis, Research Institute for Mathematical Sciences Kyoto Univer-sity, Kyoto, Japan, 1365 (2004), 156–165.

16. J. K. Kim, K. H. Kim and K. S. Kim, Convergence theorems of implicit iteration processfor a finite family of asymptotically quasi-nonexpansive mappings in convex metricspaces, Nonlinear Analysis and Convex Analysis, Research Institute for MathematicalSciences Kyoto University, Kyoto, Japan, 1484 (2006), 40-51.

17. J. K. Kim, K. S. Kim and Y. M. Nam, Convergence and stability of iterative processesfor a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convexmetric spaces, Jour. of Comput. Anal. and Appl. 9(2) (2007), 159–172.

18. L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly ac-cretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114–125.

19. Q. H. Liu, On Naimpally and Singh’s open questions, J. Math. Anal. Appl. 124 (1987),157–164.

20. Q. H. Liu, A convergence theorem of the sequence of Ishikawa iterates for quasi con-tractive mappings, J. Math. Anal. Appl. 146 (1990), 301–305.


21. Z. Liu, J. K. Kim and S.-A. Chun, Iterative approximation od fixed points for general-ized asymptotically contractive and generalized hemicontractive mappings, PanAmeri-can Math. Jour. 12(4) (2002), 67-74.

22. S. A. Naimpally and K. L. Singh, Extensions of some fixed point theorems of Rhoades,J. Math. Anal. Appl. 96 (1983), 437–446.

23. B. E. Rhoades, Comments on two fixed point iteration scheme for a generalized con-traction, J. Math. Anal. Appl. 56 (1976), 741–750.

24. B. E. Rhoades, Convergence of an Ishikawa type iteration scheme for a generalizedcontraction, J. Math. and Appl. 185 (1994), 350–355.

25. W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math.Sem. Rep. 22 (1970), 142–149.

26. H. K. Xu, A note on the Ishikawa iteration scheme, J. Math. Anal. Appl. 167 (1992),582–587.

KIM ET AL162

On the analogue of Bernoulli polynomials

Cheon Seoung Ryoo†∗and Seog-Hoon Rim‡

†Department of Mathematics,Hannam University, Daejeon 306-791, Korea

e-mail: ryoocs @hannam.ac.kr

‡ Department of Mathematics Education,Kyungpook National University, Daegu 702-701, Korea

e-mail:[email protected]

Abstract In this paper we define the analogue of Bernoulli polynomials. We inves-tigate some properties of the analogue of Bernoulli polynomials. Furthermore, some newrelations, related to Bernoulli numbers and Euler numbers, are given. Finally, we considerthe reflection symmetries of the analogue of Bernoulli polynomials.

2000 Mathematics Subject Classification - 11B68, 11S40

Key words- Bernoulli numbers, Bernoulli polynomials, Euler polynomials, Eulernumbers

1. Introduction

Throughout this paper we use the following notations. By Zp we denotethe ring of p-adic rational integers, Qp denotes the field of rational numbers,C denotes the complex number field, and Cp denotes the completion of al-gebraic closure of Qp. Let νp be the normalized exponential valuation of Cp

with |p|p = p−νp(p) = p−1. When one talks of q-extension, q is considered inmany ways such as an indeterminate, a complex number q ∈ C, or p-adicnumber q ∈ Cp. If q ∈ C one normally assume that |q| < 1. If q ∈ Cp, we

∗This work was supported by Hannam University Research Fund, 2007

1

normally assume that |q − 1|p < p−1

p−1 so that qx = exp(x log q) for |x|p ≤ 1.Throughout this paper we use the below notation:

[x] = [x : q] =1− qx

1− q.

Hence, limq→1[x] = x for any x with |x|p ≤ 1 in the present p-adic case. Letd be a fixed integer and let p be a fixed prime number. For any positiveinteger N , we set

X = lim←−N

(Z/dpNZ),

X∗ =⋃

0<a<dp(a,p)=1

a + dpZp,

a + dpNZp = x ∈ X | x ≡ a (mod dpN), cf. [1-4]

where a ∈ Z lies in 0 ≤ a < dpN , set

µq(a + dpNZp) =qa

[dpN ]

and this is known to be a distribution on X due to Kim [4, 5].Let UD(Zp) be the set of uniformly differentiable functions on Zp. Let

Tp = ∪N≥1CpN = limN→∞ CpN , where CpN = w|wpN= 1 for some N ≥ 0

is the cyclic group of order pN . For w ∈ Tp, we denote by φw : Zp → Cp thelocally constant function x 7−→ wx ( see [3, 6, 14]). Then φw has continuationto a continuous group homomorphism. For f ∈ UD(Zp), the Kim’s p-adicq-integral is defined by

Iq(f) =

∫

Zp

f(x)dµq(x) = limN→∞

1

[p]Nq

pN−1∑x=0

f(x)qx, see [2, 3, 4, 5] .

Now we consider I1(f) = limq→1 Iq(f). From this, we can derive the below

I1(f) =

∫

Zp

f(x)dµ1(x) = limN→∞

1

pN

pN−1∑x=0

f(x), see [4, 6, 14] .

From the above definition, we can also derive I1(f1) = I1(f) + f ′(0), wheref1(x) = f(x + 1) (see [2, 4, 6, 14]. By using I1(f)-integral, many authors

2

RYOO-RIM164

are studied the analogs of Bernoulli numbers and polynomials, cf.[1, 6, 7, 8,10, 11, 14]. The remainder of the paper is organized as follows: In Section2, we define the analogue of Bernoulli polynomials. We investigate someproperties of the analogue of Bernoulli polynomials. In Section 3, we considerthe reflection symmetries of the analogue of Bernoulli polynomials.

2. The analogue of Bernoulli numbers and polynomials

The purpose of this section is to introduce the analogue of Bernoullinumbers and polynomials. By using these numbers, we will give relationsbetween Bernoulli numbers and Euler numbers. First, we start from thedefinition of the analogue of Bernoulli numbers as follows:

t

wet − 1=

∞∑n=0

Bn(w)tn

n!, w ∈ Tp. (1)

where Bn(w) are called analogue of nth Bernoulli numbers. Since I1(f1) =I1(f) + f ′(0), if we take f(x) = etxwx, we easily see that

I1(wxext) =

t

wet − 1.

Hence we have ∫

Zp

wxxndµ1(x) = Bn(w).

Now we define the analogue of Bernoulli polynomials Bn(w, x) as

ext t

wet − 1=

∞∑n=0

Bn(w, x)tn

n!. (2)

By (1) and (2), it is not difficult to see that

Bn(w, x) =n∑

l=0

(n

l

)Bl(w)xn−l.

By (2), we also have∫

Zp

wt(x + t)ndµ1(t) = Bn(w, x). (3)

3

BERNOULLI POLYNOMIALS 165

Let u be algebraic in complex number field. Then Frobenius-Euler numbersare defined by

eH(u)t =1− u

et − u=

∞∑n=0

Hn(u)tn

n!, (4)

This relation can be written as

H0(u) = 1, (H(u) + 1)k − uHk(u) = 0 (1 ≤ k).

Therefore we have

uHk(u) =k∑

i=0

(k

i

)Hi(u), Hk(u) =

1

u− 1

k−1∑i=0

(k

i

)Hi(u), for u 6= 1.

By (3) and (4), we give a interesting formula on relationship between theBn(w) and Hn(w). Since

t

wet − 1=

t

w

1

et − w−1=

t

w

1

1− w−1

1− w−1

et − w−1=

t

w − 1

1− w−1

et − w−1,

we have

1

w − 1

∞∑n=0

Hn(w−1)tn

n!=

1

t

∞∑n=0

Bn(w)tn

n!=

1

t

∞∑n=1

Bn(w)tn

n!

=1

t

∞∑n=0

Bn+1(w)tn+1

(n + 1)!=

∞∑n=0

Bn+1(w)

n + 1

tn

n!.

Hence we have the following theorem.

Theorem 1. For n ≥ 1, we have

(1) Bn(w) =n

w − 1Hn−1(w

−1), w 6= 1,

(2) Bn(w) = I1(φw(x)xn),

(3) Bn(w) =1

n + 1lim

N→∞1

CpN

CpN−1∑x=0

wxxn+1.

4

RYOO-RIM166

In [6, 14], the I1-integral transform of f is the function f : Tp → Cp

defined byf(w) = I1(fφw) for all w ∈ Tp, f ∈ UD(Zp).

Now, we consider the Iq-integral transform by using p-adic q-integral on Zp

for a variable q ∈ Cp (see [7]) . For f ∈ UD(Zp) the p-adic q-integral wasdefined as

Iq(f) =

∫

Zp


1

[pN ]

pN−1∑x=0

f(x)qx, cf. [6] .

By simple calculation, we have

limN→∞

pN−1∑x=0

w−x

∫

Zp

f(x)wxqxdµ1(x)

= limN→∞

pN−1∑x=0

w−x limN→∞

1

pN

pN−1∑y=0

f(y)wyqy

= limN→∞

1

pN

pN−1∑y=0

f(y)qy

pN−1∑x=0

wy−x = f(x)qx = φq(x)f(x), see [8] .

(5)

Since I1(fφwq) =∫Zp

f(x)wxqxdµ1(x), we also have

limN→∞

pN−1∑x=0

w−x

∫

Zp

f(x)wxqxdµ1(x) = limN→∞

∑w∈C

pN

φw−1Iq(fφqw) =∑w∈Tp

φw−1I1(fφqw).

(6)By (5), (6), we obtain

log q

q − 1

∑w∈Tp

φw−1

q − 1

log qI1(fφqw) =

log q

q − 1

∑w∈Tp

φw−1Iq(fφw) = φq(x)f(x).

Therefore, we obtain the following Iq-integral transform.

Theorem 2. For f ∈ UD(Zp), w ∈ Tp, we have [7]

f(qw) =∑w∈Tp

Iq(fφw)φw−1 =q − 1

log qφq(x)f(x).

5


Now we introduce the convolution for any f, g ∈ UD(Zp,Cp) due to Wood-cock as follows [14] :

(f ⊗ g)(n) =n∑

k=0

f(k)g(n− k), n ≥ 0.

(f ∗ g)(x) =∑

w

fwgwφw−1(x),

where the Fourier transform fw = I1(fφq). From Kim and Woodcock [4, 6,7, 14], we have

∆n+1(f ⊗ g)(x) = (f ⊗∆n+1g)(x) =n∑

j=0

∆jf(x + 1)∆n−jg(0).

If g(0) = 0, then we obtain

∆(f ⊗ g)(x) = (f ⊗∆g)(x).

Since I1(f1) = I1(f) + f ′(0), we have

I1(∆f) = f ′(0).

Hence we obtain

I1(∆(f ⊗ g))(x)) = I1(f ⊗∆g)(x) = (f ⊗ g)′(0).

On the other hand, Woodcock [8] introduced the following results.

(f ⊗ g)′ = (f ⊗ g′) + (f ′ ⊗ g) + f ∗ g,

(f ∗ g)(z) = I1(f(x)g(z − x))− (f ⊗ g′).

By definition, we have (f ⊗ g)(0) = f(0)g(0). Hence

(f ⊗ g)′(0) = (f ⊗ g′)(0) + f ′(0)⊗ g(0) + (f ∗ g)(0)

= f(0)g′(0) + f ′(0) + (f ∗ g)(0)

= f(0)g′(0) + (f ∗ g)(0).

6

RYOO-RIM168

Therefore, we obtain

I1(f ⊗∆g) = (f ⊗ g)′(0)

= f(0)g′(0) + (f ∗ g)(0)

= f(0)g′(0) + I1(fg−)− (f ∗ g′)(0)

= f(0)g′(0)− f(0)g′(0) + I1(fg−)

= I1(fg−),

where g−(x) = g(−x). For w ∈ Tp, let f = zmφw(z), g = zn. Then we have

I1(f ⊗∆g)(z) = I1(zmφw(z)(−z)n)

= (−1)nI1(zm+nφw(z))

= (−1)nBn+m(w).

Since I1(φw(x)) = 0 and

etx = limN→∞

∑w∈C

PN

tφw(x)

wet − 1=

∞∑n=0

limN→∞

∑w∈C

pN

I1(xnφw(x))φw(x),

we obtain

xn = Bn(1) +∑

w∈Tp,w 6=1

1

w − 1Hn−1(w

−1)φw(x).

Therefore we have the following theorem.

Theorem 3. For m,n ≥ 1, we have

(1) (−1)nBm+n(w) =n−1∑

k=0

(n

k

)I1(z

mφw(z)⊗ zk),

(2) xn = Bn(1) +∑

w∈Tp,w 6=1

1

w − 1Hn−1(w

−1)φw(x)

= Bn(1) +∑

w∈Tp,w 6=1

Bn(w)

nφw(x).

7


3. The reflection symmetries of the analogue of Bernoulli polynomials

In this section we consider the reflection symmetries of the analogue ofBernoulli polynomials. Let R be the field of real numbers and let w be thepN -th root of unity. For x ∈ R, we consider the Bernoulli polynomials Bn(x)as follows:

F (t, x) =t

et − 1ext =

∞∑n=0

Bn(x)tn

n!, see [9-13] .

Since ∞∑n=0

Bn(1− x)(−t)n

n!= F (−t, 1− x)

=−t

e−t − 1e(1−x)(−t)

=t

et − 1ext

= F (t, x) =∞∑

n=0

Bn(x)tn

n!,

we obtain thatBn(1− x) = (−1)nBn(x). (7)

Hence Bn(x), x ∈ C, has Re(x) = 1/2 reflection symmetry in addition tothe usual Im(x) = 0 reflection symmetry analytic complex functions. Whathappens with the reflection symmetry (7), when one considers the analogueof Bernoulli polynomials ? We are going now to reflection at 1/2 of x on theanalogue of Bernoulli polynomials. Since

Fw(t, x) =t

wet − 1ext =

∞∑n=0

Bn(w, x)tn

n!,

by simple calculation, we have

Fw−1(−t, 1− x) =−t

w−1e−t − 1e(1−x)(−t)

=−t

w−1e−t − 1e(−t)ext

= wt

wet − 1ext

= wFw(t, x).

8

RYOO-RIM170

Hence we obtain the following theorem.

Theorem 4. For n ≥ 0, we have

Bn(w, x) = (−1)nw−1Bn(w−1, 1− x). (8)

We have the following corollary.

Corollary 5. If Bn(w, x) = 0, then Bn(w−1, 1− x) = 0.

Finally, we shall consider the more general problems. Prove or dis-prove: Since n is the degree of the polynomial Bn(w, x), the number ofreal zeros reBn(w,x) lying on the real plane Im(x) = 0 is then reBn(w,x) =n − cBn(w,x), where cBn(w,x) denotes complex zeros. In general, how manyroots does Bn(w, x) have ? Find the numbers of complex zeros cBn(w,x) of theBn(w, x), Im(x) 6= 0. Using numerical experiments, we hope to investigatethe structure of the complex roots of the analogue of Bernoulli polynomialsBn(w, x). For related topics the interested reader is referred to [9]. Theauthors have no doubt that investigation along this line will lead to a newapproach employing numerical method in the field of research of the analogueof Bernoulli polynomials Bn(w, x) to appear in mathematics and physics.

References

[1] M. Cenkci, M. Can, V. Kurt, ‘p-adic interpolation functions andKummer type congruences for q-twisted and q-generalized twisted Eulernumbers’, Advan. Stud. Contemp. Math., 9, 203-216 (2004).

[2] T. Kim, ‘Non-Arichimedean q-integrals associated with multipleChanghee q-Bernoulli polynomials’, Russ. J. Math. Phys., 10, 91-98(2003).

[3] T. Kim et al, ‘Introduction to non-archimedean analysis ’, KyowooPubl. Company, 2004.

[4] T. Kim, ‘On a q-analogue of the p-adic log gamma functions and relatedintegrals’, J. Number Theory,76, 320-329 (1999).

[5] T. Kim, ‘q-Volkenborn integration’, Russ. J. Math. Phys., 9, 288-299(2002).

9


[6] T. Kim, ‘An analogue of Bernoulli numbers and their congruences’, Rep.Fac. Sci. Engrg. Saga Univ. Math.,22, 7-13 (1994).

[7] T. Kim, ‘A note on the Fourier transform of p-adic q-integrals’, arXiv:math.NT/0511573(23 Nov. 2005), 1-4 (2005).

[8] N. Koblitz, ‘A new proof of certain formulas for p-adic L-function ’,Duke Math. J.,46, 455-468 (1979).

[9] C. S. Ryoo, H. song, R. P. Argawal , ‘On the real roots ofthe Changhee-Barnes’ q-Bernoulli polynomials’, Advan. Stud. Contemp.Math., 9, 153-163, (2004).

[10] K. Shiratani, S. Yamamoto , ‘On a p-adic interpolation of Eulernumbers and its derivatives’, Mem. Fac. Sci. Kyushu Univ., 39, 113-125, (1985).

[11] Y. Simsek , ‘Theorems on Twisted L-function and Twisted BernoulliNumbers’, Advan. Stud. Contemp. Math., 11, 205-218, (2005).

[12] P. G. Todorov , ‘On the theory of the Bernoulli polynomials andnumbers’, J. Math. Anal. Appl., 104, 175-180, (1984).

[13] E. T. Whittaker and G. N. Waston , ‘A Course of Morden Anal-ysis’, Cambridge Univ. Press, 1963.

[14] C. F. Woodcock, ‘Convolutions on the ring of p-adic integers’, J.London Math. Soc. 20(1979), 101-108.

10

RYOO-RIM172

TWO GENERAL FIXED POINT THEOREMS ON THREECOMPLETE METRIC SPACES

DURAN TURKOGLU

Abstract. In this paper two general xed point theorems on three completemetric spaces which generalize the results from [1] and [2] using functions areproved.

1. Introduct¬on

The following xed point theorems were proved by [1-2].

Theorem 1. [2]Let (X; d), (Y; ); (Z; ) be complete metric spaces and suppose Tis a mapping of X into Y , S is a mapping of Y into Z and R is a mapping of Zinto X satisfying the inequalities

d(RSy;RSTx)maxfd(x;RSy); d(x;RSTx)g c(Sy; STx)maxf(Sy; STx);

d(x;RSTx)g(TRz; TRSy)maxf(y; TRz); (y; TRSy)g cd(Rz;RSy)maxfd(Rz;RSy);

(y; TRSy)g

(STx; STRz)maxf(z; STx); (z; STRz)g c(Tx; TRz)maxf(Tx; TRz);(z; STRz)g

for all x in X, y in Y and z in Z, where 0 c < 1:If one of the mappings R; S; T is continuous, then RST has a unique xed pointu in X, TRS has a unique xed point v in Y and STR has a unique xed point win Z. Further,Tu = v, Sv = w; Rw = u:

Theorem 2. [1]Let (X; d), (Y; ); (Z; ) be complete metric spaces and suppose Tis a continious mapping of X into Y , S is a continuous mapping of Y into Z andR is a continuous mapping of Z into X satisfying the inequalities

d(RSTx;RSy) cmaxf(y; Tx); d(x;RSTx); d(x;RSy); (Sy; STx)g

(TRSy; TRz) cmaxf(z; Sy); (y; TRSy); (y; TRz); d(Rz;RSy)g(SRSz; STx) cmaxfd(x;Rz); (z; STRz); (z; STx); (Tx; TRz)g

for all x in X, y in Y and z in Z, where 0 c < 1: Then RST has a unique xedpoint u in X, TRS has a unique xed point v in Y and STR has a unique xedpoint w in Z. Further, Tu = v, Sv = w; Rw = u:

2000 Mathematics Subject Classication. Primary 54H25; Secondary, 47H10.Key words and phrases. Fixed points, complete metric spaces.

1

2 DURAN TURKOGLU

Throughout this paper, <+ stands for the non-negative reals. We will also denoteby = the set of all real functions f : <3+ ! <+ such that:(i) f is upper semi-continuous in each coordinate variable;(ii) If either u f(v; u; 0) or u f(v; 0; u) for all u; v 0; then there exists a

real constant 0 c < 1 such that u cv:

2. Main Results

We now generalize Theorem1 as follows:

Theorem 3. Let (X; d), (Y; ); (Z; ) be complete metric spaces and suppose T isa mapping of X into Y , S is a mapping of Y into Z and R is a mapping of Z intoX satisfying the inequalities

(2.1) d(RSy;RSTx) f((Sy; STx); d(x;RSTx); ; d(x;RSy))

(2.2) (TRz; TRSy) g(d(Rz;RSy); (y; TRSy); (y; TRz))

(2.3) (STx; STRz) h((Tx; TRz); (z; STRz); (z; STx))for all x in X, y in Y and z in Z, where f; g; h 2 =3: If one of the mappings R;S; T is continuous, then RST has a unique xed point u in X, TRS has a uniquexed point v in Y and STR has a unique xed point w in Z. Further, Tu = v,Sv = w; Rw = u:

Proof. Let x0 be an arbitrary point in X: Dene the sequence fxng; fyng andfzng in X;Y and Z; respectively by xn = (RST )nx0; yn = Txn1; zn = Syn forn = 1; 2; :::.Applying inequality (2.1) for y = yn and x = xn we have

d(xn; xn+1) f((zn; zn+1); d(xn; xn+1); 0)which implies by (ii) that

(2.4) d(xn; xn+1) c1(zn; zn+1)where c1 2 [0; 1): Applying inequality (2.3) for x = xn1 and z = zn we have

(zn; zn+1) h((yn; yn+1); (zn; zn+1); 0)which implies by (ii) that

(2.5) (zn; zn+1) c3(yn; yn+1)where c3 2 [0; 1): Applying inequality (2.2) for z = zn1 and y = yn we have

(yn; yn+1) g(d(xn1; xn); (yn; yn+1); 0)which implies by (ii) that

(2.6) (yn; yn+1) c2d(xn1; xn)where c2 2 [0; 1): It follows from inequalities (2.4), (2.5) and (2.6) that

d(xn+1; xn) c1(zn; zn+1) c1c3(yn; yn+1) ::: (c1c2c3)nd(x0; x1):Since 0 c1c2c3 < 1; fxng; fyng and fzng are Cauchy sequence with the limits u; vand w in X;Y and Z respectively.Now suppose that S is continuous. Then lim

n!1Syn = lim

n!1zn and so

(2.7) Sv = w:

174

SHORT TITLE 3

Applying inequality (2.1) we now have

d(RSv; xn1) f((Sv; zn); d(xn1; xn); d(xn1; RSv)):Letting n tend to innity and using (i), it follows

d(RSv; u) f((Sv;w); 0; d(u;RSv))using equation (2.7) we have

d(RSv; u) f(0; 0; d(u;RSv)):By (ii) follows that d(u;RSv) c:0 which implies by (2.7) that(2.8) u = RSv = Rw:

Applying inequality (2.2) we have

(Tu; yn+1) g(d(u; xn); (yn; yn+1); (yn; TRw)):Letting n tend to innity and using (i), it follows that

(Tu; v) g(0; 0; (v; TRw))which implies by (2.8) that

(Tu; v) g(0; 0; (v; Tu)):By (ii) follows that

(2.9) Tu = v:

It now follows from equations (2.7), (2.8) and (2.9)

TRSv = TRw = Tu = v;

STRw = STu = Sv = w;

RSTu = RSv = Rw = u:

The same results of course will hold if R or T is continuous instead of S:We now prove the uniqueness of the xed point u. Supoose that RST has a

second xed point u0 . Then using inequality (2.1), we have

d(RSTu;RSTu0) f((STu0; STu); d(u;RSTu); d(u;RSTu0))d(u; u) f((STu; STu0); 0; d(u; u0)):

By (ii) we have

(2.10) d(u; u0) c1(STu; STu0):Further, using inequality (2.3), we have succesively

(STRSTu; STu0) h((Tu0; TRSTu); 0; (STu; STu0))

(STu; STu0) h((Tu0; Tu); 0; (STu; STu0)):

By (ii) we have

(2.11) (STu; STu0) c2(Tu; Tu0):Finally, using inequality (2.2), we have

(2.12) (Tu; Tu0) c2d(u; u0):By (2.10), (2.11) and (2.12) we have

d(u; u0) (c1c2c3)d(u; u0)

COMPLETE METRIC SPACES 175

4 DURAN TURKOGLU

which implies u = u0: The xed point u of RST is therefore unique. Similarly, itcan be proved that v is the unique xed point of TRS and w is the unique xedpoint of STR. This completes the proof of the theorem.

Remark 1. Letting

f(t1; t2; t3) = g(t1; t2; t3) = h(t1; t2; t3) =ct1maxft1; t2gmaxft2; t3g

with (0 c < 1), then we see that Theorem1 is a consequence of Theorem 3.

We will also denote by = the set of all real functions f : <4+ ! <+ such that:(iii) f is upper semi-continuous in each coordinate variable;(iv) If either u f(v; u; 0; w) or u f(v; 0; u; w) for all u; v 0; then there

exists a real constant 0 c < 1 such that u cmaxfv; wg:We now generalize Theorem 2 as follows:

Theorem 4. Let (X; d), (Y; ); (Z; ) be complete metric spaces and suppose T isa mapping of X into Y , S is a mapping of Y into Z and R is a mapping of Z intoX satisfying the inequalities

(2.13) d(RSTx;RSy) f((y; Tx); d(x;RSTx); d(x;RSy); (Sy; STx))

(2.14) (TRz; TRSy) g((z; Sy); (y; TRSy); (y; TRz); d(Rz;RSy))

(2.15) (STRz; STx) h(d(x;Rz); (z; STRz); (z; STx); (Tx; TRz))for all x in X, y in Y and z in Z, where f; g; h 2 =4: If one of the mappings R;S; T is continuous, then RST has a unique xed point u in X, TRS has a uniquexed point v in Y and STR has a unique xed point w in Z. Further, Tu = v,Sv = w; Rw = u:

Proof. Let x0 be an arbitrary point in X. Dene the sequence fxng; fyng and fzngin X;Y and Z; respectively, by

xn = (RST )nx0 , yn = Txn1, zn = Syn for n = 1; 2; 3; :::.

Applying inequality (2.14) and using property (iv) for z = zn1 and y = yn wehave

(yn; yn+1) = (TRzn1; TRSyn) g((zn1; zn); (yn; yn+1); 0; d(xn1; xn))and it follows that

(2.16) (yn; yn+1) cmaxfd(xn1; xn); (zn1; zn)gwhich implies by (iv) and inequality (2.16) that

(2.17) (zn; zn+1) cmaxfd(xn1; xn); (yn; yn+1)g

cmaxfd(xn1; xn); (zn1; zn)g.Applying inequality (2.13) for y = yn and x = xn we have

d(xn; xn+1) = d(RSTxn; RSyn) f((yn; yn+1); d(xn; xn+1); 0; (zn; zn+1))which implies by (iv) and inequality (2.16) and (2.17) that

(2.18) d(xn; xn+1) cmaxf(yn; yn+1); (zn; zn+1)g

cmaxfd(xn1; xn); (zn1; zn)g.

176

SHORT TITLE 5

It now follows easily by induction on using inequalities (2.16), (2.17) and (2.18)that

d(xn; xn+1) cn1maxfd(x1; x2); (z1; z2)g;(yn; yn+1) cn1maxfd(x1; x2); (z1; z2)g(zn; zn+1) cn1maxfd(x1; x2); (z1; z2)g.

Since 0 c < 1; it follows that fxng; fyng and fzng are Cauchy sequences withlimits u; v and w in X;Y and Z respectively.Now suppose that S is continuous. Then limSyn = lim zn and so

(2.19) Sv = w.

Applying inequality (2.13) for y = v and x = xn we now have

d(RSv; xn+1) f((v; Txn); d(xn; xn+1); d(xn; RSv); (Sv; STxn)):Letting n tend to innity and using (iii) it follows

d(RSv; u) f(0; 0; d(RSv; u); 0)which implies by (iv) that d(RSv; u) = 0 and so

(2.20) RSv = u:

Using equation (2.19) this gives us

(2.21) Rw = u:

Using equation (2.20) and inequality (2.14) for z = Sv and y = yn, we have

(Tu; yn+1) g((Sv; Syn); (yn; TRSyn); (yn; TRSv); d(RSv;RSyn)):Letting n tend to innity and using (iii) it follows

(Tu; v) g(0; 0; (v; Tu); 0)which implies (ii) that (Tu; uv) = 0 and so

(2.22) Tu = v:

It follows from equations (2.19),(2.21) and (2.22) that

TRSv = TRw = Tu = v;

STRw = STu = Sv = w;

RSTu = RSv = Rw = u:

The same results of course hold if R or T is continuous instead of S:We now provethe uniqueness of the xed point u: Supoose that RST has a second xed point u

0:

Then using inequality (2.13) for y = Tu and x = u0we have

d(u; u0) = d(RSTu;RSTu0) f((Tu; Tu0); 0; d(u; u

0); (STu; STu

0))

which implies by (ii) that

(2.23) d(u; u0) cmaxf(Tu; Tu0); (STu; STu

0)g:

Further, using inequality (2.14) for z = STu and y = Tu0 we have

(Tu; Tu0) g((STu; STu0); 0; (Tu; Tu0); d(u; u0))

which implies by (ii) that

(2.24) (Tu; Tu0) cmaxf(STu; STu0); d(u; u0)g:

COMPLETE METRIC SPACES 177

6 DURAN TURKOGLU

inequalities (2.23) and (2.24) implies that

(2.25) d(u; u0) c(STu; STu0):Finally, using inequality (2.15) and property (ii), we have

(STu; STu0) h(d(u; u0); 0; (STu; STu0); (Tu; Tu0))which implies by (ii)

(2.26) (STu; STu0) cmaxfd(u; u0); (Tu; Tu0)g:It now follows from inequalities (2.24),(2.25) and (2.26) that

d(u; u0) c(STu; STu0) c2(STu; STu0);and so u = u0; since 0 c < 1: The xed point u of RST is therefore unique.Similarly, it can proved that v is the unique xed point of TRS and w is the uniquexed point of STR: This completes the proof of theorem. Remark 2. Letting

f(t1; t2; t3; t4) = g(t1; t2; t3; t4) = h(t1; t2; t3; t4) = cmaxft1; t2; t3; t4gwith (0 c < 1), then we see that Theorem 2 is a consequence of Theorem 4.

References

[1] Nung N. P., A xed point theorem in three metric spaces, Math Seminar Notes, Kobe Univ.,1983, 11, 77-79.

[2] Jain R. K., Shrivastava A. K., Fisher B., Fixed points on three complete metric spaces, NoviSad J. Math., 1997, 27, 27-35.

Department of Mathematics, Faculty of Science and Arts, Gazi University, 06500Teknikokullar, Ankara / TURKEY

E-mail address : [email protected], [email protected]

178

Journal of Computational Analysis and Applications manuscript No.(will be inserted by the editor)

Explicite Quasiconformal Extensionsof Planar Harmonic Mappings

Andrzej Ganczar

Received: date / Accepted: date

Abstract Harmonic mappings appear in connection with the theory of minimal sur-

faces – a nonparametric minimal surface expressed in terms of isothermal parameters

and projected onto its base plane induces a harmonic mapping. Intensive studies of

these mappings were initiated by J. Clunie and T. Sheil-Small (see [3] for details).

They have been interested in harmonic mappings as generalizations of conformal map-

pings. The aim of this paper is to find explicite quasiconformal extensions, to the

extended plane C, for some special harmonic mappings defined on the unit disk.

Keywords harmonic mapping · quasiconformal extension

Mathematics Subject Classification (2000) MSC 30C62 · 30C45

1 Introduction

A complex–valued function f defined on the unit disk D = z ∈ C : |z| < 1 which is

injective and satisfies the equation fzz = 0 will be called a harmonic map of D. The

statement that f is harmonic on D implies that fz is analytic and fz is anti–analytic.

According to the above remark, we have

f = h+ g (1)

where h and g are analytic on D. In view of Lewy’s theorem (see [7]), the Jacobian

Jf (z) = |h′(z)|2 − |g′(z)|2 doesn’t vanish on D and we may assume that Jf > 0 (i.e. f

to be sense–preserving).

The function ωf = g′/h′ = fz/fz has special significance. If f is a sense–preserving

harmonic mapping, we see that ωf is analytic and satisfies |ωf (z)| < 1 on D. Moreover,

ωf agrees in modulus with the first complex dilatation µf = fz/fz , so it may be

Andrzej GanczarInstitute of Mathematics, Maria Curie–Sk lodowska UniversityPl. Marii Curie–Sk lodowskiej 1, 20-031 Lublin, PolandTel.: +48-081-537-6264Fax: +48-081-533-3669E-mail: [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.2,179-186,2008, COPYRIGHT 2008 EUDOXUS PRESS, LLC

2

called the analytic (or second complex) dilatation of f . On each compact subset of

the unit disk ωf is bounded away form one, and so f is locally quasiconformal. It is

clear that a sense–preserving harmonic mapping f is quasiconformal on D if and only

if ‖ωf‖∞ = supz∈D

|ωf (z)| < 1.

In this paper we are going to find conditions for a given harmonic mapping f of the

disk D to have a quasiconformal extension to C, and to give an explicite construction

of an extension. To this end we need the notions of quasicircle and quasiconformal

reflection due to Ahlfors. Recall that, a Jordan curve Γ in C is called a quasicircle if it

is the image line of a circle under a quasiconformal automorphism of C. Let G, G∗ be

complementary domains of a Jordan curve Γ in C. We say that φ is a quasiconformal

reflection in Γ if φ is a sense-reversing quasiconformal mapping of G onto G∗ whose

homeomorphic extension to the closure G keeps the points on Γ fixed. It is known (see

e.g. [6]) that a Jordan curve Γ admits a quasiconformal reflection φ if and only if it is

a quasicircle.

In the last fifteen years, many papers have been published on harmonic mappings

satisfying certain coefficient conditions. For a fixed sequence (ψn)n=2,3,... of positive

real numbers, we denote by H0(ψn) the class of mappings f of the form (1) where

h(z) = z +

∞∑n=2

anzn, g(z) =

∞∑n=2

bnzn, z ∈ D, (2)

that satisfy the condition∞∑n=2

ψn(|an|+ |bn|) ≤ 1. (3)

It has been shown (see [2]) that each function of the class H0(n) is a sense–preserving

and starlike harmonic mapping. The class H0(n2) was also considered by Avci and

Z lotkiewicz ([2]) and it is well-known that f ∈ H0(n2) is a convex harmonic mapping

of D.

A natural question arises in connection with this, i.e., to characterize harmonic

mappings from H0(ψn) that admit a quasiconformal extension to C and find an explicit

construction of such an extension.

2 Main results

Consider a sequence (ψn)n=2,3,... of positive real numbers satisfying the condition

ψnn

≥ ψ2

2, n = 3, 4, . . . . (4)

Notice that if ψ2 < 2, then f ∈ H0(ψn) need not be univalent on D (see e.g. [5,

Theorem 3]), so we will restrict our attention to the case ψ2 ≥ 2. Clearly, the last

inequality implies that H0(ψn) ⊂ H0(n), and we obtain

Lemma 1 Assume that f ∈ H0(ψn), where (ψn)n=2,3,... is a sequence of positive real

numbers satisfying the condition (4) and such that ψ2 ≥ 2. Then f is a sense–preserving

harmonic mapping of the disk D onto a starlike domain.

The following technical lemma, provides some information on a sequence (ψn)n=2,3,...

satisfying (4), which will be relevant later on.

GANCZAR180

3

Lemma 2 Let (ψn)n=2,3,... be a sequence of positive real numbers satisfying the con-

dition (4).

(i) If ψ2 ∈ [4,+∞), then

ψnn

sin

(π

2 log2 ψ2

)>

ψ2

2 log2 ψ2≥ 1, n = 2, 3, . . . .

(ii) If ψ2 ∈ (2, 4), then

ψnn

sin

(π

2 log2 ψ2

)> 1 >

ψ2

2 log2 ψ2, n = 2, 3, . . . .

Proof If ψ2 ∈ [4,+∞), obviously by Jordan’s inequality

ψnn

sin

(π

2 log2 ψ2

)>

ψnn log2 ψ2

and then (i) follows from the condition (4).

Now we prove (ii). According to the condition (4), we have

ψnn

sin

(π

2 log2 ψ2

)≥ ψ2

2sin

(π

2 log2 ψ2

), n = 2, 3, . . . ,

so it suffices to proveψ2

2sin

(π

2 log2 ψ2

)> 1

for ψ2 ∈ (2, 4). Consider the function

G(x) = x sin

(π

2 log2 x

).

Clearly G(2) = 2, and it is enough to show that G is strictly increasing on the interval

(2, 4). But the inequality (see e.g. [8, p. 246])

4

π

x

π − 2x< tanx,

where x ∈ (0, π2 ), implies

G′(x) >1

2π log2 xcos

(π

2 log2 x

)(4 ln 2) log2

2 x− π2(log2 x− 1)

(ln 2)(log2 x− 1) log2 x> 0

for any x ∈ (2, 4), and the proof is complete.

utThe next step is an estimation of the modulus |ωf (z)| for f ∈ H0(ψn).

Lemma 3 Suppose that f = h+g ∈ H0(ψn) and (ψn)n=2,3,... is a sequence of positive

real numbers satisfying the condition (4).

(i) If ψ2 ∈ [4,+∞), then

‖ωf‖∞ = supz∈D

∣∣∣∣ g′(z)h′(z)

∣∣∣∣ < 2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

).

PLANAR HARMONIC MAPPINGS 181

4

(ii) If ψ2 ∈ (2, 4), then

‖ωf‖∞ < sin

(π

2 log2 ψ2

).

Proof We shall justify the case (i) only. For the case (ii) the proof may be done similarly.

Let us first observe that for f = h + g ∈ H0(ψn), where h and g are of the form (2),

we have

|ωf (z)| =

∣∣∣∣ g′(z)h′(z)

∣∣∣∣ ≤∞∑n=2

n|bn|

1−∞∑n=2

n|an|, z ∈ D.

By the above, it is enough to show that the condition (4) implies

∞∑n=2

n|bn|

1−∞∑n=2

n|an|<

2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

).

Since ψ2 ∈ [4,+∞), we conclude that

∞∑n=2

n|bn|+2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

) ∞∑n=2

n|an| ≤∞∑n=2

n(|an|+ |bn|) <

2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

) ∞∑n=2

ψn(|an|+ |bn|),

the last inequality being a consequence of Lemma 2. Condition (3) now yields

∞∑n=2

n|bn|+2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

) ∞∑n=2

n|an| <2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

),

and (i) is proved.

ut

Corollary 1 Assume that (ψn)n=2,3,... is a sequence of positive real numbers satisfying

the condition (4) and ψ2 > 2, then f ∈ H0(ψn) is quasiconformal on D.

Observe that a necessary condition for a harmonic mapping f ∈ H0(ψn) to have a

quasiconformal extension to the whole plane is the following: the image curve f(T) is

a quasicircle, where T = z ∈ C : |z| = 1. On account of the above remark, we now

prove

Theorem 1 Suppose that f ∈ H0(ψn), where (ψn)n=2,3,... is a sequence of positive

real numbers satisfying the condition (4) and ψ2 > 2. Then the curve f(T) is a quasi-

circle.

Proof Let f(z) = z+∞∑n=2

anzn+

∞∑n=2

bnzn, for all z ∈ D. By Lemma 2 and the condition

(3), we have

∞∑n=2

n(|an|+ |bn|) < sin

(π

2 log2 ψ2

) ∞∑n=2

ψn(|an|+ |bn|) ≤ sin

(π

2 log2 ψ2

).

GANCZAR182

5

Thus, for z1, z2 ∈ D such that z1 6= z2

|f(z1)− f(z2)| =

∣∣∣∣∣z1 − z2 +

∞∑n=2

an(zn1 − zn2 ) +

∞∑n=2

bn(zn1 − zn2 )

∣∣∣∣∣ ≤ (5)

|z1 − z2|

(1 +

∞∑n=2

n(|an|+ |bn|)

)< |z1 − z2|

(1 + sin

(π

2 log2 ψ2

))and

|f(z1)−f(z2)| ≥ |z1−z2|

(1−

∞∑n=2

n(|an|+ |bn|)

)> |z1−z2|

(1− sin

(π

2 log2 ψ2

))> 0.

(6)

From (5) and (6) we deduce that f has homeomorphic extension on D, which also

satisfies (5) and (6). Therefore the image line Γ = f(T) is a Jordan curve.

According to Ahlfors (see [1]), a Jordan curve γ is a quasicircle if and only if

K(γ) = sup|w1 − w2| · |w3 − w4|+ |w1 − w4| · |w2 − w3|

|w1 − w3| · |w2 − w4|

is finite, where supremum being taken over the set of all ordered quadruples w1, w2, w3, w4of points on γ. We conclude from (5) and (6) that

K(Γ ) ≤

1 + sin(

π2 log2 ψ2

)1− sin

(π

2 log2 ψ2

)2

·K(T) <∞,

and finally that Γ is a quasicircle.

ut

Remark 1. Possibility of homeomorphic extension on D in the proof above follows

immediately from the fact that f , according to Lemma 3, is quasiconformal on D (see

e.g. [6, Theorem I.8.2]).

W can now formulate our main theorem.

Theorem 2 Let f(z) = z+∞∑n=2

anzn+

∞∑n=2

bnzn belong to H0(ψn), where (ψn)n=2,3,...

is a sequence of positive real numbers satisfying the condition (4) and ψ2 > 2. Then

the mapping

F (z) =

f(z), for |z| ≤ 1,

z +∞∑n=2

anz−n +∞∑n=2

bnz−n, for |z| ≥ 1,

(7)

is a quasiconformal extension of f onto C. Moreover, its complex dilatation µF satisfies

|µF (z)| < sin(

π2 log2 ψ2

)for any ψ2 ∈ (2, 4), and |µF (z)| < 2 log2 ψ2

ψ2sin(

π2 log2 ψ2

)in

the case ψ2 ∈ [4,+∞).

6

Proof We justify the case ψ2 ∈ [4,+∞) only. On account of Lemma 3 f is quasicon-

formal on D and |µF (z)| < 2 log2 ψ2ψ2

sin(

π2 log2 ψ2

)for all z ∈ D. Moreover, this map

has a homeomorphic extension on D.

Next, in the same way as in Theorem 1, for z1, z2 ∈ C\D such that z1 6= z2, we obtain

|f(z1)− f(z2)| ≤ |z1 − z2|+ |z−11 − z−1

2 |∞∑n=2

n(|an|+ |bn|) ≤

|z1 − z2|

(1 +

∞∑n=2

n(|an|+ |bn|)

)< |z1 − z2|

(1 + sin

(π

2 log2 ψ2

)),

and

|f(z1)−f(z2)| ≥ |z1−z2|

1−

∞∑n=2

n(|an|+ |bn|)

|z1z2|

> |z1−z2|

(1−

sin ( π2 log2 ψ2

)

|z1z2|

)> 0.

In view of the part (i) of Lemma 2, for any z ∈ C \ D

|Fz | = |1−∞∑n=2

nbnz−n−1| ≥ 1−

∞∑n=2

n|bn| ≥ 1−∞∑n=2

n(|an|+|bn|) > 1−sin

(π

2 log2 ψ2

)> 0,

and

|µF (z)| =

∣∣∣∣FzFz∣∣∣∣ =

∣∣∣∣∣∣∣∣∞∑n=2

nanz−n−1

1−∞∑n=2

nbnz−n−1

∣∣∣∣∣∣∣∣ ≤∞∑n=2

n|an|

1−∞∑n=2

n|bn|<

2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

)< 1.

This leads to

JF (z) = |Fz |2 − |Fz |2 = |Fz |2(1− |µF (z)|2) > 0

for z ∈ C \ D.

By the above considerations, F given by (7) is continuous and locally univalent on C,

moreover limz→∞

f(z) = ∞. Therefore (see e.g. [9, Theorem 2.7.2]), the mapping F is a

sense-preserving homeomorphism of C onto itself. Its complex dilatation µF satisfies

|µF (z)| < 2 log2 ψ2

ψ2sin

(π

2 log2 ψ2

)for any z ∈ C \ T. Since T is a removable set for F (see [6, p. 44]) it follows that F is

a quasiconformal in the whole plane.

A trivial verification shows that f(z) = z + 12z

2 belongs to H0(n) but has no

quasiconformal extension on C. For this reason, if ψ2 = 2 then the condition (4) does

not imply the possibility of quasiconformal extension of f ∈ H0(ψn). Therefore, given

k ∈ (0, 1) we consider the class H0(ψn, k) of functions f = h+ g, where h and g are of

the form (3), that satisfying the condition

∞∑n=2

ψn(|an|+ |bn|) ≤ k < 1.

We can now state the analogy of Lemma 3 and Theorem 1.

GANCZAR184

7

Theorem 3 Suppose that f = h+ g ∈ H0(ψn, k), where (ψn)n=2,3... is a sequence of

positive real numbers satisfying the condition (4) and ψ2 = 2. Then

(i) ‖ωf‖∞ ≤ k,

(ii) f(T) is a quasicircle.

Analysis similar to that in the proof of Theorem 2 shows that

Theorem 4 Let (ψn)n=2,3,... be a sequence of positive real numbers such that the

condition (4) holds and ψ2 = 2. If f(z) = z +∞∑n=2

anzn +

∞∑n=2

bnzn ∈ H0(ψn, k), then

the function F given by (7) is a quasiconformal extension of f onto C whose complex

dilatation µF satisfies |µF (z)| ≤ k.

The following theorem gives a non–trivial way of producing quasiconformal extensions.

Theorem 5 Let f : D → C be a sense–preserving harmonic mapping such that

‖ωf‖∞ < 1 and Γ = f(T) is a quasicircle. If φ is a quasiconformal reflection in

Γ , then

G(z) =

f(z), for |z| ≤ 1,

φ(f(1/z)), for |z| ≥ 1,(8)

is a quasiconformal extension of f onto C.

Proof The proof of Theorem follows immediately from the fact that a Jordan curve

Γ admits a quasiconformal reflection φ if and only if it is a quasicircle. Moreover, G

defined by (8) is a homeomorphism of C onto itself which is quasiconformal in D and

C \ D. We omit the details.

As a direct consequence of Theorem 5, we obtain

Corollary 2 If f is a sense–preserving harmonic automorphism of the disk D such

that ‖ωf‖∞ < 1, then the mapping

G(z) =

f(z), for |z| ≤ 1,

1/f(1/z), for |z| ≥ 1,

defines a quasiconformal extension of f onto C.

It is known (see [4, Corollary 2]) that a convex Jordan curve contained in the

annulus w : r ≤ |w| ≤ R is a quasicircle. According to the above remark, we have

Lemma 4 Suppose that f : D → C is a sense-preserving harmonic mapping and f(D)

is a bounded convex region. If ‖ωf‖∞ < 1, then the mapping G given by (8), where φ

denotes a quasiconformal reflection in f(T), is a quasiconformal extension of f onto

C.

Notice that, if (ψn)n=2,3,... is a sequence of positive real numbers satisfying the

condition ψn ≥ n2, n = 2, 3, . . . , then f ∈ H0(ψn) is a convex harmonic mapping.

Remark 2. If the co-analytic part of f = h+ g ∈ H0(ψn) is zero, i.e. the function g

is identically zero, then our results describe analogous properties of analytic univalent

functions (see e.g. [4]).

8

References

1. Ahlfors, L. V., Quasiconformal reflections, Acta Math., 109, (1963), 291-301.2. Avci, Y., Z lotkiewicz, E., On harmonic univalent mappings, Ann. Univ. Mariae Curie-

Sk lodowska, Sect. A, 44, (1990), 1-7.3. Clunie, J. G., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser.

A.I., 9, (1984), 3-25.4. Fait, M., Krzyz, J., Zygmunt, J., Explicit quasiconformal extensions for some classes of

univalent functions, Comment. Math. Helvetici, 51, (1976), 279-285.5. Ganczar, A., On harmonic univalent mappings with small coefficients, Demonstratio

Math., 34, no. 3, (2001), 549-558.6. Lehto, O., Virtanen, K. I., Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin-

Heidelberg-New York, Second Edition, 1973.7. Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull.

Amer. Math. Soc. 42 (1936), 689-692.8. Mitrinovic, D. S., Analytic Inequalities, Springer-Verlag, Berlin-Heidelberg-New York,

1970.9. Sheil-Small, T.,Complex Polynomials, Cambridge University Press, 2002.

GANCZAR186

Numerical solution of integral equations by using localpolynomial regression

Hikmet Caglar(1) and Nazan Caglar(2,∗)

(1) Istanbul Kultur University, Science and Letters Faculty, Department ofMathematics-Computer,Istanbul, Turkey

(2) Istanbul Kultur University Faculty of Economic and Administrative, 34156Atakoy-Istanbul, Turkey Email : [email protected]∗To whom all correspondence should be addressed

Abstract

In this paper, we find numerical solution of

x(t) + λ∫ ba k(t, s)x(s)ds = y(t), a ≤ t ≤ b

or

x(t) + λ∫ ta k(t, s)x(s)ds = y(t), a ≤ t ≤ b, a ≤ s ≤ b

by Local Polynomial Regression(LPR). We shown that, present new methodis powerful in solving both Fredholm and Volterra integral equations. Themethod is tested on some model problems to demonstrate its usefulness. Theconvergence of the method is discusses.

Keywords: Integral equations,Local polynomial regression, Kernel functions.

1. Introduction

Consider the linear Fredholm integral equation ,

x(t) + λ∫ ba k(t, s)x(s)ds = y(t), a ≤ t ≤ b (1)

and the linear Volterra integral equation

x(t) + λ∫ ta k(t, s)x(s)ds = y(t), a ≤ t ≤ b, a ≤ s ≤ b (2)

1


Where the parameter λ , the kernel k and the function y are given and x(t)is the unknown function to be determined. Many different techniques havebeen presented so far for solving the above mentioned integral equation[4,5,6].The theory of integral equations can be found in [1,2]. The subject of thispaper is to present new method by use of local polynomial regression for thenumerical solution of Fredholm and Volterra integral equations, and finally,we shown that the method to achive the desired accuracy. Details of thestructure of the present method is explained in section 2. We apply methodfor linear Fredholm and Volterra integral equations.In Section 3 for showingthe efficiency of numerical method. Finally, in Section 4 contains conclussionand direction for future research.

2. Numerical method

In this section, we describe our method to find the approximating solutionof integral Eqs.(1) and Eqs.(2). In our study We have opted for using localpolynomial regression. The fundamential idea of this methodology appearsin the [3]. The following is the mathematical formulation of the local poly-nomial regression.

2.1. Local polynomial regression

Suppose that the (p + 1)th derivative of x(t) at point t0 exists. We approx-imate the unknown regression function x(t) locally at t0 by a polynomial oforder p. The theoretical justification is that we can approximate, in a neigh-borhood of t0 , x(t) using a Taylor expansion

x(t) ≈ ∑pk=0 βk(t − t0)

k (3)

where

βk = x(k)(t0)k!

(4)

This polynomial, used to approximate the unknown function locally at t0, isobtained by solving a locally weighted least squares regression problem, i.e.by minimizing

2

CAGLAR-CAGLAR188

∑ni=1

Yi − ∑p

k=0 βk(ti − t0)k2

K( ti−t0h

) (5)

where h is a parameter called bandwidth (also called a smoothing parame-ter), K is a weighting function called the kernel function. Let βk, k = 0, 1, ...pbe the solution of the minimizing problem. From Eqs.(4), it is clear that j!βj

is an estimator for the derivatives x(j)(t0), j = 0, 1, ...p. Thus, the estimationobtained, of both the regression function and its derivatives, is local, andtherefore, the process must be repeated at all points where an estimation isof interest. Let us see the analytical expression of the solution βk, k = 0, 1, ...pof the locally weighted least squares regression problem. Let X be the n x(p + 1) matrix

X=

1 (t1 − t0) ... (t1 − t0)p

. . . .

. . . .

. . . .1 (tn − t0) ... (tn − t0)

p

(6)

and the vectors y = (Y1, Y2, ..., Yn)′and β = (β0, β1, ...βp)

′. Finally, denote

by W the nxn diagonal matrix of weights W = diagKh(ti − t0). Then, thesolution is

β = (XT WX)−1XT Wy. (7)

The selection of K does not influence the results much. We selected thequartic kernel as follows

K(u) =

1516

(1 − u2)2 if |u| ≤ 10 otherwise

(8)

2.2. Illustration of the method

In this section the LPR method for solving Eqs(1) and Eqs(2) is outlined.Let Eqs(3) be an approximate solution of Volterra integral equation(2)

x(t) =∑p

j=0 βj(t − x0)j (9)

3

NUMERICAL SOLUTION OF INTEGRAL EQUATIONS 189

where

t1 = a, t2, ...tn = b

and It is required that the approximate solution(9) satisfies the integral equa-tion at the points t=ti. Putting (9) in (2), it is follows that

∑pj=0 βj(t − t0)

j + λ∑p

j=0 βj

∫ ta k(t, s)(s − t0)

jds = y(t),

a ≤ t ≤ b, a ≤ s ≤ b (10)

Then, matrix form(6) can be written as

X=

1 (t1 − t0) + λ∫ t

ak(t, s)(s − x0)ds ... (t1 − t0)p + λ

∫ t

ak(t, s)(s − x0)pds

. . . .

. . . .

. . . .

1 (tn − t0) + λ∫ t

ak(t, s)(s − x0)ds ... (tn − t0)p + λ

∫ t

ak(t, s)(s − x0)pds

Y=

y1

.

.

.yn

(11)

Putting (11) in (7), then estimated set of coefficients βi are obtained bysolving matrix system solution. Therefore, approximate solution (9) is ob-tained. Same procedure can be used for Eqs(1).

4

CAGLAR-CAGLAR190

3. Numerical examples

In this section we consider some examples of Fredholm and Volterra typethat demonstrate the accurancy and effectiveness of the present method. Allcomputations were carried out using MATLAB 6.5.

Example 1.

We first consider the following integral equation

x(s) = sin(s) − s +∫ π

20 stx(t)dt , 0 ≤ t ≤ π

2

with the exact solution: x(s) = sin(s)

The numerical results are shown in Table 1 and illustruate Fig.1. The max-imum absolute error is given in Table 2.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.2

0

0.2

0.4

0.6

0.8

1

1.2

Exact SolutionLPR

Figure 1: Results for Example 1 with x(s)=sin(s)

5


Table 1: yi : ExactSolution; Yi : LPRSolutionyi Yi

0 -0.000000001206640.09801714032956 0.098017163352870.19509032201613 0.195090308958610.29028467725446 0.290284661946630.38268343236509 0.382683435738620.47139673682600 0.471396750446620.55557023301960 0.555570237965610.63439328416365 0.634393271553920.70710678118655 0.707106758901480.77301045336274 0.773010437685790.83146961230255 0.831469612615480.88192126434835 0.881921272961500.92387953251129 0.923879529575800.95694033573221 0.956940310397420.98078528040323 0.980785253446020.99518472667220 0.995184736302611.00000000000000 0.99999998107762

Example 2.

x(s) = es − es+1−1s+1

+∫ 10 estx(t)dt, 0 ≤ t ≤ 1

Exact solution: x(s) = es

The maximum absolute error is given in Table.2. Also numerical results areshown Fig.2.

Example 3.

x(s) = s +∫ 10 K(s, t)x(t)dt, 0 ≤ t ≤ 1

K(s, t) =

s, s ≤ tt, s ≥ t

Exact solution: x(s) = sec(1)sin(s)

6

CAGLAR-CAGLAR192

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Exact SolutionLPR

Figure 2: Results for Example 2 with x(s)=exp(s)

The maximum absolute error is given in Table 2. See Fig.3.

Example 4.

x(s) = − ∫ s0 stx(t)dt + (2−s)e−s2+x

2, 0 ≤ t ≤ 1

Exact solution: x(s) = e−s2

The maximum absolute error is given in Table 2. See Fig.4.

4. Conclusions

In this study, we introduced new method for solve the integral equations.We shown that the method is a very fast convergent for solving integralequations.Numerical results showed that the present method approximentsthe exact solution very well. The Method can also be extended to the differ-ent bandwidth and kernel functions.

7


Table 2: The Maximum ErrorsExample n,p Absolute Error

1 11,9 1.686115x10−7

1 21,9 3.004420x10−9

2 11,8 1.716089x10−7

2 21,8 1.938185x10−8

3 11,8 1.365100x10−7

3 21,8 1.007513x10−9

4 21,9 2.188129x10−6

4 41,9 1.489571x10−7

5. References

[1] Baker,C.T.H. , 1969, The Numerical Solution of Integral Equations, (Ox-ford: Clarendon Press).[2] Delves, L.M. and Mohamed,J.L. , 1985,Computational Methods for Inte-gral Equations, (Cambridge:University Press).[3] Fontan Lopez, J.L.,Ruso, J.M., Prieto,G. and Sarmiento, F.,2004, Eur.Phys.J.E,13,133-140.[4] Golbabai,A. and Seifollahi,S.,2006,Applied Mathematics and Computa-tion, 174, 877-883.[5]Maleknejad,K. and Derili,H.,2006,Applied Mathematics and Computation,175,1235-1244[6] Maleknejad,K. and Lotfi,T.,2006,Applied Mathematics and Computa-tion,175, 347-355.

8

CAGLAR-CAGLAR194

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Exact SolutionLPR

Figure 3: Results for Example 3 with with x(s)=sec(1)sin(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Exact SolutionLPR

Figure 4: Results for Example 4 with with x(s) = e−s2

9


Bayesian Classifier based on the Multivariate

Normal Distribution

Iuliana Florentina Iatan,

Department of Mathematics and Informatics,Technical University of Civil Engineering, Bucharest, Romania,

e-mail [email protected]

Abstract

The aim of this paper is the Bayesian estimation techniques toobtain the form of the a posteriori density p(µ|D) and the desiredprobability density p(X|D) in the case when p(X|µ) ∼ N(µ,Σ). Thetreatment of the multivariate case in which Σ is known but µ is notrepresents the generalization of the univariate case (see [5]).

This approach is then used in Bayesian classification.AMS Subject Classification: 62C10, 62E17Keywords: Bayesian estimation, a posteriori probability, a priori

probability, Bayes theorem, conditional distribution, posterior distri-bution

1 The aim of the paper

Bayesian decision theory is a fundamental statistical approach to the prob-lem of pattern classification.

We consider M classes ω1, . . . , ωM with a priori probabilities (the prob-abilities of each class occurring) P (ω1), . . . , P (ωM ) assumed known.

Let be X a d- dimensional vector, called the feature vector which isnormally distributed. The general multivariate normal density is given bya d-dimensional mean vector and a d-by-d covariance matrix:

p(X) =1

(2π)d/2|Σ|1/2exp

[−1

2(X − µ)tΣ−1(X − µ)

],

where the covariance matrix Σ is assumed to be known and the mean vectorµ is unknown.

1


We wish to assign X to one of the M classes. We know that the Bayesrule is to assign X to class ωj if

P (ωj |X) > P (ωk|X), k = 1,M ; k 6= j, (1)

namely if the probability of class ωj given the feature vector X, P (ωj |X)is greatest over all classes ω1, . . . , ωM .

The a posteriori probability P (ωj |X) represents the probability so thatX belongs to class ωj , after X was classified.

The a posteriori probabilities P (ωj |X) may be expressed in terms ofthe a priori probabilities P (ωj) and the class-conditional density functionsp(X|ωj) using Bayes theorem as

P (ωj |X) =p(X|ωj)P (ωj)

p(X)(2)

so the relation (1) may be written

p(X|ωj)P (ωj) > p(X|ωk)P (ωk), k = 1,M ; k 6= j. (3)

Let us consider the sample set D = X1, X2, . . . , Xn on a randomvector X.

We consider that the a priori multivariate density for the mean µ, p(µ)is also normal

p(µ) ∼ N(µ0,Σ0),

namely

p(µ) =1

(2π)d/2|Σ0|1/2exp

[−1

2(µ − µ0)tΣ0

−1(µ − µ0)]

,

where both the mean vector µ0 and the covariance matrix Σ0 are known.Our goal is to estimate the probability density p(X|D) because it rep-

resents the desired class conditional density p(X|ωj) which multiplied bythe a prior probability P (ωj) it gives the probabilistic information neededto design the classifier.

From [2] we know that

p(X|D) =∫

Dp(X|µ)p(µ|D)dµ, (4)

where:

2

IATAN198

• the function p(X|µ) represents the parametric form of the probabilitydensity p(X)

• p(µ|D) is a posteriori density of the random vector µ,• p(X|D) is the desired class-conditional density.

Therefore, for realizing our goal are necessary two stages:Stage I. We estimate p(µ|D),Stage II. We estimate p(X|D) using the relation (4).

2 Bayesian estimation approach in the gaussiancase, used in pattern recognition: the multi-variate hypothesis

Theorem 1 [2] The desired class-conditional density is

p(X|D) ∼ N(µn,Σn + Σ),

where:

Σn−1 = Σ0

−1 + nΣ−1

µn = Σn

(Σ0−1µ0 + Σ−1

∑nk=1 Xk).

Proof:

Stage I. In order to estimate p(µ|D) we shall use the Bayes theorem [2],i.e

p(µ|D) =p(D|µ)p(µ)∫

D p(D|µ)p(µ)dµ. (5)

As X1, X2, . . . , Xn are stochastically independent, we have:

T = p(X1, X2, . . . , Xn|µ)p(µ) = p(µ)n∏

k=1

p(Xk|µ). (6)

Thus,

T =1

(2π)d/2|Σ0|1/2exp

[−1

2(µ − µ0)tΣ0

−1(µ − µ0)]·

·n∏

k=1

1(2π)d/2|Σ|1/2

exp[−1

2(Xk − µ)tΣ−1(Xk − µ)

].

3

BAYESIAN CLASSIFIER 199

The last formula can be written as

T =1

(2π)d+nd

2 |Σ0|1/2|Σ|n/2exp

−1

2[(µ − µ0)tΣ0

−1(µ − µ0) +n∑

k=1

(Xk − µ)tΣ−1(Xk − µ)]

.

Denoting

A =1

(2π)d+nd

2 |Σ0|1/2|Σ|n/2

we shall have

T = A · exp− 1

2

[µtΣ0

−1µ − µtΣ0−1µ0 − µt

0Σ0−1µ + µt

0Σ0−1µ0+

+n∑

k=1

(XtkΣ−1Xk − Xt

kΣ−1µ − µtΣ−1Xk + µtΣ−1µ)

]=

= A exp−1

2

[µtΣ0

−1µ−2µtΣ0−1µ0+µt

0Σ0−1µ0+

n∑k=1

XtkΣ−1Xk−2µtΣ−1

n∑k=1

Xk+nµtΣ−1µ]

and, finally:

T = A·exp−1

2

[µt(nΣ−1+Σ0

−1)µ−2µt(Σ0−1µ0+Σ−1

n∑k=1

Xk)+µt0Σ0

−1µ0+n∑

k=1

XktΣ−1Xk

]Using the notation

µn =1n

n∑k=1

Xk (7)

we shall obtain

T = A·exp−1

2

[µt(nΣ−1+Σ0

−1)µ−2µt(Σ0−1µ0+nΣ−1µn)+µt

0Σ0−1µ0+

n∑k=1

XktΣ−1Xk

]Using the notation

Σn−1 = Σ0

−1 + nΣ−1 (8)

we shall have

T = A·exp−1

2

[µtΣn

−1µ−2µt(Σ0−1µ0+nΣ−1µn)+µt

0Σ0−1µ0+

n∑k=1

XktΣ−1Xk

];

4

IATAN200

therefore

T = A·exp−1

2

[µtΣn

−1(µ − 2(ΣnΣ0

−1µ0 + nΣnΣ−1µn))+µt

0Σ0−1µ0+

n∑k=1

XktΣ−1Xk

].

Introducing the new notation

µn = Σn

(Σ0−1µ0 + nΣ−1µn

)(9)

it results

T = A · exp

−1

2

[µtΣn

−1(µ − 2µn) + µt0Σ0

−1µ0 +n∑

k=1

XktΣ−1Xk

],

namely

T = A·exp

−1

2

[µt

0Σ0−1µ0 +

n∑k=1

XktΣ−1Xk + (µ − µn)tΣn

−1(µ − µn) − µtnΣn

−1µn

].

Taking into account the notation

kn = µt0Σ0

−1µ0 +n∑

k=1

XktΣ−1Xk − µt

nΣn−1µn (10)

we obtain the shorter form

T = A exp[−1

2(µ − µn)tΣn

−1(µ − µn)]

exp(−kn

2

)(11)

Using equation (5) as in terms of the normal case notations

p(µ|D) =A exp

[−1

2(µ − µn)tΣn−1(µ − µn)

]exp

(−kn

2

)A exp

(−kn

2

) ∫D exp

[−1

2(µ − µn)tΣn−1(µ − µn)

]dµ

(12)

we obtain the posterior distribution as

p(µ|D) =1

(2π)d/2|Σn|1/2exp

[−1

2(µ − µn)tΣn

−1(µ − µn)]

. (13)

We used the fact that∫1

(2π)d/2|Σn|1/2exp

[−1

2(µ − µn)tΣn

−1(µ − µn)]

dµ = 1

5


implies ∫exp

[−1

2(µ − µn)tΣn

−1(µ − µn)]

dµ = (2π)d/2|Σn|1/2

From (13) it results that p(µ|D) is a normal density with mean vectorµn and covariance matrix Σn, i.e.

p(µ|X1, X2, . . . , Xn) ∼ N(µn,Σn).

Stage II. Let’s determine now, the conditional distribution p(X|X1, X2, . . . , Xn).Using (4) we obtain

B = p(X|X1, X2, . . . , Xn) =∫

Dp(X|µ)p(µ|X1, X2, . . . , Xn)dµ.

Using the formula (13), the previous relation becomes

B =1

(2π)d/2|Σ|1/2(2π)d/2|Σn|1/2·

·∫

Dexp

[−1

2(X − µ)tΣ−1(X − µ)

]exp

[−1

2(µ − µn)tΣn

−1(µ − µn)]

dµ

or

B =1

(2π)d|Σ|1/2|Σn|1/2·∫

Dexp

−1

2[(X − µ)tΣ−1(X − µ) + (µ − µn)tΣn

−1(µ − µn)]

dµ

(14)Let us consider the expression

C = (X − µ)tΣ−1(X − µ) + (µ − µn)tΣn−1(µ − µn),

which can be developed as follows

C = XtΣ−1X−2µtΣ−1X +µtΣ−1µ+µtΣn−1µ−2µtΣn

−1µn+µtnΣn

−1µn =

= XtΣ−1(Σ + Σn)(Σ + Σn)−1X − 2µtΣ−1(Σ + Σn)(Σ + Σn)−1X+

+µtΣ−1(Σ + Σn)(Σ + Σn)−1µ + µtΣ−1n (Σ + Σn)(Σ + Σn)−1µ−

−2µtΣ−1n (Σ + Σn)(Σ + Σn)−1µn + µt

nΣ−1n (Σ + Σn)(Σ + Σn)−1µn =

= Xt(Σ + Σn)−1X + XtΣ−1Σn(Σ + Σn)−1X − 2µt(Σ + Σn)−1X−

6

IATAN202

−2µtΣ−1Σn(Σ + Σn)−1X + µt(Σ + Σn)−1µ + µtΣ−1Σn(Σ + Σn)−1µ+

+µtΣ−1n Σ(Σ + Σn)−1µ + µt(Σ + Σn)−1µ − 2µtΣ−1

n Σ(Σ + Σn)−1µn−

−2µt(Σ + Σn)−1µn + µtnΣ−1

n Σ(Σ + Σn)−1µn + µtn(Σ + Σn)−1µn−

−2µtn(Σ + Σn)−1X + 2µt

n(Σ + Σn)−1X.

Thus, finally we have

C = (X−µn)t(Σ+Σn)−1(X−µn)−2µt(Σ−1X+Σ−1n µn)+µt(Σ−1+Σ−1

n )µ+

+(XtΣ−1 + µtnΣ−1

n )Σn(Σ + Σn)−1Σ(Σ−1X + Σ−1n µn)

One obtains

p(X|X1, X2, . . . , Xn) =1

(2π)d/2|Σ + Σn|1/2exp

−1

2[(X − µn)t(Σ + Σn)−1(X − µn)

].

(15)and hence p(X|D) is normally distributed with mean vector µn and covari-ance matrix Σ + Σn, i.e.

p(X|D) ∼ N(µn,Σ + Σn).

3 Conclusions

The Bayesian approach described above involves two stages. The first stageis concerned on the calculation of the posterior density p(µ|X1, X2, . . . , Xn)for a specified prior with (13). The second stage is the integration over µ toobtain the conditional density p(X|X1, X2, . . . , Xn) which may be viewedas making allowance for the variability in the estimate due to sampling.

In conclusion, as in the univariate case (see [5]), to obtain the class con-ditional density p(X|D), whose parametric form is known to be p(X|µ) ∼N(µ,Σ), we have to replace µ by µn and Σ by Σ + Σn. In effect, themean vector µn is treated as if it were the true mean vector and the knowncovariance matrix is increased to Σn .

The density p(X|D) is the desired class conditional density p(X|ωj)which multiplied by the a prior probability P (ωj) it gives the probabilisticinformation needed to design the classifier.

7


References

[1] Bishop, C.,M., 2006. Pattern Recognition and Machine Learning.Springer, Heidelberg.

[2] Duda, D.,O., Hart P., E., Stork, D., G., 2001. Pattern Classification.John Wiley, second edition.

[3] Enachescu, D., 2003. Tehnici statistice de data mining. Editura Uni-versitatii din Bucuresti.

[4] Fukunaga, K.,1990. Introduction to Statistical Pattern Recognition.Academic Press, New York.

[5] Iatan, I., 2005. ”Classification Using Bayesian Approach: GaussianCase”, Analele Universitatii Bucuresti, : 55-64.

[6] Jain, A., Duin, P., Mao, J., 2000. ”Statistical Pattern Recognition: Areview”, IEEE Transactions on Pattern Analysis and Machine Intel-ligence, 22(1): 4-37.

[7] Webb, A., 2002. Statistical Pattern Recognition. John Wiley and Sons,N.York, second edition.

8

IATAN204

On Harmonic Function Spaces II

Stevo StevicMathematical Institute of the Serbian Academy of Science,

Knez Mihailova 35/I, 11000 Beograd, SerbiaE-mail: [email protected]; [email protected]

Abstract. In this paper we investigate several spaces of harmonic functions,such as: a-Bloch, Hardy, the weighted Bergman, Besov and Dirichlet on the openunit ball in Rn.

MSC 2000: 31B05.

Keywords: Harmonic function, Hardy space, Bergman space, Besov space,Hadamard gaps.

1 Introduction and preliminaries

Throughout this paper G is a domain in the Euclidean space Rn, n ≥ 1,B(a, r) = x ∈ Rn | |x − a| < r denotes the open ball centered at a ∈ Rn

of radius r > 0, where |x| denotes the norm of x ∈ Rn and B is the open unitball in Rn. Also, S = ∂B = x ∈ Rn | |x| = 1 is the boundary of B.

Let dV denote the Lebesgue measure on Rn, vn the volume of B, dσ thesurface measure on S, σn the surface area of S, dVN the normalized Lebesguemeasure on B, dσN the normalized surface measure on S, and Va,r = V (B(a, r)).Let H(B) denote the set of complex valued harmonic functions on B.

Hardy type spaces. For p > 0, letHp(B) denote the set of harmonic functionsu on B such that

‖u‖Hp(B) = sup0<r<1

Mp(u, r) = sup0<r<1

(∫

S

|u(rζ)|pdσN (ζ))1/p

< +∞.

For p ≥ 1, Hp(B) is a Banach space and for p ∈ (0, 1) it is a complete metricspace with the following translation invariant metric

d(u, v) = sup0<r<1

∫

S

|u(rζ)− v(ζ)|pdσN (ζ).

Elements of Hp(B) theory can be found in [3, Chapter VI].The Hardy space Hp(B) is contained in the spaces Hp

β(B), consisting of allharmonic functions u on B such that

‖u‖Hpβ(B) = sup

0<r<1(1− r)β

(∫

S

|u(rζ)|pdσN (ζ))1/p

< +∞,

where β ∈ [0,∞). These spaces we call Hardy type spaces or weighted Hardyspaces.

a-Bloch space. Let a > 0. A function f ∈ C1(B) is said to be an a-Blochfunction if

‖f‖Ba = supx∈B

(1− |x|)a|∇f(x)| < +∞.

The space of a-Bloch functions is denoted by Ba(B) = Ba. If a = 1, Ba justbecomes the Bloch space B. Note that ‖ · ‖Ba is a seminorm on Ba(B), thatis, Ba(B) is a normed space, modulo constant functions. The norm is usuallydefined by the quantity |f(0)|+‖f‖Ba . With this norm Ba(B) is a Banach space.Let HBa(B) denote the space which consists of all harmonic a-Bloch functionson the unit ball, i.e., H(B) ∩ Ba(B). Basic results on Bloch functions can befound, for example, in [2, 14, 18, 20, 32, 42, 44, 46]. In [18] it was proved thatfor p ≥ 1, there is a positive constant c(p, n), depending on p and n, such thatfor every u ∈ H(B)

1c(p, n)

‖u‖BMOp ≤ ‖u‖H,n := supx∈B

12(1− |x|2)|∇u(x)| ≤ c(p, n)‖u‖BMOp .

In the case n = 2, this result was essentially obtained in [8]. For some infor-mation on BMO space, see, for example [4]. In [29, Theorems 2 and 3] weproved that Muramoto’s result is true also for p ∈ (0, 1). This Muramoto paperinspired us to calculate exactly the BMOp norm for harmonic functions (see[29]) where we essentially proved a generalization of the Hardy-Stein identity(see, for example, [10, p.42]). Further applications of this identity can be foundin [31] and [37], where among others some results for analytic functions on theunit disk in [45] and [47] are extended.

Square area space. In the main result of [37] a space of functions on B,consisting of all u ∈ H(B) such that

∫ 1

0

(1− r)γAp/2(r, u)dr < ∞,

where γ ∈ (−1,∞), p > 0 and

A(r, u) =∫

rB

|∇u(x)|2dV (x)

is introduced. We will denote this space by Sp,γ(B). It can be considered as anatural generalization of a space of analytic functions on the unit disk whichappears in [13]. When p ≥ 1 it is easy to see that

‖u‖Sp,γ = |u(0)|+(∫ 1

0

(1− r)γAp/2(r, u)dr

)1/p

,

defines a norm on Sp,γ(B) and that with this norm it is a Banach space. Forp ∈ (0, 1), Sp,γ(B) becomes a complete metric space with the following metric

dSp,γ (u, v) = |u(0)− v(0)|p +∫ 1

0

(1− r)γAp/2(r, u− v)dr.

STEVIC206

Weighted Bergman space. Let ω(r), 0 < r < 1, be a positive weight functionwhich is integrable on (0, 1). We extend ω on B by setting ω(x) = ω(|x|).

For 0 −1, we denote the norm by ‖u‖bpα

and the correspondingspace by bp

α(B). When p ≥ 1, bpω(B) is a Banach space with the norm ‖ · ‖bp

ω

and when p ∈ (0, 1) a complete metric space with the metric defined by

db(u, v) =∫

B

|(u− v)(x)|pω(x)dV (x).

Recently there has been a great interest in studying the weighted Bergman typespaces of analytic or harmonic functions, see, for example, [1], [7], [16], [21],[22], [26], [27], [30], [33], [34], [38], [39] and the references therein.

Dirichlet type space. For α ∈ (−1,∞) let Dpα(B) = Dp

α be the class of allharmonic functions u on the unit ball obeying

‖u‖pDp

α= |u(0)|p +

∫

B

|∇u(x)|p(1− |x|)αdV (x) < ∞.

For p = 2 and α = 0 we obtain the classical Dirichlet space.Harmonic Besov space. The Harmonic Besov space Bp(B) = Bp is the space

of all u ∈ H(B) such that∫

B

(1− |x|2)p|∇u(x)|pdτ(x) < ∞,

where

dτ(x) =dV (x)

(1− |x|2)n.

For p > 1, the Besov space with the following norm

‖u‖Bp= |u(0)|+

(∫

B

(1− |x|2)p|∇u(x)|pdτ(x))1/p

,

becomes a Banach space.We say that a locally integrable function f on B possesses the HL−property,

with a constant c > 0 if

f(a) ≤ c

rn

∫

B(a,r)

f(x)dV (x) whenever B(a, r) ⊂ B.

Every subharmonic function ([11]) possesses the HL−property when c = 1/vn.In [9] Hardy and Littlewood proved that |u|p, p > 0, n = 2, also possesses theHL−property whenever u is a harmonic function in B. In the case n ≥ 3 a

HARMONIC FUNCTION SPACES 207

generalization was made by Fefferman and Stein [5]. In this paper we need thefollowing generalization of the Fefferman and Stein result, see [23].

Lemma A. Let f be a non-negative subharmonic function on a proper opendomain G ⊂ Rn and p > 0. Then there is a constant C depending only on nand p, such that

fp(x) ≤ C

rn

∫

B(x,r)

fp(y)dV (y),

where 0 < r < d(x, ∂G) (the distance from the point x to the boundary of G).

The following lemma could be folklore. We omit its proof.

Lemma B. The following statements are true.

(a) If u ∈ HBa(B), then there is a positive constant C independent of u and x,such that

|u(x)| ≤

C‖u‖Ba , 0 ≤ a < 1,C‖u‖Ba ln 2

1−|x| , a = 1,

C(1− |x|)1−α‖u‖Ba , a > 1.

(b) If u ∈ Hpβ(B), p, β ∈ (0,∞), then there is a positive constant C independent

of u and x, such that

|u(x)| ≤ C‖u‖Hp

β

(1− |x|)n+pβ−1p

.

This paper can be considered as a continuation of our investigations devotedto harmonic functions on the unit ball, see [1, 29, 31, 33, 35, 37, 40, 41, 42].

The paper is organized as follows. Section 2 is devoted to the study of somerelationships among the functions which belong to the above mentioned spaces.Some new equivalent conditions for harmonic Bergman functions are presentedin Section 3. Equivalent conditions for harmonic Besov spaces are presentedin Section 4. In Section 5 we initiate the study of harmonic functions withHadamard gaps.

Throughout the paper, constants are denoted by C, they are positive andmay differ from one occurrence to the other. The notation a ¹ b means thatthere is a positive constant C such that a ≤ Cb. If both a ¹ b and b ¹ a hold,then one says that a ³ b.

2 Some relationships among spaces of harmonicfunctions

In this section we investigate some relationships among the spaces of harmonicfunctions mentioned in Section 1.

STEVIC208

2.1 Fejer-Riesz type inequality

In this subsection we will extend our Theorem 2 in [37], which is an inequalityof Riesz-Fejer type, see [6, 11, 19].

Theorem 1. Suppose that u ∈ H(B) and ε > 0. Then∫ r

0

(r − ρ)n−2+εM(ρ)pdρ ≤ cp,n,ε rn−1+ε sup0<ρ<r

Mpp (u, ρ), p > 0, (1)

for some cp,n,ε > 0, which depends only on p, n and ε, and all r ∈ (0, 1), where

M(r) = M(r, u) = sup |u(x)| | |x| = r .

Proof. Without loss of generality we may assume that r = 1. ApplyingLemma B (b) with β = 0, we have that

(1− |x|)n−1|u(x)|p ≤ C‖u‖pHp ,

which implies that(1− ρ)n−1Mp(ρ) ≤ C‖u‖p

Hp .

Multiplying the last formula by (1 − ρ)−1+ε and then integrating from 0 to 1,we obtain inequality (1), as desired.

Remark 1. Note that Theorem 1 extends our result from [37] for the casep ∈ (0, 1), and that in the proof of Theorem 2 in [37] we have used the Poissonintegral formula

u(x) =∫

∂B

1− |x|2|x− ζ|n u(ζ)dσN (ζ), x ∈ B

and Jensen’s and Harnack’s inequalities, to obtain

|u(x)|p ≤2‖u‖p

Hp(B)

(1− |x|)n−1,

however Jensen’s inequality holds only for p ≥ 1.

Remark 2. We do not know, at the moment, whether or not Theorem 1 holdsfor ε = 0. However, it is interesting that for the case of analytic functions onthe unit disk the result also holds for this case.

2.2 Inclusion theorems

In this section we prove several inclusion results regarding spaces of harmonicfunctions. Some of them could be known but we cannot find any specific ref-erences in the literature. Before formulating the main result of this section wequote the following known result, see, for example, [15].

Lemma C. Suppose 0 ≤ p < ∞ and α > −1. Then there is a constant C =C(p, ω) such that

|u(0)|p +∫

B

|∇u(x)|p(1− |x|)p+αdV (x) ³∫

B

|u(x)|p(1− |x|)αdV (x),

for all u ∈ H(B).

Now we formulate and prove the main result of this section.

Theorem 2. The following statements hold.

(a) If a 6= 1 and α > −1, then Ba ⊂ bpα if a < 1 + 1+α

p .

(b) If a = 1, then Ba ⊂ bpα for every p ∈ (0,∞) and α > −1.

(c) bpα ⊂ B n+α

p +1.

(d) If a ∈ (0, 1), then Ba ⊂ Hpβ , for every p > 0 and β ∈ [0,∞).

(e) Hpβ ⊂ B n−1

p +β+1, p ∈ (0,∞).

(f) If p > 2, β ≥ 0, and β1 = 2pβ/(2− p), then Sp,(n−2+β1+ε)(2−p)/2 ⊂ Hpβ , for

every ε > 0.

(g) If a > p + 1 then Dpα = bp

α−p.

Proof. (a) By the proof of Lemma B (a), we have that

|f(x)| ≤ |f(0)|+ ‖f‖Ba

a− 11

(1− |x|)a−1.

From this and a well-known inequality, we obtain

|f(x)|p ≤ cp

(|f(0)|p +

‖f‖pBa

(a− 1)p

1(1− |x|)p(a−1)

),

and consequently∫

B

|f(x)|p(1−|x|)αdV (x) ≤ C

(|f(0)|p +

(‖f‖Ba

a− 1

)p ∫

B

(1− |x|)α−p(a−1)dV (x))

.

The last integral converges if and only if α+1 > p(a−1), from which the resultfollows.

(b) By Lemma B (a) case a = 1 and a known inequality, it follows that∫

B

|f(x)|p(1−|x|)αdV (x) ≤ cp

(|f(0)|p + ‖f‖p

B1

∫

B

(1− |x|)α lnp 11− |x|dV (x)

).

STEVIC210

By polar coordinates, and the change of variables r = 1 − ρ, it is easy to seethat the last integral is equiconvergent to the following integral

∫ 1/2

0

tα lnp 1tdt.

Since this integral converges for every p ∈ (0,∞), and α > −1, it follows thatB1(B) ⊂ bp

α(B), for every p ∈ (0,∞), and α > −1.

(c) By HL-property of the function |∇u|p, p > 0, we have that

|∇u(z)|p ≤ C

(1− |x|)n

∫

B(x,(1−|x|)/2)

|∇u(y)|pdV (y).

For y ∈ B(x, (1− |x|)/2), we have 12 (1− |x|) < (1− |y|) < 3

2 (1− |x|). Thus

|∇u(x)|p ≤ C

(1− |x|)n+α+p

∫

B(x,(1−|x|)/2)

|∇u(y)|p(1− |y|)α+pdV (y).

From this and by Lemma C, it follows that

(1− |x|)n+αp +1|∇u(x)| ≤ C

(∫

B

|∇u(y)|p(1− |y|)α+pdV (y))1/p

,

≤ C1

(∫

B

|u(y)|p(1− |y|)αdV (y))1/p

.

Hence bpα ⊂ B n+α

p +1, from which the result follows.

(d) Assume that u ∈ Ba and a ∈ (0, 1). Then by Lemma 6.4.8 in [24],u ∈ Lip1−a(B), which implies that u ∈ Lip1−a(B).

We have

|u(r1ζ)− u(r2ζ)| =∣∣∣∣∫ r2

r1

〈∇u(tζ), ζ〉dt

∣∣∣∣ ≤∫ r2

r1

|∇u(tζ)|dt

≤ ‖u‖Ba

∫ r2

r1

1(1− |tζ|)a

dt

= ‖u‖Ba

11− a

((1− r1)1−a − (1− r2)1−a

) → 0,

uniformly on ζ ∈ S. Since the functions ur(ζ) = u(rζ) are continuous on S itfollows that the limit function u(ζ) is also continuous on S.

Of course, every continuous function on B, belongs to Hpβ(B) for every p ∈

(0,∞), which is what we need to prove.

(e) By Lemma B (b) it follows that there is a positive constant C such that

|u(x)| ≤ C

(1− |x|)n−1p

‖u‖Hp (2)

for every u ∈ Hp(B).On the other hand, by the Cauchy’s estimate we have that

|∇u(x)| ≤ C

1− |x| supy∈B(x,(1−|x|)/2)

|u(y)|. (3)

Applying (2) to the harmonic function u( (1+|x|)

2 y), (3) and the fact that

1− |x| ³ 1− |y| if y ∈ B(x, (1− |x|)/2),

it follows that there is a positive constant C independent of the function u suchthat

(1− |x|)n−1p +β+1|∇u(x)| ≤ C sup

y∈B

(1− 1 + |y|

2

)β

Mp

(u,

1 + |y|2

),

from which the result follows.

(f) This has been proved in [37, Theorem 2], case p > 2.

(g) This is a direct consequence of Lemma C.

From Theorem 2 we obtain the following corollary.

Corollary 1. If a ∈ (0, 1). Then, Ba ⊂ bpα.

Proof. Note that when a ∈ (0, 1), then a < 1 + 1+αp .

Remark 3. Note that statements (a) and (b) in Theorem 2 hold for everyf ∈ C(1)(B), that is, we have not used the harmonicity of function f in theproofs of these statements.

2.3 A little on the Hp0(B) space

By Hp0(B), we denote the subset of the Hardy space Hp(B) consisting of all

harmonic Hardy functions u which satisfy the following condition

limr→1

∫

S

|u(rζ)|pdσN (ζ) = 0.

If p ≥ 1, then in view of the monotonicity of the integral means Mp(u, r), itfollows that Hp

0(B) is trivial, that is Hp0(B) = 0. In view of this fact, a

somewhat surprising fact is that Hp0(B) is nontrivial when p ∈ (0, 1) (for the

case n = 1, see, for example, [25]). We now quote some functions which belongto the space. Before that we cite a well known fundamental integral estimateon the unit ball B.

Lemma D. Let p > n−1n , ζ ∈ S and Ip =

∫S

dσ(ζ)|x−ζ|np , then

Ip < cp,n(1− r)n−1−np, 0 ≤ r < 1 (4)

STEVIC212

for some cp,n > 0, where x = rη, η ∈ S.

It is interesting that Lemma D can be proved by a somewhat forgotten Zonalregion method (see [12]). Let γ = ∠(η, ζ), then by a simple calculation we have

Ip =∫

S

dσ(ζ)(1− 2〈x, ζ〉+ |x|2)np/2

=∫

S

dσ(ζ)(1− 2r cos γ + r2)np/2

.

Let F (γ) =∫

c(x,γ)dσ(ζ), where c(x, γ) is the polar cap centered at x and

with half angle γ (see, [12]). Then Ip can be written, in the following form

Ip =∫ π

0

dF (γ)(1− 2r cos γ + r2)np/2

.

Employing integration by parts we obtain

Ip =F (π)

(1 + r)np+ npr

∫ π

0

sin γ F (γ)dγ

(1− 2r cos γ + r2)np/2+1,

since F (0) = σ(c(x, 0)) = σ(x) = 0. Note that F (π) =∫

c(x,π)dσ(ζ) =∫

Sdσ(ζ) = σn. It is enough only to estimate the following integral

∫ δ

0

sin γ F (γ)dγ

(1− 2r cos γ + r2)np/2+1.

SinceF (γ) = σ(c(x, γ)) = σn−1

∫ γ

0

sinn−2 θ dθ,

we have F (γ) ≤ σn−1γn−1/(n− 1), by the well-known inequality sin θ ≤ θ, θ ≥

0. Thus, we have that

∫ δ

0

sin γ F (γ)dγ

(1− 2r cos γ + r2)np/2+1≤

∫ δ

0

σn−1

n− 1γn dγ

((1− r)2 + 4r

π2 γ2)np/2+1

. (5)

Using the substitution 2π

√rγ = (1− r)ϕ we obtain that the last integral is less

than

cn

rn+1

2

∫ 2√

rδπ(1−r)

0

(1− r)n+1ϕndϕ

(1− r)np+2(1 + ϕ2)np/2+1≤ C

rn+1

2

(1− r)n−1−np,

since for p > (n−1)/n the last integral converges. Thus for r ∈ [1/4, 1) we havethe desired estimates. From this and since the integrand is bounded away fromzero for r ∈ [0, 1/4], the estimate follows.

Note that by using the same method, the following (local) estimate can beproved.


Lemma D a). Let 0 0.

It may be noted that

Ip(δ) =∫ δ

0

dF (γ)(1− 2r cos γ + r2)np/2

,

where γ = ∠(η, ζ), and that by integration by parts it follows that

Ip(δ) =F (δ)

(1 + r)np+ npr

∫ δ

0

sin γ F (γ)dγ

(1− 2r cos γ + r2)np/2+1.

Remark 4. By Lemma D can be proved that for p ∈ (0, 1) there exists anonconstant function u ∈ Hp

0(B). Indeed, if u(x) = P (x, ζ) is a Poisson kernel,then by Lemma D we have that limr→1−0

∫S

P (rη, ζ)pdσ(η) = 0 for p ∈ ( (n −1)/n, 1). If p ∈ (0, (n − 1)/n] the result easily follows by Jensen’s inequality.Lemma D also can be used in proving the following known result ([3, p. 167]),that a Poisson kernel P (· , ζ) belongs to bp(B) for p < n/(n− 1), as well as thatfor p < (n + α)/(n− 1), α > −1, there exists a positive harmonic function u onB belonging to bp

α(B).The next result shows that Hp

0(B), p ∈ (0, 1) contains a large class of func-tions (this is a natural generalization of Theorem 1 in [25]).

Theorem 3. Let µ be a complex Borel measure on S, singular with respect toLebesgue measure, then P [µ](x) ∈ Hp

0(B), for 0 < p < 1.

Proof. Let ur(ζ) = u(rζ) = P [µ](rζ), r ∈ (0, 1). This family of functionsbelong to L(S) and by a well-known theorem (see [12]) lim

r→1−0u(rζ) = 0 a.e.

ζ ∈ S. By Jensen’s inequality we obtain(

1σ(E)

∫

E

|ur(η)|pdσ(η))1/p

≤ 1σ(E)

∫

E

|ur(η)|dσ(η)

for every E ⊂ S and p ∈ (0, 1).From this we have

∫

E

|ur(η)|pdσ(η) ≤ σ(E)1−p

(∫

S

|ur(η)|dσ(η))p

≤ σ(E)1−p‖u‖pH1 .

Since u = P [µ] we have u ∈ H1(B) ([3]) i.e. ‖u‖H1 < +∞. By this estimate weobtain

(∀ε > 0)(∃δ > 0)(∀E ⊂ S)(

σ(E) < δ ⇒∫

E

|ur(η)|pdσ(η) < ε

)

STEVIC214

i.e. the family of functions |ur|p, r ∈ (0, 1) is uniformly integrable. By Vitali’stheorem, we have that

limr→1−0

∫

S

|u(rη)|pdσ(η) =∫

S

limr→1−0

|u(rη)|pdσ(η) = 0,

finishing the proof of the theorem.

3 Equivalent conditions for the Bergman har-monic space

In [30], [33], [34], [38] and [39], we have investigated relationships among varioustype of integrals on the Bergman space on the unit disk, unit ball and unitpolydisc. In talk [36] we posed several open problems and conjectures concerningthis topic. Among others we posed the following conjecture.

Conjecture 1. Let p > 1, q ∈ [0, p], α ∈ (−1,∞), and u ∈ H(B). Show that∫

B

|u(x)|p(1− |x|)αdV (x) ³ |u(0)|p +∫

B

|u(x)|p−q|∇u(x)|q(1− |x|)α+qdV (x). (7)

The above means that there are finite positive constants C and C ′ indepen-dent of u such that the left and right hand sides L(u) and R(u) satisfy

CR(u) ≤ L(u) ≤ C ′R(u)

for every harmonic function u.

Remark 5. Note that for q = 0 the relationship (7) is obvious, and if p = qthen the result is contained in Lemma C.

In [41] we partially confirmed Conjecture 1, by proving the following result:

Theorem B. Suppose that u ∈ H(B), u(0) = 0, p ≥ 2 and α > 1. Then thefollowing quantities

∫

B

|u(x)|p−2|∇u(x)|2(1− |x|)αdVN (x),

and ∫

B

|u(x)|p−1|∇u(x)|(1− |x|)α−1dVN (x)

are equivalent.If α > 2, then these quantities are equiconvergent with

∫

B

|u(x)|p(1− |x|)α−2dVN (x).


Corollary 2. Let 1 −1 and u ∈ H(B), then the followingrelationship hold

‖u‖pp,α ³ |u(0)|p +

∫ 1

0

Mp−1p (u, r)Mp(∇u, r)(1− r)α+1dr

³ |u(0)|p +∫ 1

0

Mp−2p (u, r)M2

p (∇u, r)(1− r)α+2dr.

Proof. Assume that q ∈ 1, 2. From the proof of Theorem B, we obtainthat

‖u‖pp,α ≤ C

(|u(0)|p +

∫

B

|u(x)|p−q|∇u(x)|q(1− |x|)α+1dV (x))

≤ C

(|u(0)|p +

∫ 1

0

Mp−qp (u, r)Mq

p (∇u, r)(1− r)α+1dr

).

The reverse inequality, follows by applying Holder’s inequality with expo-nents p/(p− q) and p/q to the integral

∫ 1

0

Mp−qp (u, r)Mq

p (∇u, r)(1− r)α+qdr,

and by Lemma C for p = q.

In this section we will extend Theorem B. Namely, we prove the followingresult.

Theorem 4. Let G be a domain in Rn, u ∈ H(G), p ≥ 2, and d(x) =dist(x, ∂G) be the distance from x to the boundary of G. Then

∫

G

|u(x)|pd(x)αdV (x) ³∫

G

|∇u(x)|pd(x)α+pdV (x)

³∫

G

|∇u(x)|2|u(x)|p−2d(x)α+2dV (x)

³∫

G

|∇u(x)||u(x)|p−1d(x)α+1dV (x),

if all of these integrals are convergent.

In order to prove Theorem 4 we need three auxiliary results, which areincorporated in lemmas which follows. The first one is a simple consequence ofFubini’s theorem.

STEVIC216

Lemma 1. Let f ∈ L1(G) and α ∈ R. Then∫

G

d(x)α|f(x)|dV (x) ³∫

G

d(w)α−n

∫

B(x,d(x)/2)

|f(x)|dV (x)dV (w).

The following lemma was proved in [41, Theorems 1 and 2].

Lemma 2. Let u ∈ H(B). Then the following statements are true:

(a) If p > 1 and α > 0, then∫

B

|u(x)|p−2|∇u(x)|2(1− |x|)αdVN (x)

≤ C

∫

B

|u(x)|p−1|∇u(x)|(1− |x|)α−1dVN (x), (8)

for some positive constant independent of u.(b) If 2 ≤ p < ∞ and α > 1, then

∫

B

|u(x)|p−1|∇u(x)|(1− |x|)α−1dVN (x) ≤ C

∫

B

|u(x)|p(1− |x|)α−2dVN (x), (9)

for some positive constant independent of u.

Lemma 3. Let u ∈ H(B(0, r)), r > 0 and p ≥ 2. Then

rp

∫

Br/8

|∇u(x)|pdV (x) ≤ Cr2

∫

Br/4

|u(x)|p−2|∇u(x)|2dV (x)

≤ Cr

∫

Br/2

|u(x)|p−1|∇u(x)|dV (x)

≤ C

∫

Br

|u(x)|pdV (x).

Proof. Applying Lemma 2 (a), with α = 1, to the harmonic functionur/2(x) = u

(r2x

)on the unit ball, and then using the change of variable rx/2 →

x, we obtain that

C

∫

Br/2

|u(x)|p−1|∇u(x)|dVN (x) ≥ r

2

∫

Br/2

|u(x)|p−2|∇u(x)|2(1− 2|x|

r

)dVN (x)

≥ r

4

∫

Br/4

|u(x)|p−2|∇u(x)|2dVN (x),

which implies the second inequality.

By using Lemma 2 (b), with α = 2, to the harmonic function ur(x) = u(rx)on the unit ball, and then using the change of variable rx → x, we obtain that

C

∫

Br

|u(x)|pdVN (x) ≥ r

∫

Br/2

|u(x)|p−1|∇u(x)|(1− |x|

r

)dVN (x)

≥ r

2

∫

Br/2

|u(x)|p−1|∇u(x)|dVN (x),

which implies the third inequality.The first inequality was proved in [43, Lemma 3].

3.1 Proof of Theorem 4

We are now in a position to prove Theorem 4.

Proof of Theorem 4. By Lemma 1 we have that∫

G

|∇u(x)|pd(x)α+pdV (x) ³∫

G

d(x)α+p−n

∫

B(w,d(w)/8)

|∇u(x)|pdV (x)dV (w). (10)

On the other hand, applying Lemma 3 to the harmonic function u(y) = u(w+x),we have that

∫

B(w,d(w)/8)

|∇u(y)|pdV (y) =∫

B(0,d(w)/8)

|∇u(x)|pdV (x)

≤ Cd(w)2−p

∫

B(0,d(w)/4)

|u(x)|p−2|∇u(x)|2dV (x)

= Cd(w)2−p

∫

B(w,d(w)/4)

|u(y)|p−2|∇u(y)|2dV (y).

Using this inequality in (10), and then Fubini’s theorem, it follows that∫

G

|∇u(x)|pd(x)α+pdV (x)

≤ C

∫

G

d(x)α+2−n

∫

B(w,d(w)/4)

|u(y)|p−2|∇u(y)|2dV (y)dV (w)

≤ C

∫

G

d(x)α+2|u(x)|p−2|∇u(x)|2dV (x). (11)

By Lemmas 1 and 3, similar to the just proven inequality, it can be obtainedthat the following inequalities hold:

∫

G

|u(x)|p−2|∇u(x)|2d(x)α+2dV (x) ≤ C

∫

G

|u(x)|p−1|∇u(x)|d(x)α+1dV (x) (12)

STEVIC218

and

∫

G

|u(x)|p−1|∇u(x)|d(x)α+1dV (x) ≤ C

∫

G

|u(x)|pdV (x). (13)

The inequality∫

G

|u(x)|pdV (x) ≤ C

∫

G

|∇u(x)|pd(x)α+pdV (x), (14)

follows from the main result in [15]. From inequalities (11)-(14), the resultfollows.

3.2 A result concerning Conjecture 1

In this subsection we partially confirm Conjecture 1, for all positive values p, qsuch that p ≥ q. We prove the following result.

Theorem 5. Let p > 1, q ∈ [0, p], α ∈ (−1,∞), and u ∈ H(B) such that thefunctions |u(y)|p−q| ∂u

∂xk(y)|q, k ∈ 1, . . . , n are subharmonic. Then

∫

B

|u(x)|p(1− |x|)αdV (x) ³ |u(0)|p +∫

B

|u(x)|p−q|∇u(x)|q(1− |x|)α+qdV (x). (15)

The proof of Theorem 5 follows the lines of the proof of Theorem 1 in [1],hence we omit some details.

Before proving Theorem 5 we need some auxiliary results. The followinglemma was proved in [33].

Lemma E. Suppose 0 < p < ∞ and u ∈ H(B). Then∣∣∣∣

d

dr(|u(x)|p)

∣∣∣∣ ≤ p|u(x)|p−1|∇u(x)|, (16)

for almost every x = rζ ∈ B.

The following lemma can be proved by Lemma E similar to Lemma 2 in [1].

Lemma 4. Suppose 1 < q ≤ p < ∞ and α > −1. Then, there is a constantC = C(p, q, α, n) such that

Mp∞(u, 1/2) ≤ C

(|u(0)|p +

∫

B

|u(x)|p−q|∇u(x)|q(1− |x|)q+αdV (x))

,

for all u ∈ H(B).

Lemma 5. Suppose 0 < p < ∞, q ∈ [0, p], 0 ≤ r < 1 and a function u satisfiesthe conditions in Theorem 5. Then there is a constant C independent of u andr such that

∫

S

sup0≤τ<1

|u(τrζ)|p−q|∇u(τrζ)|qdσ(ζ) ≤ C

∫

S

|u(rζ)|p−q|∇u(rζ)|qdσ(ζ)

for all u ∈ H(B).

Proof. By [28, p.165] there is a positive constant C independent of thenonnegative subharmonic function u on the unit ball B ⊂ Rm such that

∫

S

sup0≤τ<1

u(τrζ) dσ(ζ) ≤ C

∫

S

u(rζ) dσ(ζ)

for every r ∈ (0, 1). From this and employing the fact that the functions|u(y)|p−q| ∂u

∂xk(y)|q, k ∈ 1, . . . , n are subharmonic, we can easily obtain the

result. ¤By using Lemmas 4 and 5, some calculations and the fact that there is a

positive constant C such that∫

B

|u(x)|p−q|∇u(x)|q(1− |x|)α+qdV (x) ≤ C

∫

B

|u(x)|p(1− |x|)αdV (x)

(see [33], with ω(x) = (1− |x|)α), Theorem 5 can be proved similar to Theorem1 in [1].

4 Characterization of the Besov space

In this section, we give some characterizations of the harmonic Besov space.The following result is our main result in this section.

Theorem 6. Assume that u ∈ H(B), r > 0, α, β ∈ (0, 1), α + β = 1 and2(n− 1) < p < ∞. Then the following statements are equivalent:

(a) u ∈ Bp;

(b)∫

B

(sup

y∈B(x,(1−|x|)/2)

|u(x)− u(y)||x− y| (1− |x|2)α(1− |y|2)β

)p

dτ(x) < ∞;

(c)∫

B

( 1Vx,(1−|x|)/2

∫

B(x,(1−|x|)/2)

|u(x)− u(y)|p|x− y|p (1−|x|2)pα(1−|y|2)pβdv(y)

)dτ(x) < ∞;

(d)∫

B

( 1Vx,(1−|x|)/2

∫

B(x,(1−|x|)/2)

|u(x)− u(y)||x− y| (1−|x|2)α(1−|y|2)βdv(y)

)p

dτ(x) < ∞;

STEVIC220

(e)∫

B

∫

B(x,

1−|x|2

)(1− |x|2)pα(1− |y|2)pβ |u(x)− u(y)|p|x− y|p dτ(y)dτ(x) < ∞.

Proof. (a) ⇔ (e) was proved in [42].(d) ⇒ (a). By the Cauchy’s inequality and subharmonicity we have that for

every ε ∈ (0, 1/2)

(1− |x|)|∇u(x)|≤ C sup

y∈B(x,(1−|x|)/4)

|u(y)− u(x)|

≤ C

(1− |x|)n

∫

B(x,(1−|x|)/2)

|u(y)− u(x)|dv(y)

≤ C

∫

B(x,(1−|x|)/2)

|u(y)− u(x)|dτ(y)

≤ C

∫

B(x,(1−|x|)/2)

|u(y)− u(x)| (1− |x|2)α(1− |y|2)β

|x− y| dτ(y), (17)

where we have used the fact that

1− |y||x− y| >

1− |x|2|x− y| > 1, when y ∈ B(x, (1− |x|)/2),

as well as the condition α + β = 1.By (17) and the fact that

1− |y| ³ 1− |x| if y ∈ B(x, (1− |x|)/2), (18)

we have that

(1− |x|)|∇u(y)|

≤ C

∫

B(x,(1−|x|)/2)

|u(y)− u(x)| (1− |x|2)α(1− |y|2)β

|x− y| dτ(y)

≤ C

Vx,(1−|x|)/2

∫

B(x,(1−|x|)/2)

|u(y)− u(x)||y − x| (1− |x|2)α(1− |y|2)βdv(y).

Therefore∫

B

(1− |x|)p|∇u(x)|pdτ(x)

≤ C

∫

B

( 1Vx,(1−|x|)/2

∫

B(x,(1−|x|)/2)

|u(y)− u(x)||y − x| (1− |x|2)α(1− |y|2)βdv(y)

)p

dτ(x)

< ∞.

(a) ⇒ (b). By (18), the mean value theorem and the subharmonicity of|∇u|p, p > 1, we have that

(sup

y∈B(x,(1−|x|)/2)

|u(x)− u(y)||x− y| (1− |x|2)α(1− |y|2)β

)p

≤ C((1− |x|) sup

ζ∈B(x,(1−|x|)/2)

|∇u(ζ)|)p

≤ C(

supζ∈B(x,(1−|x|)/2)

(1− |ζ|)|∇u(ζ)|)p

≤ C

∫

ζ∈B(x,1−|x|)

((1− |ζ|)|∇u(ζ)|

)p

dτ(ζ). (19)

Multiplying (19) by dτ(x), then integrating over B and applying Fubini’s theo-rem, we obtain

∫

B

(sup

y∈B(x,(1−|x|)/2)

|u(y)− u(x)||y − x| (1− |x|2)α(1− |y|2)β

)p

dτ(x)

≤ C

∫

B

∫

y∈B(x,1−|x|)

((1− |ζ|)|∇u(ζ)|

)p

dτ(ζ)dτ(x)

≤ C

∫

B

((1− |ζ|)|∇u(ζ)|

)p∫

x∈A(ζ)

dτ(x)dτ(ζ)

≤ C

∫

B

((1− |ζ|)|∇u(ζ)|

)p

dτ(ζ), (20)

where we have used the fact that the quantity∫

x∈A(ζ)dτ(x) is bounded, since the

set A(ζ) is contained in the ball B(y, 1−|y|). Note that (b) implies (c) is trivial,and (c) implies (d) follows from Holder inequality. The proof is completed.

5 Harmonic functions with Hadamard gaps

For u ∈ H(B), the radial derivative Ru of u is Ru(x) =∑n

i=1 xi∂u∂xi

(x) =∑∞k=0 kpk(x), where

∑+∞k=0 pk(x) is the homogeneous polynomial expansion of

u (see, [3]). For p, α ∈ (0,∞), the Dirichlet type space Bpα is defined to consist

of those u ∈ H(B) for which

‖u‖Bpα

=(∫

B

|Ru(x)|p(1− |x|)α−1dV (x))1/p

< ∞.

We say that u ∈ H(B) with the homogeneous polynomial expansion u(x) =∑+∞k=0 pmk

(x), where pmk(x) is a harmonic homogeneous polynomial of degree

mk on B, has Hadamard gaps if mk+1/mk ≥ τ > 1 for all k ∈ N.In order to prove the main result in this section we need the following lemma

from [17].

STEVIC222

Lemma F. Suppose α > 0, p > 0, n ≥ 0, an ≥ 0, In = k | 2n ≤ k < 2n+1, k ∈N, tn =

∑k∈In

ak and f(x) =∑∞

n=1 anxn. Then there is a positive constantK depending only on p and α such that

1K

∞∑n=0

tpn2nα

≤∫ 1

0

(1− x)α−1fp(x)dx ≤ K∞∑

n=0

tpn2nα

.

Theorem 7. Let p, α ∈ (0,∞), u ∈ H(B) and u(x) =∑+∞

k=0 pmk(x), have

Hadamard gaps. Then u ∈ Bpα if

∑+∞k=0 ‖pmk

‖p∞mp−α

k < ∞.

Proof. By polar coordinates and some calculations we obtain

‖u‖pBp

α=

∫

B

|Ru(x)|p(1− |x|)α−1dV (x)

=∫ 1

0

∫

S

∣∣∣∣∣+∞∑

k=0

mkpmk(rζ)

∣∣∣∣∣

p

(1− r)α−1dσ(ζ)rn−1dr

≤ σn

∫ 1

0

(+∞∑

k=0

mk‖pmk‖∞ rmk

)p

(1− r)α−1dr

≤ σnK+∞∑n=0

2−nα

( ∑

mk∈In

mk‖pmk‖∞

)p

(Lemma F)

≤ σnC2p([logλ 2] + 1)p+∞∑

k=0

‖pmk‖p∞

mα−pk

,

from which the result follows.

It is known that the norm in the a-Bloch spaces of harmonic functions isequivalent with the following norm

ba(u) = |u(0)|+ supx∈B

(1− |x|2)a|Ru(x)| = |u(0)|+ sup0<r<1

(1− r2)aM∞(Ru, r).

On the other hand, the quantity ba, can be considered as border case for thefollowing quantities bp

a(f) = sup0<r<1(1− r2)a‖Rur‖p, where p ∈ (0,∞), sinceclearly for every u ∈ H∞(B) and p ∈ (0,∞)

sup0<r<1

(1− r2)a‖Rur‖p ≤ sup0<r<1

(1− r2)a‖Rur‖∞. (21)

Now we consider harmonic functions with Hadamard gaps on the followingspaces

Ba2 =

u | sup

0<r<1(1− r2)a‖Rur‖2 < ∞, u ∈ H(B)

andBa

2,0 =

u | limr→1

(1− r2)a‖Rur‖2 = 0, u ∈ H(B)

.

Some characterization for the classes Ba2 and Ba

2,0 are given in function of thesequence (‖pmk

‖2)k∈N.

Theorem 8. Assume that a > 0 and u(x) =∑∞

k=1 pmk(x) is a harmonic func-

tion on B with Hadamard gaps. Then the following statements are equivalent:

(a) f ∈ Ba2

(b) lim supk→∞ ‖pmk‖2m1−a

k < ∞.

Proof. (a) ⇒ (b) Let u ∈ Ba2 . We have that

‖u‖2Ba2

= sup0<r<1

(1− r2)a

( ∫

S

|Ru(rζ)|2dσ(ζ))1/2

= sup0<r<1

(1− r2)a

( ∫

S

∣∣∣∞∑

k=1

mkpmk(ζ)rmk

∣∣∣2

dσ(ζ))1/2

= sup0<r<1

(1− r2)a

( ∞∑

k=1

m2k‖pmk

‖22r2mk

)1/2

≥ sup0<r<1

(1− r2)amk‖pmk‖2rmk

for every k ∈ N and r ∈ (0, 1). Choosing r = 1− 1mk

we obtain

m1−ak ‖pmk

‖2 ≤ C1,

as desired.(b) ⇒ (a) Assume that lim supk→∞ ‖pmk

‖2m1−ak < ∞. We have that

‖u‖2Ba2

= sup0<r<1

(1− r2)a

( ∞∑

k=1

m2k‖pmk

‖22r2mk

)1/2

≤ sup0<r<1

(1− r2)a∞∑

k=1

mk‖pmk‖2rmk

≤ sup0<r<1

(1− r2)a+1∞∑

m=1

( ∑

mk≤m

mk‖pmk‖2

)rm

≤ C sup0<r<1

(1− r2)a+1∞∑

m=1

( ∑

mk≤m

mak

)rm

≤ C sup0<r<1

(1− r2)a+1∞∑

m=1

marm ≤ C,

STEVIC224

where we have used the fact that there is a positive constant C independent ofm such that

∑

mk≤m

mak ≤ Cma (22)

(here is used the assumption that mk+1/mk ≥ λ > 1) and the following wellknown estimate

∞∑m=1

marm ≤ C(1− r)−(a+1), (23)

a > 0, r ∈ [0, 1), see, for example, [48].

Theorem 9. Assume that a > 0 and u(x) =∑∞

k=1 pmk(x) is a harmonic func-

tion on B with Hadamard gaps. Then the following statements are equivalent:

(a) f ∈ Ba2,0

(b) limk→∞ ‖pmk‖2m1−a

k = 0.

Proof. (a) ⇒ (b) Let u ∈ Ba2,0, then for every ε > 0 there is a δ > 0 such

that

(1− r2)a

( ∫

S

|Ru(rζ)|dσ(ζ))1/2

< ε, (24)

whenever δ < r < 1. From (24) we have that

ε > supδ<r<1

(1− r2)a

( ∫

S

|Ru(rζ)|2dσ(ζ))1/2

= supδ<r<1

(1− r2)a

( ∫

S

∣∣∣∞∑

k=1

mkpmk(ζ)rmk

∣∣∣2

dσ(ζ))1/2

= supδ<r<1

(1− r2)a

( ∞∑

k=1

m2k‖pmk

‖22r2mk

)1/2

≥ supδ<r<1

(1− r)amk‖pmk‖2rmk

for every k ∈ N and r ∈ (δ, 1). Choosing r = 1− 1mk

we obtain

m1−ak ‖pmk

‖2 ≤ εC1,

from which the implication follows.

(b) ⇒ (a) Assume that limk→∞ ‖pmk‖2m1−a

k = 0. Then for every ε > 0 thereis a k0 ∈ N such that for every k > k0

‖pmk‖2m1−a

k < ε. (25)

From (25) and some simple estimates we have that

(1− r2)a

( ∫

S

|Ru(rζ)|2dσ(ζ))1/2

= (1− r2)a

( ∞∑

k=1

m2k‖pmk

‖22r2mk

)1/2

≤ (1− r2)a∞∑

k=1

mk‖pmk‖2rmk ≤ (1− r2)a+1

∞∑m=1

( ∑

mk≤m

mk‖pmk‖2

)rm

≤ (1− r2)a+1k0∑

m=1

( ∑

mk≤m

mk‖pmk‖2

)rm

+(1− r2)a+1∞∑

m=k0+1

( ∑

mk≤m

mk‖pmk‖2

)rm

≤ (1− r2)a+1k0∑

m=1

( ∑

mk≤m

mk‖pmk‖2

)rm

+εC(1− r2)a+1∞∑

m=1

( ∑

mk≤m

mak

)rm

≤ (1− r2)a+1k0∑

m=1

( ∑

mk≤m

mk‖pmk‖2

)rm + εC(1− r2)a+1

∞∑m=1

marm

≤ (1− r2)a+1k0∑

m=1

( ∑

mk≤m

mk‖pmk‖2

)rm + εC, (26)

where we have used (22) and (23).Now note that for every ε > 0 there is a δ ∈ (0, 1) such that for every

r ∈ (δ, 1)

(1− r2)a+1k0∑

m=1

( ∑

mk≤m

mk‖pmk‖2

)rm < ε. (27)

From (26) and (27) the result follows.

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STEVIC226

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STEVIC228

CHARACTERIZATIONS OF LAGUERRE-HAHN AFFINEORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE

A. BRANQUINHO∗ AND M.N. REBOCHO∗∗

∗Departamento de Matematica da FCTUC, Universidade de Coimbra, Apartado 3008, 3000Coimbra, Portugal

E-mail:[email protected]∗∗ Departamento de Matematica, Universidade da Beira Interior, 6201-001 Covilha, Portugal

E-mail:[email protected]

Abstract. In this work we characterize a monic polynomial sequence, ortho-gonal with respect to a hermitian linear functional u that satisfies a functional

equation D (Au) = Bu + zHL , where A, B and H are polynomials and L is

the Lebesgue functional, in terms of a first order linear differential equationfor the Caratheodory function associated with u and in terms of a first order

structure relation for the orthogonal polynomials.

keywords. Orthogonal polynomials on the unit circle, hermitian functionals,measures on the unit circle, semi-classical functionals, Caratheodory function.

AMS Subject Classifications (2000).Primary 33C47,42C05.

1. Introduction

Let u be a linear functional, defined in the linear space of polynomials withreal coefficients. A linear functional u is Laguerre-Hahn affine if the correspondingformal Stieltjes function satisfies a first order linear differential equation,

φ(x)S′(x) = B(x)S(x) + C(x) (1)

with φ,B,C polynomials (cf. [11]). It is known that the Laguerre-Hahn affine classand the semi-classical class coincide. This result follows from [10, 12], where it isestablished that u is Laguerre-Hahn affine if, and only if, u satisfies a functionalPearson equation, D (φu) = ψu, where φ and ψ are polynomials (φ is the samepolynomial as in (1)).

Laguerre-Hahn affine functionals on the real line are also characterized in termsof first order structure relations for the corresponding sequence of orthogonal poly-nomials on the real line, Pn,

φ(x)P ′n+1(x) = Cn(x)Pn+1(x) +Dn(x)Pn(x), n = 0, 1, . . .

where Cn, Dn are polynomials of bounded degree ([10, 11, 12]).In [8, 18], an analogue theory for hermitian linear functionals defined in the linear

space of Laurent polynomials with complex coefficients was outlined. The conceptof semi-classical functional was extended to this set of functionals; a hermitian linearfunctional u is said to be semi-classical if it satisfies a Pearson equation D (Au) =Bu, where A,B are polynomials (see, in section 3, the definition of the derivationoperator D ), and the corresponding sequences of orthogonal polynomials, semi-classical orthogonal polynomials on the unit circle, were defined; the Laguerre-Hahn

1


2 BRANQUINHO AND M.N. REBOCHO

affine class on the unit circle was defined in terms of a first order linear differential

equation with polynomial coefficients for the formal seriesG(z) =+∞∑

n=−∞cnz

n, where

cn is the n-th moment of the hermitian functional u,

A(z)G′(z) = B1(z)G(z) +H(z). (2)

Since then, the comparison between both theories, namely the characterization ofthe functionals in terms of differential properties for the corresponding sequencesof orthogonal polynomials, as well the generating functions for the moments, hasbeen the central theme in some works (see [1, 2, 9, 13, 16, 18]).

In [2, 13, 18], it is established that u is Laguerre-Hahn affine and the corres-ponding G satisfies (2) if, and only if, the corresponding u satisfies the generalizedPearson equation D (Au) = Bu+zHL, where L is the Lebesgue operator and B is apolynomial depending on A,B1. Moreover, in [2, 13], the authors obtain conditionson the coefficients of equation (2) and, also, on the polynomial coefficients of adifferential equation for the formal Caratheodory function F ,

A(z)F ′(z) = B1(z)F (z) + C(z) (3)

in order to establish the semi-classical character of the corresponding functional.Some examples of functionals and the corresponding sequences of orthogonal poly-nomials that are not semi-classical are given, thus showing that, in the complexcase, the Laguerre-Hahn affine class and the semi-classical class do not coincide.

In this work, following a different approach from the referred works, we study therelation between a first order differential equation for the Caratheodory function,(3), and the distributional equation for the corresponding u (we remind that, inmany ways, the Caratheodory function is the analogue of the Stieltjes function (see[14])). We prove that if F satisfies a first order differential equation (3) in |z| < 1,then the corresponding linear functional u satisfies a generalized Pearson equationD (Au) = Bu +HL, where L is the Lebesgue operator and B,H are polynomialsgiven explicitly in terms of A,B1, C. Then, we deduce first order structure relationsfor the corresponding sequences of orthogonal polynomials on the unit circle (ana-logue of the structure relations for orthogonal polynomials on the real line, studiedin [10, 11, 12]). Finally, using these structure relations, we obtain a differentialsystem for semi-classical orthogonal polynomials on the unit circle (the analogue ofthe result established for semi-classical orthogonal polynomials on the real line in[7], by Magnus).

This paper is organized as follows: in section 2 we give the definitions andstate the main results which will be used in the next sections. In section 3 westudy the relation between the first order linear differential equation for F , andthe generalized Pearson equation for u (see theorem 3). In section 4, we establishthe equivalence between a first order differential equation for F and a system ofdifferential relations for the sequence of orthogonal polynomials, for the sequenceof associated polynomials of the second kind and for the sequence of functions ofthe second kind. We deduce a differential system for sequences of semi-classicalorthogonal polynomials on the unit circle.

230

LAGUERRE-HAHN AFFINE ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE 3

2. Preliminary results

Let Λ = span zk : k ∈ Z be the space of Laurent polynomials with complexcoefficients, Λ′ its algebraic dual space, P = span zk : k ∈ N the space of complexpolynomials and T = z ∈ C : |z| = 1 (or, using the parametrization z = eiθ,T = eiθ : θ ∈ [0, 2π[ ) the unit circle.

Given a sequence of moments (cn) of a linear functional u : Λ → C, cn =〈u, ξ−n〉, n ∈ Z, c0 = 1, the minors of the Toeplitz matrix are defined by

∆k =

∣∣∣∣∣∣∣∣∣c0 c1 · · · ckc−1 c0 · · · ck−1

......

...c−k c−k+1 · · · c0

∣∣∣∣∣∣∣∣∣ , ∆0 = c0, ∆−1 = 1, k ∈ N

Definition 1. (cf. [17]). The linear functional u is:a) hermitian if c−n = cn,∀n ≥ 0,b) regular or quasi-definite if ∆n 6= 0,∀n ≥ 0,c) positive definite if ∆n > 0,∀n ≥ 0.

If u is a positive definite hermitian functional there exists a non-trivial probabilitymeasure µ supported on the unit circle such that

〈u, ξ−n〉 =12π

∫ 2π

0

ξ−ndµ(θ), n ∈ Z, ξ = eiθ.

Hereafter we will use the notation 〈uθ, . 〉 to denote the action of the linear func-tional u over the variable θ, θ ∈ [0, 2π[.

Definition 2. Let φn be a sequence of complex polynomials with deg(φn) = nand u a hermitian linear functional. We say that φn is a sequence of orthogonalpolynomials with respect to u (or φn is a sequence of orthogonal polynomials onthe unit circle) if

〈u, φn(ξ)φm(1/ξ)〉 = Knδn,m, Kn 6= 0, n,m ∈ N, ξ = eiθ.

If the leading coefficient of each φn is 1, then φn is said to be a sequence of monicorthogonal polynomials.

It is well known (see [3, 4, 5]) that a given hermitian linear functional u is regularif, and only if, there exists a sequence φn of orthogonal polynomials with respectto u. Sequences of monic orthogonal polynomials φn satisfy each of the followingrecurrence relations, for n ≥ 1,

(R1) φn(z) = zφn−1(z) + anφ∗n−1(z)

(R2) φ∗n(z) = φ∗n−1(z) + anzφn−1(z)

with an = φn(0), and initial conditions φ0(z) = 1, φ−1(z) = 0, and the polynomialsφ∗n are defined by φ∗n(z) = znφn(1/z), n = 0, 1, . . . , where n = deg(φn). Also,|an| 6= 1, ∀n ∈ N, in the regular case and |an| < 1, ∀n ∈ N, in the positive-definitecase.

We consider the formal series associated with the hermitian linear functional u(whose sequence of moments is (cn) and c0 = 1), and denote it by F,

F (z) = 1 + 2+∞∑k=1

ckzk, |z| < 1 , F (z) = −1− 2

+∞∑k=1

ckz−k , |z| > 1 (4)

231

Since, for each θ ∈ [0, 2π[, the following expansions take place,

eiθ + z

eiθ − z= 1 + 2

+∞∑k=1

(eiθ)−kzk, |z| < 1 ,eiθ + z

eiθ − z= −1− 2

+∞∑k=1

(e−iθ)kz−k, |z| > 1

then, formally,

〈uθ,eiθ + z

eiθ − z〉 = F (z) (5)

Thus, we will also say that the series in (4) correspond (formally) to the functionF defined by (5). In the positive definite case, F is the Caratheodory functioncorresponding to u, and is represented by

F (z) =∫ 2π

0

eiθ + z

eiθ − zdµ(θ), z ∈ C \ T

where µ is the probability measure associated with u.Given a sequence of monic orthogonal polynomials φn with respect to u, the

sequence of associated polynomials of the second kind Ωn are defined by

Ωn(z) = 〈uθ,eiθ + z

eiθ − z

(φn(eiθ)− φn(z)

)〉, n = 1, 2, . . .

Ω0(z) = 1.

The associated polynomials Ωn also satisfy recurrence relations,

Ωn(z) = zΩn−1(z)− anΩ∗n−1(z), n = 1, 2, . . .

with initial conditions Ω0(z) = 1, Ω−1(z) = 0.The functions of the second kind associated with φn are defined by

Qn(z) = 〈uθ,eiθ + z

eiθ − zφn(eiθ)〉, n = 1, 2, . . .

Q0(z) = F (z)

and Qn satisfy the following recurrence relations (cf. [15]),

Qn(z) = zQn−1 − anQ∗n−1(z), n = 1, 2, . . .

with Q0(z) = F (z) and Q∗0(z) = −F (z).

Theorem 1 (cf. [3, 4, 5]). Let φn be a sequence of monic orthogonal polynomialson the unit circle and Ωn, Qn the sequence of the associated polynomials andthe functions of the second kind, respectively. Then the following equations hold,∀n ≥ 1,

Qn(z) = Ωn(z) + F (z)φn(z), (6)Q∗n(z) = Ω∗n(z)− F (z)φ∗n(z) (7)φ∗n(z)Ωn(z) + φn(z)Ω∗n(z) = 2hnz

n (8)φ∗n(z)Qn(z) + φn(z)Q∗n(z) = 2hnz

n (9)

with hn =∏n

k=1(1− |ak|2) and Q∗n = znQn(1/z).

As a consequence we get the following results (see [15]).

232

Corollary 1. Let Qn be the sequence of functions of the second kind associatedwith φn. Then, the following holds, ∀n ≥ 1,

Qn(z) = 2hnzn +O(zn+1), |z| < 1 (10)

Qn(z) = 2an+1hnz−1 +O(z−2), |z| > 1 . (11)

Corollary 2. Let φn be a sequence of monic orthogonal polynomials on the unitcircle and Ωn the sequence of associated polynomials of the second kind. Then,the following holds:

a) If there exists k ∈ N such that φk(α) = Ωk(α) = 0, then α = 0;b) If there exists k ∈ N such that φk(α) = Qk(α) = 0, then α = 0.

3. The first order differential equation for the Caratheodoryfunction

Let u ∈ Λ′ be a regular hermitian functional and f ∈ Λ. We define the linearfunctional fu ∈ Λ′ as

〈fu, g(ξ)〉 = 〈u, f(ξ)g(ξ)〉, g ∈ Λ,

and the derivative Du ∈ Λ′ as

〈Du, f(ξ)〉 = −i〈ξu, f ′(ξ)〉 = −i〈u, ξf ′(ξ)〉.

In [2, 13, 18] it is established the equivalence between the Laguerre-Hahn affinecharacter of a hermitian linear functional u, and the distributional equation

D(Au) = Bu+ zHL (12)

where L is the Lebesgue operator and A,B,H are polynomials. We remark thatwhen H = 0 and A 6≡ 0 in (12), u is said to be semi-classical.

In this section we study the relation between regular hermitian functionals uthat satisfy (12) and a first order differential equation for the corresponding F.

We begin by establishing some properties for the function F . Throughout thissection we will use the representation (5) for F.

Lemma 1. If A and B are polynomials and u is a hermitian linear functional, thefollowing relations hold, for |z| 6= 1:

〈B(eiθ)uθ,eiθ + z

eiθ − z〉 = P (z) +B(z)F (z) , (13)

A(z)F ′(z) = −A′(z)F (z) +Q(z) +1zi〈D (Au),

eiθ + z

eiθ − z〉 (14)

where P and Q are the polynomials defined by

P (z) = 〈uθ,eiθ + z

eiθ − z

(B(eiθ)−B(z)

)〉 (15)

Q(z) = −A′(z)− 〈uθ, 2eiθ

deg(A)∑k=2

A(k)(z)k!

(eiθ − z)k−2〉 (16)

Proof. Since

〈B(eiθ)uθ,eiθ + z

eiθ − z〉 = 〈u, e

iθ + z

eiθ − z

(B(eiθ)−B(z)

)〉+B(z)〈uθ,

eiθ + z

eiθ − z〉

233

and 〈uθ,eiθ + z

eiθ − z

(B(eiθ)−B(z)

)〉 is a polynomial, we get (13) with P defined as

referred.To obtain (14) we proceed as follows:

A(z)F ′(z) = 〈uθ,2eiθ

(eiθ − z)2A(z)〉

= −〈uθ,2eiθ

(eiθ − z)2(A(eiθ)−A(z))〉+ 〈uθ,

2eiθA(eiθ)(eiθ − z)2

〉

= −〈uθ, 2eiθ

deg(A)∑k=1

A(k)(z)k!

(eiθ − z)k−2〉+ 〈uθ,2eiθA(eiθ)(eiθ − z)2

〉

Butdeg(A)∑

k=1

A(k)(z)k!

(eiθ − z)k−2 =A′(z)eiθ − z

+deg(A)∑

k=2

A(k)(z)k!

(eiθ − z)k−2

and, multiplying and dividing by z in the second term and taking into account that

〈uθ,2zeiθA(eiθ)(eiθ − z)2

〉 = −〈A(eiθ)uθ, eiθ ∂

∂θ

(eiθ + z

eiθ − z

)〉,

we get

A(z)F ′(z) = −A′(z) (F (z) + c0)− 〈uθ, 2eiθ

deg(A)∑k=2

A(k)(z)k!

(eiθ − z)k−2〉

− 1z〈A(eiθ)u, eiθ ∂

∂θ

(eiθ + z

eiθ − z

)〉

Now we use the definition of D and assume that c0 = 1, to get

A(z)F ′(z) = −A′(z)F (z) +Q(z) +1zi〈D(Au),

eiθ + z

eiθ − z〉,

with Q given by (16).

Remark 1. Throughout this section we will use the notation

PA,B(z) = zQ(z)− iP (z)

where the polynomials P and Q are defined in terms of A and B by (15) and (16),respectively.

Next, we recover a result established in [2, 13], but here a different approach isused.

Theorem 2. Let u be a regular hermitian functional. If u satisfies D (Au) =Bu + ξH(ξ)L, where L is the Lebesgue functional, and A,B,H are polynomials,then F satisfies the first order differential equations

zA(z)F ′(z) = (−iB(z)− zA′(z))F (z) + PA,B(z)− 2izH(z), |z| < 1 (17)zA(z)F ′(z) = (−iB(z)− zA′(z))F (z) + PA,B(z), |z| > 1 (18)

Conversely, if F satisfies (17) and (18), then u satisfies the functional equation

D (Au) = Bu+ ξH(ξ)L.

234

Proof. Let u satisfy D (Au) = Bu+ξH(ξ)L. By substituting D (Au) = Bu+ξH(ξ)Lin (14) we obtain

A(z)F ′(z) = −A′(z)F (z) +Q(z) +1iz〈Bu, e

iθ + z

eiθ − z〉+

1iz〈eiθH(eiθ)L, e

iθ + z

eiθ − z〉

From (13) follows

A(z)F ′(z)

= −A′(z)F (z) +Q(z) +P (z) +B(z)F (z)

iz+

1iz〈eiθH(eiθ)L, e

iθ + z

eiθ − z〉 (19)

Let eiθH(eiθ) = h1eiθ + · · ·+ hl(eiθ)l, for some l ∈ N.

Since, for |z| < 1,

〈eiθH(eiθ)L, eiθ + z

eiθ − z〉 = 〈(h1e

iθ + · · ·+ hl(eiθ)l)L, 1 + 2+∞∑k=1

(eiθ)−kzk〉

= 2(h1z + h2z

2 + · · ·+ hlzl),

= 2zH(z)

then, for |z| < 1, (19) is equivalent to

A(z)F ′(z) = −A′(z)F (z) +Q(z) +1zi

(P (z) +B(z)F (z)) +1iz

2zH(z)

and we obtain the equation

zA(z)F ′(z) = (−iB(z)− zA′(z))F (z) + C1(z), |z| < 1,

with C1(z) = zQ(z)− iP (z)− 2izH(z) = PA,B(z)− 2izH(z).On the other hand, for |z| > 1,

〈eiθH(eiθ)L, eiθ + z

eiθ − z〉 = 〈(h1e

iθ + · · ·+ hl(eiθ)l)L,−1− 2+∞∑k=1

(eiθ)kz−k〉 = 0.

Therefore, for |z| > 1, (19) is equivalent to

zA(z)F ′(z) = (−iB(z)− zA′(z))F (z) + C2(z),

with C2(z) = zQ(z)− iP (z) = PA,B(z).Conversely, let F satisfy equations (17) and (18). We observe that, if F satisfies

a differential equation with polynomial coefficients

zA(z)F ′(z) = (−iB(z)− zA′(z))F (z) + C(z),

then, from (14) and (13), we obtain

〈Bu, eiθ + z

eiθ − z〉 − P (z) + iC(z) = izQ(z) + 〈D (Au),

eiθ + z

eiθ − z〉

and the following equation follows

〈D(Au)−Bu,eiθ + z

eiθ − z〉 = R(z) (20)

where R(z) = iC(z)− P (z)− izQ(z) = iC(z)− iPA,B(z).

235

We now study equation (20) in both domains, |z| < 1 and |z| > 1:a) since, if |z| < 1, equation (17) holds, then C(z) = PA,B(z) − 2izH(z), and (20)becomes


eiθ − z〉 = 2zH(z)

Sinceeiθ + z

eiθ − z= 1 + 2

+∞∑k=1

(eiθ)−kzk, |z| < 1, last equation is equivalent to

〈D (Au)−Bu, 1〉+ 2+∞∑k=1

〈D (Au)−Bu, (eiθ)−k〉zk = 2zH(z) (21)

b) since, if |z| > 1, equation (18) holds, then C(z) = PA,B(z) and (20) becomes


eiθ − z〉 = 0

Sinceeiθ + z

eiθ − z= −1− 2

+∞∑k=1

(eiθ)kz−k, last equation is equivalent to

−〈D (Au)−Bu, 1〉 − 2+∞∑k=1

〈D (Au)−Bu, (eiθ)k〉z−k = 0 (22)

Finally, from (21) and (22), we have

〈D (Au)−Bu, (eiθ)k〉 = 0, ∀k ≥ 0

〈D (Au)−Bu, (eiθ)−k〉 = 0, ∀k > deg(H) + 1

〈D (Au)−Bu, (eiθ)−k〉 = hk, k = 1, . . . ,deg(H) + 1

with zH(z) = h1z + h2z2 + · · ·+ hlz

l.Therefore, we obtain the functional equation D (Au)−Bu = ξH(ξ)L.

Note that if u is a semi-classical functional such that D (Au) = Bu, then thefunction F associated with u satisfies a first order linear differential equation withpolynomial coefficients,

zA(z)F ′(z) = (−iB(z)− zA′(z)))F (z) + C(z), |z| 6= 1,

as is stated in [18].

Corollary 3. Let F satisfy

zA(z)F ′(z) = (−iB(z)− zA′(z))F (z) + C(z), |z| 6= 1 .

Then, the corresponding linear functional u is semi-classical and satisfies D (Au) =Bu if and only if C(z) = PA,B(z).

Finally, we study the case of one differential equation for F . We will need thefollowing lemma, which is a generalization of a result from [1].

Lemma 2. Let u be a regular hermitian functional. If there exist polynomialsA,B,H such that D (Au) = Bu+HL, where L is the Lebesgue functional, then thefollowing equation holds,

D (A+A)u = (B +B)u+ (H +H)L (23)

236

Conversely, if (23) holds, then there exist polynomials A, B, H such that D (Au) =Bu+ HL.

Proof. If D (Au) = Bu+HL, then

〈D (Au), ξk〉 = 〈Bu+HL, ξk〉, ∀k ∈ Z.

Applying conjugates, follows

〈D (Au), ξ−k〉 = 〈Bu+HL, ξ−k〉, ∀k ∈ Z.

Therefore, we get

〈D ((A+A)u), ξn〉 = 〈(B +B)u+ (H +H)L, ξn〉, ∀n ∈ Z

and (23) follows.Conversely, if u satisfies (23), then

〈D ((A+A)u), ξk〉 = 〈(B +B)u, ξk〉+ 〈(H +H)L, ξk〉, ∀k ∈ Z.

From the definition of D , we obtain for all k ∈ Z ,

−ik〈u, (A(ξ) +A(1/ξ))ξk〉 = 〈u, (B(ξ) +B(1/ξ))ξk〉+ 〈L, (H(ξ) +H(1/ξ))ξk〉.

Let s = maxdeg(A),deg(B),deg(H) . Last equation can be written as

− ik〈u, ξs(A(ξ) +A(1/ξ))ξk−s〉

= 〈u, ξs(B(ξ) +B(1/ξ))ξk−s〉+ 〈L, ξs(H(ξ) +H(1/ξ))ξk−s〉 , k ∈ Z .

If we write k = s+m and

A1(ξ) = ξs(A(ξ) +A(1/ξ))

B1(ξ) = ξs(B(ξ) +B(1/ξ))

H1(ξ) = ξs(H(ξ) +H(1/ξ))

then A1, B1,H1 are polynomials, and last functional equation is

−i(s+m)〈u,A1(ξ)ξm〉 = 〈u,B1(ξ)ξm〉+ 〈L,H1(ξ)ξm〉

which is equivalent to

−im〈u,A1(ξ)ξm〉 = 〈u, (B1(ξ) + isA1(ξ))ξm〉+ 〈L,H1(ξ)ξm〉.

From the definition of D , follows

〈D (A1u), ξm〉 = 〈(B1 + isA1)u, ξm〉+ 〈H1L, ξm〉, ∀m ∈ Z,

and we obtain the required result with A = A1, B = B1 + isA1, H = H1.

Theorem 3. Let u be a regular hermitian functional. If F satisfies a first orderdifferential equation with polynomial coefficients

zA(z)F ′(z) = (−iB(z)− zA′(z))F (z) + C(z), |z| < 1 (24)

then there exist polynomials A, B, H such that u satisfies D (Au) = Bu+ HL, withL the Lebesgue functional.

237

Proof. If F satisfies (24), following the same steps as in the second part of the proofof theorem 2 (see (20)), we obtain the equation


eiθ − z〉 = i(C(z)− PA,B(z)), |z| < 1 (25)

Applying conjugates and the transformation z → 1/z, follows

〈D (Au)−Bu,e−iθ + 1/ze−iθ − 1/z

〉 = −i(C(1/z)− PA,B(1/z)) .

Sincee−iθ + 1/ze−iθ − 1/z

= −eiθ + z

eiθ − z, last equation is equivalent to

〈D (Au)−Bu,eiθ + z

eiθ − z〉 = i(C(1/z)− PA,B(1/z)), |z| > 1 (26)

Now, since there exists an analytic continuation outside the unit disk, in (25), andinside the unit disk, in (26), we get, for 1− ε2 < |z| < 1 + ε1, ε1, ε2 > 0,

〈D ((A+A)u)− (B +B)u,eiθ + z

eiθ − z〉 = i(C(z)− PA,B(z)) + i(C(1/z)− PA,B(1/z)).

By computing the moments of the hermitian functional D ((A+A)u)− (B +B)u,from last equation follows D ((A + A)u) − (B + B)u = (H + H)L, with H(ξ) =i(C(ξ)− PA,B(ξ))/2. From previous lemma, we obtain the required result.

4. First order structure relations for orthogonal polynomials onthe unit circle

In this section we establish the equivalence between the first order differentialequation

zAF ′ +BF + C = 0,

for the Caratheodory function associated with a hermitian functional u, and firstorder structure relations for the corresponding orthogonal polynomials, the associ-ated polynomials of the second kind and for the functions of the second kind. Thiswill be done using the same ideas of [6].

Theorem 4. Let u be a regular and hermitian functional, φn the sequence ofmonic orthogonal polynomials with respect to u, Ωn the associated polynomials ofthe second kind and Qn the functions of the second kind. If there exist polynomialsA,B,C such that F satisfies

zA(z)F ′(z) +B(z)F (z) + C(z) = 0, |z| < 1

then there exist polynomials Gn and Hn with degrees not depending on n, such thatthe following relations holds, for all n ∈ N,

zA(z)φ′n(z) = (Gn(z) +B

2(z))φn(z) +Hn(z)φ∗n(z) (27)

zA(z)Ω′n(z) = (Gn(z)− B

2(z))Ωn(z)−Hn(z)Ω∗n(z) + C(z)φn(z) (28)

zA(z)Q′n(z) = (Gn(z)− B

2(z))Qn(z)−Hn(z)Q∗n(z) (29)

238

Conversely, if equations (27), (28) and (29) hold, for all n ∈ N, then F satisfies afirst order linear differential equation with polynomial coefficients

zAF ′ +BF + C = 0.

Proof. Before going into the proof we remark that if φn(0) = 0, ∀n ∈ N, thenφn(z) = Ωn(z) = zn, Qn(z) = 2zn, φ∗n(z) = Ω∗n(z) = 1, Q∗n(z) = 0, F (z) = 1, andthe result holds with A = 1, B = 0, C = 0, and the differential relations (27), (28)and (29) with Gn = n,Hn = 0, ∀n ∈ N. So, in what follows we will assume thatwe are not in the case φn(0) = 0, ∀n ∈ N.

Using (6) in zAF ′ +BF + C = 0 we get

zA

Ωnφ

′n − Ω′nφn

φ2n

+ zA

(Qn

φn

)′

−BΩn

φn+B

Qn

φn+ C = 0.

Therefore the following equation holds,

zA (Ωnφ′n − Ω′nφn)−BΩnφn + Cφ2

n = Θn(z) (30)

with

Θn(z) =

−zA(z)

(Qn(z)φn(z)

)′

−B(z)Qn(z)φn(z)

φ2

n(z) (31)

From the asymptotic expansion of Qn in |z| < 1 (see (10)), and since the left sideof (30) is a polynomial, we get Θn = znΘn,1, for some polynomial Θn,1. Moreover,using the asymptotic expansion of Qn in |z| > 1 (see (11)), we conclude that Θn,1

has bounded degree,

deg(Θn,1) = maxdeg(A)− 1,deg(B)− 1, ∀n ∈ N.

Thus, (30) becomes

−φn zAΩ′n +BΩn − Cφn+ Ωn(zAφ′n) = znΘn,1(z).

Using (8) follows

−φn zAΩ′n +BΩn − Cφn+ Ωn(zAφ′n) = Θn,2(z) (φ∗nΩn + φnΩ∗n) ,

with Θn,2(z) = Θn,1(z)/(2hn), and we obtain

φn

zAΩ′n +

B

2Ωn − Cφn + Θn,2Ω∗n

= Ωn

zAφ′n −

B

2φn −Θn,2φ

∗n

(32)

We distinguish the following cases (see corollary 2):a) φn and Ωn have no common roots, ∀n ∈ N, i.e., φn(0) 6= 0,∀n ∈ N;b) There exists a finite number of indexes k ∈ N such that φk and Ωk have commonroots, i.e., φk(0) = Ωk(0) = 0 for a finite number of k’s;c) There exists n0 > 1 such that φn(0) = 0, ∀n ≥ n0.

Case a): If φn and Ωn do not have common roots, then we conclude that thereexists a polynomial ln, ∀n ∈ N, such that

zAφ′n − B2 φn −Θn,2φ

∗n = lnφn

zAΩ′n + B2 Ωn − Cφn + Θn,2Ω∗n = lnΩn

(33)

and we obtain (27) and (28) with Gn = ln and Hn = Θn,2. Moreover, as deg(Hn)is bounded, then deg(Gn) is bounded,

deg(Gn) = maxdeg(A),deg(B), ∀n ∈ N.

239


Case b): We first suppose φ1(0) 6= 0, . . . , φk−1(0) 6= 0, and k is the first indexsuch that φk(0) = 0. Then, φn and Ωn have no common roots, for n = 1, . . . , k− 1.From case a), equations (33) hold for n = 1, . . . , k − 1. Now we write equations(33) for k − 1 and multiply by z, to obtain

zAzφ′k−1 − B2 zφk−1 − zΘk−1,2φ

∗k−1 = lk−1zφk−1

zAzΩ′k−1 + B2 zΩk−1 − Czφk−1 + zΘk−1,2Ω∗k−1 = lk−1zΩk−1

By substituting

φk(z) = zφk−1(z), φ∗k(z) = φ∗k−1(z), zφ′k−1(z) = φ′k(z)− φk−1(z)

and

Ωk(z) = zΩk−1(z), Ω∗k(z) = Ω∗k−1(z), zΩ′k−1(z) = Ω′k(z)− Ωk−1(z)

in previous equations followszAφ′k − B

2 φk − zΘk−1,2φ∗k = (lk−1 +A)φk

zAΩ′k + B2 Ωk1 − Cφk + zΘk−1,2Ω∗k = (lk−1 +A)Ωk

and we obtain (27) and (28) for n = k with Gk = lk−1 + A and Hk = zΘk−1,2.Further, if φk+1(0) = · · · = φk+k0(0) = 0, and φk+k0+1(0) 6= 0 for some k0 ∈ N,using the same procedure as before, the differential relations (27) and (28) areobtained for n = k + 1, . . . , k + k0, with

Gn = lk−1 + (n− k + 1)A, Hn = zn−k+1Θk−1,2.

Case c): If φn(0) = 0, ∀n ≥ n0, then φn and Ωn are polynomials of the Bernstein-Szego type,

φn(z) = zn−n0+1φn0−1(z), Ωn(z) = zn−n0+1Ωn0−1(z).

Applying the same procedure as before we conclude that equations (27) and (28)hold for n ∈ N, and, ∀n ≥ n0, the polynomials Gn and Hn are given by

Gn = ln0−1 + (n− n0 + 1)A, Hn = zn−n0+1Θn0−1,2.

On the other hand, if we replace Θn by 2hnznΘn,2 in (31) we get

−zA(Qn

φn

)′

−B

(Qn

φn

)φ2

n = Θn,2(z)2hnzn.

Using (9) we get−zA

(Qn

φn

)′

−B

(Qn

φn

)φ2

n = Θn,2(z)(φ∗nQn + φnQ∗n).

Therefore, ∀n ∈ N,zAQ′n +

B

2Qn + Θn,2Q

∗n

φn =

zAφ′n −

B

2φn −Θn,2φ

∗n

Qn.

If we distinguish the two cases (see corollary 2):a) φn and Qn have no common roots, ∀n ∈ N, i.e., φn(0) 6= 0,∀n ∈ N;b) φn and Qn have the common root z = 0;

240


then, applying the same procedure as before, we conclude that, in both cases, thereexists a polynomial Ln such that

zAφ′n − B2 φn −Θn,2φ

∗n = Lnφn

zAQ′n + B2 Qn + Θn,2Q

∗n = LnQn.

Since Ln = ln, we obtain (29) with Gn = ln and Hn = Θn,2.To prove the converse result we use (6) and (7) in equation (28), thus obtaining

zA (Q′n − φ′nF − φnF′) = (Gn −

B

2) (Qn − φnF )−Hn (Q∗n + φ∗nF ) + Cφn,

i.e.,

zAQ′n + (B

2−Gn)Qn +HnQ

∗n

=zAφ′n −Gnφn +

B

2φn −Hnφ

∗n

F + zAF ′ + Cφn.

From (27) and (29) we obtain zAF ′ +BF + Cφn = 0, and zAF ′ +BF +C = 0follows.

Remark 2. Moreover, from (7) and using the same reasoning as before, we deducethe equations for φ∗n and Q∗n, ∀n ∈ N,

zA(φ∗n)′ = (Sn +B/2)φ∗n − Tnφn (34)zA(Q∗n)′ = TnQn + (Sn −B/2)Q∗n (35)

where Sn, Tn are bounded degree polynomials.

From the differential equations (27), (29), (34) and (35), we obtain a differentialsystem for semi-classical orthogonal polynomials on the unit circle (analogue to theequations deduced in [7], for the real case).

Theorem 5. Let φn be a sequence of monic orthogonal polynomials with respectto a semi-classical functional u, such that D (Au) = Bu. If u is positive definite andw is the absolutely continuous part of the corresponding measure, then the followingequations hold,

zA

[φn Qn/wφ∗n −Q∗n/w

]′=

[Gn − B/2 Hn

−Tn Sn − B/2

] [φn Qn/wφ∗n −Q∗n/w

], ∀n ∈ N

where B(z) = −iB(z)−zA′(z), and Gn,Hn, Sn, Tn are bounded degree polynomials.

Proof. If u satisfies D (Au) = Bu then the corresponding F satisfies zAF ′ = BF +C, with B = −iB − zA′, and C a polynomial (see corollary 3 of theorem 2).

From theorem 4 and the subsequent remark we have the following equations,

zA

[Q′n/w

−(Q∗n)′/w

]= (Bn +

B

2I)

[Qn/w−Q∗n/w

](36)

and

zA

[φn

φ∗n

]′= (Bn −

B

2I)

[φn

φ∗n

](37)

with Bn =[

Gn Hn

−Tn Sn

]and I the identity matrix of order two.

241


On the other hand, since w′(z)/w(z) = B(z)/(zA(z)), (see [18]) we obtain

zA

[Qn/w−Q∗n/w

]′= zA

[Q′n/w

−(Q∗n)′/w

]− B

[Qn/w−Q∗n/w

](38)

Substituting (36) in (38) we get

zA

[Qn/w−Q∗n/w

]′= (Bn −

B

2I)

[Qn/w−Q∗n/w

](39)

Finally, from (37) and (39) we obtain the required differential system.

Acknowledgments: This work was supported by CMUC, Department of Ma-thematics, University of Coimbra, and Intas Network Constructive Complex Ap-proximation ref. 03-51-6637. The second author was supported by FCT, Fundacaopara a Ciencia e Tecnologia, with grant ref. SFRH/BD/25426/2005.

References

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tugaliae Mathematica, 51 (1), 47-62 (1994).

[2] A. Cachafeiro and C. Perez, A study of the Laguerre-Hahn affine functionals on the unit circle,J. Comput. Anal. Appl., 6, n. 2, 107-123 (2004).

[3] J. Geronimus, On the trignometric moment problem, Annals of Mathematics, 47, (4), 742-761(1946).

[4] Ya. L. Geronimus, Polynomials orthogonal on a circle and interval, vol. 18, International

Series on Applied Mathematics, Consultants Bureau, New York, 1961.[5] Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, American Math-

ematical Society Translations, Series 1, vol. 3, Providence R. I., 1962.

[6] E. Laguerre, Sur la reduction en fractions continues dune fraction qui satisfait a une equationdifferentielle lineaire du premier ordre dont les coefficients sont rationnels, J. Math. Pures

Appl. (4) 1 (1885), 135-165, pp. 685-711, in Oeuvres, Vol. II, Chelsea, New-York 1972.

[7] A. P. Magnus, Painleve-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comput. Appl. Math., 57, 215-237 (1995).

[8] F. Marcellan, Orthogonal polynomials and Toeplitz matrices: some applications in rational

approximation and orthogonal polynomials, Seminario Matematico Garcıa Galdeano, Univ.Zaragoza, Ed. M. Alfaro, 31-57, (1989).

[9] F. Marcellan and P.Maroni, Orthogonal Polynomials on the unit circle and their derivatives,Constr. Approx., 7, 341-348 (1991).

[10] P. Maroni, Prolegomenes a l’etude des polynomes orthogonaux semi-classiques, Ann. Mat.Pura Appl., 149, n. 4, 165-184 (1987).

[11] P. Maroni, Le calcul des formes lineaires et les polynomes orthogonaux semi-classiques, in

Lect. Notes in Math., vol. 1329, Springer Verlag, Berlin, 1988, pp. 279-290.

[12] P.Maroni, Une theorie algebrique des polynomes orthogonaux. Application aux polynomesorthogonaux semi-classiques. In ”Orthogonal polynomials and their applications”, C. Brezinski,

L. Gori and A. Ronveux Eds. J.C.Baltzer AG. Basel IMACS Annals on Computing and Applied

Mathematics, 9 (1991), pp. 95-130.[13] C. Perez, Polinomios ortogonales de Laguerre-Hahn afin sobre la circunferencia unidad, PhD

Thesis, Universidade de Vigo, 2002 (is spanish).[14] B. Simon, Analogs of the m-function in the theory of orthogonal polynomials on the unit

circle, J. Comp. Appl. Math., 171, 1-26 (2004).

[15] B. Simon, Orthogonal Polynomials on the Unit Circle, Amer. Math. Soc. Colloq. Publ., vol.54, Providence Rhode Island, 2005.

[16] C. Suarez, About semi-classical orthogonal polynomials of the class (2, 2), J. Comp. Appl.

Math., 131, 457-472 (2001).[17] G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Providence

Rhode Island, 1975 (fourth edition).

[18] C. Tasis, Propriedades diferenciales de los polinomios ortogonales relativos a la circunferen-cia unidad, PhD Thesis, Universidad de Cantabria, 1989 (in spanish).

242

Weighted Composition Operator

between the Little α-Bloch Spaces and

the Logarithmic Bloch

Shanli YeDepartment of Mathematics, Fujian Normal University, Fuzhou 350007, China

E-mail: ye [email protected]; [email protected]

Abstract: We characterize the boundedness and compactness of the weightedcomposition operator uCϕ between the logarithmic Bloch space βL and the littleα-Bloch spaces β0

α on the unit disk. Some necessary and sufficient conditionsare given for which uCϕ is a bounded or compact operator between βL and β0

α.

MSC 2000: 47B38; 30H05; 30D05.

Keywords: Weighted composition operators, Boundedness, Compactness,Weighted Bloch spaces.

1 Introduction

Let D = z : |z| < 1 be the open unit disk in the complex plane C, and H(D)denote the set of all analytic functions on D. For f ∈ H(D), let

‖f‖βα= sup(1− |z|2)α|f ′(z)| : z ∈ D, 0 < α < +∞,

‖f‖βL= sup(1− |z|2) ln(

21− |z|

)|f ′(z)| : z ∈ D.

As in [7, 10], the α-Bloch space βα consists of all f ∈ H(D) satisfying ‖f‖βα<

+∞ and the little α-Bloch space β0α consists of all f ∈ H(D) satisfying lim

|z|→1(1−

|z|2)α|f ′(z)| = 0; the logarithmic Bloch space βL consists of all f ∈ H(D)satisfying ‖f‖βL

< +∞ and the little logarithmic Bloch space β0L consists of all

f ∈ H(D) satisfying lim|z|→1

(1 − |z|2) ln(2

1− |z|)|f ′(z)| = 0. The βL is a Banach

space under the norm‖f‖L = |f(0)|+ ‖f‖βL

. (1)

In [7], the author proved that β0L is a closed subspace and coincides with the

closure of polynomials under the norm and also studied the multiplication op-erator on the βL space and β0

L space. It is well known that with the norm

‖f‖α = |f(0)|+ ‖f‖βα , βα is a Banach space and β0α is a closed subspace of βα.

It is easily proved that for 0 < α < 1, βα $ βL $ β1.Let ϕ be a holomorphic self-map of D and u ∈ H(D). The weighted com-

position operator uCϕ is defined by:

uCϕf = uf ϕ, for f ∈ H(D).

It is easy to see that this operator is linear. We can regard the operator asa generalization of a multiplication operator Mu, which denote by Muf = uffor f ∈ H(D), and a composition operator Cϕ, which denote by Cϕf = f ϕfor f ∈ H(D). It is interesting to provide a function theoretic characterizationof when ϕ and u induce a bounded or compact weighted composition operatorbetween different function spaces. The weighted composition operator maybe first studied on the Bloch space and the little Bloch space in [4]. In [5],Ohno, Stroethoff and Zhao got the characterization on ϕ and u for the weightedcomposition operator is bounded or compact between the α-Bloch spaces. Later,the author [7, 8] study the multiplication operator Mu in βL, β0

L, between β0α1

and βα2 for α1 6= α2. In this paper, we will consider the boundedness andthe compactness of the weighted composition operator uCϕ between the littleα-Bloch spaces β0

α and logarithmic Bloch space βL on the unit disk.Recall that if X, Y are Banach spaces and T : X −→ Y is a linear operator,

T is said to be compact if for every bounded sequence xn in X, T (xn) hasa convergent subsequence. T is weakly compact, if for every bounded sequencexn in X, T (xn) has a weakly convergent subsequence. Every compact oper-ator is weakly compact operator. A useful characterization of weak compactnessis Gantmacher’s Theorem [1]: T is weakly compact if and only if T ∗∗(X∗∗) ⊂ Ywhere T ∗∗ is the second adjoint of T and X is identified with its image underthe natural embedding into its second dual X∗∗. In this paper, C denotes thepositive constant depending only on the index α,, the C may differ at differentplaces.

2 uCϕ from β0α to βL

Lemma 2.1 (see [7]) If f ∈ βL, then |f(z)| ≤ (2 + ln(ln 21−|z| ))‖f‖L.

Lemma 2.2 (see [8, 10]) Let α > 0 and f ∈ βα. Then(1) ‖ft‖α ≤ ‖f‖α , 0 < t < 1, where ft(z) = f(tz);(2) |f(z)| ≤ C‖f‖α, where α < 1;

(3) |f(z)| ≤ C ln2

1− |z|2‖f‖α, where α = 1;

(4) |f(z)| ≤ C

(α− 1)(1− |z|)α−1‖f‖α, where α > 1.

Lemma 2.3 Let α > 0 and X be a Banach space. Then uCϕ : β0α → X is a

weakly compact operator if and only if uCϕ is compact.

SHANLI YE244

Proof By Theorem 14 and 15 in [10] we know that (L1a)∗ ∼= βα and (β0

α)∗ ∼= L1a,

where L1a is the Bergman space of analytic function on D. We also have L1

a∼= l1,

and l1 has the Schur property[6]. Then we easily complete the proof.

Lemma 2.4 (see [8]) Let t > 0 and f ∈ H(D). Then(1) sup

z∈D(1− |z|)t|f(z)| < +∞ if and only if sup

z∈D(1− |z|)t+1|f ′(z)| < +∞.

(2) lim|z|→1

(1− |z|)t|f(z)| = 0 if and only if lim|z|→1

(1− |z|)t+1|f ′(z)| = 0.

Theorem 2.1 Let u be an analytic function on the unit disk D and ϕ an ana-lytic self-map of D. Let α > 0. Then the following statements are equivalent:

(a) uCϕ : β0α → βL is bounded;

(b) uCϕ : βα → βL is bounded;(c) uCϕ : β0

α → βL is weakly compact;(d) uCϕ : β0

α → βL is compact;

(e) (i) If α > 1, supz∈D

(1− |z|2) ln 21−|z|

(1− |ϕ(z)|2)α−1|u′(z)| < +∞ and

supz∈D

(1− |z|2) ln 21−|z|

(1− |ϕ(z)|2)α|ϕ′(z)u(z)| < +∞.

(ii) If α = 1, supz∈D

(1− |z|2) ln(2

1− |z|) ln(

21− |ϕ(z)|2

)|u′(z)| < +∞ and

supz∈D

(1− |z|2) ln 21−|z|

1− |ϕ(z)|2|ϕ′(z)u(z)| < +∞.

(iii) If 0 < α < 1, u ∈ βL and supz∈D

(1− |z|2) ln 21−|z|

(1− |ϕ(z)|2)α|ϕ′(z)u(z)| < +∞.

Proof (e) =⇒ (a). First we consider the case that α > 1. Suppose that u andϕ satisfy the condition (i) in (e). For f ∈ β0

α, we have supz∈D

(1 − |z|)α−1|f(z)| ≤

C‖f‖α < +∞ by Lemma 2.4. It follows that

‖uCϕf‖βL≤ sup

z∈D(1− |z|2) ln(

21− |z|

)|u′(z)f(ϕ(z))|

+ supz∈D

(1− |z|2) ln(2

1− |z|)|u(z)||f ′(ϕ(z))ϕ′(z)|

= supz∈D

(1− |ϕ(z)|2)α−1|f(ϕ(z))|(1− |z|2) ln 2

1−|z|

(1− |ϕ(z)|2)α−1|u′(z)|

+ supz∈D

(1− |ϕ(z)|2)α|f ′(ϕ(z))|(1− |z|2) ln 2

1−|z|

(1− |ϕ(z)|2)α|ϕ′(z)u(z)| ≤ C‖f‖α,

WEIGHTED COMPOSITION OPERATOR 245

and by Lemma 2.3, |u(0)f(ϕ(0))| ≤ C|u(0)|(α− 1)(1− |ϕ(0)|)α−1

‖f‖α. Hence uCϕ is

bounded from β0α to βL.

Secondly, let α = 1. Assume that f ∈ β01 and that u and ϕ satisfy the

condition (ii) in (e). From Lemma 2.2, we have

‖uCϕf‖βL≤ C‖f‖1 sup

z∈D(1− |z|2) ln

21− |z|

ln(2

1− |ϕ(z)|2)|u′(z)|

+‖f‖β1 supz∈D

(1− |z|2) ln 21−|z|

1− |ϕ(z)|2|ϕ′(z)u(z)| ≤ C‖f‖1,

and by Lemma 2.3, |u(0)f(ϕ(0))| ≤ C|u(0)| ln(2

1− |ϕ(0)|2)‖f‖1. Hence uCϕ is

bounded from β01 to βL.

Finally, let 0 < α < 1. Similarly, Suppose that u and ϕ satisfy the condition(iii) in (e) . By Lemma 2.1 we have sup

z∈D|f(z)| ≤ C‖f‖α for every f ∈ β0

α. It

follows that

‖uCϕf‖βL≤ sup

z∈D(1− |z|2) ln(

21− |z|

)|u′(z)f(ϕ(z))|

+ supz∈D

(1− |z|2) ln(2

1− |z|)|u(z)||f ′(ϕ(z))ϕ′(z)|

≤ C‖u‖L + ‖f‖βα supz∈D

(1− |z|2) ln 21−|z|

(1− |ϕ(z)|2)α|ϕ′(z)u(z)| ≤ C‖f‖α.

Hence uCϕ is bounded from β0α to βL.

(a) =⇒ (b). Suppose that uCϕ : β0α → βL is bounded. It is clear that for

any f ∈ βα, we have ft ∈ β0α for all 0 < t < 1. Then, according Lemma 2.2,

‖uCϕ(ft)‖L ≤ ‖uCϕ‖‖ft‖α ≤ ‖uCϕ‖‖f‖α < +∞.

Hence ‖uCϕ(f)‖L ≤ ‖uCϕ‖‖f‖α < +∞, which shows uCϕ is bounded from βα

to βL.(b) =⇒ (c). Since (β0

α)∗∗ ∼= βα, by Gantmacher’s Theorem, we know thatuCϕ : β0

α → βL is weakly compact if and only if uCϕ : βα → βL is bounded. Itfollows uCϕ is weakly compact from β0

α into βL.(c) =⇒ (d). By Lemma 2.3, it is obvious.(d) =⇒ (e). Suppose that uCϕ is compact from β0

α to βL. It is obviousthat uCϕ is bounded from β0

α to βL. Taking the constant function and f(z) = zin β0

α respectively, we get u ∈ βL and

supz∈D

(1− |z|2) ln(2

1− |z|)|ϕ′(z)u(z)| < +∞. (2)

SHANLI YE246

First, let α > 1. Fixed w ∈ D. Let the test function

fw(z) =α

(1− ϕ(w)z)α−1− (α− 1)(1− |ϕ(w)|2)

(1− ϕ(w)z)α

for z ∈ D. It is obvious that fw ∈ β0α for every w ∈ D. Since

f ′w(z) =α(α− 1)ϕ(w)(1− ϕ(w)z)α

− α(α− 1)(1− |ϕ(w)|2)ϕ(w)(1− ϕ(w)z)α+1

,

We get supw∈D

‖fw‖α ≤ α + 3(α − 1)α2α by a direct calculation. Noting that

f ′w(ϕ(w)) = 0 and fw(ϕ(w)) =1

(1− |ϕ(w)|2)α−1, we have

(1− |w|2) ln 21−|w|

(1− |ϕ(w)|2)α−1|u′(w)| = (1− |w|2) ln(

21− |w|

)|u′(w)fw(ϕ(w))|

= (1− |w|2) ln2

1− |w||(uCϕfw)′(w)|

≤ ‖uCϕfw‖L ≤ ‖uCϕ‖‖fw‖α ≤ C‖uCϕ‖ < +∞,

which showing that the first condition in (i) holds.Next, fix w ∈ D for ϕ(w) 6= 0. We take the test function

gw(z) =1

ϕ(w)(

1− |ϕ(w)|2

(1− ϕ(w)z)α− 1

(1− ϕ(w)z)α−1)

for z ∈ D. It is clear that gw ∈ β0α for every w ∈ D such that ϕ(w) 6= 0. By

a direct calculation, we get supw∈D

‖gw‖α ≤ 1 + α2α+2. Since gw(ϕ(w)) = 0 and

g′w(ϕ(w)) =1

(1− |ϕ(w)|2)α, it follows that

(1− |w|2) ln(2

1− |w|)|u(w)g′w(ϕ(w))ϕ′(w)| = (1− |w|2) ln

21− |w|

|(uCϕgw)′(w)|

≤ ‖uCϕgw‖L ≤ ‖uCϕ‖‖gw‖α ≤ C‖uCϕ‖ < +∞.

So,

supw∈D

(1− |w|2) ln 21−|w|

(1− |ϕ(w)|2)α|ϕ′(w)u(w)| < +∞.

For w ∈ D with ϕ(w) = 0, by (2), we have

(1− |w|2) ln 21−|w|

(1− |ϕ(w)|2) ln 21−|ϕ(w)|

|u(w)ϕ′(w)| = 1ln 2

(1−|w|2) ln(2

1− |w|)|u(w)ϕ′(w)| < +∞.

This proves that the second condition in (i) is also necessary.Secondly, let α = 1. Fix w ∈ D, we take the test function

hw(z) = 2 ln2

1− ϕ(w)z− 1

ln 21−|ϕ(w)|2

(ln2

1− ϕ(w)z)2

for z ∈ D. Since

h′w(z) =2ϕ(w)

1− ϕ(w)z− 2 ln(

21− ϕ(w)z

)ϕ(w)

(1− ϕ(w)z)1

ln 21−|ϕ(w)|2

,

then hw ∈ β01 for every w ∈ D and sup

w∈D‖hw‖1 ≤ 13 < +∞. Since h′w(ϕ(w)) = 0

and hw(ϕ(w)) = ln2

1− |ϕ(w)|2, it follows that

supw∈D

(1− |w|2) ln2

1− |w|ln(

21− |ϕ(w)|2

)|u′(w)|

= supw∈D

(1− |w|2) ln(2

1− |w|)|(uCϕhw)′(w)|

≤ ‖uCϕhw‖βL≤ ‖uCϕ‖‖hw‖1 ≤ C‖uCϕ‖ < +∞,

which proves that the first condition in (ii) holds. That the second condition in(ii) holds is also proved as before.

Finally, let 0 < α < 1. The proof is the similar to that in (i). The detailsare omitted. This completes the proof of Theorem 2.1.

In the following corollary, we need some notion. Suppose X and Y are twoBanach spaces of analytic functions on D. A complex valued function u in Dis called a multiplier from X into Y , if uf ∈ Y for every f ∈ X. The set ofall multipliers from X into Y will be denoted by M(X, Y ). Using the closedgraph theorem, it is easily seen that every u ∈ M(X, Y ) defines a boundedmultiplication operator Mu : f → uf from X into Y . We also define the spacesof functions WM(X, Y ) and CM(X, Y ) as follows:

WM(X, Y ) = u ∈ M(X, Y ) : Mu is weakly compact,

CM(X, Y ) = u ∈ M(X, Y ) : Mu is compact.

Corollary 2.1 (1) If α ≥ 1, then M(β0α, βL) = CM(β0

α, βL) = WM(β0α, βL) =

M(βα, βL) = u : u = 0;(2) If 0 < α < 1, then M(β0

α, βL) = CM(β0α, βL) = WM(β0

α, βL) =M(βα, βL) = βL.

Proof Assume that α ≥ 1. Then u ∈ M(β0α, βL) implies that sup

z∈D|u(z)|(1 −

|z|2)1−α ln2

1− |z|< +∞ if α > 1 and sup

z∈D|u(z)| ln 2

1− |z|< ∞ if α = 1 by

Theorem 2.1. They both show that lim|z|→1

|u(z)| = 0.

SHANLI YE248

Next, suppose that 0 < α < 1. Then u ∈ M(β0α, βL) if and only if u ∈ βL

and supz∈D

(1 − |z|2)1−α ln2

1− |z||u(z)| < ∞ by Theorem 2.1. Using Lemma 2.1,

one may easily prove that u ∈ βL implies supz∈D

(1−|z|2)1−α ln(2

1− |z|)|u(z)| < ∞.

Corollary 2.2 Let α > 0 and ϕ an analytic self-map of D. Then the followingstatements are equivalent:

(1) Cϕ : β0α → βL is bounded;

(2) Cϕ : β0α → βL is compact;

(3) Cϕ : β0α → βL is weakly compact;

(4) Cϕ : βα → βL is compact;

(5) supz∈D

(1− |z|2) ln 21−|z|

(1− |ϕ(z)|2)α|ϕ′(z)| < +∞.

3 uCϕ from βL to β0α

Now we study the boundedness and compactness of the weighted compositionoperator uCϕ from the logarithmic Bloch-type spaces βL to the little α-Blochspaces β0

α on the unit disk.

Lemma 3.1 Let α > 0 and U ⊂ β0α. Then U is compact if and only if it is

closed, bounded and satisfies lim|z|→1

supf∈U

(1− |z|2)α|f ′(z)| = 0.

The proof is similar to Lemma 1 in [3], we omitted it. The following twolemmas can be found in [7] or [9].

Lemma 3.2 Let f(z) =(1− |z|) ln 2

1−|z|

|1− z| ln 4|1−z|

, z ∈ D. Then |f(z)| < 2.

Lemma 3.3 Let 0 ≤ t ≤ 1, f(z) =(1− |z|) ln 2

1−|z|

(1− |tz|) ln 21−|tz|

, z ∈ D. Then |f(z)| < 2.

Theorem 3.1 Let u be an analytic function on the unit disk D and ϕ an ana-lytic self-map of D. Let α > 0. Then the following statements are equivalent:

(a) uCϕ : βL → β0α is compact;

(b) uCϕ : β0L → β0

α is compact;

(c) lim|z|→1

(1− |z|2)α ln(ln2

1− |ϕ(z)|)|u′(z)| = 0

and lim|z|→1

(1− |z|2)α

(1− |ϕ(z)|2) ln 21−|ϕ(z)|

|ϕ′(z)u(z)| = 0.

Proof (c) =⇒ (a). Suppose that f ∈ βL. By Lemma 2.1, we get

(1− |z|2)α|(uCϕ(f))′(z)|

≤ (1− |z|2)α|u′(z)f(ϕ(z))|+ (1− |z|2)α|u(z)f ′(ϕ(z))ϕ′(z)|

≤ (1− |z|2)α(2 + ln(ln2

1− |ϕ(z)|))|u′(z)|‖f‖L

+‖f‖βL

(1− |z|2)α

(1− |ϕ(z)|2) ln 21−|ϕ(z)|

|ϕ′(z)u(z)| −→ 0 (|z| −→ 1−).

So uCϕ(f) ∈ β0α, thus uCϕ is bounded from βL to β0

α. Then, by Lemma 3.1, weenough show that

lim|z|→1

sup(1− |z|2)α|(uCϕf)′(z)| : f ∈ βL, ‖f‖L ≤ 1 = 0.

However, it has just been proved above. Thus uCϕ is compact from βL to β0α.

(a) =⇒ (b). It is obvious.(b) =⇒ (c). Assume that uCϕ is compact from β0

L to β0α. Then u = uCϕ1 ∈

β0α. Also uϕ = uCϕz ∈ β0

α, thus (1 − |z|2)α|u′(z)ϕ(z) + u(z)ϕ′(z)| −→ 0 as|z| → 1. Since |ϕ| ≤ 1 and u ∈ β0

α, we have

lim|z|→1

(1− |z|2)α|ϕ′(z)u(z)| = 0. (3)

On the other hand, by Lemma 3.1 we have

lim|z|→1

sup(1− |z|2)α|(uCϕf)′(z)| : f ∈ β0L, ‖f‖L ≤ M = 0 (4)

for some M > 0. Fix w ∈ D, we take the test functions

fw(z) = 2 ln ln4

1− ϕ(w)z− 1

ln ln 41−|ϕ(w)|2

(ln ln4

1− ϕ(w)z)2

for z ∈ D. It is clear that fw ∈ β0L for every w ∈ D. By Lemma 3.2 and

3.3 we know that supw∈D

‖fw‖L ≤ 16. Since f ′w(ϕ(w)) = 0 and fw(ϕ(w)) =

ln ln4

1− |ϕ(w)|2, we get

lim|w|→1

(1− |w|2)α ln(ln2

1− |ϕ(w)|)|u′(w)| = 0 (5)

by (4). Similarly, we take another functions

gw(z) =∫ z

0

(1− w2

|w|2z2)−1(ln

41− w2

|w|2 z2)−1 dz.

SHANLI YE250

It is clear that gw ∈ β0L for every w ∈ D \ 0. By Lemma 3.2, we have

supz1∈D

(1− |z1|2)(ln2

1− |z1|2)|1− z2

1 |−1| ln 41− z2

1

|−1 < 2 < +∞,

applying z1 = w|w|z, we have

supz∈D

(1− |z|2)(ln 21− |z|2

)|1− w2

|w|2z2|| ln 4

1− w2

|w|2 z2|−1 < 2 < +∞.

Hence we have supw∈D\0

‖gw‖L < 4. Then by Lemma 2.1 we get

(1− |z|2)α

(1− |ϕ(z)|2) ln 21−|ϕ(z)|

|u(z)ϕ′(z)|

≤ 2(1− |z|2)α|(uCϕfw)′(z)|+ 2(1− |z|2)α(2 + ln(ln2

1− |ϕ(z)|))|u′(z)|

for ϕ(z) 6= 0. Thus by (4) and (5) it follows that

lim|z|→1

(1− |z|2)α

(1− |ϕ(z)|2) ln 21−|ϕ(z)|

|u(z)ϕ′(z)| = 0

for ϕ(z) 6= 0. However, if ϕ(z) = 0, by (3), we get lim|z|→1

(1−|z|2)α|u(z)ϕ′(z)| = 0.

This completes the proof of Theorem 3.1.

Remark 1 If the βL with the norm (1) is isometric to the second dual (β0L)∗∗,

we can prove that uCϕ : βL → β0α is bounded if and only if uCϕ : βL → β0

α iscompact.

However, we have the following result.

Theorem 3.2 Let α > 0 and ϕ an analytic self-map of D. Then the followingstatements are equivalent:

(a) Cϕ : βL → β0α is bounded;

(b) Cϕ : β0L → β0

α is compact;(c) Cϕ : βL → β0

α is compact;

(d) lim|z|→1

(1− |z|2)α

(1− |ϕ(z)|2) ln 21−|ϕ(z)|

|ϕ′(z)| = 0 .

For the proof of this theorem, we need another lemma.

Lemma 3.4 (see [2]) There exist two functions, f, g ∈ βL such that

|f ′(z)|+ |g′(z)| ≥ C

(1− |z|) ln 21−|z|

.

The proof of Theorem 3.2 It is obvious that (b) ⇐⇒ (c) ⇐⇒ (d) byTheorem 2.2. We only prove (a) ⇐⇒ (d). Assume that Cϕ is bounded from βL

to β0α. Then using the functions of Lemma 3.4, we get

(1− |z|2)α|(Cϕ(f))′(z)|+ (1− |z|2)α|(Cϕ(g))′(z)|

≥ (1− |z|2)α(|f ′(ϕ(z))|+ |g′(ϕ(z))|)|ϕ′(z)|

≥ C(1− |z|2)α

(1− |ϕ(z)|2) ln 21−|ϕ(z)|

|ϕ′(z)|.

On the other hand, one may easily prove (d) =⇒ (a). This completes theproof of Theorem 3.2.

Acknowledgments

The author is partially supported by National Science Foundation of FujianProvince (No: 2006J0201).

References

[1] J. Diestel, Sequences and Series in Banach Space, Spring-Verlag, New York,1984.

[2] P. Galanopulos, On Blog to Qplog pullbacks, J. Math. Anal. Appl. (In press).

[3] K. Madigan, A. Matheson, Compact composition operators on the Blochspace, Trans. Amer. Math. Soc. 347(7), 2679-2687(1995).

[4] S. Ohno and R. H. Zhao, Weighted composition operators on the Blochspace, Bull. Austral. Math. Soc. 63(2), 177-185(2001).

[5] S. Ohno, K. Stroethoff and R. H. Zhao, Weighted composition operatorsbetween Bloch-type spaces, Rocky Mountain J. Math. 33(1), 191-215(2003).

[6] P. Wojtaszczyk, Banach space for analysis, Cambridge: Cambridge Uni-versity Press, 1991.

[7] S. L. Ye, Multipliers and cyclic vectors on weighted Bloch space, Math. J.Okayama Univ. 48, 135-143(2006).

[8] S. L. Ye, Multiplication operators between the little Bloch type and theBloch type spaces, J. Fujian Normal Univ. 22(2) 1-4(2006)(in Chinese).

[9] S. L. Ye, Weighted composition operator between different on weightedBloch-type spaces, Acta Math. Sin. Chinese Series 50(4), 727-742(2007).

[10] K. H. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J.Math. 23(3), 1143-1177(1993).

SHANLI YE252

DERIVATIVE-FREE

CHARACTERIZATIONS OF BLOCH

SPACES

Songxiao LiDepartment of Mathematics, Shantou University, 515063, Shantou, ChinaDepartment of Mathematics, JiaYing University, 514015, Meizhou, China

E-mail: [email protected]; [email protected]

Abstract: We obtain some new characterizations for Bloch spaces on theunit ball of Cn in terms of the function |f(z)− f(w)|/|z − w|.

MSC 2000: 32A18; 32A37.

Keywords: Bloch space, little Bloch space, derivative-free.

1 Introduction

Let B = z ∈ Cn : |z| < 1 be the unit ball in the complex space Cn of dimensionn, dν be the normalized Lebesgue measure of B. The class of all holomorphicfunctions on B is denote by H(B). Let z = (z1, . . . , zn) and w = (w1, . . . , wn)be points in the complex vector space Cn and 〈z, w〉 = z1w1 + · · ·+ znwn. Theball centered at z with radius r will be denoted by B(z, r). Let Aut(B) be thegroup of all biholomorphic selfmaps of B. It is known that Aut(B) is generatedby the unitary operators on Cn and the involutions ϕa of the form

ϕa(z) =a− Paz − saQaz

1− 〈z, a〉 ,

where sa = (1−|a|2)1/2, Pa is the orthogonal projection into the space spannedby a, i.e., Paz = (〈z, a〉a)/|a|2, |a|2 = 〈a, a〉, P0z = 0, and Qa = I − Pa (see [8],[11]).

Let dλ(z) = (1−|z|2)−n−1dν(z). Then dλ(z) is a Mobius invariant measure,that is, for any ψ ∈ Aut(B) and f ∈ L1(B),

∫

B

f(z)dλ(z) =∫

B

f ψ(z)dλ(z).

Let E(z, r) = w ∈ B : |ϕz(w)| < r. Then E(z, r) is an ellipsoid and its

volume is given by

ν(E(z, r)) =r2n(1− |z|2)n+1

(1− r2|z|2)n+1. (1)

Set |E(z, r)| = ν(E(z, r)). For r > 0 and w ∈ E(z, r), we have (see [4, 8])

11 + r

≤ |1− 〈z, w〉|1− |w|2 ≤ 2

1− r2,

12≤ |1− 〈z, w〉|

1− |z|2 ≤ 21− r

. (2)

It follows from (1) and (2) that

(1− |z|2)n+1 ³ (1− |w|2)n+1 ³ |1− 〈z, w〉|n+1 ³ |E(z, r)|, (3)

when w ∈ E(z, r).For f ∈ H(B), let ∇f denote the complex gradient of f , i.e.

∇f(z) =(

∂f

∂z1(z), . . . ,

∂f

∂zn(z)

).

For f ∈ H(B) and z ∈ B, set

Qf (z) = supw∈Cn\0

|〈∇f(z), w〉|(Hz(w, w))1/2

,

where Hz(w, w) is the Bergman metric on B, i.e.

Hz(w, w) =n + 1

2(1− |z|2)|w|2 + |〈w, z〉|2

(1− |z|2)2 .

The Bloch space B(B), which was introduced by Timoney (see [9]), is thespace of all f ∈ H(B) such that

‖f‖B = supz∈B

Qf (z) < ∞.

The little Bloch space B0(B) is the space of all f ∈ H(B) such that lim|z|→1 Qf (z)= 0.

For f ∈ H(Bn), Nowak [4] proved that f ∈ B if and only if

M1 = supz,w∈Bn

z 6=w

(1− |z|2)1/2(1− |w|2)1/2 · |f(z)− f(w)||w − Pwz − (1− |w|2)1/2Qwz| < ∞. (4)

In [7], Ren and Tu replaced the above condition by

M2 = supz,w∈Bn

z 6=w

(1− |z|2)1/2(1− |w|2)1/2 · |f(z)− f(w)||z − w| < ∞. (5)

Moreover, they proved that f ∈ B0 if and only if

lim|z|→1

supw∈Bn

z 6=w

(1− |z|2)1/2(1− |w|2)1/2 · |f(z)− f(w)||z − w| = 0. (6)

S.LI254

If n = 1, the conditions (4) and (5) become the same thing which was establishedby Holland and Walsh [2]. Since

|w−Pwz− (1− |w|2)1/2Qwz| = |Pw(w− z)+ (1− |w|2)1/2Qw(w− z)| ≤ |w− z|,

it is easy to see that (4) implies (5). Hence (5) is weaker than (4) if n > 1.See [1, 3, 4, 5, 6, 7, 9, 10, 11] for more characterizations of the Bloch space ofholomorphic functions.

The purpose of this paper is to establish some equivalent derivative-free char-acterizations of Bloch functions. The following theorems are our main results.

Theorem 1 Suppose 0 < r < 1, 0 < p < ∞, nn+1 < β < ∞, 1 < q < ∞ and

f(z) ∈ H(B). Then the following conditions are equivalent:

(a) f ∈ B;

(b) supz∈B1

ν(E(z,r))

∫E(z,r)

|f(w)−f(z)|p|w−z|p (1− |w|2)p/2(1− |z|2)p/2dν(w) < ∞;

(c) supz∈B

∫E(z,r)

|f(w)−f(z)|p|w−z|p (1− |w|2)p/2(1− |z|2)p/2dλ(w) < ∞;

(d) supz∈B

∫B|f(w)−f(z)|p|w−z|p (1− |w|2)p/2(1− |z|2)p/2(1− |ϕz(w)|2)nqdλ(w) < ∞;

(e) supz∈B

∫B|f(w)−f(z)|p|w−z|p (1−|w|2)p/2(1−|z|2)p/2(JRϕz(w))β(1−|w|2)(n+1)βdλ(w)

< ∞.

Theorem 2 Suppose 0 < r < 1, 0 < p < ∞, nn+1 < β < ∞, 1 < q < ∞ and

f(z) ∈ H(B). Then the following conditions are equivalent:

(a) f ∈ B0;

(b) lim|z|→11

ν(E(z,r))

∫E(z,r)

|f(w)−f(z)|p|w−z|p (1− |w|2)p/2(1− |z|2)p/2dν(w) = 0;

(c) lim|z|→1

∫E(z,r)

|f(w)−f(z)|p|w−z|p (1− |w|2)p/2(1− |z|2)p/2dλ(w) = 0;

(d) lim|z|→1

∫B|f(w)−f(z)|p|w−z|p (1− |w|2)p/2(1− |z|2)p/2(1− |ϕz(w)|2)nqdλ(w) = 0;

(e) lim|z|→1

∫B|f(w)−f(z)|p|w−z|p (1−|w|2)p/2(1−|z|2)p/2(JRϕz(w))β(1−|w|2)(n+1)βdλ(w)

= 0.

Throughout this paper, constants are denoted by C, they are positive andmay differ from one occurrence to the other. The notation A ³ B means thatthere is a positive constant C such that B/C ≤ A ≤ CB.

BLOCH SPACES 255

2 Proof of Main Results

Proof of Theorem 1. (b)⇒(a). For every ε ∈ (0, 1/2),

1− |w||z − w| > (1− 2ε)

1− |z||z − w| > 1, when w ∈ B(z, 2ε(1− |z|)).

By the Cauchy’s estimate and the subharmonicity we have that

(1− |z|)p|∇f(z)|p ≤ C supw∈B(z,ε(1−|z|))

|f(w)− f(z)|p

≤ C

(1− |z|)n+1

∫

B(z,2ε(1−|z|))|f(w)− f(z)|pdν(w)

≤ C

∫

B(z,2ε(1−|z|))|f(w)− f(z)|p(1− |w|2)−n−1dν(w)

≤ C

∫

B(z,2ε(1−|z|))|f(w)− f(z)|p (1− |w|2)−n−1+p

|w − z|p dν(w).

(7)

For a fixed r > 0, choose a ball B(z, 2ε(1 − |z|)) ⊂ E(z, r). It follows from (3)and (7) that

(1− |z|)p|∇f(z)|p

≤ C

∫

E(z,r)

|f(w)− f(z)|p|w − z|p (1− |w|2)−n−1+pdν(w)

≤ C

ν(E(z, r))

∫

E(z,r)

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2dν(w). (8)

Taking supz∈B at both sides we get the desired result.(a)⇒(d). By (5) and the Mobius invariance of dλ(z), we have

∫

B

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2(1− |ϕz(w)|2)nqdλ(w)

≤ supz,w∈B

z 6=w

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2

∫

B

(1− |ϕz(w)|2)nqdλ(w)

≤ C

∫

B

(1− |ϕz(w)|2)nqdλ(w) ≤ C

∫

B

(1− |w|2)nqdλ(w) ≤ CB(n, n(q − 1)) < ∞,

where the condition q > 1 and the property of Beta function are used.(d)⇒(b). For a fixed r > 0, since 1 − |ϕz(w)|2 > 1 − r2 when w ∈ E(z, r),

we obtain

1ν(E(z, r))

∫

E(z,r)

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2dν(w)

≤ (1− r2)−nq

ν(E(z, r))

∫

E(z,r)

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2(1− |ϕz(w)|2)nqdν(w)

S.LI256

≤ C

∫

E(z,r)


≤ C

∫

B

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2(1− |ϕz(w)|2)nqdλ(w).

(b)⇔(c). For a fixed r ∈ (0, 1), ν(E(z, r)) ³ (1 − |z|2)n+1 ³ (1 − |w|2)n+1

when w ∈ E(z, r), then the desired result follows.(e)⇔(d). Let q = n+1

n β. Then∫

B

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2(JRϕz(w))β(1− |w|2)(n+1)βdλ(w)

=∫

B

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2(JRϕz(w)(1− |w|2)n+1)βdλ(w)

=∫

B


It follows that (e) and (d) are equivalent.

Proof of Theorem 2. By the proof of Theorem 1,

(1− |z|2)p|∇f(z)|p

≤ C

ν(E(z, r))

∫

E(z,r)

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2dν(w)

≤ C

∫

B


Let |z| → 1 in above inequality, we obtain (d)⇒(b)⇒(a).(a)⇒(d). Let f ∈ B0. By (6), for arbitrary ε > 0 there exists r0 ∈ (0, 1),

when z ∈ B \Br0 ,

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2 < ε (9)

for any w ∈ B with w 6= z, where Br0 = z ∈ B : |z| ≤ r0. Write∫

B


=( ∫

Br0

+∫

B\Br0

) |f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2(1− |ϕz(w)|2)nqdλ(w)

= I1 + I2. (10)


I1 ≤ supz,w∈B

z 6=w

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2

∫

Br0

(1− |ϕz(w)|2)nqdλ(w)

≤ C

∫

E(z,r0)

(1− |w|2)nq−n−1dν(w)

≤ C(1− |z|2)nq. (11)

BLOCH SPACES 257

On the other hand,

I2 ≤ ε

∫

B\Br0

(1− |ϕz(w)|2)nqdλ(w) ≤ ε

∫

B

(1− |w|2)nq−n−1dν(w) ≤ Cε. (12)

Hence, for given ε above, there exists r1 ∈ (0, 1) such that (1−|z|2)nq < ε whenz ∈ B \Br1 . Therefore, (10)-(12) give that

lim|z|→1

∫

B

|f(w)− f(z)|p|w − z|p (1− |w|2)p/2(1− |z|2)p/2(1− |ϕz(w)|2)nqdλ(w) = 0.

(b)⇔(c). The desired result follows from (3).(e)⇔(d). Let q = n+1

n β. As in the proof of Theorem 1, we know that theyare equivalent. The proof of Theorem 2 is finished.

References

[1] D. Clahane and S. Stevic, Norm equivalence and composition operatorsbetween Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl. Volume2006 Article ID 61018, (2006), pp. 11.

[2] F. Holland and D. Walsh, Criteria of membership of Bloch space and itssubspace BOMA, Math. Ann. 273 (1986), 317-335.

[3] B. Li and C. H. Ouyang, Higher radial derivative of Bloch type functions,Acta Math Scientia, 22B (4) (2002), 433-445.

[4] M. Nowak, Bloch and Mobius invariant Besov spaces on the unit ball ofCn, Complex Variables, 44 (2001), 1-12.

[5] M. Nowak, Bloch space on the unit ball of Cn. Ann. Acad. Sci. Fenn. 23(1998), 461-473.

[6] C. Ouyang, W. Yang, and R. Zhao, Characterizations of Bergman spacesand the Bloch space in the unit ball of Cn, Trans. Amer. Math. Soc. 374(1995), 4301-4312.

[7] G. Ren and C. Tu, Bloch space in the unit ball of Cn, Proc. Amer. Math.Soc. 133 (3) (2005), 719-726.

[8] W. Rudin, Function Theory in the Unit Ball of Cn, Springer-Verlag, NewYork, 1980.

[9] R. M. Timoney, Bloch functions in several complex variables I, Bull. Lon-don Math. Soc. 12 (1980), 241-267.

[10] R. M. Timoney, Bloch functions in several complex variables II, J. ReineAngew. Math. 319 (1980), 1-22.

[11] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, New York,2005.

S.LI258


259


260


261


262

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.10,NO.2,2008 A SEQUENTIAL PROBABILITY RATIO TEST,I.IATAN,…………………………139 CONVERGENCE THEOREMS OF ITERATIVE SEQUENCES FOR GENERALIZED P-QUASICONTRACTIVE MAPPINGS IN P-CONVEX METRIC SPACES,J.KIM,S-A.CHUN,Y.M.NAM,…………………………………………………………………..147 ON THE ANALOGUE OF BERNOULLI POLYNOMIALS,C.S.RYOO,S-H.RIM,…163 TWO GENERAL FIXED POINT THEOREMS ON THREE COMPLETE METRIC SPACES,D.TURKOGLU,……………………………………………………………...173 EXPLICIT QUASICONFORMAL EXTENSIONS OF PLANAR HARMONIC MAPPINGS,A.GANCZAR,……………………………………………………………179 NUMERICAL SOLUTION OF INTEGRAL EQUATIONS BY USING LOCAL POLYNOMIAL REGRESSION,H.CAGLAR,N.CAGLAR,…………………………..187 BAYESIAN CLASSIFIER BASED ON THE MULTIVARIATE NORMAL DISTRIBUTION,I.IATAN,…………………………………………………………….197 ON HARMONIC FUNCTION SPACES II,S.STEVIC,……………………………….205 CHARACTERIZATIONS OF LAGUERRE-HAHN AFFINE ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE,A.BRANQUINHO,M.REBOCHO,……..229 WEIGHTED COMPOSITION OPERATOR BETWEEN THE LITTLE a-BLOCH SPACES AND THE LOGARITHMIC BLOCH,S.YE,………………………………..243 DERIVATIVE-FREE CHARACTERIZATIONS OF BLOCH SPACES,S.LI,……….253

Volume 10,Number 3 July 2008 ISSN:1521-1398 PRINT,1572-9206 ONLINE







10) Ding-Xuan Zhou





28) T. E. Simos











33) Xin-long Zhou

A new interpolatory type quadrature rule for weightedCauchy principal value integrals

Bong-Gyu Jang

Department of Mathematics, KAIST, 373-1 Guseong-Dong, Yuseong-Gu, Daejeon, 305-701, Korea

[email protected]

Hoseok Lee


[email protected]

And

Kum Hwan Roh


[email protected]

Abstract. This paper presents a new interpolatory type quadrature rule for approximating

the weighted Cauchy principal value integrals∫ 1

−1(1− t2)λ−1/2f(t)/(t− c)dt where −1/2 < λ < 1.

We prove that the rule has almost optimal stability property behaving in the form O(K log n+L),

where K and L are constants depending only on c. Also, when f(t) possesses continuous derivatives

up to order p ≥ 0 and the derivative f (p)(t) satisfies Holder continuity of order ρ, we obtain that

the rule has the convergence rate of O((A + B log n + n2ν)n−p−ρ), where ν is as small as we like

and A and B are constants depending on c.

Key words: Cauchy principal value integral, quadrature rule, trigonometric interpolation,

singular integral, interpolatory type

AMS subject classifications: 30E20, 41A05, 42A15, 45E05

1. Introduction and results

The weighted Cauchy principal value integral(CPV) to be considered in this paper is of the

form

(1.1) Q(wf ; c) =∫ 1

−1

− w(t)f(t)t− c

dt = limε→0

∫ c−ε

−1

+∫ 1

c+ε

w(t)f(t)

t− cdt , |c| < 1 ,

where

(1.2) w(t) = (1− t2)λ−1/2, −1/2 < λ < 1.

It have been extensively used to solving singular integral equations (SIE) occurring in several

boundary value problems, particularly in aerodynamics, fluid and fracture mechanics and many

other fields of physics and the engineering sciences(cf. [3, 8, 14] and the literature cited there in).

There exist many bibliographies on quadratures for the CPV integral (1.1). Among those, we

specially mention the papers [1, 2, 10, 11, 12] concerning quadrature rules of interpolatory type.

Usually, quadrature rules of interpolatory type for the CPV integral (1.1) are of the form

(1.3) Q(wf ; c) =n∑

i=0

wni(c)f(tni) + Rn(f)

and obtained by interpolating the function f(t) at n + 1 distinct nodes tni with a polynomial

of degree n.

In [2], Elliott and Paget considered a rule of type (1.3) with the Jacobi weight w(t) = (1 −t)α(1+t)β , for α, β > −1, and they set tni to be zeros of the Jacobi polynomial of degree (n+1).

They have obtained the bound

(1.4)n∑

i=0

|wni(c)| ≤ K + L log n,

where K and L are constants depending only on c. By using it, they derived a pointwise conver-

gence property of Rn(f) = O(n−s) for the function f(t) Holder continuous of order µ, where s is

any positive real number less than µ. As well as a bound similar to (1.4), Monegato obtained the

boundn∑

i=0

|wni(c)| ≤ C log2 n,

where C is a constant depending on c in [11], provided that one takes the zeros of the function Tn(t)

(classical nodes) or (1 − t2)Un−1(t) (practical nodes) as the nodes tni. Here, Tn(t) and Un(t)

denote Chebyshev polynomials of first and second kind, respectively. In [1], Dagnino and Palamara

Orsi also obtained a bound of type (1.4) with the weight function w(t) ∈ Hµ(−1, 1)∩Lp[−1, 1], p >

1, when the nodes tni are the zeros of Tn(t) or (1− t2)Un−1(t).

The main contribution of this article is to show that the inequality (1.4) holds for the practical

nodes when the weight function w(t) is given by (1.2). More specifically, by modifying the rule

presented in [4, 5, 6, 7] we construct a new rule of interpolatory type such as (1.3).

First we change the variables of the integral in (1.1) by letting t = cos y and c = cos x, then

(1.1) becomes

Q(w(cos ·)f(cos ·); cos x) =∫ π

0

− sin2λ yf(cos y)cos y − cosx

dy

=∫ π

0

sin2λ y h(y)− sin2λ xh(x)cos y − cosx

dy

:= H(h;x), say,

where h(x) = f(cos x). In the second equation(see [4]) we will use the identity

(1.5)∫ π

0

− 1cos y − cos x

= 0, x ∈ (0, π).

JANG ET AL272

Define a trigonometric quadrature rule of interpolatory type for approximating H(h; x) by

(1.6) Hn(h; x) = H(phn;x) =

n∑′′

k=0

bnkJk(x) =n∑′′

k=0

h(xnk)ωnk(x),

where

(1.7)

phn(x) =

n∑′′

k=0

bnk cos kx, bnk =2n

n∑′′

j=0

h(xnj) cos kxnj , xnk =πk

n,

Jk(x) = H(cos(k·); x), ωnk(x) = H(lnk ; x),

lnk (x) =2n

n∑′′

j=0

cos jxnk cos jx =(−1)k

n

sin nx sinx

cos xnk − cosx, x 6= xnk.

A summation symbol with double primes denotes a sum with first and last terms halved. The last

identity of (1.7) yields that

(1.8) ωnk(x) =(−1)k+1

n

qn(x)− qn(xnk)cos x− cos xnk

, x 6= xnk,

q∗n(x), x = xnk,

where

(1.9) qn(x) =∫ π

0

− sin2λ+1 y sin ny

cos y − cosxdy, q∗n(x) =

∫ π

0

− sin2λ+1 y sin ny

(cos y − cosx)2dy.

The quadrature rule (1.6) is exact when the density function h(x) is a trigonometric polynomial

of degree ≤ n, so it is interpolatory type. The reader can find the details of above in [4, 6].

Numerically, Λn(x) which is defined by

(1.10) Λn(x) =n∑′′

k=0

|ωnk(x)|

is an important factor. Indeed, if h1 and h2 are two functions such that supx∈[0,π] |h1(x)−h2(x)| <ε, then (1.6) gives

|Hn(h1; x)−Hn(h2; x)| ≤ εΛn(x).

Thus we can say that Λn(x) gives the stability of the quadrature rule. In other words, when n

increases, the magnitude of the left hand side is subject to Λn(x). For this reason, we call Λn(x)

the stability factor.

In this article, we show that the stability factor Λn(x) has the bound such as

(1.11) Λn(x) = O(K log n + L),

where K and L are constants depending only on fixed x ∈ (0, π). The result of (1.11) is used to

prove that the error Rn(h;x) of the quadrature rule has the following behavior:

|Rn(h; x)| = O((A + B log n + n2ν)n−p−ρ),

CAUCHY PRINCIPAL VALUE INTEGRALS 273

where ν > 0 is as small as we like and constants A and B depend only on x, when the function f(t)

has continuous derivatives up to order p ≥ 0 and its derivative f (p)(t) satisfies Holder continuity

of order ρ for 0 < ρ ≤ 1.

2. Recurrence relations for the quadrature weights

If we use the identity (1.5), then Jk(x) in (1.7) can be written as follows:

(2.1) Jk(x) = Ik(x) + cos kxq0(x),

where

(2.2) Ik(x) =∫ π

0

cos ky − cos kx

cos y − cosxsin2λ ydy, q0(x) =

∫ π

0

− sin2λ y

cos y − cosxdy.

For λ > −1/2, and λ 6= 0, 1/2, q0(x) satisfies

q0(x) = −π tan(λπ) sin2λ−1 x

− 22λ−1 Γ(λ− 1/2)Γ(λ + 1/2)Γ(2λ)

F (1, 1− 2λ;32− λ; sin2 x

2),

where F (·, ·; ·; ·) is the hypergeometric function and Γ(·) the gamma function (see, e.g., [15]). Also

if λ = 0 or λ = 1/2, then from (1.5) and the definition of q0(x), we directly get

q0(x) =

0, λ = 0,

log tan2 x

2, λ = 1/2.

The terms Ik(x) in (2.2) can be calculated by the three-term recurrence relation(see [5])

Ik+2(x)− 2 cos xIk+1(x) + Ik(x) = 2dk+1, k ≥ 0

with initial values I0(x) = 0, I1(x) = d0. Further, In(x) has the explicit expression

(2.3) In(x) = 2n−1∑′

j=0

djsin(n− j)x

sin x.

Here, dk are defined by dk =∫ π

0

cos ky sin2λ ydy and satisfy the identities(see [5, (3.3), (3.4)])

(2.4) d2k =(−1)kΓ(λ + 1/2)Γ(λ + 1)

√π

Γ(λ + k + 1)Γ(λ− k + 1), (λ 6= 0), and d2k+1 = 0.

It also satisfy that

dk = 22λ Γ(λ + 1/2)2

Γ(2λ + 1)gk,

for gk satisfying

(2λ + 1 + k)gk+1 + (2λ + 1− k)gk−1 = 0

and having initials g0 = 1 and g1 = 0. For details, we refer to [13].

We devote the rest of this section to find the asymptotic behavior of d2k and a bound of In(x).

By some manipulation of gamma function, we can write d2k as

d2k = −√

πΓ(λ + 1/2)Γ(λ + 1)Γ(λ)Γ(1− λ)

Γ(k − λ)Γ(λ + k − 1)

1(λ + k − 1)(λ + k)

.

JANG ET AL274

Using the asymptotic behavior (see [9, p. 15])

Γ(z + α)Γ(z + β)

= zα−β(1 +

(α− β)(α + β − 1)2z

+ O(|z|−2)),

for z > 0 and arbitrary constants α and β, we see that for k ≥ 2

(2.5) d2k = −√

πΓ(λ + 1/2)Γ(λ + 1)Γ(λ)Γ(1− λ)

ck(λ)(1 + O((k − 1)−2)),

where

ck(λ) =(k − 1)1−2λ

(λ + k − 1)(λ + k).

Lemma 2.1. Assume that ak is a bounded sequence. Then

∣∣∣m∑

k=2

d2kak

∣∣∣ ≤ C maxk=2,··· ,m

∣∣∣k∑

i=2

ai

∣∣∣,

where C is a constant depending only on λ.

Proof. Consider

A =m∑

k=2

ck(λ)ak,

where ck(λ) is in (2.5). And let e1 = 0 and ek =k∑

i=2

ai, for k ≥ 2, then we know that ek−ek−1 = ak,

k ≥ 2. Thus A becomes

A =m∑

k=2

ck(ek − ek−1) = cmem +m−1∑

k=2

(ck − ck+1)ek.

By using the monotonicity of ck, we can complete the proof. ¤Using this lemma, we can obtain that In(x) given in (2.3) is bounded in the case of x = xnk.

Corollary 2.2. For x = xnk, 0 < k < n, In(x) given in (2.3) has the bounds

|In(x)| ≤ C

1

sin2 x, −1/2 < λ < 0 ,

1

sin x, 0 < λ <

1

2,

1,1

2< λ < 1

for some constant C depending only on λ.

Proof. From (2.3) and the fact x = xnk,

In(x) = 2(−1)k+1n−1∑

j=1

djsin jx

sin x.

Hence, (2.5) and Lemma 1 yield immediately

|In(x)| ≤ C1

∣∣∣[n/2]−1∑

j=2

sin 2jx

sin x

∣∣∣, −1

2< λ < 0,

1

sin x+

1

sin x

[n/2]−1∑

j=2

cj(λ), 0 < λ <12,

1 +[n/2]−1∑

j=2

cj(λ)j,12

< λ,

where cj(λ) is in (2.5) and C1 is a constant depending only on λ. Since 2 sinx

2

n−1∑

j=1

sin jx =

cosx

2− cos(n− 1

2)x, we can derive the first bound C/sin2 x when −1/2 < λ < 0.

On the other hand, the inequality

cj(λ) ≤

(j − 1 + λ)−1−2λ, 0 < λ < 1/2,

(j − 1)−1−2λ, 1/2 < λ < 1,

and the Riemann lower summation rule give

[n/2]−1∑

j=2

cj(λ) ≤∫ [n/2]

1

1(x− 1 + λ)1+2λ

dx < C2, 0 < λ < 1/2.

Here, C2 is a constant depending only on λ. The argument similar to this gives a constant bound

for the case λ > 1/2. The proof is complete. ¤

3. Asymptotic behavior of Λn(x)

In this section, we shall estimate the asymptotic behavior of the stability factor Λn(x) of (1.10)

using the expression (1.8). Note that if p1n denotes the trigonometric interpolation polynomial of

degree n for the constant function 1, then

Λn(x) ≥∣∣∣

n∑′′

k=0

ωnk(x)∣∣∣ = |H(p1

n; x)| = |q0(x)|,

where q0(x) is defined in (2.2).

Before estimating the upper bound of Λn(x), we consider an explicit expression of qn(x) in

(1.9). Note that2π

n∑′′

j=0

dj cos jx is a truncated cosine series of sin2λ x and it converges uniformly

to sin2λ x as n →∞. Thus (2.1) and (2.3) yield

(3.1)

qn(x) =Jn−1(x)− Jn+1(x)

2= sin nx sin xq0(x)− 2

n∑′′

j=0

dj cos(n− j)x

∼ sin nx sin x(q0(x)− 2

n∑′′

j=0

djsin jx

sin x

)− π cosnx sin2λ x

JANG ET AL276

for sufficiently large n. Using this relation and the first equation of (1.8), the weights ωnk(x) have

the bounds

(3.2) |ωnk(x)| ≤ C(|ω1

nk(x)||S1n(x)|+ |ω2

nk(x)| sin2λ x + |ω3nk(x)|

), 0 < k < n,

where C is a constant independent of n, and

ω1nk(x) =

(−1)k+1

n

sin nx sin x

cos x− cosxnk, ω2

nk(x) =2(−1)k

n

cosnx− (−1)k

cosx− cos xnk,

ω3nk(x) =

2n

sin2λ x− sin2λ xnk

cos x− cos xnk, S1

n(x) = q0(x)− 2n∑′′

j=0

djsin jx

sin x.

Theorem 3.1. The stability factor Λn(x) defined in (1.10) has the asymptotic behavior

(3.3) Λn(x) = O(K log n + L),

where K and L are constants depending only on x.

Proof. Assume that x ∈ (xnl, xn l+1) for some fixed 0 ≤ l < n/2. Also set n to be even for

convenience sake.

Now consider the estimation of A = |ωn0(x)| + |ωnn(x)|. By (1.8), (3.1), and (2.4), we obtain

a upper bound of A in the following way:

|A| ≤ 1n

(|q0(x)|| sin nx| sin x + 2

∣∣∣n/2∑′′

j=0

d2j(1− cos(n− 2j)x)∣∣∣) 2

sin2 x

≤ 2|q0(x)|+ C1

n

(2− cos nx− cos(n− 2)xsin2 x

+∣∣∣

n/2−1∑

j=2

d2j

sin2(n2 − j)x

sin2 x

∣∣∣)

for some constant C1. Since

2− cosnx− cos(n− 2)x = 2 sin2 n

2x + sin2 x cos nx + sin x sin nx cos x

and | sin kx/ sin x| ≤ k, Lemma 1 gives

(3.4)|A| ≤ 2|q0(x)|+ C1

( 1sinx

+ 1 +1n

n/2−1∑

j=1

sin2(n2 − j)x

sin2 x

)

≤ 2|q0(x)|+ C1 sin−2 x.

The remainder Λn(x)−A/2 is estimated by using (3.2). In order to do this, let

n−1∑

k=1

|ω1nk(x)| = A1(x) + A2(x),

where

A1(x) = |ω1nl(x)|+ |ω1

nl+1(x)|, A2(x) =n−1∑k=1

k 6=l, l+1

|ω1nk(x)|.

The inequalities

(3.5) | cos y − cosx| ≥

2x

π2|x− y|, y ∈ [0, π/2),

√2

π|x− y|, y ∈ [π/2, π]

yield

| cos x− cos xnk| ≥

2x

nπ(l − k), k = 1, 2, · · · , l − 1,

2x

nπ(k − l − 1), k = l + 2, · · · ,

n

2,

√2

n(k − l − 1), xnl < x < xn l+1 ≤ π

2< xnk.

Thus,

A2(x) ≤ π sin x

2x

[ l−1∑

k=1

1l − k

+n/2∑

k=l+2

1k − l − 1

]+

1√2

n−1∑

k=n/2+1

1k − l − 1

≤ (π +√

2)n∑

k=1

1k≤ (π +

√2) log(en).

To estimate A1(x), we use the generalized law of mean and the fact that sin x is increasing on

(0, π/2). So the term |ω1nl(x)| can be bounded as

|ω1nl(x)| = sin x

n

x− xnl

cos xnl − cos x

| sin nx|x− xnl

≤ sin x| cos nx|sin xnl

, x ∈ (xnl,xnl + xn l+1

2)

≤ sin xn l+1

sin xnl= cos

π

n+ cot xnl sin

π

n≤ 2 cos

π

n≤ 2.

Through the similar way, we can obtain the same bound for |ω1nl+1(x)|. In other words, we can

get

(3.6)n−1∑

k=1

|ω1nk(x)| ≤ C2 log(en)

for some constant C2 independent of n. Also the analogous argument leads us to the bound of

ω2nk(x):

(3.7)n−1∑

k=1

|ω2nk(x)| ≤ C3

sin xlog(en)

for some constant C3 independent of n.

Furthermore, the inequalities in (3.5) and the Riemann lower summation rule give

(3.8)

n−1∑

k=1

|ω3nk(x)| ≤ π2

xn

n−1∑

k=1

∣∣∣ sin2λ x− sin2λ xnk

x− xnk

∣∣∣

≤ C4

x

∫ π

0

∣∣∣ sin2λ x− sin2λ y

x− y

∣∣∣dy = C5 < ∞,

where C4 and C5 are independent of n. The bound of S1n(x) is immediately derived from the

bound of |q0(x)| and the proof of Corollary 2. Hence, by combining (3.4) and (3.6)-(3.8), we can

complete the proof for the case x ∈ (xnl, xn l+1).

JANG ET AL278

Finally consider the case of x = xnl for some 0 < l < n. We can easily verify that, for qn(x)

and q∗n(x) in (1.9),

limx→xnk

qn(x)− qn(xnk)cosx− cosxnk

= q∗n(xnk).

Thus the proof can be done by using some limiting arguments in the proof for the case x ∈(xnl, xn l+1). The details left to the reader. ¤

Table 1. Evaluation of Λn(x) according to (1.8) and (1.9) when λ = 1.

x stability

Λ4(x) Λ8(x) Λ16(x) Λ32(x) Λ64(x) Λ128(x) Λ256(x)

4π/7 4.62154 5.15211 7.32288 8.31478 9.26716 11.289 12.3456

5π/7 3.16745 4.54696 5.7359 7.19066 8.11555 8.93716 10.5329

6π/7 3.1627 3.51303 4.10565 4.83477 5.61028 6.1445 6.60688

In Table 1, we list the stability factor Λn(x)’s for the node number n varying from 4 to 256 and

fixed pole values πi/7, i = 4, 5, 6. In it, we can observe that the stability factor Λn(x) increases

about by 1 when the node number n varies as n = 2k, k = 2, 3, · · · , 8, for each fixed pole value

x. Therefore, we can say that the estimation of (3.3) substantiates the actual growth presented in

Table 1.

4. Convergence analysis

In this section, we derive a bound for the error Rn(h; x) = H(h; x)−Hn(h;x) when the function

f(cos x) is Holder continuous.

Using Theorem 4 and Lemma 3.2 of [6], we obtain the following convergence theorem for the

quadrature rule (1.6).

Theorem 4.1. Let us consider the quadrature rule (1.6). Suppose the function f(τ) possesses

continuous derivatives up to order p ≥ 0 and the derivative f (p)(τ) satisfies Holder continuity of

order ρ. Then the remainder term Rn(h; x) satisfies

|Rn(h; x)| = O((A + B log n + n2ν)n−p−ρ),

where A and B are constants depending only on x, and ν > 0 is as small as we like.

Proof. Let pn be any trigonometric polynomial of degree ≤ n. Then, since Hn(h; x) is an

interpolatory type rule, we find that

(4.1) |Rn(h; x)| ≤ |H(rn; x)|+ |Hn(rn; x)|,

where rn(x) = h(x)− pn(x). The quadrature rule (1.6) shows that

|Hn(rn; x)| ≤ maxτ∈[−1,1]

|f(τ)− pn(cos−1 τ)|Λn(x),

where Λn(x) is the stability factor defined in (1.10). Now the behavior of (3.3) and Lemma 3.2 of

[6] give

(4.2) |Hn(rn;x)| = O((K1 + L1 log n)n−p−ρ

),

where K1 and L1 are constants depending on x. For estimating |H(rn;x)|, we make use of (1.5)

and the change of variables y = cos−1 τ , x = cos−1 t. If we let rn(τ) = f(τ)− pn(cos−1 τ), then

|H(rn;x)| ≤ |rn(t)||q0(x)|+∫ 1

−1

(1− τ2)λ−1/2

|τ − t|1−ν

|rn(τ)− rn(t)||τ − t|ν dτ,

where q0(x) are defined in (2.2). Since for each t ∈ (−1, 1),∫ 1

−1

(1− τ2)λ−1/2

|τ − t|1−νdτ < ∞, ν > 0,

Lemma 3.2 of [6] shows that

(4.3) |H(rn; x)| = O((K2 + L2n

ν)n−p−ρ),

where K2 and L2 are constants depending on x and ν > 0 is as small as we like. Finally, by

substituting (4.2) and (4.3) for (4.1), we can complete the proof. ¤

References

[1] C. Dagnino and A. Palamara Orsi, On the Evaluation of Certain Two-Dimensional Singular Integrals with

Cauchy Kernels, Numer. Math. 46, 121-130(1985).

[2] D. Elliott and D.F. Paget, On the convergence of a quadrature rule for evaluating certain Cauchy principal

value integrals, Numer. Math. 23, 311-319(1975).

[3] A. Gerasoulis, Piecewise-polynomial quadratures for cauchy singular integrals, SIAM J. Numer. Anal. 23,

891-902(1986).

[4] P. Kim and U.J. Choi, A quadrature rule of interpolatory type for Cauchy integrals, J. Comput. Appl. Math.

126, 207-222(2000).

[5] P. Kim and U.J. Choi, A quadrature rule for weighted Cauchy integrals, J. Comput. Appl. Math. 126, 223-

232(2000).

[6] P. Kim and B.I. Yun, On the convergence of interpolatory type quadrature rules for evaluating Cauchy inte-

grals,J. Comput. Appl. Math. 149, 381-395(2002).

[7] P. Kim, A trigonometric quadrature rule for Cauchy integrals with Jacobi weight,J. Approx. Theory 108,

18-35(2001).

[8] R.D. Kurtz, T.N. Farris and C.T. Sun, The numerical solution of Cauchy singular integral equations with

application to fracture, Int. J. Fracture 66, 139-154(1994).

[9] N.N. Lebedev, Special functions and their applications, Dover Publications Inc., New York, 1972.

[10] G. Monegato, The numerical evaluation of one-dimensional Cauchy principal value integrals, Computing 29

(1982) 337-354.

[11] G. Monegato, On the weight of certain quadratures for the numerical evaluation of Cauchy principal value

integrals and their derivatives, Numer. Math. 50, 273-281(1987).

[12] D.F. Paget and D. Elliott, An algorithm for the numerical evaluation of certain Cauchy principal value integrals,

Math. Comput. 19, 373-385(1972).

[13] R. Piessens and M. Branders, The evalutation and application of some modified moments, BIT 13, 443-

450(1973).

JANG ET AL280

[14] H.W. Stolle and R. Strauss, On the numerical integration of certain singular integral, Computing 48, 177-

189(1992).

[15] F.G. Tricomi, On the finite Hilbert transform, Q. J. Math. 2, 199-211(1951).


Some Characteristics of

Logistic and Bessel Random Variablesby

Saralees NadarajahSchool of Mathematics

University of ManchesterManchester M60 1QD

United [email protected]

Abstract: Products, ratios and sums of random variables arise explicitly in many areas of thesciences, engineering and medicine. This has increased the need to have available the widest possiblerange of statistical results on products, ratios and sums of random variables. In this note, the exactdistributions of XY , X/Y and X + Y are derived when X and Y are logistic and Bessel randomvariables distributed independently of each other. Tabulations of the associated percentage pointsobtained by inverting the derived distributions are also provided.

AMS 1990 Subject Classification: 62E15 (Primary) and 33C90 (Secondary).

Keywords and Phrases: Bessel function distribution; Logistic distribution; Products of randomvariables; Ratios of random variables; Sums of random variables.

1 Introduction

For given random variables X and Y , the distributions of the product XY , the ratio X/Y andthe sum X + Y arise explicitly in many areas of the sciences, engineering and medicine. Thedistributions of XY , X/Y and X +Y have been studied by several authors especially when X andY are independent random variables and come from the same family. With respect to products ofrandom variables, see Sakamoto (1943) for uniform family, Harter (1951) and Wallgren (1980) forStudent’s t family, Springer and Thompson (1970) for normal family, Stuart (1962) and Podolski(1972) for gamma family, Steece (1976), Bhargava and Khatri (1981) and Tang and Gupta (1984)for beta family, Abu-Salih (1983) for power function family, and Malik and Trudel (1986) forexponential family (see also Rathie and Rohrer (1987) for a comprehensive review of known results).With respect to ratios of random variables, see Marsaglia (1965) and Korhonen and Narula (1989)for normal family, Press (1969) for Student’s t family, Basu and Lochner (1971) for Weibull family,Shcolnick (1985) for stable family, Hawkins and Han (1986) for non-central chi-squared family,Provost (1989a) for gamma family, and Pham-Gia (2000) for beta family. With respect to sumsof random variables, see Fisher (1935) and Chapman (1950) for Student’s t family, Christopeitand Helmes (1979) for normal family, Davies (1980) and Farebrother (1984) for chi-squared family,Ali and Obaidullah (1982) for exponential family, Moschopoulos (1985) and Provost (1989b) forgamma family, Dobson et al. (1991) for Poisson family, Pham-Gia and Turkkan (1993) and Phamand Turkkan (1994) for beta family, Kamgar-Parsi et al. (1995) and Albert (2002) for uniformfamily, Hitczenko (1998) and Hu and Lin (2001) for Rayleigh family, and Witkovsky (2001) for

1


inverted gamma family.

However, there is relatively little work of the above kind when X and Y belong to differentfamilies. In practical applications, it is quite possible that X and Y could arise from different butsimilar distributions. Nadarajah (2005) considered the exact distributions of XY and X/Y whenX and Y are independent random variables having the Laplace and Bessel function distributions.In this note, we consider the exact distributions of XY , X/Y and X + Y when X and Y areindependent random variables having the logistic and Bessel function distributions specified by theprobability density functions (pdfs)

fX(x) =λ exp (−λx)

1 + exp (−λx)2 (1)

and

fY (y) =

∣∣1− c2∣∣m+1/2 | y |m

√π2mbm+1Γ (m + 1/2)

exp(−cy

b

)Km

(∣∣∣yb

∣∣∣) , (2)

respectively, for −∞ < x < ∞, −∞ < y < ∞, λ > 0, b > 0, | c |< 1 and m > 1, where

Km(x) =√

πxm

2mΓ (m + 1/2)

∫ ∞

1

(t2 − 1

)m−1/2 exp (−xt) dt

is the modified Bessel function of the third kind. Note that (1) can be reexpressed as the mixtureof Laplace pdfs:

fX(x) =∞∑

k=0

2k + 1

(−2k

)λ(k + 1)

2exp −λ(k + 1) | x |

=∞∑

k=0

(−1)k(k + 1)λ exp −λ(k + 1) | x | . (3)

This representation will be crucial for the calculations of this note. Note that (3) holds for allx 6= 0. When x = 0, the right hand side of (3), λ

∑∞k=0(−1)k(k + 1), is a diverging series. This will

not affect the subsequent results because (3) can be applied over all sets A\(−δ, δ) and the limittaken as δ ↓ 0. However, since both sides of (3) should integrate to 1, i.e.∫ ∞

−∞fX(x)dx =

∞∑k=0

(−1)k(k + 1)λ∫ ∞

−∞exp −λ(k + 1) | x | dx,

we will make the convention that∑∞

k=0(−1)k = 1/2.

Logistic and Bessel function distributions have found applications in a variety of areas that rangefrom image and speech recognition and ocean engineering to finance. Both are rapidly becomingdistributions of first choice whenever “something” with heavier than Gaussian tails is observed inthe data. Some examples where these two distributions could arise simultaneously are:

1. in communication theory, X and Y could represent the random noise corresponding to twodifferent signals.

2. in ocean engineering, X and Y could represent distributions of navigation errors.

2

NADARAJAH284

3. in finance, X and Y could represent distributions of log-returns of two different commodities.

4. in image and speech recognition, X and Y could represent “input” distributions.

For further examples, the readers are referred to Balakrishnan (1992) and Kotz et al. (2001).

The results of this note are organized as follows: exact expressions for the pdf and the cumulativedistribution function (cdf) of XY are given in Section 2; the same for X/Y and X +Y are given inSections 3 and 4, respectively; moment properties of XY , X/Y and X +Y including characteristicfunctions, moments, factorial moments, skewness and kurtosis are considered in Section 5; finally,tabulations of the percentile points of XY , X/Y and X +Y obtained by inverting the derived cdfsare provided in Section 6.

The calculations of this note involve several special functions, including the Bessel function ofthe first kind defined by

Jν(x) =xν

2νΓ (ν + 1)

∞∑k=0

1(ν + 1)kk!

(−x2

4

)k

,

the modified Bessel function of the first kind defined by

Iν(x) =xν

2νΓ (ν + 1)

∞∑k=0

1(ν + 1)kk!

(x2

4

)k

,

the 0F3 hypergeometric function defined by

0F3 (a, b, c;x) =∞∑

k=0

1(a)k (b)k (c)k

xk

k!,

and the Gauss hypergeometric function defined by

2F1 (a, b; c;x) =∞∑

k=0

(a)k (b)k

(c)k

xk

k!,

where (e)k = e(e + 1) · · · (e + k − 1) denotes the ascending factorial. The properties of the abovespecial functions can be found in Prudnikov et al. (1986) and Gradshteyn and Ryzhik (2000).

2 Exact Distribution of XY

Theorem 1 derives an explicit expression for the cdf of | XY | in terms of the 0F3 hypergeometricfunction. The corresponding expression for the pdf of | XY | is given by Theorem 2.

Theorem 1 Suppose X and Y are independent random variables distributed according to (1) and(2), respectively, with c = 0. The cdf of Z =| XY | can be expressed as

FZ(z) = 1−∞∑

k=0

(−1)k

√

π2mbm+1Γ(

m +12

)0F3

(12,12−m,

12;(k + 1)2λ2z2

16b2

)

+(k + 1)λz2m+1

(2b)m Γ(−m)Γ(−2m− 1) 0F3

(1 + m,

32

+ m, 1 + m;(k + 1)2λ2z2

16b2

)

3

LOGISTIC AND BESSEL RANDOM VARIABLES 285

+(

3C

2− 1

)(2b)m(k + 1)λzΓ(m) 0F3

(1−m,

32, 1;

(k + 1)2λ2z2

16b2

) /

√π2m−1bm+1Γ

(m +

12

) , (4)

where C denotes the Euler’s constant.

Proof: Using the relationship (3), one can write

Pr (| XY |≤ z) = 2∞∑

k=0

(−1)k Pr (| XkY |≤ z) , (5)

where Xk are Laplace random variables with parameter λ(k+1). The cdf Fm,k(z) = Pr(| XkY |≤ z)can be expressed as

Fm,k(z) = Pr (| Xk |≤ z/ | Y |)

=1√

π2mbm+1Γ (m + 1/2)

∫ ∞

−∞

[1− exp

−(k + 1)λz

| y |

]| y |m Km

(∣∣∣yb

∣∣∣) dy

= 1− 1√π2mbm+1Γ (m + 1/2)

∫ ∞

−∞exp

−(k + 1)λz

| y |

| y |m Km

(∣∣∣yb

∣∣∣) dy

= 1− 1√π2m−1bm+1Γ (m + 1/2)

∫ ∞

0exp

−(k + 1)λz

y

ymKm

(y

b

)dy

= 1−

√

π2mbm+1Γ(

m +12

)0F3

(12,12−m,

12;(k + 1)2λ2z2

16b2

)+(k + 1)λz2m+1

(2b)m Γ(−m)Γ(−2m− 1) 0F3

(1 + m,

32

+ m, 1 + m;(k + 1)2λ2z2

16b2

)+

(3C

2− 1

)(2b)m(k + 1)λzΓ(m) 0F3

(1−m,

32, 1;

(k + 1)2λ2z2

16b2

) /

√π2mbm+1Γ

(m +

12

) , (6)

where the last step follows by direct application of equation (2.16.8.9) in Prudnikov et al. (1986,volume 2). The result of the theorem follows by using the convention

∑∞k=0(−1)k = 1/2 (see Section

1 below equation (3)), after combining (5) and (6).

Theorem 2 Suppose X and Y are independent random variables distributed according to (1) and(2), respectively, with c = 0. The pdf of Z =| XY | can be expressed as

fZ(z) = λ

∞∑k=0

(−1)k(k + 1)

√

π2m−1bm−1(k + 1)λzΓ(

m− 12

)0F3

(32,32−m,

32;(k + 1)2λ2z2

16b2

)+2−mb−m (k + 1)λz2m Γ(−m)Γ(−2m) 0F3

(1 + m, 1 + m,

12

+ m;(k + 1)2λ2z2

16b2

)−3C2m−1bmΓ(m) 0F3

(1−m,

12, 1;

(k + 1)2λ2z2

16b2

) /

√π2m−1bm+1Γ

(m +

12

) , (7)

4

NADARAJAH286

where C denotes the Euler’s constant.

Proof: Follows by differentiating (4) with respect to z and using properties of the 0F3 hypergeo-metric function.

Using special properties of the 0F3 hypergeometric function, one can derive simpler forms forthe distribution of | XY | when m takes half integer values. This is illustrated in the corollarybelow.

Corollary 1 If m = 3/2, 5/2, 7/2, 9/2, 11/2 then (4) can be expressed as

FZ(z) = 2∞∑

k=0

(−1)kFm,k(z),

where

F3/2,k(z) = −1/(8y)− 8y − 4I0 (2y) y − 3I0 (2y) y3C + 2I0 (2y) y3 − 4J0 (2y) y − 3J0 (y) y3C

+2J0 (2y) y3 + 8J1 (2y) + 6I1 (2y) y2C − 4I1 (2y) y2 + 8J1 (2y)

+6J1 (2y) y2C − 4J1 (2y) y2

,

F5/2,k(z) = −1/(96y)− 96y − 80I0 (2y) y − 45I0 (2y) y3C + 30I0 (2y) y3 − 80J0 (2y) y

−45J0 (2y) y3C + 30J0 (2y) y3 + 128I1 (2y) + 72I1 (2y) y2C − 32I1 (2y) y2

+9I1 (2y) y4C − 6I1 (2y) y4 + 128J1 (2y) + 72J1 (2y) y2C − 64J1 (2y) y2

−9J1 (2y) y4C + 6J1 (2y) y4

,

F7/2,k(z) = −1/(960y)− 960y − 1056I0 (2y) y − 495I0 (2y) y3C + 298I0 (2y) y3

−15I0 (2y) y5C + 10I0 (2y) y5 − 1056J0 (2y) y − 495J0 (2y) y3C + 362J0 (2y) y3

+15J0 (2y) y5C − 10J0 (2y) y5 + 1536I1 (2y) + 720I1 (2y) y2C − 160I1 (2y) y2

+150I1 (2y) y4C − 100I1 (2y) y4 + 1536J1 (2y) + 720J1 (2y) y2C − 800J1 (2y) y2

−150J1 (2y) y4C + 100J1 (2y) y4

,

F9/2,k(z) = −1/(53760y)− 1120J0 (2y) y5 + 23626J0 (2y) y3 + 15434I0 (2y) y3

−512I1 (2y) y2 − 6954I1 (2y) y2 − 53248J1 (2y) y2 + 7466J1 (2y) y4

−70I1 (2y) y6 − 70J1 (2y) y6 − 71424I0 (2y) y − 71424J0 (2y) y

−29295I0 (2y) y3 − 1680I0 (2y) y5C − 29295J0 (2y) y3C + 1680J0 (2y) y5C

+105I1 (2y) y6C − 10815J1 (2y) y4C + 105J1 (2y) y6C + 40320I1 (2y) y2C

+40320J1 (2y) y2C + 10815I1 (2y) y4C + 1120I0 (2y) y5 + 98304I1 (2y)

−53760y + 98304J1 (2y)

,

F11/2,k(z) = −1/(967680y)

1966080I1 (2y) + 1966080J1 (2y)− 967680y + 483042J0 (2y) y3

−29618J0 (2y) y5 + 126J0 (2y) y7 + 227934I0 (2y) y4C + 4536I1 (2y) y6C

−227934J1 (2y) y4C + 4536J1 (2y) y6C + 725760I1 (2y) y2C

+725760J1 (2y) y2C − 547155I0 (2y) y3C − 43659I0 (2y) y5C − 189I0 (2y) y7C

−547155J0 (2y) y3C + 43659J0 (2y) y5C − 189J0 (2y) y7C + 28594I0 (2y) y5

5


−1101312J1 (2y) y2 + 164244J1 (2y) y4 + 246498I0 (2y) y3 + 126I0 (2y) y7

−3024I1 (2y) y6 − 1482240I0 (2y) y − 3024J1 (2y) y6 − 1482240J0 (2y) y

+133632I1 (2y) y2 − 139668I (2y) y4

,

where y =√

(k + 1)λz/b and C denotes the Euler’s constant.

3 Exact Distribution of X/Y

Theorem 3 derives an explicit expression for the cdf of | X/Y | in terms of the Gauss hypergeometricfunction. The corresponding expression for the pdf of | X/Y | is given by Theorem 4.

Theorem 3 Suppose X and Y are independent random variables distributed according to (1) and(2), respectively. The cdf of Z =| X/Y | can be expressed as:

FZ(z) = 1−∣∣1− c2

∣∣m+1/2 Γ(2m + 1)√

π22m+1Γ (m + 1/2) Γ (m + 3/2)

∞∑k=0

2k + 1

(−2k

)

×

1

(k + 1)λbz − c2F1

(12, 1;m +

32; 1− 1

(k + 1)λbz − c2

)

+1

(k + 1)λbz + c2F1

(12, 1;m +

32; 1− 1

(k + 1)λbz + c2

) . (8)

Proof: Using the relationship (3), one can write

Pr (| X/Y |≤ z) =∞∑

k=0

2k + 1

(−2k

)Pr (| Xk/Y |≤ z) , (9)

where Xk are Laplace random variables with parameter λ(k +1). The cdf Fm,k(z) = Pr(| Xk/Y |≤z) can be expressed as

Fm,k(z) =

∣∣1− c2∣∣m+1/2

√π2mbm+1Γ (m + 1/2)

×∫ ∞

−∞F (z | y |)− F (−z | y |) | y |m exp

(−cy

b

)Km

(∣∣∣yb

∣∣∣) dy, (10)

where F (·) inside the integral denotes the cdf – corresponding to a Laplace random variable withparameters (λ(k + 1), θ) – given by

F (x) =

12

exp (k + 1)λx , if x ≤ 0,

1− 12

exp −(k + 1)λx , if x > 0.(11)

Substituting (11) for F (·), one can rewrite (10) as

Fm,k(z) =

∣∣1− c2∣∣m+1/2

√π2mbm+1Γ (m + 1/2)

∫ 0

−∞[1− exp (k + 1)λzy] | y |m exp

(−cy

b

)Km

(∣∣∣yb

∣∣∣) dy

6

NADARAJAH288

+∫ ∞

0[1− exp −(k + 1)λzy] | y |m exp

(−cy

b

)Km

(∣∣∣yb

∣∣∣) dy

= 1−∣∣1− c2

∣∣m+1/2

√π2mbm+1Γ (m + 1/2)

[ ∫ ∞

0ym exp

−(k + 1)λzy +

cy

b

Km

(y

b

)dy

+∫ ∞

0ym exp

−(k + 1)λzy − cy

b

Km

(y

b

)dy

]

= 1−∣∣1− c2

∣∣m+1/2 Γ(2m + 1)√

π22m+1Γ (m + 1/2) Γ (m + 3/2)

×

1

(k + 1)λbz − c2F1

(12, 1;m +

32; 1− 1

(k + 1)λbz − c2

)

+1

(k + 1)λbz + c2F1

(12, 1;m +

32; 1− 1

(k + 1)λbz + c2

) , (12)

where the last step follows by direct application of equation (2.16.6.3) in Prudnikov et al. (1986,volume 2). The result of the theorem follows by using the convention

∑∞k=0(−1)k = 1/2 after

combining (9) and (12).

Theorem 4 Suppose X and Y are independent random variables distributed according to (1) and(2), respectively. The pdf of Z =| X/Y | can be expressed as

fZ(z) = 1−bλ

∣∣1− c2∣∣m+1/2 Γ(2m + 2)

√π22m+1Γ (m + 1/2) Γ (m + 5/2)

∞∑k=0

(−2k

)

×

1

(k + 1)λbz − c2 2F1

(1,

32;m +

52; 1− 1

(k + 1)λbz − c2

)

+1

(k + 1)λbz + c2 2F1

(1,

32;m +

52; 1− 1

(k + 1)λbz + c2

) .(13)

Proof: Follows by differentiating (8) with respect to z and using properties of the Gauss hyperge-ometric function.

Using special properties of the Gauss hypergeometric function (see, for example, Section 7.3 ofPrudnikov et al. (1986, volume 3)), one can derive elementary forms for the distribution of | X/Y |when m takes integer or half integer values. This is illustrated in the corollaries below.

Corollary 2 If m ≥ 2 is an integer then (8) reduces to

FZ(z) = 1−∣∣1− c2

∣∣m+1/2 Γ(2m + 1)√

π22m+1Γ (m + 1/2) Γ (m + 3/2)

∞∑k=0

2k + 1

(−2k

)

×

[1

(k + 1)λbz − ch

(1− 1

(k + 1)λbz − c2

)

+1

(k + 1)λbz + ch

(1− 1

(k + 1)λbz + c2

) ],

7


where

h(z) =(1/2)m+1(z − 1)m

m!zm+1

2√

zarctanh(√

z)

+m∑

k=1

(k − 1)!(1/2)k

(z

z − 1

)k

.

Corollary 3 If m ≥ 3/2 is a half-integer then (8) reduces to

FZ(z) = 1−∣∣1− c2

∣∣m+1/2 Γ(2m + 1)√

π22m+1Γ (m + 1/2) Γ (m + 3/2)

∞∑k=0

2k + 1

(−2k

)

×

[1

(k + 1)λbz − ch

(1− 1

(k + 1)λbz − c2

)

+1

(k + 1)λbz + ch

(1− 1

(k + 1)λbz + c2

) ],

where

h(z) =(z − 1)m+1/2

(−1/2)m+3/2zm+1/2

√π(1− z)−1/2 +

1z

m+1/2∑k=1

(−1/2)k

(z − 1

z

)−k .

4 Exact Distribution of X + Y

If Z = X + Y then its pdf can be written as

fZ(z) =

∣∣1− c2∣∣m+1/2

λ√

π2mbm+1Γ (m + 1/2)

∫ ∞

−∞| y |m exp

(−cy

b

)Km

(∣∣∣yb

∣∣∣) exp −λ(z − y)[1 + exp −λ(z − y)]2

dy

=

∣∣1− c2∣∣m+1/2

λ√

π2mbm+1Γ (m + 1/2)

∫ ∞

−∞| y |m exp

(−cy

b

)Km

(∣∣∣yb

∣∣∣) ∞∑k=0

(−1)k(k + 1)

× exp −λ(k + 1) | z − y | dy

=

∣∣1− c2∣∣m+1/2

λ√

π2mbm+1Γ (m + 1/2)

∞∑k=0

(−1)k(k + 1)∫ ∞

−∞| y |m exp

(−cy

b

)Km

(∣∣∣yb

∣∣∣)× exp −λ(k + 1) | z − y | dy. (14)

Unfortunately, the integral in (14) cannot be reduced to an analytical form even for the particularcase c = 0. Thus, the pdf and the cdf of Z = X + Y will have to be obtained by numerical means.

5 Moment Properties of XY , X/Y and X + Y

The moment properties of XY , X/Y and X + Y can be derived by knowing the same for X andY . It is well known (see, for example, Johnson et al. (1995)) that

E (| X |n) = 2λ−nn!∞∑

k=0

(−1)k

(1 + k)n

8

NADARAJAH290

and

E (| Y |n) =(2b)nΓ (m + (n + 1)/2) Γ ((n + 1)/2)√

πΓ (m + 1/2)

provided that c = 0 in (2). Thus, the nth moment of Z =| XY | is

E (Zn) =n!2n+1bnΓ (m + (n + 1)/2) Γ ((n + 1)/2)

λn√πΓ (m + 1/2)

∞∑k=0

(−1)k

(1 + k)n .

In particular,

E (Z) =4bΓ (m + 1)

λ√

πΓ (m + 1/2)

∞∑k=0

(−1)k

1 + k,

E(Z2

)= 4b2λ−2 (2m + 1)

∞∑k=0

(−1)k

(1 + k)2,

E(Z3

)=

96b3Γ (m + 2)λ3√πΓ (3 + 1/2)

∞∑k=0

(−1)k

(1 + k)3

and

E(Z4

)= 144b4λ−4(2m + 3)(2m + 1)

∞∑k=0

(−1)k

(1 + k)4.

Moments of | X/Y | do not exist because moments of negative order are undefined for Y . The nthmoment of Z = X + Y is

E (Zn) =n∑

k=0

(n

k

)E

(Xk

)E

(Y n−k

),

where

E(Xk

)=

0, if k odd,E

(| X |k

), if k even

and

E(Y n−k

)=

0, if n− k odd,E

(| Y |n−k

), if n− k even.

In particular,

E (Z) = 0,

E(Z2

)= b2(2m + 1) + 4λ−2

∞∑k=0

(−1)k

(1 + k)2,

9


E(Z3

)= 0

and

E(Z4

)= 48λ−4

∞∑k=0

(−1)k

(1 + k)4+ 3b4(2m + 3)(2m + 1) + 24b2λ−2 (2m + 1)

∞∑k=0

(−1)k

(1 + k)2.

The factorial moments, skewness and the kurtosis can be calculated by using the relationships that

E [(Z)n] = E [Z(Z − 1) · · · (Z − n + 1)] ,

Skewness(Z) =E

(Z3

)− 3E(Z)E

(Z2

)+ 2E3(Z)

E(Z2

)− E2 (Z)

3/2,

and

Kurtosis(Z) =E

(Z4

)− 4E(Z)E

(Z3

)+ 6E

(Z2

)E2(Z)− 3E4(Z)

E(Z2

)− E2 (Z)

2 .

Finally, using the facts that the characteristic functions (chfs) of X and Y are

E [exp(itX)] = B

(1 +

it

λ, 1− it

λ

)and

E [exp(itY )] =(1 + b2t2

)−m−1/2,

where i =√−1 denotes the complex unit and B(a, b) denotes the beta function defined by

B(a, b) =∫ 1

0ta−1(1− t)b−1dt,

the chfs of XY , X/Y and X + Y can be expressed as

E [exp(itXY )] =1√

π2mbm+1Γ (m + 1/2)

∫ ∞

−∞B

(1 +

ity

λ, 1− ity

λ

)| y |m Km

(∣∣∣yb

∣∣∣) dy,

E [exp(itX/Y )] =1√

π2mbm+1Γ (m + 1/2)

∫ ∞

−∞B

(1 +

it

λy, 1− it

λy

)| y |m Km

(∣∣∣yb

∣∣∣) dy

and

E [exp(it(X + Y ))] = B

(1 +

it

λ, 1− it

λ

) (1 + b2t2

)−m−1/2,

respectively. The two integrals do not appear to have tractable analytical solutions.

6 Percentiles of XY , X/Y and X + Y

In this section, we provide tabulations of percentage points associated with the distributions of| XY |, | X/Y | and X + Y . These values are obtained numerically by solving the equation∫ zp

−∞ fZ(w)dw = p when fZ(·) is given by (7), (13) and (14), respectively. Evidently, this involvescomputation of the hypergeometric functions and routines for this are widely available. We usedthe function hypergeom (· · ·) in the algebraic manipulation package, MAPLE. Tables 1, 2 and 3provide the numerical values of zp for m = 2, 3, . . . , 50. We have assumed that b = 1 and λ = 1.

10

NADARAJAH292

Table 1. Percentage points of Z =| XY |.m p = 0.6 p = 0.7 p = 0.8 p = 0.9 p = 0.95 p = 0.992 1.723 2.474 3.648 5.937 8.549 15.7683 2.103 3.005 4.407 7.109 10.161 18.4894 2.426 3.458 5.055 8.116 11.552 20.8525 2.711 3.860 5.630 9.013 12.794 22.9716 2.969 4.223 6.153 9.829 13.926 24.9107 3.207 4.558 6.635 10.582 14.973 26.7078 3.428 4.870 7.084 11.286 15.952 28.3919 3.636 5.163 7.506 11.948 16.874 29.98010 3.833 5.441 7.906 12.576 17.749 31.48911 4.020 5.705 8.287 13.174 18.582 32.92912 4.199 5.957 8.651 13.746 19.380 34.30813 4.370 6.200 9.001 14.295 20.146 35.63414 4.536 6.433 9.337 14.824 20.884 36.91215 4.695 6.658 9.662 15.335 21.597 38.14716 4.849 6.875 9.976 15.829 22.287 39.34417 4.998 7.086 10.280 16.308 22.956 40.50518 5.143 7.291 10.576 16.774 23.607 41.63419 5.284 7.490 10.864 17.227 24.240 42.73320 5.421 7.684 11.144 17.668 24.856 43.80521 5.555 7.874 11.418 18.099 25.458 44.85122 5.686 8.059 11.685 18.520 26.046 45.87323 5.814 8.239 11.946 18.931 26.622 46.87324 5.939 8.416 12.201 19.334 27.184 47.85125 6.062 8.589 12.452 19.728 27.736 48.81126 6.182 8.759 12.697 20.115 28.277 49.75427 6.299 8.926 12.938 20.494 28.778 50.67528 6.415 9.089 13.174 20.866 29.328 51.58229 6.529 9.250 13.406 21.232 29.840 52.47330 6.640 9.408 13.634 21.592 30.344 53.34931 6.750 9.563 13.859 21.946 30.839 54.21132 6.858 9.716 14.080 22.294 31.326 55.05933 6.965 9.866 14.297 22.637 31.806 55.89534 7.069 10.014 14.511 22.975 32.278 56.71835 7.173 10.160 14.723 23.308 32.744 57.53036 7.274 10.304 14.931 23.636 33.203 58.33037 7.375 10.446 15.136 23.959 33.656 59.12038 7.474 10.586 15.338 24.279 34.104 59.89939 7.571 10.725 15.538 24.594 34.545 60.66840 7.668 10.861 15.735 24.905 34.980 61.42741 7.763 10.996 15.930 25.213 35.411 62.17742 7.857 11.129 16.123 25.516 35.836 62.91943 7.950 11.261 16.313 25.816 36.256 63.65144 8.042 11.391 16.501 26.113 36.671 64.37545 8.133 11.519 16.687 26.406 37.082 65.09246 8.223 11.646 16.871 26.697 37.488 65.800

11


47 8.312 11.772 17.053 26.984 37.890 66.50148 8.400 11.897 17.233 27.267 38.288 67.19449 8.487 12.020 17.411 27.549 38.681 67.88150 8.573 12.142 17.587 27.827 39.071 68.560

Table 1. Percentage points of Z =| XY |.m p = 0.6 p = 0.7 p = 0.8 p = 0.9 p = 0.95 p = 0.992 1.723 2.474 3.648 5.937 8.549 15.7683 2.103 3.005 4.407 7.109 10.161 18.4894 2.426 3.458 5.055 8.116 11.552 20.8525 2.711 3.860 5.630 9.013 12.794 22.9716 2.969 4.223 6.153 9.829 13.926 24.9107 3.207 4.558 6.635 10.582 14.973 26.7078 3.428 4.870 7.084 11.286 15.952 28.3919 3.636 5.163 7.506 11.948 16.874 29.98010 3.833 5.441 7.906 12.576 17.749 31.48911 4.020 5.705 8.287 13.174 18.582 32.92912 4.199 5.957 8.651 13.746 19.380 34.30813 4.370 6.200 9.001 14.295 20.146 35.63414 4.536 6.433 9.337 14.824 20.884 36.91215 4.695 6.658 9.662 15.335 21.597 38.14716 4.849 6.875 9.976 15.829 22.287 39.34417 4.998 7.086 10.280 16.308 22.956 40.50518 5.143 7.291 10.576 16.774 23.607 41.63419 5.284 7.490 10.864 17.227 24.240 42.73320 5.421 7.684 11.144 17.668 24.856 43.80521 5.555 7.874 11.418 18.099 25.458 44.85122 5.686 8.059 11.685 18.520 26.046 45.87323 5.814 8.239 11.946 18.931 26.622 46.87324 5.939 8.416 12.201 19.334 27.184 47.85125 6.062 8.589 12.452 19.728 27.736 48.81126 6.182 8.759 12.697 20.115 28.277 49.75427 6.299 8.926 12.938 20.494 28.778 50.67528 6.415 9.089 13.174 20.866 29.328 51.58229 6.529 9.250 13.406 21.232 29.840 52.47330 6.640 9.408 13.634 21.592 30.344 53.34931 6.750 9.563 13.859 21.946 30.839 54.21132 6.858 9.716 14.080 22.294 31.326 55.05933 6.965 9.866 14.297 22.637 31.806 55.89534 7.069 10.014 14.511 22.975 32.278 56.71835 7.173 10.160 14.723 23.308 32.744 57.53036 7.274 10.304 14.931 23.636 33.203 58.33037 7.375 10.446 15.136 23.959 33.656 59.12038 7.474 10.586 15.338 24.279 34.104 59.89939 7.571 10.725 15.538 24.594 34.545 60.668

12

NADARAJAH294

40 7.668 10.861 15.735 24.905 34.980 61.42741 7.763 10.996 15.930 25.213 35.411 62.17742 7.857 11.129 16.123 25.516 35.836 62.91943 7.950 11.261 16.313 25.816 36.256 63.65144 8.042 11.391 16.501 26.113 36.671 64.37545 8.133 11.519 16.687 26.406 37.082 65.09246 8.223 11.646 16.871 26.697 37.488 65.80047 8.312 11.772 17.053 26.984 37.890 66.50148 8.400 11.897 17.233 27.267 38.288 67.19449 8.487 12.020 17.411 27.549 38.681 67.88150 8.573 12.142 17.587 27.827 39.071 68.560

Table 2. Percentage points of Z =| X/Y |.m p = 0.6 p = 0.7 p = 0.8 p = 0.9 p = 0.95 p = 0.992 1.169 1.715 2.761 5.783 11.714 58.8253 0.954 1.392 2.228 4.640 9.380 47.0624 0.825 1.201 1.916 3.982 8.042 40.3395 0.737 1.071 1.707 3.542 7.150 35.8586 0.673 0.976 1.554 3.221 6.500 32.5987 0.623 0.902 1.435 2.974 6.001 30.0908 0.582 0.843 1.340 2.776 5.601 28.0849 0.549 0.794 1.262 2.613 5.272 26.43210 0.520 0.753 1.196 2.476 4.994 25.04111 0.496 0.718 1.140 2.358 4.757 23.84912 0.475 0.687 1.090 2.256 4.550 22.81213 0.456 0.660 1.047 2.166 4.368 21.90014 0.439 0.635 1.008 2.086 4.206 21.08815 0.424 0.614 0.974 2.014 4.061 20.36116 0.411 0.594 0.942 1.949 3.930 19.70417 0.399 0.576 0.914 1.890 3.811 19.10718 0.387 0.560 0.888 1.836 3.702 18.56119 0.377 0.545 0.864 1.786 3.602 18.06020 0.367 0.531 0.842 1.741 3.510 17.59721 0.358 0.518 0.821 1.698 3.424 17.16822 0.350 0.506 0.802 1.659 3.345 16.76823 0.342 0.495 0.785 1.622 3.270 16.39624 0.335 0.484 0.768 1.587 3.201 16.04925 0.328 0.475 0.752 1.555 3.136 15.71926 0.322 0.465 0.738 1.525 3.074 15.41127 0.316 0.457 0.724 1.496 3.016 15.12028 0.310 0.448 0.711 1.469 2.961 14.84529 0.305 0.440 0.698 1.443 2.909 14.58530 0.300 0.433 0.686 1.418 2.860 14.33831 0.295 0.426 0.675 1.395 2.813 14.10332 0.290 0.419 0.664 1.373 2.769 13.879

13


33 0.286 0.413 0.654 1.352 2.726 13.66534 0.282 0.407 0.644 1.332 2.685 13.46135 0.278 0.401 0.635 1.312 2.646 13.26636 0.274 0.395 0.626 1.294 2.609 13.07937 0.270 0.390 0.618 1.276 2.573 12.90038 0.266 0.385 0.609 1.259 2.539 12.72839 0.263 0.380 0.602 1.243 2.506 12.56340 0.260 0.375 0.594 1.227 2.474 12.40441 0.256 0.370 0.587 1.212 2.444 12.25142 0.253 0.366 0.580 1.197 2.414 12.10343 0.250 0.361 0.573 1.183 2.386 11.96144 0.247 0.357 0.566 1.170 2.359 11.82345 0.245 0.353 0.560 1.157 2.332 11.69146 0.242 0.349 0.554 1.144 2.306 11.56247 0.239 0.346 0.548 1.132 2.282 11.43848 0.237 0.342 0.542 1.120 2.258 11.31749 0.234 0.339 0.536 1.108 2.234 11.20150 0.232 0.335 0.531 1.097 2.212 11.088

Table 3. Percentage points of Z = X + Y .m p = 0.6 p = 0.7 p = 0.8 p = 0.9 p = 0.95 p = 0.992 0.675 1.407 2.288 3.575 4.712 7.0683 0.762 1.586 2.572 4.002 5.254 7.8144 0.840 1.747 2.830 4.390 5.745 8.4895 0.913 1.896 3.068 4.747 6.197 9.1126 0.980 2.035 3.288 5.079 6.619 9.6937 1.043 2.165 3.496 5.391 7.015 10.2408 1.102 2.288 3.692 5.687 7.390 10.7599 1.159 2.404 3.878 5.968 7.747 11.25310 1.213 2.515 4.056 6.236 8.088 11.72611 1.264 2.622 4.226 6.494 8.416 12.18112 1.314 2.725 4.390 6.741 8.731 12.61813 1.362 2.824 4.549 6.980 9.035 13.04114 1.408 2.919 4.701 7.211 9.330 13.45115 1.453 3.012 4.849 7.435 9.615 13.84816 1.496 3.102 4.993 7.653 9.892 14.23417 1.538 3.189 5.133 7.864 10.162 14.61018 1.580 3.274 5.269 8.070 10.425 14.97619 1.620 3.357 5.401 8.271 10.681 15.33420 1.659 3.438 5.531 8.467 10.931 15.68321 1.697 3.517 5.657 8.659 11.176 16.02522 1.734 3.594 5.781 8.846 11.415 16.36023 1.771 3.670 5.902 9.030 11.650 16.68724 1.807 3.744 6.021 9.210 11.880 17.00925 1.842 3.816 6.137 9.386 12.105 17.324

14

NADARAJAH296

26 1.876 3.888 6.252 9.559 12.327 17.63427 1.910 3.958 6.364 9.730 12.544 17.93928 1.944 4.027 6.474 9.897 12.758 18.23829 1.976 4.094 6.583 10.061 12.968 18.53330 2.009 4.161 6.689 10.223 13.175 18.82331 2.040 4.226 6.794 10.383 13.379 19.10832 2.071 4.291 6.898 10.540 13.580 19.38933 2.102 4.355 7.000 10.694 13.777 19.66734 2.132 4.417 7.100 10.847 13.972 19.94035 2.162 4.479 7.199 10.997 14.165 20.21036 2.192 4.540 7.297 11.145 14.354 20.47637 2.221 4.600 7.393 11.292 14.542 20.73938 2.250 4.660 7.488 11.436 14.726 20.99839 2.278 4.718 7.582 11.579 14.909 21.25440 2.306 4.776 7.675 11.720 15.089 21.50741 2.334 4.833 7.767 11.859 15.268 21.75842 2.361 4.890 7.858 11.997 15.444 22.00543 2.388 4.946 7.947 12.133 15.618 22.25044 2.415 5.001 8.036 12.267 15.790 22.49245 2.441 5.056 8.123 12.401 15.961 22.73146 2.467 5.110 8.210 12.532 16.129 22.96847 2.493 5.163 8.296 12.663 16.296 23.20248 2.519 5.216 8.381 12.792 16.461 23.43549 2.544 5.269 8.465 12.919 16.625 23.66450 2.569 5.321 8.548 13.046 16.787 23.892

We expect these numbers could be of use to the practitioners mentioned in Section 1. Similartabulations could be easily derived for other values of m, b, λ and p by using the hypergeom (· · ·)function in MAPLE.

References

Abu-Salih, M. S. (1983). Distributions of the product and the quotient of power-function randomvariables. Arab Journal of Mathematics, 4, 77–90.

Albert, J. (2002). Sums of uniformly distributed variables: a combinatorial approach. CollegeMathematics Journal, 33, 201–206.

Ali, M. M. and Obaidullah, M. (1982). Distribution of linear combination of exponential variates.Communications in Statistics—Theory and Methods, 11, 1453–1463.

Balakrishnan, N. (1992). Handbook of the Logistic Distribution. Marcel Dekker, New York, NY.

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15


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18

NADARAJAH300

On the Moments of the B Distributionby

Saralees NadarajahSchool of Mathematics

University of ManchesterManchester M60 1QD

United [email protected]

Abstract: Explicit expressions are derived for the moments and the moment generating functionof the B distribution introduced by Bousquet et al. (2006).

Keywords and Phrases: B distribution, Generalized hypergeometric function, Moment gener-ating function, Moments.

1 Introduction

The recent paper by Bousquet et al. (2006) introduced the B distribution specified by the cumu-lative distribution function (cdf)

FX(x) = 1− exp (−axr − sx) (1)

for x > 0, a > 0, −∞ < r < ∞ and s > 0. The paper discussed various properties of (1) withapplications. However, little was presented in terms of basic mathematical properties. The onlymathematical properties of substance discussed in Bousquet et al. (2006) were the expected value(E(X)), variance (V ar(X)) and the moment generating function (mgf) (M(t) = E[exp(tX)]) allfor the particular case r = 2. Here, I would like to point out that much more general expressionscan be derived for the moments and the mgf associated with (1). The main results are presentedin Section 2. The calculations involve the generalized hypergeometric function defined by

pFq (a1, . . . , ap; b1, . . . , bq;x) =∞∑

k=0

(a1)k (a2)k · · · (ap)k

(b1)k (b2)k · · · (bq)k

xk

k!,

where (c)k = c(c+1) · · · (c+k−1) denotes the ascending factorial. The properties of the generalizedhypergeometric function can be found in Prudnikov et al. (1986) and Gradshteyn and Ryzhik(2000).

2 Main Results

Theorem 1 derives explicit expressions for E(Xγ) associated with (1), where r and γ can be any realnumbers. The corresponding mgf is given by Theorem 2. An essential assumption of the theoremsis that the parameter r is a rational number.

1

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.10,NO.3,301-303,2008, COPYRIGHT 2008 EUDOXUS PRESS, LLC

Theorem 1 Suppose a random variable X has the cdf (1). Then, the γth moment of X is given by

E (Xγ) =

γI(γ, a, r, s), if r > 0 and γ > 0,γJ(γ, a,−r, s), if r < 0

for a > 0 and s > 0, where I(· · ·) and J(· · ·) are given by Lemmas 1 and 2, respectively.

Proof: The result follows by writing

E (Xγ) = γ

∫ ∞

0xγ−1 exp (−axr − sx) dx

and using the results of Lemmas 1 and 2.

Theorem 2 Suppose a random variable X has the cdf (1). Then, the mgf of X is given by

M(t) =

arI(r, a, r, s− t) + sI(1, a, r, s− t), if r > 0,arJ(r, a,−r, s− t) + sJ(1, a,−r, s− t), if r < 0

for a > 0 and s > t, where I(· · ·) and J(· · ·) are given by Lemmas 1 and 2, respectively.

Proof: The probability density function (pdf) corresponding to (1) is:

fX(x) =(arxr−1 + s

)exp (−axr − sx) .

Thus, the mgf can be expressed as

M(t) = ar

∫ ∞

0xr−1 exp −axr − (s− t)x dx + s

∫ ∞

0exp −axr − (s− t)x dx.

The result follows by Lemmas 1 and 2.

Appendix

The proofs of Theorems 1 and 2 require the following technical lemmas.

Lemma 1 (Equation (2.3.1.13), Prudnikov et al., 1986, volume 1) For r > 0, γ > 0, a > 0 ands > 0, ∫ ∞

0xγ−1 exp (−axr − sx) dx = I(γ, a, r, s),

where

I =

q−1∑j=0

(−a)j

j!sγ+rjΓ (γ + rj) p+1Fq (1,∆ (p, γ + rj) ;∆(q, 1 + j); (−1)qz) , if 0 < r < 1,

p−1∑h=0

(−s)h

rh!a(γ+h)/rΓ

(γ + h

r

)q+1Fp

(1,∆

(q,

γ + h

r

);∆(p, 1 + h);

(−1)p

z

), if r > 1,

Γ(γ)(a + s)γ , if r = 1

provided that r = p/q, where p ≥ 1 and q ≥ 1 are co-prime integers, where z = ppaq/spqq and∆(k, a) = (a/k, (a + 1)/k, . . . , (a + k − 1)/k).

2

NADARAJAH302

Lemma 2 (Equation (2.3.1.14), Prudnikov et al., 1986, volume 1) For r > 0, a > 0 and s > 0,∫ ∞

0xγ−1 exp

(−ax−r − sx

)dx = J(γ, a, r, s),

where

J =q−1∑j=0

(−a)j

j!sγ−rjΓ (γ − rj) 1Fp+q (1;∆ (p, 1− γ + rj) ,∆(q, 1 + j); z)

+p−1∑h=0

(−s)ha(γ+h)/r

rh!Γ

(−γ + h

r

)1Fp+q

(1;∆

(q, 1 +

γ + h

r

),∆(p, 1 + h); z

)provided that r = p/q, where p ≥ 1 and q ≥ 1 are co-prime integers, where z = (−1)p+qspaq/ppqqand ∆(k, a) = (a/k, (a + 1)/k, . . . , (a + k − 1)/k).

References

Bousquet, N., Bertholon, H. and Celeux, G. (2006). An alternative competing risk model to theWeibull distribution for modelling aging in lifetime data analysis. Lifetime Data Analysis, 12,481–504.

Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products, sixth edition.Academic Press, San Diego.

Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series, volumes 1, 2and 3. Gordon and Breach Science Publishers, Amsterdam.

3

B-DISTRIBUTION 303

The Beta-Laplace Distribution

Tomasz J. Kozubowski, Department of Mathematics & Statistics, University of Nevada,

Reno, NV 89557, USA, [email protected]

Saralees Nadarajah, Department of Statistics, University of Nebraska, Lincoln, NE

68583, USA, [email protected]

Abstract. Motivated by the recent work of Eugene et al. [3] and Gupta and

Nadarajah [5], we introduce the beta Laplace distribution generated from the

logit of a beta random variable. Our focus are the basic theoretical properties of

this distribution, including modality and concavity of the density, moments and

related parameters, and stochastic representations that aid in random variate

generation from this model.

Keywords: Beta-normal distribution, Gauss hypergeometric exponential distribution, Gener-

alized exponential distribution, Mixture representation, Unimodality

AMS 2000 Subject Classifications: 60E05, 62E10

1 Introduction

Every cumulative distribution function (c.d.f.) G generates a generalized class of distributions

with c.d.f.’s

FG(x) =BG(x)(a, b)

B(a, b), x ∈ R, a, b > 0, (1)

where

By(a, b) =∫ y

0wa−1(1 − w)b−1dw (2)

is the incomplete beta function and

B(a, b) = B1(a, b) =Γ(a)Γ(b)Γ(a + b)

.


Note that if a random variable X has the c.d.f. given by (1) then it admits the representation

Xd= G−1(W ), (3)

where W has the beta distribution with parameters a > 0, b > 0, denoted by Be(a, b). In the

special case a = b = 1 the variable W is standard uniform, and (3) becomes the probability

integral transformation.

Eugene et al. [3] introduced what is known as the beta normal distribution by taking

G to be the c.d.f. of the normal distribution with parameters µ and σ. The properties of

this distribution have been studied in more detail in [5]. In this paper, we introduce the beta

Laplace (BL) distribution by taking G in (1) to be the c.d.f. of the Laplace distribution.

Thus, the c.d.f. of the BL distribution is given by (1), where

G(x) =

12

exp(

x − θ

σ

), if x < θ,

1 − 12

exp(

θ − x

σ

), if x ≥ θ,

(4)

and −∞ < θ < ∞ and σ > 0. The corresponding probability density function (p.d.f.) and

the hazard rate function are

fa,b,θ,σ(x) =1

2σB(a, b)exp

(−| x − θ |

σ

)Ga−1(x) 1 − G(x)b−1 (5)

and

λa,b,θ,σ(x) =1

2σ[B(a, b)]2exp

(−| x − θ |

φ

)Ga−1(x) 1 − G(x)b−1

B1−G(x)(b, a), (6)

respectively. We shall denote this distribution by BL(a, b, θ, σ). Since θ and σ are location

and scale parameters, we shall restrict attention to the standard case with θ = 0 and σ = 1,

denoted by BL(a, b). In this case the p.d.f. (5) takes the form

fa,b(x) =(

12

)a+b−1 Γ(a + b)Γ(a)Γ(b)

·

eax(2 − ex)b−1, if x ≤ 0,

e−bx(2 − e−x)a−1, if x > 0.(7)

Note that X ∼ BL(a, b, θ, σ) if and only if −X ∼ BL(b, a,−θ, σ). In particular, for the

standard variable we have X ∼ BL(a, b) if and only if −X ∼ BL(b, a).

KOZUBOWSKI-NADARAJAH306

The standard Laplace distribution is contained as the particular case of (1) for a = 1

and b = 1. Another special case is

FG(x) =n∑

i=a

(n

i

)Gi(x) 1 − G(x)n−i

for b = n − a + 1 and integer values of a. This is the distribution of the ath order statistic

connected with a random sample of size n = a + b − 1 from the Laplace distribution with

c.d.f. G, see [2]. Basic properties of this distribution can be found in Section 2.5 of [7]. Other

special cases include

FG(x) = 1 − 1 − G(x)b

Γ(b)

a∑

i=1

Γ(b + i − 1)Γ(i)

Gi−1(x)

for integer values of a,

FG(x) =Ga(x)Γ(a)

b∑

i=1

Γ(a + i − 1)Γ(i)

1 − G(x)i−1

for integer values of b, and

FG(x) =2π

arctan

√G(x)

1 − G(x)

for a = 1/2 and b = 1/2.

In this paper we derive basic theoretical properties of (5), deferring practical issues

of estimation and testing to future work. In particular, in Section 2 we derive stochastic

representations of the corresponding random variables, which aid in simulation from this

distribution, characterize the corresponding densities in terms of modality and concavity in

Section 3, and provide expressions for the moments and related characteristics in Section 4.

Proofs and technical lemmas are collected in Section 5.

2 Representations

Note that if X ∼ BL(a, b), then the distribution of X|X > 0 (the distribution of X truncated

below at zero) is given by the p.d.f.

ga,b(x) =(1/2)a+b−1

B1/2(b, a)e−bx(2 − e−x)a−1, x ≥ 0, a, b > 0, (8)

BETA-LAPLACE DISTRIBUTION 307

where B1/2(b, a) is the incomplete beta function given by (2). This “one-sided” beta-Laplace

distribution is a generalization of the standard exponential distribution, similar in spirit to

the generalized exponential distribution introduced in [4]. The latter can be defined through

the representation (3), where W has the power function distribution with the c.d.f. wα,

0 < w ≤ 1, α > 0, while G(x) = 1 − e−x, 0 < x ≤ 1, is the c.d.f. of the standard exponential

distribution (so it reduces to the exponential when α = 1). Similarly, a variable with p.d.f.

(8) admits such a representation with the same G and W having the p.d.f.

wa,b(x) =(1/2)a+b−1

B1/2(b, a)(1 − x)b−1(1 + x)a−1, 0 ≤ x ≤ 1, a, b > 0. (9)

The distribution with density (9) is a special case of Gauss hypergeometric distribution studied

in [1] (see also [6], p. 253). For this reason we shall refer to the distribution with density (8)

as the Gauss hypergeometric exponential distribution with parameters a, b > 0, denoted by

GHE(a, b). Note that when a = b = 1 this distribution reduces to the standard exponential

as in this case W is standard uniform.

Similarly, when we truncate X ∼ BL(a, b) above at zero, we obtain a distribution on

(−∞, 0) corresponding to a r.v. −Y , where Y has a GHE(b, a) distribution on (0,∞). This

leads to the following representation of X in terms of its “one-sided” counterparts.

Proposition 1 If X ∼ BL(a, b) then we have

Xd= IY1 + (I − 1)Y2, (10)

where Y1 ∼ GHE(a, b), Y2 ∼ GHE(b, a), I takes on the values 0 and 1 with probabilities

qa,b(x) =Γ(a + b)Γ(a)Γ(b)

B1/2(a, b) and pa,b = 1 − qa,b(x) =Γ(a + b)Γ(a)Γ(b)

B1/2(b, a), (11)

respectively, and all the variables on the right-hand-side of (10) are mutually independent.

As we stated before, the r.v.’s Y1 and Y2 in the above representation have the same distribu-

tions as F−1(W1) and F−1(W2), respectively, where F is the standard exponential c.d.f. and

W1 and W2 have densities given by wa,b and wb,a, respectively. Further, W1 and W2 have the

same distributions as 2V1 − 1 and 1− 2V2, respectively, where V1 and V2 have truncated beta

distributions, that is

V1d= W |W > 1/2, and V2

d= W |W < 1/2 (12)

with W ∼ Be(a, b). Moreover, the quantities qa,b and pa,b in (11) are the probabilities

P (W < 1/2) and P (W > 1/2), respectively. This result admits a generalization, where X is

still given by (3) but G and W are not necessarily the Laplace c.d.f. and the beta variable,

respectively. Indeed, let V have a continuous distribution on (0,∞) with density f and c.d.f.

F , and let Y be the corresponding “symmetrization” with p.d.f. g(x) = f(|x|), x ∈ R, and

c.d.f.

G(x) =

12

(1 − F (−x)) , if x < 0,

1 − 12

(1 − F (x)) , if x ≥ 0.(13)

[If V is standard exponential then Y is standard Laplace (4) with θ = 0 and σ = 1.] Further,

define a r.v. X via (3) where W has some continuous distribution on (0, 1). Then the

following representation holds.

Proposition 2 The r.v. X ∼ G−1(W ) admits the representation (10), where

Y1d= F−1(2V1 − 1) and Y2

d= F−1(1 − 2V2) (14)

with V1, V2 given by (12), I takes on the values 0 and 1 with probabilities P (W < 1/2) and

P (W > 1/2), respectively, and all the variables on the right-hand-side of (10) are mutually

independent.

3 Characterization of the density

Before we describe the density (7) of the standard beta-Laplace distribution in terms of

modality and concavity, we start with one-sided distributions given by (8). Thus, we consider

the function

ha,b(x) = e−bx(2 − e−x)a−1, x ≥ 0, a, b > 0, (15)

which appears in the expression for the density. The relevant properties of ha,b are as follows.

Proposition 3 The function ha,b given by (15) admits the following properties:

(i) The function ha,b is continuous on [0,∞) and differentiable on (0,∞) with ha,b(0) = 1,

limx→∞ ha,b(x) = 0, and limx→0+ h′a,b(x) = a − b − 1.

(ii) The function ha,b is monotonically decreasing on (0,∞) whenever a > 0 and b ≥ a − 1,

and is monotonically increasing on (0, xa,b) and decreasing on (xa,b,∞) whenever a > 1 and

b < a − 1, where

xa,b = lna + b − 1

2b, 0 3 and b−(a) ≤ b < b+(a) or 1 < a ≤ 3 and b < b+(a) then ha,b is concave

down on (0, x+a,b) and concave up on (x+

a,b,∞). Here,

b±(a) = a − 1 ±√

2(a − 1) (18)

and

x±a,b = ln

1 + t±a,b

2, (19)

where

t±a,b =(a − 1)(1 + 2b) ±

√(a − 1)2(1 + 2b)2 − 4b2(a − 1)(a − 2)

2b2. (20)

(iv) If a > 3 and b < b−(a) then ha,b is concave down on (x−a,b, x

+a,b) and concave up on

(0, x−a,b) and (x+

a,b,∞).

(v) If either 1 < a and b ≥ b+(a) or a ≤ 1 then ha,b is concave up on (0,∞).

Using the results above we obtain the following four distinct cases of the one-sided BL density

ga,b given by (8):

• If either 0 < a ≤ 1 or a > 1 but b ≥ b+(a) then ga,b(x) is concave up and decreasing in

x on (0,∞).

• If a > 1 and a − 1 ≤ b < b+(a) [note that b+(a) > a − 1 whenever a > 1] then ga,b(x)

is decreasing in x on (0,∞). Moreover, it is concave down on (0, x+a,b) and concave up

on (x+a,b,∞) with x+

a,b as in (19).

• If either 1 < a ≤ 3 and 0 3 and b−(a) ≤ b < a − 1 [note that

b−(a) < a − 1 whenever a > 3] then ga,b(x) is increasing in x on (0, xa,b), decreasing in

x on (xa,b,∞), concave down on (0, x+a,b), and concave up on (x+

a,b,∞), with 0 < xa,b <

x+a,b < ∞.

• If 3 < a and b < b−(a) then ga,b(x) is increasing in x on (0, xa,b), decreasing in x on

(xa,b,∞), concave down on (x−a,b, x

+a,b), and concave up on (0, x−

a,b) and (x+a,b,∞), with

0 < x−a,b < xa,b < x+

a,b < ∞.

Using the above proposition and the fact that the BL(a, b) density is related to the

function (15) via

fa,b(x) =(

12

)a+b−1 Γ(a + b)Γ(a)Γ(b)

·

hb,a(|x|), if x ≤ 0,

ha,b(x), if x > 0.(21)

we obtain the following result.

Proposition 4 For any a, b > 0 the BL(a, b) distribution is unimodal and we have the

following three distinct cases:

(i) If a − 1 ≤ b ≤ a + 1 then the density (21) is monotonically increasing on (−∞, 0) and

decreasing on (0,∞) (so that the mode occurs at x = 0).

(ii) If 0 0 given by (16)).

(iii) If 0 < a < b − 1 then the density (21) is monotonically increasing on (−∞,−xb,a) and

decreasing on (−xb,a,∞) (so that the mode occurs at x = −xb,a < 0), where

xb,a = lna + b − 1

2a, 0 < a < b − 1. (22)

Moreover, in cases (ii) and (iii) the maximum values of fa,b are

fa,b(xa,b) =Γ(a + b)Γ(a)Γ(b)

(a − 1)a−1bb

(a + b − 1)a+b−1, and fa,b(−xb,a) =

Γ(a + b)Γ(a)Γ(b)

(b − 1)b−1aa

(a + b − 1)a+b−1, (23)

respectively.

Remark 1. Note that by Part (i) of Proposition 3 the BL density is continuous on R but

differentiable only at x 6= 0 (regardless of whether the mode is at zero or not).

Remark 2. Note that, in contrast, the beta normal distribution introduced in [3] is not always

unimodal.

Remark 3. Combining Propositions 3 and 4 and using the representation (21) one can derive

a number of distinct cases of the density fa,b in terms of its modality and concavity.

4 Moments and related measures

We start with the moment generating function (m.g.f.) corresponding to the BL(a, b) distri-

bution. By (7) we have

M(t) = E(etX ) = C

∫ ∞

0

e−(a+t)x

2a

(1 − e−x

2

)b−1

dx +∫ ∞

0

e−(b−t)x

2b

(1 − e−x

2

)a−1

dx

,

(24)

where C = Γ(a + b)/[Γ(a)Γ(b)]. Both integrals above are convergent whenever −a < t < b.

Since e−x/2 ∈ (0, 1/2) when x ∈ (0,∞), using the binomial expansion

(1 + z)p =∞∑

j=0

pjzj , |z| < 1, p ∈ R, (25)

we can write (24) as

C

∫ ∞

0

∞∑

j=0

(−1)j

2a+jb − 1je−(a+t+j)xdx +

∫ ∞

0

∞∑

j=0

(−1)j

2b+ja − 1je−(b−t+j)xdx

. (26)

Since the quantities under the summations are absolutely integrable (when −a < t < b),

interchanging the order of integration and summation leads to

M(t) = C

∞∑

j=0

(−1)j

2a+jb − 1j

1a + t + j

+∞∑

j=0

(−1)j

2b+ja − 1j

1b − t + j

, −a < t < b. (27)

If a = k + 1 and b = m + 1, where k and m are nonnegative integers, then

b − 1j = 0 for j > m and a − 1j = 0 for j > k (28)

so that the series above have only finite number of terms and we have

M(t) = C

m∑

j=0

(−1)j

2k+1+jmj

1k + 1 + t + j

+k∑

j=0

(−1)j

2m+1+jkj

1m + 1 − t + j

(29)

for −k − 1 < t < m + 1. Note that when k = m = 0 (so that a = b = 1) the above

expression simplifies to (1− t2)−1, which is the m.g.f. corresponding to the standard Laplace

distribution.

Similar derivations lead to the characteristic function (ch.f.), which is of the form

φ(t) = EeitX = C

∞∑

j=0

(−1)j

2a+jb − 1j

1a + it + j

+∞∑

j=0

(−1)j

2b+ja − 1j

1b − it + j

, t ∈ R. (30)

Differentiating either the m.g.f. or the ch.f. n times and evaluating at t = 0 we obtain the

following expression for the nth moment µn = EXn of X ∼ BL(a, b), where n = 1, 2, . . .,

µn =Γ(a + b)Γ(a)Γ(b)

n!

∞∑

j=0

(−1)j+n

2a+jb − 1j

1(a + j)n+1

+∞∑

j=0

(−1)j

2b+ja − 1j

1(b + j)n+1

. (31)

As before, if a = k + 1 and b = m + 1, where k and m are nonnegative integers, the series

above have only finite number of terms and we have

µn =(

12

)r r!n!k!m!

m∑

j=0

mj(−1)n+m−j2j

(r − j)n+1+

k∑

j=0

kj(−1)k−j2j

(r − j)n+1

, (32)

where r = k + m + 1.

5 Proofs

Proof of Proposition 1. This is a special case of Proposition 2.

Proof of Proposition 2. We proceed by showing that for each x ∈ R the c.d.f. of G−1(W )

coincides with the c.d.f. of the right-hand-side of (10) with Y1, Y2, and I as stated above.

Assume first that x > 0. Since

G−1(p) =

−F−1 (1 − 2p) , if p ∈ (0, 1/2),

F−1 (2p − 1) , if p ∈ [1/2, 1),(33)

and G(0) = 1/2, we have

P (G−1(W ) > x) = P (W > 1/2 and F−1(2W − 1) > x) = P (W > 1/2 and 2W − 1 > F (x))

= P (W > 1/2 and W > (1 + F (x))/2) = P (W > (1 + F (x))/2)

as W > (1 + F (x))/2 implies W > 1/2. On the other hand, we have

P (IY1 + (I − 1)Y2 > x) = P (IF−1(2V1 − 1) > x and I = 1) = P (2V1 − 1 > F (x))P (I = 1)

= P (V1 > (1 + F (x))/2)P (I = 1) = P (W > (1 + F (x))/2|W > 1/2)P (W > 1/2),

which also simplifies to P (W > (1+F (x))/2). We thus established the equality of the c.d.f.’s

when x > 0. The case x ≤ 0 is analogous. This concludes the proof.

The following lemma is needed to prove Proposition 3.

Lemma 1 Consider the function

u(t) = b2t2 − (a − 1)(1 + 2b)t + (a − 1)(a − 2), t ∈ R, (34)

where a > 1 and b > 0. Then

(i) The function u is decreasing on (−∞, vab) and increasing on (va,b,∞), where

va,b =(a − 1)(1 + 2b)

2b2> 0. (35)


Moreover, the minimum value of u on (−∞,∞) is equal to

u(va,b) = −(a − 1)

(a − 1)

[(1 + 2b

2b

)2

− 1

]+ 1

< 0. (36)

(ii) We have u(0) = (a − 1)(a − 2), so that u(0) < 0 when a < 2, u(0) = 0 when a = 2, and

u(0) > 0 when a > 2.

(iii) The equation u(t) = 0 admits the roots t±a,b given by (20).

(iv) For any fixed a > 1, as a function of b the larger root, t+a,b, is continuous and decreasing

on (0,∞) with limb→0+ t+a,b = ∞ and limb→∞ t+a,b = 0. Moreover, t+a,b = 1 when b = b+(a)

with b+(a) given by (18).

(v) For any fixed a > 2, as a function of b the smaller root, t−a,b, is continuous and decreasing

on (0,∞) with limb→0+ t−a,b = t−a,0 = a − 2 and limb→∞ t−a,b = 0. Moreover, t−a,b = 1 when

b = b−(a) with b−(a) given by (18).

Proof. Calculations needed to prove Parts (i) - (iii) are elementary. To establish Part (iv),

write

t+a,b = (a − 1)(

12b2

+1b

)+

12b

√(a − 1)2

(1b

+ 2)2

− 4(a − 1)(a − 2) (37)

to see that it is decreasing in b on (0,∞) and the limits are as stated. With some tedious

albeit routine algebra one can check that the solution of the equation t+a,b = 1 is given by

b+(a) in (18). Similarly, straightforward algebra leads to

t−a,b =2(a − 1)(a − 2)

(a − 1)(1 + 2b) +√

(a − 1) (a − 1)(1 + 4b) + 4b2, (38)

which shows that this is a decreasing function of b on (0,∞) with limits as stated. Further

routine calculations show that the solution of the equation t−a,b = 1 is given by b−(a) in (18).

This concludes Part (v).

Proof of Proposition 3. The continuity and differentiability of ha,b, as well as the values at

zero and infinity, are clear. To check the limit of the derivative, write ha,b(x) = exp(sa,b(x)),

where

sa,b(x) = lnha,b(x) = −bx + (a − 1) ln(2 − e−x), x ∈ [0,∞), (39)

so that for x > 0 we have h′a,b(x) = ha,b(x)s′a,b(x) with

s′a,b(x) = −b +a − 1

2ex − 1, x ∈ (0,∞). (40)

Since sa,b(x) and s′a,b(x) converge to 0 and a − b − 1, respectively, as x → 0+, we conclude

that the corresponding limit of h′a,b(x) is a − b − 1. This concludes Part (i).

To study the monotonicity of the function ha,b we look at the derivate (40) since the

signs of s′a,b(x) and h′a,b(x) coincide. It is clear that s′a,b(x) < 0 for all x > 0 whenever α ≤ 1,

in which case ha,b is monotonically decreasing on (0,∞). Further, if a > 1, simple algebra

shows that the derivative is negative if and only if

ex >a + b − 1

2b. (41)

This inequality is always true whenever b ≥ a− 1, while for b < a− 1 (and a > 1) its solution

is x > xa,b, where xa,b given by (16). Moreover, routine calculations show that in the latter

case the maximum value of ha,b is given by (17). This concludes Part (ii).

Next, we study the concavity of ha,b. Observe that

h′′a,b(x) = ha,b(x)(s′′a,b(x) + [s′a,b(x)]2), x ∈ (0,∞), (42)

so that h′′a,b(x) > 0 if and only if

s′′a,b(x) + [s′a,b(x)]2 > 0. (43)

Since

s′′a,b(x) =−2(a − 1)ex

(2ex − 1)2, x ∈ (0,∞), (44)

the inequality (43) holds for all x ∈ (0,∞) whenever a ≤ 1, in which case the function ha,b

is concave up on this interval. We thus established the second part of (v). In the sequel, we

shall study the inequality (43) assuming that a > 1. Utilizing the expressions for the first

and the second derivatives of sa,b, given by (40) and (44), respectively, after some routine

algebra we find that this inequality is equivalent to

u(2ex − 1) > 0, (45)

where u(·) is the quadratic function defined in Lemma 1. Since for any x > 0 we have

2ex − 1 > 1, we need to check how the roots of the equation u(t) = 0, which by Part (iii) of

Lemma 1 are given by (20), relate to t = 1. Observe that by Part (iv) of Lemma 1 the larger

root (denoted by t+a,b in (20)) is less than or equal to 1 whenever b ≥ b+(a), where b+(a) is

given by (18). Consequently, for these values of b we have u(t) > 0 for all t > 1, so that the

inequality (45) is satisfied by all x > 0. This shows that whenever a > 1 and b ≥ b+(a) the

function ha,b is concave up on the interval (0,∞), which is the first part of (v). To establish

Parts (iii) and (iv) we shall proceed by separately considering the cases 1 < a ≤ 3 and a > 3.

Case 1: 1 < a ≤ 3. If a ≤ 2, then by Parts (i) and (ii) of Lemma 1, we conclude that the

smaller root of the equation u(t) = 0, denoted by t−a,b in (20), is less than or equal to zero.

Thus, when b < b+(a), in which case we have t+a,b > 1 by Part (iv) of Lemma 1, the function u

is negative for t ∈ (0, t+a,b) and positive for t ∈ (t+a,b,∞). In turn, the inequality (45) holds for

all x > x+a,b with x+

a,b defined in (19). The same conclusion is reached when 2 < a ≤ 3, since

in this case a − 2 ≤ 1 so that by Part (v) of Lemma 1 we have t−a,b ≤ 1. We thus obtained

the second part of (iii).

Case 2: 3 < a. We shall restrict attention to the case b < b+(a) (so that the larger root

of the equation u(t) = 0 is greater than one), as otherwise we get the case t+a,b ≤ 1 already

considered (where the function ha,b is concave up on (0,∞)). By Part (v) of Lemma 1 we

have t−a,b > 1 if and only if b < b−(a) with b−(a) given by (18). Consequently, with these

values of b the equation (45) holds whenever 1 < 2ex − 1 < t−a,b(a) or t+a,b(a) < 2ex − 1 < ∞.

Equivalently, the function ha,b is concave up on the intervals (0, x−a,b) and (x+

a,b,∞), and is

concave down on the interval (x−a,b, x

+a,b), where x±

a,b are given by (19). This is Part (iv) of our

result. In turn, when b−(a) ≤ b < b+(a) then t−a,b ≤ 1 and we similarly conclude that now

the function ha,b is concave up on the interval (x+a,b,∞) and concave down on the interval

(0, x+a,b), which the first part of (iii). This concludes the proof.

References

[1] C. Armero and M.J. Bayarri, Prior assessments for prediction in queues, The Statistician,

43, 139-153 (1994).

[2] H.A. David, Order Statistics, John Wiley & Sons, New York, 1981.

[3] N. Eugene, C. Lee and F. Famoye, Beta-normal distribution and its applications, Comm.

Statist. Theory Methods, 31, 497–512 (2002).

[4] R.D. Gupta and D. Kundu, Generalized exponential distributions, Austral. N. Zealand

J. Statist., 41(2), 173-188 (1999).

[5] A.K. Gupta and S. Nadarajah, On the moments of the beta normal distribution, Comm.

Statist. Theory Methods, 33, 1–13 (2004).

[6] N.L. Johnson, S. Kotz and B. Balakrishnan, Continuous Univariate Distributions, Vol

2, Second Edition, Wiley, New York, 1995.

[7] S. Kotz, T.J. Kozubowski and K. Podgorski, The Laplace Distribution and Generaliza-

tions: A Revisit with Applications to Communications, Economics, Engineering,

and Finance, Birkhauser, Boston, 2001.


On the Stability and Asymptotic Behavior of GeneralizedQuadratic Mappings ∗

Hark-Mahn Kim

Department of MathematicsChungnam National University

220 Yuseong-Gu, Daejeon, 305-764, [email protected]

ABSTRACT

Let E1 and E2 be linear spaces. In this paper we extend a classical quadratic functional

equation to more general equations of two types. In addition we solve the generalized

Hyers-Ulam-Rassias stability problem for the functional equations, and thus obtain asymp-

totic properties of quadratic mappings as an application.

Keywords: stability, functional equation, generalized quadratic mappings.

2000 Mathematics Subject Classification: 39B82, 39B72.

1 Introduction

In 1960 and in 1964 S.M. Ulam [24] proposed the general Ulam stability problem: “When is it

true that by slightly changing the hypotheses of a theorem one can still assert that the thesis

of the theorem remains true or approximately true?” The concept of stability for a functional

equation arises when we replace the functional equation by an inequality which acts as a

perturbation of the equation. Thus one can ask the following question for general functional

equations: If we replace a given functional equation by a functional inequality, when can one

assert that the solutions of the inequality must be close to the solutions of the given equation?

If the answer is affirmative, we would say that a given functional equation is stable. In 1978

P.M. Gruber [8] remarked that Ulam’s problem is of particular interest in probability theory and

in the case of functional equations of different types. We wish to note that stability properties

of different functional equations can have applications to unrelated fields. For instance, Zhou

[25] used a stability property of the functional equation f(x− y) + f(x + y) = 2f(x) to prove a

conjecture of Z. Ditzian about the relationship between the smoothness of a mapping and the

degree of its approximation by the associated Bernstein polynomials.

The Ulam’s problem for ε-additive mappings f : E1 → E2 between Banach spaces i.e., ‖f(x +y) − f(x) − f(y)‖ ≤ ε for all x, y ∈ E1, was solved by D.H. Hyers [9] and then generalized by

D.G. Bourgin [4], Th.M. Rassias [17] and P. Gavruta [7] who permitted the Cauchy difference

to become unbounded.

Now, a square norm on an inner product space satisfies the important parallelogram equality

‖x + y‖2 + ‖x − y‖2 = 2(‖x‖2 + ‖y‖2) for all vectors x, y. If 4ABC is a triangle in a finite


dimensional Euclidean space and I is the center of the side BC, then the following identity

‖−−→AB‖2 + ‖−→AC‖2 = 2(‖−→AI‖2 + ‖−→CI‖2) holds for all vectors A,B and C. The following functional

equation which was motivated by these equations

Q(x + y) + Q(x− y) = 2Q(x) + 2Q(y) (1.1)

is called a quadratic functional equation, and every solution of the equation (1.1) is said to be

a quadratic mapping. The quadratic functional equation and several other functional equations

are useful to characterize inner product spaces [1, 2, 18, 22]. A Hyers-Ulam stability theorem

for the quadratic functional equation was proved by a lot of authors [5, 19]. C. Borelli and G.L.

Forti [3] generalized the stability result as follows: Let G be an abelian group, and E a Banach

space. Assume that a mapping f : G → E satisfies the functional inequality

‖f(x + y) + f(x− y)− 2f(x)− 2f(y)‖ ≤ ϕ(x, y)

for all x, y ∈ G, and ϕ : G×G → [0,∞) is a function such that

Φ(x, y) :=∞∑

i=0

14i+1

ϕ(2ix, 2iy) < ∞

for all x, y ∈ G. Then there exists a unique quadratic mapping Q : G → E satisfying

‖f(x)− f(0)3

−Q(x)‖ ≤ Φ(x, x)

for all x ∈ G. In 1983 F. Skof [23] was the first author to solve the Ulam problem for additive

mappings on a restricted domain. In 1998 S. Jung [12] and in 2004 J.M. Rassias [16] inves-

tigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains.

The stability problems of several functional equations have been extensively investigated by a

number of authors and there are many interesting results concerning this problem [10, 11, 14,

15, 20, 21].

Now we are going to extend the equation (1.1) to more generalized equations with (d + 1)-variables. For this purpose, we employ the operator

⊎x2

f(x1), which is defined in [13] as

follows⊎x2

f(x1) = f(x1 + x2) + f(x1 − x2)

for a given mapping f : E1 → E2 between vector spaces. Similarly, we define⊎2

x2,x3f(x1) =

⊎x3

(⊎x2

f(x1))

and inductively

d⊎x2,··· ,xd+1

f(x1) =⊎

xd+1

( d−1⊎x2,··· ,xd

f(x1))

for all natural number d. Then it is easy to see that the operation⊎d

x2,··· ,xd+1f(x1) can be

expressed in the form

d⊎x2,··· ,xd+1

f(x1) =d∑

k=0

∑

2≤i1<i2<···<ik≤d+1

f(−xi1 − xi2 · · · − xik +∑

j 6=i1,i2,··· ,ikxj)

.

KIM320

Thus it follows from definition that

2⊎x2,x3

f(x1) =2⊎

x3,x2

f(x1),d⊎

x2,··· ,xk+1,0, · · · , 0︸︷︷︸d−k

f(x1) = 2d−kk⊎

x2,··· ,xk+1

f(x1), (1.2)

andd−1⊎

x2,··· ,xd

f(x1 + xd+1) +d−1⊎

x2,··· ,xd

f(x1 − xd+1) =d⊎

x2,··· ,xd+1

f(x1).

In [13], the author introduced and determined the general solution of the equation

d⊎x2,··· ,xd+1

f(x1) + 2d(d− 1)d+1∑

i=1

f(xi) = 2d−1∑

1≤i<j≤d+1

⊎

xj

f(xi)

,

and then investigated the generalized Hyers-Ulam-Rassias stability problem for the equations.

Now, we consider the following new functional equation,

d⊎x2,··· ,xd+1

f(x1) +∑

1≤i<j≤d+1

⊎

xj

f(xi)

= (2d + 2d)

d+1∑

i=1

f(xi), d ≥ 1, (1.3)

d⊎x2,··· ,xd+1

f(x1) =∑

1≤i<j≤d+1

⊎

xj

f(xi)

+ (2d − 2d)

d+1∑

i=1

f(xi), d ≥ 2 (1.4)

for all (d + 1)-variables x1, · · · , xd+1 ∈ E1, where d is a natural number. As a special case, the

equation (1.3) reduces to the equation (1.1) in the case d = 1. In this paper, it will be verified that

the general solutions of the above functional equations (1.3) and (1.4) are quadratic mappings

in the class of functions between vector spaces. Besides we establish new theorems about

the Ulam stability for the general equations and apply our results on restricted domains to the

asymptotic behavior of functional equations.

2 Generalized quadratic mappings

Let E1 and E2 be vector spaces. First, we present the general solutions of the functional

equations (1.3) and (1.4).

Lemma 2.1. If a mapping f : E1 → E2 satisfies the functional equation (1.3) or (1.4), then the

mapping f is quadratic.

Proof. Let f be a solution of the functional equation (1.3). Set xi := 0 in (1.3) for all i =1, · · · , d + 1 to get f(0) = 0. Putting xi := 0 in (1.3) for all i = 3, · · · , d + 1, by (1.2) we get

(2d−1 +1)[f(x1 +x2)+ f(x1−x2)] = (2d + 2)[f(x1) + f(x2)] for all x1, x2 ∈ E1. So the mapping

f is quadratic.

Now if a mapping f is a solution of the functional equation (1.4), then we get f(0) = 0 by setting

xi := 0 in (1.4) for all i = 1, · · · , d + 1. Putting xi := 0 in (1.4) for all i = 3, · · · , d + 1, by (1.2)

we get (2d−1 − 1)[f(x1 + x2) + f(x1 − x2)] = (2d − 2)[f(x1) + f(x2)] for all x1, x2 ∈ E1. So the

mapping f is quadratic. ¤

We now investigate the generalized Hyers-Ulam-Rassias stability problem for the equation

(1.3). Thus we give conditions in order for a true mapping near an approximate mapping of the

GENERALIZED QUADRATIC MAPPINGS 321

equation (1.3) to exist. From now on, let X be a normed linear space and Y a Banach space

unless we give any specific reference. Let R+ denote the set of all nonnegative real numbers.

Now before taking up the main subject, given a mapping f : X → Y , we define the difference

operator Df : Xd+1 → Y by

Df(x1, x2, · · · , xd+1) :=d⊎

x2,··· ,xd+1

f(x1) +∑

1≤i<j≤d+1

⊎

xj

f(xi)

−(2d + 2d)d+1∑

i=1

f(xi), d ≥ 1

for all (d + 1)-variables x1, · · · , xd+1 ∈ X, which acts as a perturbation of the equation (1.3).

Theorem 2.2. Suppose that a mapping f : X → Y satisfies

‖Df(x1, x2, · · · , xd+1)‖ ≤ ε(x1, · · · , xd+1) (2.1)

for all (d + 1)-variables x1, · · · , xd+1 ∈ X, and that ε : Xd+1 → R+ is a mapping such that the

series∞∑

i=0

ε(2ix1, · · · , 2ixd+1)22i

(2.2)

converges for all x1, · · · , xd+1 ∈ X. Then there exists a unique quadratic mapping Q : X → Y

which satisfies the equation (1.3) and the inequality∥∥∥∥f(x) +

2d−1(2d− 3) + (d2 + d− 3)3(2d−1 + 1)

f(0)−Q(x)∥∥∥∥ (2.3)

≤ 14(2d−1 + 1)

∞∑

i=0

φ(2ix, 2ix)4i

for all x ∈ X, where the mapping φ : X2 → Y is given by

φ(x, y) := min

ε(x, 0, · · · , 0,

i︷︸︸︷y , 0, · · · , 0

) ∣∣∣ 2 ≤ i ≤ d + 1

. (2.4)

The mapping Q is defined by

Q(x) = limn→∞

f(2nx)4n

for all x ∈ X. Moreover, if f is measurable or f(tx) is continuous in t ∈ R for each fixed x ∈ X,

then the mapping Q is homogeneous of degree 2 over R.

Proof. If we take(x, 0, · · · , 0,

i︷︸︸︷y , 0, · · · , 0

)instead of (x1, · · · , xd+1) in (2.1), we obtain by

virtue of (1.2)∥∥∥(2d−1 + 1)[f(x + y) + f(x− y)]− 2(2d−1 + 1)[f(x) + f(y)]

−(d− 1)(2d + d + 2)f(0)∥∥∥

≤ ε(x, 0, · · · , 0,

i︷︸︸︷y , 0, · · · , 0

)

KIM322

for all x, y ∈ X, and all i with 2 ≤ i ≤ d + 1, which can be written in the form

‖q(x + y) + q(x− y)− 2[q(x) + q(y)]− q(0)‖ ≤ 12d−1 + 1

φ(x, y) (2.5)

for all x, y ∈ X, where a mapping q : X → Y is defined by q(x) := f(x)+ 2d−1(2d−3)+(d2+d−3)3(2d−1+1)

f(0)and a mapping φ : X2 → Y is given by (2.4). Taking y := x in (2.5), we get

∥∥∥∥q(2x)

4− q(x)

∥∥∥∥ ≤1

4(2d−1 + 1)φ(x, x) (2.6)

for all x ∈ X. Hence

∥∥∥∥q(2nx)

4n− q(2mx)

4m

∥∥∥∥ ≤1

4(2d−1 + 1)

n−1∑

i=m

φ(2ix, 2ix)4i

(2.7)

for all nonnegative integers m and n with n > m and all x ∈ X. It follows from (2.2) and (2.7)

that the sequence

q(2nx)4n

is a Cauchy sequence for all x ∈ X. So one can define a mapping

Q : X → Y by

Q(x) = limn→∞

q(2nx)4n

= limn→∞

f(2nx)4n

for all x ∈ X.

Now by (2.1) and (2.2), we have

‖DQ(x1, x2, · · · , xd+1)‖ = limn→∞

14n‖Df(2nx1, · · · , 2nxd+1)‖

≤ limn→∞

14n

ε(2nx1, · · · , 2nxd+1) = 0

for all (d + 1)-variables x1, · · · , xd+1 ∈ X. Hence by Lemma 2.1, the mapping Q is a quadratic

mapping satisfying the equation (1.3). Moreover, letting m = 0 and passing the limit n →∞ in

(2.7), we get the desired inequality (2.3).

To prove the afore-mentioned uniqueness, let ε1 : X3 → R+ be a mapping such that a func-

tional inequality

‖Df(x1, x2, · · · , xd+1)‖ ≤ ε1(x1, · · · , xd+1)

holds for all x1, · · · , xd+1 ∈ X and the series

∞∑

i=0

ε1(2ix1, · · · , 2ixd+1)4i

converges for all x1, · · · , xd+1 ∈ X, and assume that there exists a quadratic mapping Q1 :X → Y which satisfies the equation (1.3) and the inequality

∥∥∥∥f(x) +2d−1(2d− 3) + (d2 + d− 3)

3(2d−1 + 1)f(0)−Q1(x)

∥∥∥∥ (2.8)

≤ 14(2d−1 + 1)

∞∑

i=0

φ1(2ix, 2ix)4i


for all x ∈ X, where a mapping φ1 : X2 → Y is given by

φ1(x, y) := min

ε1

(x, 0, · · · , 0,

i︷︸︸︷y , 0 · · · , 0

) ∣∣∣ 2 ≤ i ≤ d + 1

.

Since Q and Q1 are quadratic, we see the identities Q(x) = 2−2nQ(2nx), Q1(x) = 2−2nQ1(2nx)hold for all x ∈ X and all n ∈ N. Thus it follows from inequalities (2.3) and (2.8) that

‖Q(x)−Q1(x)‖ =1

22n‖Q(2nx)−Q1(2nx)‖

≤ 122n

(∥∥∥∥Q(2nx)− f(2nx)− 2d−1(2d− 3) + (d2 + d− 3)3(2d−1 + 1)

f(0)∥∥∥∥

+∥∥∥∥f(2nx) +

2d−1(2d− 3) + (d2 + d− 3)3(2d−1 + 1)

f(0)−Q1(2nx)∥∥∥∥

)

≤ 14(2d−1 + 1)

∞∑

i=n

φ(2ix, 2ix)4i

+1

4(2d−1 + 1)

∞∑

i=n

φ1(2ix, 2ix)4i

for all x ∈ X and all n ∈ N. Therefore letting n → ∞, one has Q(x) −Q1(x) = 0 for all x ∈ X,

which completes the proof of uniqueness.

The last assertion of homogeneous of degree two of Q in the theorem follows by the same

reasoning as the proof of [6]. The proof is complete. ¤


‖Df(x1, x2, · · · , xd+1)‖ ≤ ε(x1, · · · , xd+1)


series

∞∑

i=1

4iε(x1

2i, · · · ,

xd+1

2i

)


which satisfies the equation (1.3) and the inequality

‖f(x)−Q(x)‖ ≤ 14(2d−1 + 1)

∞∑

i=1

4iφ( x

2i,

x

2i

)

for all x ∈ X, where the mapping φ : X2 → Y is given by (2.4). The mapping Q is defined by

Q(x) = limn→∞ 4nf

( x

2n

)



Note that one has f(0) = 0 in the above theorem because ε(0, · · · , 0) = 0 by the convergence

of the series.

KIM324

Corollary 2.4. Suppose that there exist a real number ε ≥ 0 and a positive real p 6= 2 such that

a mapping f : X → Y satisfies

‖Df(x1, x2, · · · , xd+1)‖ ≤ ε(‖x1‖p + · · ·+ ‖xd+1‖p)

for all (d + 1)-variables x1, · · · , xd+1 ∈ X. Then there exists a unique quadratic mapping Q :X → Y which satisfies the equation (1.3) and the inequality

‖f(x)−Q(x)‖ ≤ 2ε‖x‖p

(2d−1 + 1)|4− 2p|for all x ∈ X. The mapping Q is defined by

Q(x) =

limn→∞f(2nx)

4n , if 0 2,

for all x ∈ X.

Corollary 2.5. Suppose that there exists a nonnegative real number ε for which a mapping

f : X → Y satisfies

‖Df(x1, x2, · · · , xd+1)‖ ≤ ε

for all (d + 1)-variables x1, · · · , xd+1 ∈ X. Then there exists a unique quadratic mapping Q :X → Y which satisfies the equation (1.3) and the inequality

∥∥∥∥f(x) +2d−1(2d− 3) + (d2 + d− 3)

3(2d−1 + 1)f(0)−Q(x)

∥∥∥∥ ≤ε

3(2d−1 + 1)

for all x ∈ X.

For a given mapping f : X → Y , we define the difference operator Ef : Xd+1 → Y by

Ef(x1, x2, · · · , xd+1) :=d⊎

x2,··· ,xd+1

f(x1)−∑

1≤i<j≤d+1

⊎

xj

f(xi)

−(2d − 2d)d+1∑

i=1

f(xi), d ≥ 2

for all (d + 1)-variables x1, · · · , xd+1 ∈ X.

We are going to investigate the generalized Hyers-Ulam-Rassias stability problem for the equa-

tion (1.4). The proof of the following theorems goes through by the same way as that of Theo-

rem 2.2.


‖Ef(x1, x2, · · · , xd+1)‖ ≤ ε(x1, · · · , xd+1) (2.9)


series∞∑

i=0

ε(2ix1, · · · , 2ixd+1)22i

(2.10)


which satisfies the equation (1.4) and the inequality∥∥∥∥f(x) +

2d−1(2d− 3)− (d2 + d− 3)3(2d−1 − 1)

f(0)−Q(x)∥∥∥∥ (2.11)

≤ 14(2d−1 − 1)

∞∑

i=0

φ(2ix, 2ix)4i


Q(x) = limn→∞

f(2nx)4n




‖Ef(x1, x2, · · · , xd+1)‖ ≤ ε(x1, · · · , xd+1)


series

∞∑

i=1

4iε(x1

2i, · · · ,

xd+1

2i

)



‖f(x)−Q(x)‖ ≤ 14(2d−1 − 1)

∞∑

i=1

4iφ( x

2i,

x

2i

)


Q(x) = limn→∞ 4nf

( x

2n

)



3 Approximately quadratic mappings on restricted domains

In this section we are going to investigate the Hyers-Ulam stability problem for the equations

(1.3) and (1.4) on an unbounded restricted domain. As results we have corollaries concerning

an asymptotic property of the equation (1.3).

Theorem 3.1. Let r > 0 be fixed. Suppose that there exists a nonnegative real number ε for

which a mapping f : X → Y satisfies

‖Df(x1, x2, · · · , xd+1)‖ ≤ ε (3.1)

KIM326

for all (d + 1)-variables x1, · · · , xd+1 ∈ X with∑d+1

i=1 ‖xi‖ ≥ r. Then there exists a unique

quadratic mapping Q : X → Y which satisfies the equation (1.3) and the inequality∥∥∥∥f(x) +

(d− 1)(2d + d + 2)f(0)2d + 2

−Q(x)∥∥∥∥ ≤

3ε

2(2d−1 + 1)(3.2)

for all x ∈ X.

Proof. Taking (x1, · · · , xd+1) as (x, y, 0, · · · , 0) in (3.1) with ‖x‖+ ‖y‖ ≥ r, we obtain by (2.5)

‖q(x + y) + q(x− y)− 2[q(x) + q(y)]‖ ≤ ε

2d−1 + 1(3.3)

where a mapping q : X → Y is defined by q(x) := f(x) + (d−1)(2d+d+2)f(0)2d+2

for all x, y ∈ X with

‖x‖+ ‖y‖ ≥ r. Specially, we have ‖q(0)‖ ≤ ε2(2d−1+1)

by setting y := 0 and x := t with ‖t‖ ≥ r

in (3.3). Now, assume ‖x‖+ ‖y‖ < r. And choose a t ∈ X with ‖t‖ ≥ 2r. Then it holds clearly

‖x± t‖ ≥ r, ‖y ± t‖ ≥ r, and ‖x± t‖+ ‖y ± t‖ ≥ r.

Therefore from (3.3) and the following functional identity

2[q(x + y) + q(x− y)− 2q(x)− 2q(y)− q(0)

]

=[q(x + y + 2t) + q(x− y)− 2q(x + t)− 2q(y + t)

]

+[q(x + y − 2t) + q(x− y)− 2q(x− t)− 2q(y − t)

]

+[− q(x + y + 2t)− q(x + y − 2t) + 2q(x + y) + 2q(2t)

]

+[2q(x + t) + 2q(x− t)− 4q(x)− 4q(t)

]

+[2q(y + t) + 2q(y − t)− 4q(y)− 4q(t)

]

+[− 2q(2t)− 2q(0) + 4q(t) + 4q(t)

],

we get

‖q(x + y) + q(x− y)− 2q(x)− 2q(y)− q(0)‖ ≤ 9ε

2(2d−1 + 1)(3.4)

for all x, y ∈ X with ‖x‖ + ‖y‖ < r. Consequently, the last functional inequality holds for all

x, y ∈ X in view of (3.3) and (3.4) because of ‖q(0)‖ ≤ ε2(2d−1+1)

. Now letting y := x in (3.4),

we obtain

‖q(2x)− 4q(x)‖ ≤ 9ε

2(2d−1 + 1).

Now applying a standard procedure of direct method [9] to the last inequality, we see that

there exists a unique quadratic mapping Q : X → Y which satisfies the equation (1.3) and the

inequality

‖q(x)−Q(x)‖ ≤ 3ε

2(2d−1 + 1)

for all x ∈ X. ¤

The proof of the following theorem is verified by the same way as that of Theorem 3.1.

Theorem 3.2. Let r > 0 and ε ≥ 0 be fixed. Suppose that there exists a nonnegative real

number ε for which a mapping f : X → Y satisfies

‖Ef(x1, x2, · · · , xd+1)‖ ≤ ε (3.5)

for all (d + 1)-variables x1, · · · , xd+1 ∈ X with∑d+1

i=1 ‖xi‖ ≥ r. Then there exists a unique

quadratic mapping Q : X → Y which satisfies the equation (1.4) and the inequality∥∥∥∥f(x) +

(d− 1)(2d − d− 2)f(0)2d − 2

−Q(x)∥∥∥∥ ≤

3ε

2(2d−1 − 1)(3.6)

for all x ∈ X.

We note that if we define Sd+1 = (x1, · · · , xd+1) ∈ Xd+1 : ‖xi‖ < r,∀i = 1, · · · , d+1 for some

fixed r > 0, then we have

(x1, · · · , xd+1) ∈ Xd+1 :d+1∑

i=1

‖xi‖ ≥ (d + 1)r

⊂ Xd+1 \ Sd+1.

Thus the following corollaries are immediate consequences of Theorem 3.1 and Theorem 3.2.

Corollary 3.3. If a mapping f : X → Y satisfies the functional inequality (3.1) for all vectors

(x1, · · · , xd+1) ∈ Xd+1 \ Sd+1, then there exists a unique quadratic mapping Q : X → Y which

satisfies the equation (1.3) and the inequality (3.2)

Corollary 3.4. If a mapping f : X → Y satisfies the functional inequality (3.5) for all vectors

(x1, · · · , xd+1) ∈ Xd+1 \ Sd+1, then there exists a unique quadratic mapping Q : X → Y which

satisfies the equation (1.4) and the inequality (3.6)

From Theorem 3.1 and Theorem 3.2, we have the following corollaries concerning asymptotic

properties of quadratic mappings.

Corollary 3.5. A mapping f : X → Y with f(0) = 0 is quadratic if and only if either

‖Df(x1, · · · , xd+1)‖ → 0

or

‖Ef(x1, · · · , xd+1)‖ → 0

as∑d+1

i=1 ‖xi‖ → ∞.

Proof. According to our asymptotic condition, there is a sequence (εm) decreasing to zero such

that ‖Df(x1, · · · , xd+1)‖ ≤ εm for all (d + 1)-variables x1, · · · , xd+1 ∈ X with∑d+1

i=1 ‖xi‖ ≥ m.

Hence, it follows from Theorem 3.1 that there exists a unique quadratic mapping Qm : X → Y


‖f(x)−Qm(x)‖ ≤ 3εm

2(2d−1 + 1)

for all x ∈ X. Let m and l be positive integers with m > l. Then, we obtain

‖f(x)−Qm(x)‖ ≤ 3εm

2(2d−1 + 1)≤ 3εl

2(2d−1 + 1)

KIM328

for all x ∈ X. The uniqueness of Ql implies that Qm = Ql for all m > l, and so

‖f(x)−Ql(x)‖ ≤ 3εm

2(2d−1 + 1)

for all x ∈ X. By letting m →∞, we conclude that f is itself quadratic.

The reverse assertion is trivial. ¤

Corollary 3.6. A mapping f : X → Y with f(0) = 0 is quadratic if and only if there exists a

positive real r > 0 such that either

supx1,··· ,xd+1

‖Df(x1, x2, · · · , xd+1)‖ :

d+1∑

i=1

‖xi‖ ≥ r

or

supx1,··· ,xd+1

‖Ef(x1, x2, · · · , xd+1)‖ :

d+1∑

i=1

‖xi‖ ≥ r

is bounded for all d ≥ 1.

Proof. Let supx1,··· ,xd+1‖Df(x1, x2, · · · , xd+1)‖ ≤ M < ∞ for all d ≥ 1. Then for each d ≥ 1,

there exists a unique quadratic mapping Qd : X → Y which satisfies the equation (1.3) and

the inequality

‖f(x)−Qd(x)‖ ≤ 3M

2(2d−1 + 1)

for all x ∈ X by Theorem 3.1. Let m be a positive integer with m > d. Then, we obtain

‖f(x)−Qm(x)‖ ≤ 3M

2(2m−1 + 1)≤ 3M

2(2d−1 + 1)

for all x ∈ X. The uniqueness of Qd implies that Qm = Qd for all m with m > d, and so

‖f(x)−Qd(x)‖ ≤ M

3(2m−1 + 1)

for all x ∈ X. By letting m →∞, we conclude that f is itself quadratic.

The reverse assertion is trivial. ¤

Acknowledgment

∗ This work was supported by the Brain Korea 21 Project in 2006.

References

[1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ.

Press, 1989.

[2] Dan Amir, Characterizations of inner product spaces, Birkhauser-Verlag, Basel, 1986.

[3] C. Borelli and G.L. Forti, On a general Hyers-Ulam stability result, Internat. J. Math. Math.

Sci. 18(1995), 229-236.

[4] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer.

Math. Soc. 57(1951), 223-237.

[5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem.

Univ. Hamburg, 62(1992), 59-64.

[6] S. Czerwik, The stability of the quadratic functional equation, in ‘Stability of Mappings of

Hyers-Ulam Type’ (edited by Th. M. Rassias and J. Tabor), Hadronic Press, Florida, 1994,

pp 81-91.

[7] P. Gavruta, A generalization of the Hyers-Ulam-Rassias Stability of approximately additive

mappings, J. Math. Anal. Appl. 184(1994), 431-436.

[8] P.M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245(1978), 263-277.

[9] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci.

27(1941), 222-224.

[10] K. Jun and H. Kim, On the Hyers-Ulam stability of a difference equation, J. Comput. Anal.

Appl. 7(4)(2005), 397-407.

[11] K. Jun, H. Kim and I. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type cubic

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[12] S. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic

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Funct. Anal. 46(1982), 126-130.

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Math. Soc. 72(1978), 297-300.

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Math. 62(2000), 23-130.

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ON STATISTICAL FUZZY TRIGONOMETRICKOROVKIN THEORY

OKTAY DUMAN AND GEORGE A. ANASTASSIOU

Abstract. In this study, we use regular matrix transformations in the ap-proximation by fuzzy positive linear operators, where the test functions aretrigonometric. So we prove a trigonometric fuzzy Korovkin theorem by meansof A-statistical convergence, where A is a non-negative regular summabilitymatrix. We also study rates of A-statistical convergence of a sequence of fuzzypositive linear operators in the trigonometric environment.

1. Introduction

The study of the Korovkin type approximation theory is an area of active re-search, which deals with the problem of approximating a function by means of asequence of positive linear operators. In recent years, this theory has been improvedwith the help of two di¤erent ways. The rst one, statistical approximation, is touse the notion of statistical convergence in the approximation by operators. Earlierstudies show that the statistical approximation enables us to obtain more powerfulresults than the classical aspects (see, for instance, [1, 2, 3, 4, 5]). The second oneis to obtain fuzzy approximation by fuzzy positive linear operators via the conceptof fuzzy set theory (see, [6, 7, 8, 9, 10]). Our primary interest of the present paperis to combine these ways: statistical approximation and fuzzy approximation. So,we obtain a statistical fuzzy Korovkin-type approximation theorem and computeits statistical rates in the approximation. Here the test functions are trigonometric.We should note that, as a rule, neither limits nor statistical limits can be calculatedor measured with absolute precision. To reect this imprecision several approachesin mathematics have been developed: fuzzy set theory, fuzzy logic, interval analysis,set valued analysis, etc.We rst collect some basic denitions and results used in the paper.A fuzzy number is a function : R ! [0; 1]; which is normal, convex, upper

semi-continuous and the closure of the set supp() is compact, where supp() :=fx 2 R : (x) > 0g: The set of all fuzzy numbers are denoted by RF . Let

[]0 := fx 2 R : (x) > 0g and []r := fx 2 R : (x) rg; (0 < r 1):Then, it is well-known [11] that, for each r 2 [0; 1]; the set []r is a closed andbounded interval of R. For any u; v 2 RF and 2 R, it is possible to deneuniquely the sum u v and the product u v as follows:

[u v]r = [u]r + [v]r and [ v]r = [u]r; (0 r 1):

Key words and phrases. Statistical convergence, statistical rates, fuzzy positive linear opera-tors, trigonometric functions, trigonometric fuzzy Korovkin theory, fuzzy modulus of continuity.

2000 Mathematics Subject Classication. 26E50, 41A25, 41A36.

1

2 OKTAY DUMAN AND GEORGE A. ANASTASSIOU

Now denote the interval [u]r by [u(r)_ ; u(r)+ ]; where u

(r)_ u(r)+ and u(r)_ ; u

(r)+ 2 R for

r 2 [0; 1]: Then, for u; v 2 RF ; dene

u v , u(r)_ v(r)_ and u(r)+ v(r)+ for all 0 r 1:Dene also the following metric D : RF RF ! R+ by

D(u; v) = supr2[0;1]

maxnu(r)_ v(r)_

; u(r)+ v(r)+o :

In this case, (RF ; D) is a complete metric space (see [12]). Let f; g : R ! RF befuzzy number valued functions. Then, the distance between f and g on R is givenby

D(f; g) = supx2R

supr2[0;1]

maxnf (r)_ g(r)_

; f (r)+ g(r)+o :

In this article we consider that j ! 1: Let K be a subset of N. Then, the(asymptotic) density of K is dened by

(K) := limj

# fn j : n 2 Kgj

provided the limit exists, where the symbol # fBg denotes the cardinality of a setB: Using this density Fast [13] introduced the notion of statistical convergence ofnumber sequences as follows: (xn)n2N is statistically convergent to a number L if,for every " > 0; the set fn 2 N : jxn Lj "g has density zero, i.e.,

(fn 2 N : jxn Lj "g) = limj

# fn j : jxn Lj "gj

= 0:

Now let A = (ajn) be an innite summability matrix. Then, the A-transform ofx, denoted Ax := ((Ax)j); is given by (Ax)j =

P1n=1 ajnxn; provided the series

converges for each j:We say that A is regular if limj(Ax)j = L whenever limj xj =L [14]. Assume now that A is a nonnegative regular summability matrix and K isa subset of N. The A-density of K is dened by

A(K) := limj

Xn2K

ajn

provided the limit exists. Observe that if we take A = C1 = (cjn); the Cesáromatrix of order one, dened by

cjn =

8<:1

j; if 1 n j0; otherwise

then C1(K) = (K) for any subset K of N. With the help of the A-density,Freedman and Sember [16] introduced the notion of A-statistical convergence,which is a more general method of statistical convergence. Recall that the se-quence (xn)n2N is said to be A-statistically convergent to L if, for every " > 0;Afn 2 N : jxn Lj "g = 0; or equivalently

limj

Xn: jxnLj"

ajn = 0:

This limit is denoted by stA limn xn = L: It is not hard to see that if we takeA = C1; then C1-statistical convergence coincides with the statistical convergencementioned above. If A is replaced by the identity matrix, then we get the ordinary

334

STATISTICAL TRIGONOMETRIC FUZZY APPROXIMATION 3

convergence of number sequences. We also note that if A = (ajn) is any nonnega-tive regular summability matrix for which limj maxnfajng = 0; then Astatisticalconvergence is stronger than convergence (see [17]). Actually, every convergentsequence is A-statistically convergent to the same value for any non-negative reg-ular matrix A, but its converse is not always true. Some other results regardingstatistical and A-statistical convergences may be found in the papers [18, 19].Now let (n)n2N be a fuzzy number valued sequence. Then, Nuray and Savas [20]

introduced the fuzzy analog of statistical convergence by using the fuzzy metric D

instead of the classical absolute value in the above denition. So, by a similar idea,one can obtain the following denition of A-statistical convergence of fuzzy valuedsequences. We say that (n)n2N is A-statistically convergent to 2 RF ; which is de-noted by stA limnD(n; ) = 0; if for every " > 0; A (fn 2 N : D(n; ) "g) =0; i.e.,

limj

Xn:D(n;)"

ajn = 0

holds. Of course, the case of A = C1 immediately reduces to the statistical conver-gence of fuzzy valued sequences. Also, replacing A with the identity matrix, we getthe classical fuzzy convergence introduced by Matloka [21].

2. Statistical Fuzzy Trigonometric Korovkin Theory

In this section we prove a fuzzy trigonometric Korovkin-type approximationtheorem by means of A-statistical convergence. In order to show that our resultis stronger than its classical case we display an example of fuzzy positive linearoperators by using fuzzy Fejer operators.Let f : R ! RF be fuzzy number valued functions. Then f is said to be fuzzy

continuous at x0 2 R provided that whenever xn ! x0, then D (f(xn); f(x0))!1as n ! 1: Also, we say that f is fuzzy continuous on R if it is fuzzy continuousat every point x 2 R: The set of all fuzzy continuous functions on R is denoted byCF (R) (see, for instance, [7, 9]). Notice that CF (R) is only a cone not a vector space.By C(F)2 (R) we mean the space of all fuzzy continuous and 2-periodic functions onR. Also the space of all real valued continuous and 2-periodic functions is denotedby C2(R):Assume that f : [a; b]! RF is a fuzzy number valued function. Then, f is said

to be fuzzy-Riemann integrable (or, FR-integrable) to I 2 RF if, for given " > 0;there exists a > 0 such that, for any partition P = f[u; v]; g of [a; b] with thenorms (P ) < ; we have

D

MP

(v u) f(); I!< ":

In this case, we write

I := (FR)

bZa

f(x)dx:

By Corollary 13.2 of [10, p. 644], we conclude that if f 2 CF [a; b] (fuzzy continuouson [a; b]), then f is FR-integrable on [a; b]:

335

Now let L : CF (R) ! CF (R) be an operator. Then L is said to be fuzzy linearif, for every 1; 2 2 R, f1; f2 2 CF (R); and x 2 R;

L (1 f1 2 f2;x) = 1 L(f1;x) 2 L(f2;x)

holds. Also L is called fuzzy positive linear operator if it is fuzzy linear and, thecondition L(f ;x) L(g;x) is satised for any f; g 2 CF (R) and all x 2 R withf(x) g(x):Throughout the paper we use the test functions fi (i = 0; 1; 2) dened by

f0(x) = 1; f1(x) = cosx; f2(x) = sinx:

Then, we get the following result.

Theorem 2.1. Let A = (ajn) be a non-negative regular summability matrix and letfLngn2N be a sequence of fuzzy positive linear operators dened on C(F)2 (R). As-sume that there exists a corresponding sequence f~Lngn2N of positive linear operatorsdened on C2(R) with the property

(2.1) fLn(f ;x)g(r) = ~Ln

f(r) ;x

for all x 2 [a; b]; r 2 [0; 1]; n 2 N and f 2 C(F)2 (R):Assume further that

(2.2) stA limn

~Ln(fi) fi = 0 for each i = 0; 1; 2;

the symbol kgk denotes the usual supremum norm of g 2 C2(R). Then, for allf 2 C(F)2 (R); we have

stA limnD (Ln(f); f) = 0:

Proof. Suppose that I is a closed bounded interval with length 2 of R. Now letf 2 C(F)2 (R); x 2 I and r 2 [0; 1]: Taking [f(x)](r) =

hf (r)_ (x); f

(r)+ (x)

iwe get

f(r) 2 C2(R_): Hence, for every " > 0; there exists a > 0 such that

(2.3)f (r) (y) f (r) (x)

< "for all y satisfying jy xj < : On the other hand, by the boundedness of f (r) ,

(2.4)f (r) (y) f (r) (x)

2 f (r)

holds for all y 2 R. Now consider the subinterval (x ; 2 + x ] with length2: Then, by (2.3) and (2.4), it is not hard to see that

(2.5)f (r) (y) f (r) (x)

"+ 2M (r)

'(y)

sin2 2

holds for all y 2 (x ; 2 + x ]; where '(y) := sin2yx2

and M (r)

:= f (r)

:Observe that inequality (2.5) also holds for all y 2 R because of the periodicityof f (r) (see, for instance, [15]). Now using the linearity and the positivity of the

336


operators ~Ln and considering inequality (2.5), we may write, for each n 2 N, that

~Ln f (r) ;x f (r) (x)

~Ln

f (r) (y) f (r) (x) ;x

+M(r)

~Ln (f0;x) f0(x) "+

"+M

(r)

~Ln (f0;x) f0(x)+2M

(r)

sin2 2

~Ln (';x) :Hence, we obtain that

~Ln f (r) ;x f (r) (x)

"+

"+M

(r) +

2M(r)

sin2 2

!~Ln (f0;x) f0(x)+2M

(r)

sin2 2

~Ln (f1;x) f1(x)+2M

(r)

sin2 2

~Ln (f2;x) f2(x) :Setting K(r)

(") := " +M(r) +

2M(r)

sin2 2

and taking supremum over x 2 R; we easilysee that

(2.6)

~Ln f (r)

f (r)

"+K(r) (")f

~Ln (f0) f0 + ~Ln (f1) f1 + ~Ln (f2) f2 g:

Now it follows from (2.6) that

D (Ln(f); f) = supx2R

D (Ln(f ;x); f(x))

= supx2R

supr2[0;1]

maxn~Ln f (r)_ ;x

f (r)_ (x)

; ~Ln f (r)+ ;x f (r)+ (x)

o= sup

r2[0;1]max

n ~Ln f (r)_

f (r)_

; ~Ln f (r)+

f (r)+

o :Therefore, combining the above equality with (2.6), we have

(2.7)D (Ln(f); f) "+K(")f

~Ln (f0) f0 + ~Ln (f1) f1 + ~Ln (f2) f2 g;

337

where K(") := supr2[0;1]

maxnK(r)_ (");K

(r)+ (")

o. Now, for a given "0 > 0; chose " > 0

such that 0 < " < "0; and consider the following sets

U : = fn 2 N : D (Ln(f); f) "0g ;

U0 : =

n 2 N :

~Ln (f0) f0 "0 "3K(")

;

U1 : =

n 2 N :

~Ln (f1) f1 "0 "3K(")

;

U2 : =

n 2 N :

~Ln (f2) f2 "0 "3K(")

:

Then inequality (2.4) gives

U U0 [ U1 [ U2;which guarantees that, for each j 2 N,X

n2Uajn

Xn2U0

ajn +Xn2U1

ajn +Xn2U2

ajn:

If we take limit as j !1 on the both sides of inequality (2.6) and use the hypothesis(2.2), we immediately see that

limj

Xn2U

ajn = 0;

whence the result.

Concluding Remarks.1. If we replace the matrix A in Theorem 2.1 by the Cesáro matrix C1; we

immediately get the following statistical fuzzy Korovkin result in the trigonometriccase.

Corollary 2.2. Let fLngn2N be a sequence of fuzzy positive linear operators de-ned on C(F)2 (R), and let f~Lngn2N be a corresponding sequence of positive linearoperators dened on C2(R) with the property (2:1). Assume that

st limn

~Ln(fi) fi = 0 for each i = 0; 1; 2:

Then, for all f 2 C(F)2 (R); we have

st limnD (Ln(f); f) = 0:

2. Replacing the matrix A by the identity matrix, one can obtain the classicalfuzzy Korovkin result which was introduced by Anastassiou and Gal [9].

Corollary 2.3 ([9]). Let fLngn2N be a sequence of fuzzy positive linear operatorsdened on C(F)2 (R), and let f~Lngn2N be a corresponding sequence of positive lin-ear operators dened on C2(R) with the property (2:1). Assume that the sequencef~Ln(fi)gn2N is uniformly convergent to fi on the whole real line (in the ordinarysense). Then, for all f 2 C(F)2 (R); the sequence fLn(f)gn2N is uniformly conver-gent to f on the whole real line (in the fuzzy sense).

338

3. Now the following application shows that our A-statistical fuzzy Korovkin-type approximation theorem in the trigonometric case (Theorem 2.1) is a non-trivialgeneralization of its classical case (Corollary 2.3) given by Anastassiou and Gal [9].Let A = (ajn) be any non-negative regular summability matrix. Assume that K

is any subset of N satisfying A(K) = 0: Then dene a sequence (un)n2N by:

(2.8) un =

pn; if n 2 K

0; if n 2 NnK.

In this case, observe that (un)n2N is non-convergent (in the ordinary sense). How-ever, since for every " > 0

limj

Xn:junj"

ajn = limj

Xn2K

ajn = A(K) = 0,

we have

(2.9) stA limnun = 0;

although the sequence (un)n2N is unbounded from above. Now dene the fuzzyFejer operators Fn as follows:

(2.10) Fn(f ;x) =1

n

8<:(FR)Z

f(y)sin2

n2 (y x)

2 sin2

yx2

dy9=; ;

where n 2 N, f 2 C(F)2 (R) and x 2 R. Then observe that the operators Fn arefuzzy positive linear. Also, the corresponding real Fejer operators have the followingform:

fFn(f ;x)g(r) = ~Fn

f(r) ;x

:=

1

n

Z

f(y)sin2

n2 (y x)

2 sin2

yx2

dywhere f (r) 2 C2(R) and r 2 [0; 1]: Then, we obtain that (see [15])

~Fn (f0;x) = 1;

~Fn (f1;x) =n 1n

cosx;

~Fn (f2;x) =n 1n

sinx:

Now using the sequence (un)n2N given by (2.8) we introduce the following fuzzypositive linear operators dened on the space C(F)2 (R) :

(2.11) Tn(f ;x) = (1 + un) Fn(f ;x);

where n 2 N, f 2 C(F)2 (R) and x 2 R. So, the corresponding real positive linearoperators are given by

~Tn

f(r) ;x

:=1 + unn

Z

f(y)sin2

n2 (y x)

2 sin2

yx2

dy;

339


where f (r) 2 C2(R): Then we get, for all n 2 N and x 2 R, that ~Tn (f0) f0 = un; ~Tn (f1) f1 un +1 + unn

; ~Tn (f2) f2 un +1 + unn

:

It follows from (2.9) that

(2.12) stA limn

~Tn (f0) f0 = 0:Also, by the denition of (un)n2N we have

limn

1 + unn

= 0;

which yields, for any non-negative regular matrix A = (ajn); that

(2.13) stA limn

1 + unn

= 0:

Now by (2.9) and (2.13) we easily see that, for every " > 0;

limj

Xn:k ~Tn(f1)f1k"

ajn limj

Xn:junj "

2

ajn + limj

Xn:j 1+unn j "

2

ajn = 0:

So we get

(2.14) stA limn

~Tn (f1) f1 = 0:By a similar idea, one can obtain that

(2.15) stA limn

~Tn (f2) f2 = 0:Now, with the help of (2.12), (2.14), (2.15), all hypotheses of Theorem 2.1 hold.Then, we conclude, for all f 2 C(F)2 (R), that

stA limnD (Tn(f); f) = 0:

However, since the sequence (un)n2N is non-convergent and also unbounded fromabove, the sequence fTn(f)gn2N is not fuzzy convergent to f: Hence, Corollary 2.3does not work for the operators Tn dened by (2.11).

3. Statistical Fuzzy Rates

In the classical summability settings rates of summation have been introducedin several ways (see for instance, [22, 23, 24]). The concept of statistical rates ofconvergence, for nonvanishing two null sequences, is studied in [23]. Furthermore,various ways of dening rates of convergence in the A-statistical sense have beenintroduced in [3] as the following way.Let A = (ajn) be a non-negative regular summability matrix and let (pn)n2N be

a positive non-increasing sequence of real numbers. Then

340

(a) A sequence x = (xn) is A-statistically convergent to the number L with therate of o(pn) if for every " > 0;

limj

1

pj

Xn:jxnLj"

ajn = 0:

In this case we write xn L = stA o(pn) as n!1:(b) If for every " > 0;

supj

1

pj

Xn:jxnj"

ajn <1;

then (xn)n2N is Astatistically bounded with the rate of O(pn) and it isdenoted by xn = stA O(pn) as n!1:

(c) (xn)n2N is A-statistically convergent to L with the rate of om(pn), denotedby xn L = stA om(pn) as n!1; if for every " > 0;

limj

Xn:jxnLj"pn

ajn = 0:

(d) (xn)n2N is A-statistically bounded with the rate of Om(pn) provided thatthere is a positive number M satisfying

limj

Xn:jxnjMpn

ajn = 0;

which is denoted by xn = stA Om(pn) as n!1:Notice that, in denitions (a) and (b); the rateis more controlled by the entries

of the summability method rather than the terms of the sequence (xn)n2N. But, inorder to see the e¤ect on the terms of the sequence we need the denitions (c) and(d), respectively. We should also remark that, for the convergence of fuzzy numbervalued sequences or fuzzy number valued function sequences, we have to use themetrics D and D instead of the absolute value metric in all denitions mentionedabove.Let f 2 C

(F)2 (R). Then the (rst) fuzzy modulus of continuity of f , which is

introduced by [10] (see also [7, 9]), is dened by

w(F)1 (f; ) := sup

x;y2R; jxyjD (f(x); f(y))

for any > 0: With this terminology, we have the following result.

Theorem 3.1. Let A = (ajn) be a non-negative regular summability matrix and letfLngn2N be a sequence of fuzzy positive linear operators dened on C(F)2 (R): Assumethat there exists a corresponding sequence f~Lngn2N of positive linear operators onC2(R) with the property (2:1). Suppose that (an)n2N and (bn)n2N are positive non-increasing sequences and also that the operators ~Ln satisfy the following conditions:

(i) ~Ln(f0) f0 = stA o(an) as n!1;

(ii) w(F)1 (f; n) = stA o(bn) as n ! 1; where n =

r ~Ln(') and '(y) =sin2

yx2

for each x 2 R:

341



D (Ln(f); f) = stA o(cn) as n!1,

where cn := maxfan; bng for each n 2 N. Furthermore, similar results hold whenlittle \o" is replaced by big \O".

Proof. Let f 2 C(F)2 (R): Then, using the property (2.1) and applying Theorem 4of [9], we immediately see, for each n 2 N, that

D (Ln(f); f) M ~Ln (f0) f0 + ~Ln (f0) + f0 w(F)1 (f; n);

where M := Df; f0g

and f0g denotes the neutral element for : It follows

from the above inequality that(3.1)

D (Ln(f); f) M ~Ln (f0) f0 + ~Ln (f0) f0 w(F)1 (f; n) + 2w

(F)1 (f; n)

holds for each n 2 N. Now, for a given " > 0; consider the following sets:

V : = fn 2 N : D (Ln(f); f) "g ;

V0 : =nn 2 N :

~Ln (f0) f0 "

3M

o;

V1 : =

n 2 N :

~Ln (f0) f0 r"

3

;

V2 : =

n 2 N : w(F)1 (f; n)

r"

3

;

V3 : =nn 2 N : w(F)1 (f; n)

"

6

o:

Hence, inequality (3.1) implies that V V0 [ V1 [ V2 [ V3: Then we may write, foreach j 2 N, that

(3.2)1

cj

Xn2V

ajn 1

cj

Xn2V0

ajn +1

cj

Xn2V 0

1

ajn +1

cj

Xn2V 00

1

ajn +1

cj

Xn2V2

ajn:

Also using the fact cj = maxfaj ; bjg; we obtain from (3.2) that

(3.3)1

cj

Xn2V

ajn 1

aj

Xn2V0

ajn +1

aj

Xn2V 0

1

ajn +1

bj

Xn2V 00

1

ajn +1

bj

Xn2V2

ajn:

Therefore, letting j ! 1 on the both sides of inequality (3.3) and using the hy-potheses (i) and (ii), we conclude that

limj

1

cj

Xn2V

ajn = 0;

which means that

stA limnD (Ln(f); f) = 0

for all f 2 C(F)2 (R):

The above proof can easily be modied to prove the following analog.

342


Theorem 3.2. Let A = (ajn) be a non-negative regular summability matrix andlet fLngn2N be a sequence of fuzzy positive linear operators on C(F)2 (R). Assumethat there exists a corresponding sequence f~Lngn2N of positive linear operators onC2(R) with the property (2:1). Suppose that (an)n2N and (bn)n2N are positive non-increasing sequences and also that the operators ~Ln satisfy the following conditions:

(i) ~Ln(f0) f0 = stA om(an) as n!1;

(ii) w(F)1 (f; n) = stA om(bn) as n ! 1; where n is given as in Theorem3:1.


D (Ln(f); f) = stA o(dn) as n!1,where dn := maxfan; bn; anbng for each n 2 N. Furthermore, similar results holdwhen little \om" is replaced by big \Om".

References

[1] O. Duman, Statistical approximation for periodic functions, Demonstratio Math. 36 (2003)873-878.

[2] O. Duman and E. Erkus, Approximation of continuous periodic functions via statistical con-vergence, Comput. Math. Appl. 52 (2006) 967-974.

[3] O. Duman, M.K. Khan, C. Orhan, A-Statistical convergence of approximating operators,Math. Inequal. Appl. 6 (2003) 689-699.

[4] E. Erkus and O. Duman, A-Statistical extension of the Korovkin type approximation theorem,Proc. Indian Acad. Sci. Math. Sci. 115 (2005) 499-508.

[5] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, RockyMountain J. Math. 32 (2002) 129-138.

[6] G.A. Anastassiou, Fuzzy random Korovkin theory and inequalities, Math. Inequal. Appl. 10(2007) 63-94.

[7] G.A. Anastassiou, On basic fuzzy Korovkin theory, Studia Univ. Babes-Bolyai Math. 50(2005) 3-10.

[8] G.A. Anastassiou, Fuzzy approximation by fuzzy convolution type operators, Comput. Math.Appl. 48 (2004) 1369-1386.

[9] G.A. Anastassiou and S.G. Gal, On fuzzy trigonometric Korovkin theory, Nonlinear Funct.Anal. Appl. 11 (2006) 385-395.

[10] S.G. Gal, Approximation theory in fuzzy setting, Handbook of Analytic-Computational Meth-ods in Applied Mathematics, 617666, Chapman & Hall/CRC, Boca Raton, FL, 2000.

[11] R.J. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986)3143.

[12] C.X. Wu, M. Ma, Embedding problem of fuzzy number space I, Fuzzy Sets and Systems 44(1991) 3338.

[13] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241-244.[14] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, UK, 2000.[15] P.P. Korovkin, Linear Operators and Theory of Approximation, Hindustan Publ. Corp.,

Delhi, 1960.[16] A.R. Freedman, J.J. Sember, Densities and summability, Pacic J. Math. 95 (1981) 293-305.[17] E. Kolk, Matrix summability of statistically convergent sequences, Analysis 13 (1993) 77-83.[18] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301-313.[19] H.I. Miller, A measure theoretical subsequence characterization of statistical convergence,

Trans. Amer. Math. Soc. 347 (1995) 1811-1819.[20] F. Nuray, E. Savas, Statistical convergence of sequences of fuzzy numbers, Math. Slovaca 45

(1995) 269-273.[21] M. Matloka, Sequences of fuzzy numbers, BUSEFAL 28 (1986) 28-37.

343


[22] J.A. Fridy, Minimal rates of summability, Canad. J. Math. 30 (1978) 808-816.[23] J.A. Fridy, H.I. Miller and C. Orhan, Statistical rates of convergence, Acta Sci. Math. (Szeged)

69 (2003) 147-157.[24] H.I. Miller, Rates of convergence and topics in summability theory, Akad. Nauka Umjet.

Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka 22 (1983) 39-55.

Oktay DumanTOBB Economics and Technology University,Faculty of Arts and Sciences,Department of Mathematics,Sö¼gütözü TR-06530, Ankara,TURKEYE-Mail: [email protected]

George A. AnastassiouDepartment of Mathematical SciencesThe University of MemphisMemphis, TN 38152,USAE-Mail: [email protected]

344

THE LAGRANGE - POINCARE EQUATIONS FOR AREFINEMENT OF A PRINCIPAL GP (n;R) - BUNDLE

GHEORGHE IVAN, MIHAI IVAN and DUMITRU OPRIS

ABSTRACT. In this paper the geometric structures defined on the refinement of a principalbundle determined by the factorization of the projective group are studied. For this refinementthe Poisson structure, the Lagrange - Poincare and Wong equations are written. 1

Introduction

The paper consists of three sections. In the first section we present some definitions andresults developed in [2] concerning the geometry of the manifold TQ/G. The manifoldTQ/G is isomorphic with T (Q/G) ⊕ GG by the isomorphism αAG associated to aconnection AG. The brackets on the vector fields of X (Q/G) ⊕ Sect(GG) and thePoisson structure on T ∗(Q) ⊕ GG are presented. We associate the reduced of a G -invariant Lagrangian for which the Lagrange - Poincare equations are written.

The second section deals with refinements of a differentiable principal bundle definedby closed subgroups of the structure group. Also, we define the reduced bundles associatedto a refinement of a principal bundle. In the Section 3 is used the theory of reductiondescribed in the first section for the fibre bundles which constitute a refinement of aprincipal bundle having the projective group as structure group.

Throughout the paper all manifolds are of finite dimension and whitout boundary. Allmaps are differentiable of C∞ - class. We use the definitions and results about the fibrebundles and connections defined on manifolds given in [1], [4] and [5]. Notations used arethe same as those in [2] and [6].

1. The reduced bundle of a principal G - bundle and the reducedLagrangian.

We assume that we have the following set up: a manifold Q and an action of the Liegroup G on Q, say ρ : G×Q → Q.

If πG : Q → Q/G is a left principal bundle and the Lie group G acts differentiablyon the manifold F on the left, then the associated fibre bundle with standard fibre Fis, by definition, Q×G F = (Q× F )/G, where the action of G on Q× F is given bya(q, y) = (aq, ay), (∀) q ∈ Q, y ∈ F, a ∈ G. Also, πF : Q×G F → Q/G is a ( left ) fibrebundle with structure group G.

Let G the Lie algebra of the Lie group G. The associated bundle with standard fibreG, where the action of G on G is the adjoint action is called the adjoint bundle ( here

12000 Mathematics Subject Classification : 55R05,53C05, 58A30.Key words and phrases: adjoint bundle, invariant bundle, connection, refinement

1


F = G, ρg = Adg, g ∈ G ) and it is denoted by GG = AdG(Q). We let πG : G → Q/Gdenote the projection given by πG([q, ξ]G = [q]G. Each fibre GG

x of GG carries a naturalLie algebra structure defined by [ [q, ξ]G, [q, η]G ] = [ q, [ξ, η]G ].

Let TQ be the tangent bundle on Q. An element of TqQ will be denoted byvq, uq, . . . or by (q, q). The tangent lift of the action of G on Q defines an action of Gon TQ and we can form the quotient (TQ)/G =: TQ/G. There is a well defined mapτQ : TQ/G → Q/G induced by the tangent of the projection map πG : Q → Q/Gand given by [vq]G → [q]G. The rules [vq] + [uq]G = [vq + uq]G and λ[vq]G = [λvq]G,where λ ∈ R, vq, uq ∈ TqQ and [vq]G and [uq]G are their equivalence classes in thequotient TQ/G define a vector bundle structure on TQ/G having the base Q/G. Thefibre (TQ/G)x is isomorphic, as vector space, to TqQ, for each x = [q]G.

Let a connection AG on Q given by one form AG : TQ → G with the properties:

(1) AG(ξq) = ξ, for all ξ ∈ G;

(2) AG(Tqρg · v) = Adg(AG(v)), where Adg is the adjoint action of G on G.

The restriction of a connection to TqQ is denoted AGq and the vertical and horizontal

space defined at q ∈ Q is V erq = KerTqπG, Horq = AGq .

The curvature of AG, denoted by BAGis a Lie algebra valued two form on Q

given by:

BAG(uq, vq) = dAG(Horq(uq),Horq(vq)).

We assume that we have a connection AG on the bundle πG : Q → Q/G. The mapαAG : TQ/G → T (Q/G)⊕ GG defined by

αAG([q, q]G) = TπG(q, q)⊕ [q, AG(q, q)]G

is a well defined vector bundle isomorphism.Let TQ = Hor(TQ) ⊕ V er(TQ) be the decomposition into horizontal and ver-

tical parts. Since the bundles Hor(TQ) and V er(TQ) are G- invariant we haveTQ/G = Hor(TQ)/G⊕ V er(TQ)/G. We have that

αAG(V er(TQ)/G) = T (Q/G) and αAG(V er(TQ)/G) = GG.

Let ιG(TQ) : IG(TQ) → Q/G be the vector bundle whose fibre (ιG(TQ))−1(x) atan element x = [q]G ∈ Q/G is the vector space of all invariant vector fields on Q alongπ−1

G . That is, an element of IG(TQ) is a vector field, say, Z, defined only at pointsq ∈ π−1

G (x), that is Z(q) ∈ TQ for all q ∈ π−1G (x) such that gZ = Z, (∀) g ∈ G.

We also let ιG(TQ)V : IVG (TQ) → Q/G ( resp., ιG(TQ)H : IH

G (TQ) → Q/G) be the vector bundle whose fibre (ιG(TQ)V )−1(x) ( resp., (ιG(TQ)H)−1(x) ) at anelement x = [q]G ∈ Q/G is the vector space of all vertical ( resp., horizontal ) invariantvector fields on π−1

G . That is, an element of IVG (TQ) ( resp., IH

G (TQ) ) is a vector

2

IVAN ET AL346

field, say, Y ( resp., X ) on the manifold π−1G (x) such that gY = Y, (∀) g ∈ G

( resp., gX = X, (∀) g ∈ G ). We call ιG(TQ)V : IVG (TQ) → Q/G ( resp.,

ιG(TQ)H : IVG (TQ) → Q/G ) the vertical ( resp., horizontal ) invariant bundle.

Let Sect(IHG (TQ)) and Sect(IV

G (TQ)) the Lie algebra of sections of the horizontaland vertical invariant bundle, respectively. The map TπG establishes a well defined iso-morphism between Sect(IH

G (TQ)) and X (Q/G). We can deduce that there are naturalidentification

Sect(T (Q/G)⊕ GG) = X (Q/G)⊕ Sect(GG).

Given the basis εa | a = 1, p for the Lie algebra G for which Cabc are the

structure constants, we obtain the local basis ∂

∂xi, ea for πG : TQ/G → Q/G and

ea for IVG (TQ) → Q/G such that [ea, eb] = Cc

abec.Let Aa

i (x) the local functions on Q/G for the connection AG defined forπG : Q → Q/G. The corresponding covariant derivative ∇AG

ξ of a section ξ = ξaea

of IVG (TQ) reads ∇AG

ξ : Q/G → T ∗(Q/G)⊕ IVG (TQ),

∇AGξ = (

∂ξa

∂xi+ Ca

bcAGbi ξc)dxi ⊗ ea

and if X ∈ X (Q/G), then ∇AG

X ξ is given by

∇AG

X ξ = Xi(∂ξa

∂xi+ Ca

bcAGbi ξc)ea.

In particular, we have

(1.1) ∇AG

i ea = CcbaA

Gbi ec.

The curvature BAGis given by BAG

=12BGa

ij dxi ∧ dxj ⊗ ea, where

(1.2) BGaij =

∂AGaj

∂xi− AGa

i

∂xj+ Ca

bcAGbi AGc

j .

Let Xi ⊕ ξi ∈ Sect(T (Q/G)⊕ GG), i = 1, 2 be given two sections. Then[X1 ⊕ ξ1, X2 ⊕ ξ2] = [X1, X2]⊕ ∇AG

X1ξ2 − ∇AG

X2ξ1 − BAG

(X1, X2) + [ξ1, ξ2].

For ∂

∂xi⊕ ea, i = 1.n, a = 1, p we have

(1.3) [∂

∂xi⊕ ea,

∂

∂xj⊕ eb] = (Cd

cbAGci − Cd

caAGcj − BAGd

ij + Cdab)ed.

Let (xi, xi, ξa) the local coordinates of TQ⊕ GG and (xi, pi, µa) the local coordi-nates of T ∗Q⊕ G∗. The structure Poisson on T ∗Q⊕ G∗ is given by

3

LAGRANGE-POINCARE EQUATIONS 347

(1.4) ΛG =∂

∂qi∧ ∂

∂pi− BAGc

ij µc∂

∂pi∧ ∂

∂µj−

−CdcaµdA

Gci

∂

∂pi∧ ∂

∂µa+ Cc

abµc∂

∂µa∧ ∂

∂µb.

Let L : TQ → R be an invariant Lagrangian, that is L(g(q, q)) = L(q, q),for all (q, q) ∈ TQ, g ∈ G. Because holds this invariance, we get a well defined reducedLagrangian lG : TQ/G → R satisfying the relation lG([q, q]G) = L(q, q).

We will consider lG as a function defined on T (Q/G)⊕ GG or TQ/G interchange-ably, using the isomorphism αAG . Also, we will write lG(qG, qG, ξ), to emphasize thedependence of lG on (qG, qG) ∈ T (Q/G) and ξ ∈ GG.

The vertical resp., horizontal Lagrange-Poincare equation for lG is given by (1.5.1)resp., (1.5.2):

(1.5.1)DAG

Dt

∂lG

∂ξ(qG, qG, ξ) = ad∗ξ

∂lG

∂ξ(qG, qG, ξ), ξ ∈ GG

(1.5.2)∂lG

∂qG(qg, qG, ξ)− DAG

Dt

∂lG

∂qG(qG, qG, ξ) =

=<∂lG

∂ξ(qG, qG, ξ, iqGBAG

(qG) >, ξ ∈ GG.

In local coordinates the equation (1.5.1) resp., (1.5.2) becomes :

(1.6.1)d

dt(∂lG

∂ξa) =

∂lG

∂ξb(Cb

caξc − Cb

caAGci xi)

(1.6.2)∂lG

∂xi− d

dt(∂lG

∂xi) =

∂lG

∂ξa(BGa

ji xj + CacbA

Gbi ξc)

Let µ a metric G - invariant on Q and mG the metric on Q/G given by :

(1.7) mG(uqG , vqG) = µ(uq, vq), qG ∈ G/G, q ∈ Q,

where uq, vq are the horizontal vectors for the connection AG.Let κG the bi-invariant Riemannian on G and the Lagrangian

L : TQ → R given by:

(1.8) LG(q, q) = κG(AG(q, q), AG(q, q) +12µ((q, q), (q, q))).

This Lagrangian is G -invariant. An element of G has the form (q, ξ)G, where q ∈ Qand ξ ∈ G. since κG is bi-invariant follows that its restriction to G is Ad -invariantand so we can define the fiber metric κG on G by κG((q, ξ)G, (q, η)G) = κG(ξ, η).

The reduced bundle is T (Q/G)⊕GG and a typical element of it is denoted (x, x, ξ).The reduced Lagrangian is given by:

(1.9) lG(x, x, ξ) =12κG(ξ, ξ) +

12µ(x)(x, x).

4

IVAN ET AL348

The vertical resp., horizontal Lagrange- Poincare equation is (1.10.1) resp., (1.10.2):

(1.10.1)D

DtκG(ξ, ξ) = 0

(1.10.2)∂lG

∂x(x, x, ξ)− D

Dt

∂lG

∂x(x, x, ξ) =<

∂lG

∂ξ(x, x, ξ), ixBG(x) >

The equations (1.10.1) and (1.10.2) are the first and the second Wong’s equation.Locally, the expression of the Lagrangian lG is the following:

(1.11) lG(x, x, ξ) =12κG

abξaξb +

12gij(x)xixj .

The local expression of the vertical resp., horizontal Lagrange-Poincare equation isgiven by (1.12.1) resp., (1.12.2):

(1.12.1)dpb

dt= −paC

adbA

Gdi xi, where pa =

∂lG

∂ξa= κabξ

b

(1.12.2)dpi

dt= −paB

Gaji xj − 1

2∂gjk

∂xipjpk, where pi = gijx

j .

2. The reduced bundles associated to a refinement of a principal G - bundle.

Let G be a Lie group and Nq = (G = H0 ⊃ H1 ⊃ . . . ⊃ Hq−1 ⊃ Hq = e) ( e isthe identity element of G ) a sequence of Lie groups such that Hj is a closed subgroupof Hj−1 for 1 ≤ j ≤ q. Let (ξ,Nq) be a structure consisting of a differentiable principalG - bundle ξ = (E, p, B, G) and Nq a sequence of closed subgroups of G.

Let Ej = E/Hj , j = 0, q, Hjk = Hj/Hk and Gj

k = Hj/Njk for 0 ≤ j < k ≤ q,where Njk is the largest normal subgroup of Kj included in Hk and Hj/Njk is thefactor group of Hj by Njk. Finally, let pjk : zHk ∈ Ek → zHj ∈ Ej , (∀)z ∈ E for0 ≤ i < j ≤ 2, the canonical map.

The pair (ξ,Nq) defines the fibre bundles ( see [6], MR 53 # 4058 ):ξjk = (Ek, pjk, Ej ,H

jk, Gj

k) , 0 ≤ j < k ≤ q.

The triplet (ξ; ξ0j , ξjq), where ξ0j = (Ej , p0j , B, H0j , G0

j ), ξjq = (E, pjq, Ej ,Hj) iscalled the refinement of ξ defined by Hj . We have that ξjq is a principal Hj - bundlefor all 0 < j < q.

When q = 2 and H1 = H, the refinement of ξ defined by H is the triple(ξ; ξ01, ξ12) with ξ01 = (E/H, p01, B, G/H, G/N) and ξ12 = (E, p12, E/H, H), whereN is the largest normal subgroup of G included in H. This structure can be find in [4].

EXAMPLE 2.1. ([6]) Let ξ∗ = (LnM,p = πL,M, GL(n;R) be the principal bundleof tangent linear frames to n - manifold M. We consider the sequence N ∗

2 = (G =GL(n;R) ⊃ H = GT (n;R) ⊃ e), where GT (n;R) = (ai

j) ∈ GL(n;R) | aij =

0 for i > j is the subgroup of upper triangular matrices.

5

The refinement of ξ∗ defined by GT (n;R) is (ξ∗; ξ∗01, ξ∗12), where ξ∗01 =

(DnM, p01, M, GL(n;R)/GT (n;R), GP (n − 1;R)) is the fibre bundle of tangent flagsto M and ξ∗12 = (LnM,p12,DnM, GT (n;R)). Here DnM = LnM/GT (n;R) is themanifold of tangent flags to M and GP (n − 1;R)) = GL(n;R)/D(n;R) is the realprojective group of order n−1, since the largest subgroup normal subgroup of GL(n;R)included in GT (n;R) is D(n;R) = (λδi

j) | (∀) λ ∈ R . 2

REMARK 2.1. (i) If H is a closed subgroup of G such that N = e, thenthe refinement of ξ = (E, p,B,G) defined by H is (ξ; ξ01, ξ12), where ξ01 =(E/H, p01, B, G/H, G) and ξ12 = (E, p12, E/H,H).

(ii) If H is a normal closed subgroup of G, then the refinement of ξ = (E, p, B, G)defined by H is a refinement (ξ; ξ01, ξ12), where ξ01 = (E/H, p01, B,G/H) andξ12 = (E, p12, E/H, H) are principal bundles .2

EXAMPLE 2.2. Let M be a n - manifold and (T (M), πT ,M,Rn, GL(n;R)) be thefibre bundle of tangent vectors on M.

By affine frame at x ∈ M we mean a triple u = (x, y, z), where (y, z) ∈ π−1T (x)×

π−1L (x). We denote by AnM the set of affine frames on M endowed with the differentiable

structure canonically induced from the differentiable structure of M. In a local chart ofM, an affine frame at x ∈ M is given by:

(2.1) y = yi(∂

∂xi)x, zj = zi

j(∂

∂xi)x, det(zi

j) 6= 0

and the local coordinates of u = (x, y, z) ∈ AnM are (xi, yi, zij), i, j = 1, n.

Let be the affine group GA(n;R) = (

1 0ai gi

j

)∈ GL(n + 1;R) | det(gi

j) 6= 0 .The action of the Lie group G = GA(n;R) on the manifold AnM is defined by theright translations τ(a,g) : G×AnM → AnM, (∀) (a, g) ∈ G with a = (ai) ∈ Rn, g =(gi

j) ∈ GL(n;R) where:

(2.2) ((a, g), u) → (τ(a,g)(u) = (x, y + za, zg) for all u = (x, y, z) ∈ AnM.

Let η = (AnM, pA,M,GA(n;R)) be the principal bundle of affine frames to ann - manifold M, where pA : ANM → M is the canonical projection given bypA(u) = x, (∀) u = (x, y, z) ∈ AnM. We consider the sequence N 1

2 = (GA(n;R) ⊃T (n;R) ⊃ e), where T (n;R) =

(1 0ai δi

j

). Since T (n;R) is normal in G, apply-

ing Remark 2.1.(ii), the refinement of η defined by T (n;R) is (η; η01, η12), where η01 =(AnM/T (n;R), p01,M, GA(n;R)/T (n;R)) and η12 = (AnM,p2

12,AnM/T (n;R), T (n;R)). 2

Let Q be a manifold and G a Lie group which acts differentiably on Q. We oonsiderπG : Q → Q/G the principal bundle with the structure group G. We assume that isgiven a sequence N2 = (G ⊃ K ⊃ e) of closed subgroups of G.

If we denote η = (Q, πG, Q/G, G), then the pair (η;N2) determines a refinement(η; η01, η12) of η defined by K, whereη01 = (Q/K, πGK , Q/G, G/K,G/N) and η12 = (Q, πK , Q/K,K), and N is the largest

6

IVAN ET AL350

normal subgroup of G included in K.Let AG and AK two connections on Q given by the forms AG : TQ → G, AK :

TQ → K, where G resp., K is the Lie algebra of G resp., K.Let the adjoint bundles G = AdG(Q), K = AdK(Q) and the isomorphisms αAG :

TQ/G → T (Q/G)⊕ GG, αAK : TQ/K → T (Q/K)⊕ KK .The vector bundles T (Q/G)⊕GG → Q/G and T (Q/K)⊕KK → Q/K are called

the reduced bundles associated to refinement defined by (η;N2).

3. Geometric structures on the fibre bundles of a refinement of a principalGP (n;R) - bundle.

Let us we apply the above considerations in the case the group G is the projectivegroup GP (n;R) and K is the affine group GA(n;R).

Let G = GL(n + 1,R)/D(n + 1,R) the projective group of order n. The class [aij ]

determined by the matrix (aij) is denoted by a. The subgroup K of G is determined

by all classes [aij ] for which the matrix (ai

j) satisfy the condition an+1h = 0, h = 1, n

and it may be identified of the affine group of order n. We obtain thus a sequence N2 =(G ⊃ K ⊃ e), where G = GP (n;R) and K = GA(n;R).

A base for the Lie algebra G of G is εij , ε

i, εj and we have

[εij , ε

lk] = δilp

kqjεqp, [εi

j , εk] = δikεj , [εi

j , εk] = −δk

j εi, [εi, εj ] = 0. [εi, εj ] = 0,

[εi, εj ] = −γipjqε

qp, where δilp

kqj = δikδ

lqδ

pj − δl

jδiqδ

pk, γip

jq = δijδ

pq + δi

qδpj .

A base for the Lie algebra K of K is εij , εj and we have

[εij , ε

lk] = δilp

kqjεqp, [εi

j , εk] = δikεj , [εi, εj ] = 0.

Let πG : Q → Q/G the principal bundle having the projective group G as structuregroup. Let hn+i

n+i(Un+in+i , χ

n+in+i) the local charts of Q with the coordinates (qi, xi

j , xi, xj).

The base of sections of the vector bundle GG → Q/G is

eij = xh

j

∂

∂xhi

+ xj∂

∂xi, ej = xh

j

∂

∂xh− xj(xh

k

∂

∂xhk

+ xh ∂

∂xh+ xk

∂

∂xk), ei = xh ∂

∂xhi

+∂

∂xi.

Let AG a connection on the principal bundle πG : Q → Q/G given by the functions(P i

jl, Pil , Pjl) on Q/G. The following relations hold:

7


(3.1)

∇AG

∂

∂qi

elk = δlps

qkrPrpie

qs + γlq

pkPpi eq + Pkie

l

∇AG

∂

∂qi

el = P lhie

h + γlspkP

ki ep

s, ∇AG

∂

∂qi

ek = P jkiej − γsq

pkPqieps

BAG=

12Bl

kijdqi ∧ dqj ⊗ ekl +

12Blijdqi ∧ dqj ⊗ el +

12Bl

ijdqi ∧ dqj ⊗ el.

Let (qi, qi, ξlk, ξk, ξ

l) the local coordinates on TQ ⊕ GG and (qi, pi, µlk, µ

k, µl) thelocal coordinates on T ∗Q⊕GG∗ . The structure Poisson is given by the following relations:

(3.2.1) qi, qj = 0, qi, pj = δij , qi, µl

k = 0, qi, µl = 0, qi, µl = 0;

(3.2.2) pi, pj = −Blkijµ

kl −Blijµ

l −Blijµl; pi, µ

lk = µl

jPjik − µq

p(δpi Pqk + δl

qPki);

(3.2.3) pi, µl = µk

p(δpkP

li + δl

kPpi ) + µhP l

hi; µij , µ

lk = δi

kµlj − δl

jµik;

(3.2.4) µij , µ

l = −δljµ

i, µij , µl = δi

lµj , µl, µk = −(δlkδ

pq + δl

qδpk)µ

qp;

(3.2.5) µl, µk = 0, µl, µk = 0.

Let κG the bi-invariant Riemannian metric on G and the Lagrangian L : TQ → Rgiven by (1.8). The reduced Lagrangian lG on T (Q/G)⊕ GG is given by :

(3.3) lG(qi, qi, ξij , ξj , ξ

i) =12κijξ

iξj +12κjl

ikξijξ

kl +

12κijξiξj+

+κjilξ

lξji + κjl

i ξlξij +

12κihξiξh + κl

iξlξi +

12gij q

iqj .

and we have

(3.4)

pij =

∂lG

∂ξij

= κjlihξk

l + κjilξ

l + κjli ξl, pi =

∂lG

∂ξi= κl

ihξhl + κihξh + κl

iξl

pi =∂lG

∂ξi= κil

hξhl + κi

hξh + κihξh, pi =∂lG

∂qi= gij q

j .

The Lagrange-Poincare equation and Wong’s equations for the projective group aregiven by the relations (3.5.1)-(3.5.4):

(3.5.1)dpj

i

dt= −[(pl

h(δjl P

hik − δh

i P jlk + (δh

i P jk + δj

i Phk )ph + Pikp

j ] · qk

(3.5.2)dpi

dt= −[P h

ikph − (δlhPik + δl

kPhi)phl ] · qk

(3.5.3)dpi

dt= −[P i

khpk + (δlkP

ih + δi

kPlh)pk

l ]qh

(3.5.4)dpi

dt= −(pl

hBhlij + pijB

hij + plBlij)qj − 1

2∂gjl

∂qipjpl.

8

IVAN ET AL352

Let πK : Q → Q/K the principal bundle having the affine group K as structuregroup and the local coordinates (qi, ηi) on Q/K. The base of sections of the vector

bundle KK → Q/K is eij = xh

j

∂

∂xhi

, ej = xhj

∂

∂xh.

Let AK a connection on the principal bundle πK : Q → Q/K given by the functions(Ahr

k , Ahr, Ahkr, A

hr ) on Q/K. The following relations (3.6) hold:

(3.6)

∇AK

∂

∂qr

elk = (Ap

krδlq −Al

qrδpk)e

qp −Al

rek, ∇AK

∂

∂ηr

elk = (Apr

k δlq −Alr

q δpk)e

qp −Alrek

∇AK

∂

∂qr

ek = Aikrei, ∇AK

∂

∂ηr

ek = (Airk ei

BAK=

12(Bl

kijdqi ∧ dqj + Blhkidqi ∧ dηh + Blhi

k dηh ∧ dηi)⊗ ekl +

+12(Bl

ijdqi ∧ dqj + Blhi dqi ∧ dηh + Blhidηh ∧ dηi)⊗ el.

Let (qi, ηi, qi, ηi, ξ

lk, ξ

l) the local coordinates on TQ⊕ KK and (qi, ηi, pi, λi, µl

k, µl)the local coordinates on T ∗Q⊕ KK∗

. The structure Poisson is given by the relations:

qi, pj = δji , ηi, λ

j = δji , pi, pj = −1

2(Bl

kijµkl + Bl

ijµl);

pi, λj = −1

2(Blj

kiµkl + Blj

i µl), λi, λj = −12(Blji

k µkl + Bljiµl);

pi, µlk = µq

p(Apkiδ

lq −Al

qiδpk)− µkA

li; λr, µl

k = µqp(A

prk δl

q −Alpq δr

k)− µkAlr;

pr, µk = Aikrµi, λr, µk = Air

k µi; µij , µ

lk = −δi

kµlj + δl

jµik, µi

k, µj = δikµj .

Let κK the bi-invariant Riemannian metric on K and the Lagrangian L : TQ → Rgiven by (1.8). The reduced Lagrangian lK on T (Q/K)⊕ KK is given by:

(3.7) lK(qi, ηi, qi, ηi, ξ

ab , ξa) =

12κabξ

aξb +12κc

baξaξb

c+

+12κcd

abξac ξb

d +12gij q

iqj +12gji q

iηj +12gijηiηj .

and we have :

(3.8)

pba =

∂lK

∂ξab

= κbcadξ

dc + κb

acξc, pa =

∂lK

∂ξa= κc

adξdc + κabξ

b

pi =∂lK

∂ηi= gi

j qj + gij ηj , pi =

∂lK

∂qi= gij q

j + gji ηj .

The Lagrange-Poincare equations and Wong’s equations for the affine group are given

9


by the relations (3.9.1)-(3.9.4):

(3.9.1)dpb

a

dt= −[pq

p(Abqiδ

pq −Ab

qiδpa)q

i −Abipaq

i + ppq(A

pia δp

q −Abiq δp

a) · ηi −Abipaηi]

(3.9.2)dpa

dt= −[pbA

baiq

i + pbAbia ηi]

(3.9.3)dpi

dt= −(pb

aBahib ηh + pb

aBaih qh + paB

ahiηh + pbaB

aibhqh)−

−12

∂gjl

∂ηipjpl − ∂gj

l

∂ηipjp

l − 12

∂gjl

∂ηipjpl

(3.9.4)dpi

dt= −(pb

aBabjiq

j + pbaB

ajbi ηj + paB

ajiq

i + paBaji ηj)−

−12

∂gjl

∂qipjpl − ∂gj

l

∂qipjp

l − 12

∂gjl

∂qipjpl,

where gjl are the elements of the inverse of matrix (gij).

References

[1] R. Abraham, J.E. Marsden, T.S. Ratiu, Manifolds, Tensor Analysis and Applica-tions.Second Edition. Mathematical Sciences 75, Springer - Verlag (1988).

[2] H. Cendra, J.E. Marsden, T.S. Ratiu, Lagrangian Reduction by Stages. Mem.Amer.Math. Soc. 152,no.722(2001), 1-108.

[3] Gh. Ivan,D.Opris Old and new aspects in the study of refinements of a principalbundle. Tensor N.S.,63(2002), p.160 - 175.

[4] S. Kobayashi, K. Nomizu,Foundations of differential geometry. Interscience Publ.,NewYork-London, Vol.I (1963).

[5] L. Mangiarotti, G. Sardanashvily,Connections in classical and quantum field theory.World Scientific, Singapore, (2000).

[6] D. I. Papuc, Sur les raffinements d’un espace fibre principal differentiable. Anal.St. Univ. ”Al. I. Cuza” Iasi, Sect. a I-a, Mat. ( N.S.), 18 (1972), p. 367-387.

West University of TimisoaraSeminarul de Geometrie si Toplogie4, Bd. V. Parvan, 300223, Timisoara, RomaniaE-mail : [email protected],[email protected] and [email protected]

10

IVAN ET AL354

EXPLICIT p-ADIC q-EXPANSION FOR

THE ALTERNATING SUMS OF POWERS

Leechae Jang and Taekyun Kim

Department of Mathematics and Computer Science,KonKuk University, Chungju, S. Korea

e-mail: [email protected], [email protected]

EECS, Kyungpook National University,Taegu 702-701, S. Korea e-mail: [email protected] ( or [email protected])

Abstract. In this paper, we give an explicit p-adic expansion of

npX

j=1(j,p)=1

(−1)j

[j]rq

as a power series in n. The coefficients are values of p-adic q-l-function for q-Euler

numbers.

§1. Introduction

Let p be a fixed prime. Throughout this paper Zp, Qp, C and Cp will, respectively,denote the ring of p-adic rational integers, the field of p-adic rational numbers, thecomplex number field and the completion of algebraic closure of Qp, cf.[1, 4, 6, 14]. Letvp be the normalized exponential valuation of Cp with |p|p = p−vp(p) = p−1. When onetalks of q-extension, q is variously considered as an indeterminate, a complex numberq ∈ C, or a p-adic number q ∈ Cp. If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp,

Key words and phrases. p-adic q-integrals, Euler numbers, p-adic l-function.

2000 Mathematics Subject Classification: 11S80, 11B68, 11M99 .

Typeset by AMS-TEX

1

2 L.C. JANG AND T. KIM

then we assume |q − 1|p < p−1

p−1 , so that qx = exp(x log q) for |x|p ≤ 1. Kubotaand Leopoldt proved the existence of meromorphic functions, Lp(s, χ), defined overthe p-adic number field, that serve as p-adic equivalents of the Dirichlet L-series, cf.[1,14,15,16]. These p-adic L-functions interpolate the values

Lp(1− n, χ) = − 1n

(1− χn(p)pn−1)Bn,χn, for n ∈ N = 1, 2, · · · , ,

where Bn,χ denote the nth generalized Bernoulli numbers associated with the primitiveDirichlet character χ, and χn = χw−n, with w the Teichmuller character, cf.[8,14]. In[14,15], L. C. Washington have proved the following interesting formula:

np∑j=1

(j,p)=1

1jr

= −∞∑

k=1

(−r

k

)(pn)kLp(r + k, w1−k−r), where

(−r

k

)is binomial coefficient.

To give the q-extension of the above Washington result, the author derived the sumsof powers of consecutive q-integers as follows:

(*)n−1∑l=0

ql[l]m−1q =

1m

m−1∑l=0

(m

l

)qmlβl[n]m−l

q +1m

(qmn − 1)βm, see [7,8] ,

where βm are q-Bernoulli numbers. By using (*), we gave an explicit p-adic expansion

np∑j=1

(j,p)=1

qj

[j]rq= −

∞∑k=1

(−r

k

)[pn]kqLp,q(r + k,w1−r−k)

− (q − 1)∞∑

k=1

(−r

k

)[pn]kqTp,q(r + k, w1−r−k)− (q − 1)

p−1∑a=1

B(n)p,q (r, a : F ),

where Lp,q(s, χ) is p-adic q-L-function (see [7,12] ). Indeed, this is a q-extension resultdue to Washington, corresponding to the case q = 1, see [14]. Recently, the secondauthor described algorithms to deal with nested symbolic sums over combinations ofharmonic sum, binomial coefficients and denominators[13]. In addition, he treatedMellin transforms and the inverse Mellin transformation for functions that are encoun-tered in Feynman diagram calculations. Together with results for the values of thehigher harmonic sum at infinity the presented algorithm can be used for the symbolic

356

p-ADIC q-EXPANSION FOR THE ALTERNATING SUMS OF POWERS 3

evaluation of whole classes of integrals that were thus far intractable [13]. The com-putation of Feynman diagrams has confronted physicists with classes of integrals thatusually hard to be evaluated, both analytically and numerically. In [10], Rim andKim treated explicit p-integral alternating harmonic sums. Harmonic sum is criticalto explain the resonating phenomenon in a nature. It is also important as a solutionof simple harmonic oscillating system in classical mechanics and quantum mechanics(see [17]). p-adic harmonic sum can be applied to these physical phenomena. Withthis application, p-adic harmonic sum also can be used for quantum statistical physicsor quantum transportation theory( see [17]). Euler numbers and polynomials in alter-nating harmonic sum are used for Langevine equation of magnetism which is in thesystem with viscosity. For a fixed positive integer d with (p, d) = 1, set

X = Xd = lim←−N

Z/dpNZ,

X1 = Zp, X∗ =

⋃0<a<dp(a,p)=1

a + dpZp,

a + dpNZp = x ∈ X|x ≡ a (mod pN ),

where a ∈ Z satisfies the condition 0 ≤ a < dpN , (cf.[1,3,4,9,16]). We say that f isa uniformly differentiable function at a point a ∈ Zp, and write f ∈ UD(Zp), if thedifference quotients Ff (x, y) = f(x)−f(y)

x−y have a limit f ′(a) as (x, y) → (a, a), cf.[3].For f ∈ UD(Zp), let us begin with the expression

1[pN ]q

∑0≤j<pN

qjf(j) =∑

0≤j<pN

f(j)µq(j + pNZp), cf.[3,4,6,7,8,9,10],

which represents a q-analogue of Riemann sums for f . The integral of f on Zp is definedas the limit of those sums(as n→∞) if this limit exists. The q-Volkenborn integral ofa function f ∈ UD(Zp) is defined by

(1) Iq(f) =∫

X

f(x)dµq(x) =∫

Xd


1[dpN ]q

dpN−1∑x=0

f(x)qx, cf. [6,7] .

It is well known that the familiar Euler polynomials En(z) are defined by means of thefollowing generating function:

F (z, t) =2

et + 1ezt =

∞∑n=0

En(z)tn

n!, cf.[3,11].

357


We note that, by substituting z = 0, En(0) = En are the familiar n-th Euler numbers.Over five decades ago, Carlitz defined q-extension of Euler numbers and polynomials,cf.[1, 4, 5]. Recently, author gave another construction of q-Euler numbers and polyno-mials (see [1, 5, 9]). By using author’s q-Euler numbers and polynomials, we gave thealternating sums of powers of consecutive q-integers as follows: For m ≥ 1, we have

2n−1∑l=0

(−1)l[l]mq = (−1)n+1m−1∑l=0

(m

l

)qnlEl,q[n]m−l

q +((−1)n+1qnm + 1

)Em,q,

where El,q are q-Euler numbers (see [3] ). From this result, we can study the p-adic interpolating function for q-Euler numbers and sums of powers due to author [7].Throughout this paper, we use the following notation:

[x]q =1− qx

1− q, and [x]−q =

1− (−q)x

1− qcf.[3,11,5,9].

Note that when p is prime [p]q is an irreducible polynomial in Q[q]. Furthermore, thismeans that Q[q]/[p]q is a field and consequently rational functions r(q)/s(q) are welldefined mod [p]q if (r(q), s(q)) = 1. In a recent paper [3] the author constructedthe new q-extensions of Euler numbers and polynomials. In Section 2, we introducethe q-extension of Euler numbers and polynomials. In Section 3 we construct a newq-extension of Dirichlet’s type l-function which interpolates the q-extension of general-ized Euler numbers attached to χ at negative integers. The values of this function atnegative integers are algebraic, hence may be regarded as lying in an extension of Qp.We therefore look for a p-adic function which agrees with at negative integers. Thepurpose of this paper is to construct the new q-extension of generalized Euler numbersattached to χ due to author and prove the existence of a specific p-adic interpolat-ing function which interpolate the q-extension of generalized Bernoulli polynomials atnegative integer. Finally, we give an explicit p-adic q-expansion

np∑j=1

(j,p)=1

(−1)j

[j]rq,

as a power series in n. The coefficients are values of p-adic q-l-function for q-Eulernumbers.

358


2. Preliminaries

For any non-negative integer m, the q-Euler numbers, Em,q, were represented by(2)

2[2]q

∫Zp

q−x[x]mq dµ−q(x) = Em,q = 2(

11− q

)m m∑i=0

(m

i

)(−1)i 1

1 + qi, see [3,10] .

Note that limq→1 Em,q = Em. From Eq.(2), we can derive the below generating func-tion:

(3) Fq(t) = 2et

1−q

∞∑j=0

11 + qj

(−1)j(1

1− q)j tj

j!=

∞∑j=0

En,qtn

n!.

By using p-adic q-integral, we can also consider the q-Euler polynomials, En,q(x), asfollows:(4)

En,q(x) =2

[2]q

∫Zp

q−t[x + t]nq dµ−q(t) = 2(1

1− q)n

n∑k=0

(n

k

)(−qx)k

1 + qk, cf.[3,6,7,8,9,10].

Note that

(5) En,q(x) =2

[2]q

∫Zp

([x]q + qx[t]q)nq−tdµ−q(x) =n∑

j=0

(n

j

)qjxEj,q[x]n−j

q .

By (4), we easily see that

(6)∞∑

n=0

En,q(x)tn

n!= Fq(x, t) = 2e

t1−q

∞∑j=0

(−1)j

1 + qjqjx(

11− q

)j tj

j!.

From (6), we derive

(7) Fq(x, t) = 2∞∑

n=0

(−1)ne[n+x]qt =∞∑

n=0

En,q(x)tn

n!.

359

3. On the q-analogue of Hurwitz’s typeζ-function associated with q-Euler numbers

In this section, we assume that q ∈ C with |q| < 1. It is easy to see that

En,q(x) = [m]nqm−1∑a=0

(−1)aEn,qm(a + x

m), see [3,6,10] ,

where m is odd positive integer. From (7), we can easily derive the below formula:

(8) Ek,q(x) =dk

dtkFq(x, t)|t=0 = 2

∞∑n=0

(−1)n[n + x]kq .

Thus, we can consider a q-ζ-function which interpolates q-Euler numbers at negativeinteger as follows:

Definition 1. For s ∈ C, define

ζE,q(s, x) = 2∞∑

m=1

(−1)n

[n + x]sq.

Note that ζE,q(s, x) has a meromorphic function in whole complex plane.

By using Definition 1 and Eq.(8), we obtain the following:

Proposition 2. For any positive integer k, we have

ζE,q(−k, x) = Ek,q(x).

Let χ be the Dirichlet character with conductor f ∈ N. Then we define the gener-alized q-Euler numbers attached to χ as

(9) Fq,χ(t) = 2∞∑

n=0

e[n]qtχ(n)(−1)n =∞∑

n=0

En,χ,qtn

n!.

Note that

(10) En,χ,q = [f ]nq

f−1∑a=0

χ(a)(−1)aEn,qf (a

f), where f(= odd) ∈ N .

By (9), we easily see that

(11)dk

dtkFq,χ(t)|t=0 = Ek,χ,q = 2

∞∑n=1

χ(n)(−1)n[n]kq

360

Definition 3. For s ∈ C, we define Dirichlet’s type l-function as follows:

lq(s, χ) = 2∞∑

n=1

χ(n)(−1)n

[n]sq.

From (11) and Definition 3, we can derive the below theorem.

Theorem 4. For k ≥ 1, we have

lq(−k, χ) = Ek,χ,q.

In [3], it was known that

(12) 2n−1∑l=0

(−1)l[l]mq =((−1)n+1qnEm,q(n) + Em,q

), where m,n ∈ N.

From (4) and (12), we derive

(13)

2n−1∑l=0

(−1)l[l]mq

= (−1)n+1m−1∑l=0

(m

l

)qnlEl,q[n]m−l

q +((−1)n+1qnm + 1)

)Em,q.

Let s be a complex variable, and let a and F (= odd) be the integers with 0 < a < F.We now consider the partial q-zeta function as follows:

(14) Hq(s, a : F ) =∑

m≡a(F )m>0

(−1)m

[m]sq= (−1)a

[F ]−sq

2ζE,qF (s,

a

F).

For n ∈ N, we note that Hq(−n, a : F ) = (−1)a [F ]nq2 En,qF ( a

F ). Let χ be the Dirichlet’scharacter with conductor F (= odd). Then we have

(15) lq(s, χ) = 2F∑

a=1

χ(a)Hq(s, a : F ).

The function Hq(s, a : F ) will be called the q-extension of partial zeta function whichinterpolates q-Euler polynomials at negative integers. The values of lq(s, χ) at negativeintegers are algebraic, hence may be regarded as lying in an extension of Qp. Wetherefore look for a p-adic function which agrees with lq(s, χ) at the negative integersin Section 4.

361

§4. p-adic q-l-functions and sums of powers

We define < x >=< x : q >= [x]qw(x) , where w(x) is the Teichmuller character.

When F (= odd) is multiple of p and (a, p) = 1, we define a p-adic analogue of (14) asfollows:

(16) Hp,q(s, a : F ) =(−1)a

2< a >−s

∞∑j=0

(−s

j

)qja

([F ]q[a]q

)j

Ej,qF , for s ∈ Zp.

Thus, we note that(17)

Hp,q(−n, a : F ) =(−1)a

2< a >n

n∑j=0

(n

j

)qja

([F ]q[a]q

)j

Ej,qF

=(−1)a

2w−n(a)[F ]nq En,qF (

a

F) = w−n(a)Hq(−n, a : F ), for n ∈ N.

We now construct the p-adic analytic function which interpolates q-Euler number atnegative integer as follows:

(18) lp,q(s, χ) = 2F∑

a=1(a,p)=1

χ(a)Hp,q(s, a : F ).

In [5, 9], it was known that

Ek,χ,q =2

[2]qf

∫X

χ(x)[x]kqq−xdµ−q(x), for k ∈ N.

For f(= odd) ∈ N, we note that

En,χ,q = [f ]nq

f−1∑a=0

χ(a)(−1)aEn,qf (a

f).

Thus, we have(18-1)

lp,q(−n, χ) = 2F∑

a=1(p,a)=1

χ(a)Hp,q(−n, a : F ) =2

[2]qf

∫X∗

χw−n(x)[x]nq q−xdµ−q(x)

= En,χw−n,q − [p]nq χw−n(p)En,χw−n,qp .

362

In fact,

(19) lp,q(s, χ) = 2F∑

a=1

(−1)a < a >−s χ(a)∞∑

j=0

(−s

j

)qja

([F ]q[a]q

)j

Ej,qF , for s ∈ Zp.

This is a p-adic analytic function and has the following properties for χ = wt:

(20) lp,q(−n, wt) = En,q − [p]nq En,qp , where n ≡ t ( mod p− 1),

(21) lp,q(s, t) ∈ Zp for all s ∈ Zp when t ≡ 0( mod p− 1).

If t ≡ 0( mod p − 1), then lp,q(s1, wt) ≡ lp,q(s2, w

t)( mod p) for all s1, s2 ∈ Zp,lp,q(k,wt) ≡ lp,q(k + p, wt)( mod p). It is easy to see that

(22)1

r + k − 1

(−r

k

)(1− r − k

j

)=−1

j + k

(−r

k + j − 1

)(k + j

j

),

for all positive integers r, j, k with j, k ≥ 0, j + k > 0, and r 6= 1 − k. Thus, we notethat

(22-1)1

r + k − 1

(−r

k

)(1− r − k

j

)=

1r − 1

(−r + 1k + j

)(k + j

j

).

From (22) and (22-1), we derive

(23)r

r + k

(−r − 1

k

)(−r − k

j

)=

(−r

k + j

)(k + j

j

).

By using (13), we see that(24)

n−1∑l=0

(−1)Fl+a

[Fl + a]rq=

n−1∑l=0

(−1)l(−1)a([a]q + qa[F ]q[l]qF

)−r

= −∞∑

s=1

[a]−rq

([F ]q[a]q

)s

qas(−1)a

(−r

s

)(−1)n

2

s−1∑l=0

(s

l

)qnFlEl,qF [n]s−l

qF

−∞∑

s=1

[a]−rq

([F ]q[a]q

)s

qas(−1)a

(−r

s

)((−qFs)n − 1)

2Es,qF +

(1− (−1)n)[a]−rq

2(−1)a.

363

For s ∈ Zp, we define the below T -Euler polynomials:

(25) Tn,q(s, a : F ) = (−1)a < a >−s∞∑

k=1

(−s

k

)[a

F]−kqF qak

((−1)nqnFk − 1

)Ek,qF .

Note that limq→1 Tn,q(s, a : F ) = 0, if n is even positive integer. From (23) and (24),we derive

(26)

n−1∑l=0

(−1)Fl+a

[Fl + a]rq

= −∞∑

s=1

(−r

s

)[a]−r

q

([F ]q[a]q

)s (−qs)a

2

s−1∑l=0

(s

l

)qnFlEl,qF [n]s−l

qF

− w−r(a)2

Tn,q(r, a : F ), where n is positive even integer .

Let n be positive even integer. Then, we evaluate the right side of Eq.(26) as follows:(27)∞∑

s=1

(−r

s

)[a]−r

q

([F ]q[a]q

)s (−qs)a(−1)n

2

s−1∑l=0

(s

l

)qnFlEl,qF [n]s−l

qF

=∞∑

k=1

r

r + k

(−r − 1

k

)[a]−k−r

q qak(−1)n[Fn]kq(−1)a

2

∞∑l=0

(−r − k

l

)qal

([F ]q[a]q

)l

El,qF .

It is easy to check that

(28) qnFl =l∑

j=0

(l

j

)[nF ]jq(q − 1)j = 1 +

l∑j=1

(l

j

)[nF ]jq(q − 1)j .

Let(29)

Kp,q(s, a : F ) =(−1)a

2< a >−s

∞∑l=1

(−s

l

)qal

([F ]q[a]q

)l

El,qF

l∑j=1

(l

j

)[nF ]jq(q − 1)j .

Note that limq→1 Kp,q(s, a;F ) = 0. For F = p, r ∈ N, we see that

(30) 2p−1∑a=1

n−1∑l=0

(−1)a+pl

[a + pl]rq= 2

np∑j=1

(j,p)=1

(−1)j

[j]rq.

364


For s ∈ Zp, we define p-adic analytically continued function on Zp as

(31)

Kp,q(s, χ) = 2p−1∑a=1

χ(a)Kp,q(s, a : F ),

Tp,q(s, χ) = 2p−1∑a=1

χ(a)Tn,q(s, a : F ), where k, n ≥ 1 .

From (24)-(31), we derive

2np∑

j=1(j,p)=1

(−1)j

[j]rq= −

∞∑k=1

r

r + k

(−r − 1

k

)(−1)n[pn]kq lp,q(r + k,w−r−k)

−∞∑

k=1

r

r + k

(−r − 1

k

)(−1)n[pn]kqKp,q(r + k,w−r−k)− Tp,q(r, w−r).

Therefore we obtain the following theorem:

Theorem 5. Let p be an odd prime and let n ≥ 1 be positive even integer. Then wehave

(32)

2np∑

j=1(j,p)=1

(−1)j

[j]rq= −

∞∑k=1

r

r + k

(−r − 1

k

)(−1)n[pn]kq lp,q(r + k,w−r−k)

−∞∑

k=1

r

r + k

(−r − 1

k

)(−1)n[pn]kqKp,q(r + k,w−r−k)− Tp,q(r, w−r),

where r is positive integer.

Remark. When r is non-positive integer, we can easily derive the value of the leftside of Eq.(32) from Eq.(13).

For q = 1 in (32), we have

2np∑

j=1(j,p)=1

(−1)j

jr= −

∞∑k=1

r

k + r

(−r − 1

k

)(−1)n(pn)klp(r + k,w−r−k),

where n is positive even integer (see [10]).

365


References

[1] M. Cenkci, Y. Simsek, V. Kurt, Further remarks on multiple p-adic q-L-function of two

variables,, Advan. Stud. Contemp. Math. 14 (2007), 49-68.

[2] T. Kim, S.-H. Rim, Y. Simsek, A note on the the alternating sums of powers of consecutiveq-integers, Advan. Stud. Contemp. Math. 13 no.2 (2006), 340-348.

[3] T. Kim, Power series and asymptotic series associated with the q-analogue of the two variable

p-adic L-function, Russian J. Math. Phys. 12 (2005), 189-196.

[4] T. Kim, q-Volkenborn Integration, Russ. J. Math. Phys. 9 (2005), 288-299.

[5] T. Kim, On a q-analogue of the p-adic log gamma functions, J. Number Theory 76 (1999),

320-329.

[6] T.Kim, Sums powers of consecutive q-integers, Advan. Stud. Contemp. Math. 9 (2004),

15-18.

[7] T. Kim, On p-adic q-L-functions and sums of powers, Discrete Math. 252 (2002), 179-187.

[8] T. Kim, Multiple p-adic L-functions, Russian J. Math. Phys. 13 (2) (2006).

[9] T. Kim, A note on q-Volkenborn integration, Proc. Jangjeon Math. Soc. 8 (2005), 13-17.

[10] S.-H. Rim, T. Kim, Explicit p-adic expansion for the alternating sums of powers, Advan.

Stud. Contemp. Math. 14 (2007), 241-250.

[11] Y. Simsek, On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers,

Russian J. Math. Phys. 13 (2006), 340-348.

[12] T. Kim, J. Y. Choi, J. Y. Sug, Extended q-Euler numbers and polynomials associated with

fermionic p-adic q-integral on Zp, Russian J. Math. Phys. 14 (2007), 160-163.

[13] L. C. Washington, p-adic L-functions and sums of powers, J. Number Theory 69 (1998),

50-61.

[14] L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag(1’st Ed.), 1982.

[15] M. Schork, Ward’s “calculus of sequences”, q-calculus and the limit q → −1, Advan. Stud.

Contemp. Math. 13 (2006), 131–141.

[16] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers, Advan. Stud.Contemp. Math. 11 (2005), 314-321.

[17] J.Y. Sug, S. D. Choi,, Quantum yransport theory based on the equilibrium density projection

technique, Physical Review E 55 (1997), 205-218.

366

Fuzzy multi-metric spacesHakan Efe1, Cihangir Alaca2, Cemil Yildiz1

1Department of Mathematics, Faculty of Science and Arts, GaziUniversity, Teknikokullar, 06500 Ankara, Turkey.

[email protected], [email protected] of Mathematics, Faculty of Science and Arts, Ondokuz

Mayis University, Kurupelit, 55139 Samsun, [email protected]

AbstractThe purpose of this paper is to introduce the concept of fuzzy multi-

metric space by combining Smarandache multi-spaces with fuzzy met-ric space. Also some characteristics of fuzzy multi-metric space areobtained. Furthermore, we extend the Banach fixed point theorem to(fuzzy) contractive mappings in fuzzy multi-metric spaces.Keywords. Multi space; multi-metric space; triangular norm; fuzzy

metric space.M.S.C. (2000). 54A40; 54E35; 54E40; 54E45.

1. Introduction

In 1965, the concept of fuzzy set was introduced by Zadeh [13]. Manyauthors have introduced the concept of fuzzy metric space in differentways [1-3,5-8]. George and Veeramani [3,4] modified the concept offuzzy metric space introduced by Kramosil andMichalek [8] and defineda Hausdorff topology on this fuzzy metric space. They also showed thatevery metric induces a fuzzy metric.The notion of multi-spaces is introduced by Smarandache in [10]

under his idea of hybrid mathematics: combining different fields intoa unifying field [11]. The definition of multi-metric space is given byMao [9], combining Smarandache multi-spaces with the classical metricspaces. He also give some characteristics of a multi-metric space.In this paper, we give the notion of fuzzy multi-metric space by com-

bining Smarandache multi-spaces with the fuzzy metric space in thesense of George and Veeremani [3]. We give some theorems on cover-gence and continuity in fuzzy multi-metric space. Furthermore, we ex-tend the Banach fixed point theorem to (fuzzy) contractive mappings,in our sense and Grabiec’s sense [5], on complete fuzzy multi-metricspaces (in George and Veeramani’s sense).

2. Preliminaries

Definition 1 ([12]). A binary operation ∗ : [0, 1] × [0, 1] −→ [0, 1] iscontinuous t-norm if ∗ is satisfying the following conditions:

(i) ∗ is commutative and associative;(ii) ∗ is continuous;


(iii) a ∗ 1 = a for all a ∈ [0, 1];(iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1].

Definition 2 ([12]). A 3-tuple (X,M, ∗) is said to be a fuzzy metricspace if X is an arbitrary set, ∗ is a continuous t-norm and M is afuzzy set on X2 × (0,∞) satisfying the following conditions: for all x,y, z ∈ X, s, t > 0,

(i) M(x, y, t) > 0,(ii) M(x, y, t) = 1 if and only if x = y,(iii) M(x, y, t) =M(y, x, t),(iv) M(x, y, t) ∗M(y, z, s) ≤M(x, z, t+ s),(v) M(x, y, .) : (0,∞)→ [0, 1] is continuous.

Remark 1. In fuzzy metric space X, M(x, y, .) is non-decreasing forall x, y ∈ X.

Example 1. Let (X, d) be a metric space. Denote a ∗ b = ab for alla, b ∈ [0, 1] and let Md be a fuzzy set on X2× (0,∞) defined as follows:

Md(x, y, t) =ktn

ktn +md(x, y)

for all k,m, n ∈ R+. Then (X,Md, ∗) is a fuzzy metric space.

Remark 2. Note the above example holds even with the t-norm a∗ b =mina, b and henceM is a fuzzy metric with respect to any continuoust-norm. In the above example by taking k = m = n = 1, we get

Md(x, y, t) =t

t+ d(x, y)

We call this fuzzy metric induced by a metric d the standard fuzzymetric.

Definition 3 ([3]). Let (X,M, ∗) be a fuzzy metric space and let r ∈(0, 1), t > 0 and x ∈ X. The set

B(x, r, t) = y ∈ X :M(x, y, t) > 1− ris called the open ball with center x and radius r with respect to t.

Theorem 1 ([3]). Every open ball B(x, r, t) is an open set.

Remark 3. Let (X,M, ∗) be a fuzzy metric space. Define τ = A ⊂X : for each x ∈ X, there exist t > 0, r ∈ (0, 1) such that B(x, r, t) ⊂A. Then τ is a topology on X.

Remark 4.

(i) Since B(x, 1n, 1n) : n = 1, 2, ... is a local base at x, the topology

τ is first countable.(ii) Every fuzzy metric space is Hausdorff.

EFE ET AL368

(iii) Let (X,M, ∗) be an fuzzy metric space and τ be the topologyon X induced by the fuzzy metric. Then for a sequence (xn)nin X, xn −→ x if and only if M(xn, x, t) −→ 1 as n −→∞.

(iv) In a fuzzymetric space every compact set is closed and bounded.

Definition 4 ([3]). Let (X,M,N, ∗,♦) be a fuzzy metric space. Then,(i) A sequence (xn)n inX is said to be Cauchy if for each ε > 0 andeach t > 0, there exist n0 ∈ N such that M(xn, xm, t) > 1− εfor all n,m ≥ n0.

(ii) (X,M, ∗) is called complete if every Cauchy sequence conver-gent with respect to M .

Definition 5 ([10]). A multi-metric space is a union eX = ∪mi=1Xi suchthat each Xi is a metric space with metric di for all i, 1 ≤ i ≤ m. Whenwe say a multi-metric space eX = ∪mi=1Xi, it means that a multi-metricspace with metrics d1, d2, ..., dm such that (Xi, di) is a metric space forany integer i, 1 ≤ i ≤ m.

3. Main results

Definition 6. A fuzzy multi-metric space is a union eX = ∪mi=1Xi suchthat each Xi is a fuzzy metric space with fuzzy metric Mi and t-norm∗, for all i, 1 ≤ i ≤ m.

When we say a fuzzy multi-metric space eX = ∪mi=1Xi, it means thata fuzzy multi-metric space with fuzzy metricsM1,M2, ...,Mm such that(Xi,Mi, ∗) is a fuzzy metric space for any integer i, 1 ≤ i ≤ m.

Remark 5. The following two extremal cases are permitted in Defini-tion 6:

(i) There are integers i1, i2, ..., is such that Xi1 = Xi2 = · · · = Xis,where ij ∈ 1, 2, ...,m, 1 ≤ j ≤ s.

(ii) There are integers l1, l2, ..., ls such that Ml1 =Ml2 = · · · =Mls,where lj ∈ 1, 2, ...,m, 1 ≤ j ≤ s.

Definition 7. Let eX = ∪mi=1Xi be a fuzzy multi-metric space. Wedefine open ball B(x, r, t) with centre x ∈ eX and radius r, r ∈ (0, 1),t > 0 as B(x, r, t) = y ∈ eX : there exists an integer k, 1 ≤ k ≤ msuch that Mk(x, y, t) > 1− r.Remark 6. Let eX = ∪mi=1Xi be a fuzzy multi-metric space. Defineτ = A ⊂ eX : for each x ∈ eX, there exist t > 0, r ∈ (0, 1) such thatB(x, r, t) ⊂ A. Then τ is a topology on eX.Remark 7. Let eX = ∪mi=1Xi be a multi-metric space such that each(Xi, di) is a metric space for all i, 1 ≤ i ≤ m. We define a ∗ b = aband

Mi(x, y, t) =t

t+ di(x, y)

FUZZY MULTI-METRIC SPACES 369

then eX is a fuzzy multi-metric space. We call those Mi as the standardfuzzy multi-metric induced by di. Even if we take a ∗ b = min(a, b), eXwill be a fuzzy multi-metric space. The metric space (Xi, di) is complete

iff the standard fuzzy metric space³Xi,Midi

, ∗´is complete for all i,

1 ≤ i ≤ m. Then the multi-metric space eX is complete if and only ifthe standard fuzzy multi-metric space is complete.

Corollary 1. If M1,M2, ...,Mm are m fuzzy metrics on a space X,then M1 ∗ M2 ∗ ... ∗ Mm is a fuzzy metric on X, where ∗ is min orproduct t-norm.

Corollary 2. If d1, d2, ..., dm are m metrics on a space X, thent

t+ d1(x, y)∗ t

t+ d2(x, y)∗ · · · ∗ t

t+ dm(x, y)

is a fuzzy metric on X where a ∗ b = ab.

Definition 8. Let eX = ∪mi=1Xi be a fuzzy multi-metric space and (xn)nbe a sequence in eX. (xn)n is said converge to a point x, x ∈ eX if forany ε, ε ∈ (0, 1) there exist numbers n0 and i, 1 ≤ i ≤ m such that ifn ≥ n0 then Mi(xn, x, t) > 1− ε, for each t > 0.

For (xn)n convergence to a point x, x ∈ eX, we denote it by limn xn =x or xn −→ x.

Theorem 2. eX = ∪mi=1Xi be a fuzzy multi-metric space and (xn)n bea sequence in eX. Then xn −→ x iff there exist integers i, 1 ≤ i ≤ msuch that Mi(xn, x, t) −→ 1 as n −→∞.

Theorem 3. A sequence (xn)n in a fuzzy multi-metric space eX =∪mi=1Xi is convergent if and only if there exist integers n0 and k, 1 ≤k ≤ m, such that the subsequence xn : n ≥ n0 is a convergent se-quence in (Xk,Mk, ∗).Proof. If (xn)n is a convergent sequence in the fuzzy multi-metric spaceeX, by definition for any ε, ε ∈ (0, 1), there exist a point x, x ∈ eX andnatural numbers n0(ε) and k, 1 ≤ k ≤ m, such that if n ≥ n0(ε), thenMk(xn, x, t) > 1− ε for all t > 0. That is, xn : n ≥ n0(ε) ⊂ Xk andxn : n ≥ n0(ε) is a convergent sequence in (Xk,Mk, ∗).If there exist integer n0 and k, 1 ≤ k ≤ m, such that xn : n ≥ n0

is a convergent subsequence in (Xk,Mk, ∗), then for any ε, ε ∈ (0, 1),by definition there exists a positive integer p0 and a point x, x ∈ eXsuch that Mk(xn, x, t) > 1 − ε, for all t > 0, where n ≥ maxn0, p0.Hence, (xn)n is a convergent sequence in the fuzzy multi-metric spaceeX. ¤Theorem 4. Let eX = ∪mi=1Xi be a fuzzy multi-metric space, (xn)n,(yn)n are sequences in eX and (tn)n ⊂ (0,∞). If xn −→ x0, yn −→ y0,

EFE ET AL370

tn −→ t and there is an integer p, 1 ≤ p ≤ m such that x0, y0 ∈ Xp,t > 0, then limnMp(xn, yn, tn) =Mp(x0, y0, t).

Proof. Since xn −→ x0 and yn −→ y0 there exist integers n1 and n2such that if n ≥ maxn1, n2, then xn, yn ∈ Xp. Fix δ > 0 such thatδ < t/2. Then, there exists an integer n3 such that |tn − t| < δ for alln ≥ n0 = maxn1, n2, n3. Hence,

Mp(xn, yn, tn) ≥Mp (xn, x0, δ) ∗Mp(x0, y0, t− 2δ) ∗Mp (yn, y0, δ)

and

Mp(x0, y0, t+ 2δ) ≥Mp (xn, x0, δ) ∗Mp(xn, yn, tn) ∗Mp (yn, y0, δ)

for all n ≥ n0. By taking limits when n −→∞, we getlimnMp(xn, yn, tn) ≥ 1 ∗Mp(x0, y0, t− 2δ) ∗ 1

andMp(x0, y0, t+ 2δ) ≥ 1 ∗ lim

nMp(xn, yn, tn) ∗ 1

respectively. So, by continuity of the function t −→ Mp(x, y, t) weobtain

limnMp(xn, yn, tn) =Mp(x0, y0, t).

¤Theorem 5. If (xn)n is a convergent sequence in a fuzzy multi metricspace eX = ∪mi=1Xi, then (xn)n has only one limit point.

Proof. Let xn −→ x1, xn −→ x2 and x1, x2 ∈ eX. Then there existinteger n0 and i, 1 ≤ i ≤ m such that xn ∈ Xi for all n ≥ n0. Hence,

1 ≥Mi(x1, x2, t) ≥Mi

µx1, xn,

t

2

¶∗Mi

µxn, x2,

t

2

¶.

By taking limits when n −→∞, we get1 ≥Mi(x1, x2, t) ≥ 1 ∗ 1 = 1

which implies x1 = x2. ¤Theorem 6. Any convergent sequence in a fuzzy multi metric space isa bounded points set.

Proof. It is clear from Theorem 5. ¤

Definition 9. A sequence (xn)n in a fuzzy multi-metric space eX =∪mi=1Xi is called Cauchy sequence if for each ε, ε ∈ (0, 1), t > 0, thereexist integers n0(ε) and s, 1 ≤ s ≤ m such that Ms(xn, xm, t) > 1− εfor all n,m ≥ n0(ε).

Theorem 7. A Cauchy sequence (xn)n in a fuzzy multi-metric spaceeX = ∪mi=1Xi is convergent if and only if for all k, 1 ≤ k ≤ m, |(xn)n∩Xk| is finite or infinite but (xn)n ∩Xk is convergent in (Xk,Mk, ∗).

Proof. The necessity of conditions is clear from Theorem 3.Now, we prove the sufficiency. By definition, there exist integers s,

1 ≤ s ≤ m and n1 such that xn ∈ Xs for n ≥ n1. If |(xn)n ∩ Xk|is infinite and limn→∞ ((xn)n ∩Xk) = x, then there must be k = s.Denoted by (xn)n ∩ Xk = xk1, xk2, ..., xkn, .... For any δ, δ ∈ (0, 1),there exists an integer n2, n2 ≥ n1 such that Mk(xm, xn,

t2) > 1 − δ

and Mk(xkn, x,t2) > 1 − δ for all m,n ≥ n2 and for all t > 0. Since

δ ∈ (0, 1), we can find a ε, ε ∈ (0, 1), such that (1− δ)∗ (1− δ) > 1− ε.Then, by using Theorem 4, we get that

Mk(xn, x, t) ≥ Mk

µxn, xkn,

t

2

¶∗Mk

µxkn , x,

t

2

¶≥ (1− δ) ∗ (1− δ) > 1− ε.

Hence, limn xn = x which completes the proof. ¤Definition 10. A fuzzy multi-metric space is said to be complete ifevery Cauchy sequence is convergent.

Theorem 8. Let eX = ∪mi=1Xi be a complete fuzzy multi-metric space.For a ball sequence (B(xn, εn, t))n, where 0 < εn < 1 for n = 1, 2, ...,the following conditions hold:

(i) B(x1, ε1, t) ⊃ B(x2, ε2, t) ⊃ ... ⊃ B(xn, εn, t) ⊃ ...(ii) limn εn = 0.Then, ∩∞n=1B(xn, εn, t) only has one point.

Proof. First, we prove that the sequence (xn)n is a Cauchy sequencein eX. By the condition (i), we know that if m ≥ n, then xm ∈B(xm, εm, t) ⊂ B(xn, εn, t) for all t > 0. Whence, for all i, 1 ≤ i ≤ m,Mi(xm, xn, t) > 1 − εn for xm, xn ∈ Xi. For any ε, ε ∈ (0, 1), sincelimn εn = 0, there exists an integer n0(ε) such that if n > n0(ε), thenεn < ε. Therefore, if xn ∈ Xl, then limm xm = xn. Whence, thereexists an integer n0 such that m ≥ n0, xm ∈ Ml by Theorem 3. Takeintegers m,n ≥ maxn0, n0(ε). We know that

Ml(xm, xn, t) > 1− εn > 1− ε.

So, (xn)n is a Cauchy sequence.By the assumption, eX is complete. We know that the sequence (xn)n

is convergence to a point x0, x0 ∈ eX. By conditions (i) and (ii), we havethat Ml(x0, xn, t) > 1− εn as m −→∞. Hence, x0 ∈ ∩∞n=1B(xn, εn, t).Now if there is a point y ∈ ∩∞n=1B(xn, εn, t), then there must be

y ∈ Xl. We get that

1 ≥Ml(y, x0, t) = limnMl(y, xn, t) ≥ lim

n(1− εn) = 1

for all t > 0, by Theorem 4. Therefore, Ml(y, x0, t) = 1 which impliesy = x0. ¤

EFE ET AL372

Definition 11. Let fX1 and fX2 be two fuzzy multi-metric spaces and fbe a mapping from fX1 to fX2, x0 ∈ fX1 and f(x0) = y0. For ε, ε ∈ (0, 1)if there exists a number δ, δ ∈ (0, 1) such that for all x ∈ B(x0, δ, t),f(x) = y ∈ B(y0, ε, t) ⊂ fX2 , i.e., f(B(x0, δ, t)) ⊂ B(y0, ε, t), for allt > 0, then we say f is continuous at point x0. If f is continuous atevery point of fX1, then f is said to a continuous mapping from fX1 tofX2.

Proposition 1. Let fX1 and fX2 be two fuzzy multi-metric spaces, f bea continuous mapping from fX1 to fX2 and (xn)n be a sequence in fX1.If xn −→ x, then f(xn) −→ f(x).

4. Fixed points for fuzzy multi-metric spaces

Definition 12. Let eX = ∪mi=1Xi be a fuzzy multi-metric space andT : eX → eX be a mapping. x∗ ∈ eX is called a fixed point of T ifTx∗ = x∗. Denote the number of fixed points of a mapping T in eX byΦ‡(T ).

Definition 13. Let eX = ∪mi=1Xi be a fuzzy multi-metric space. Wewill say the mapping f : eX −→ eX is fuzzy contractive if there existk ∈ (0, 1), 1 ≤ i, j ≤ m, such that

1

Mj(f(x), f(y), t)− 1 ≤ k

µ1

Mi(x, y, t)− 1¶

for each x, y ∈ eX and t > 0. k ∈ (0, 1), is called contractive constantof f .

Proposition 2. Let eX = ∪mi=1Xi be a multi-metric space where each(Xi, di) is a metric space for any integer i, 1 ≤ i ≤ m. The mappingf : eX −→ eX is contractive (a contraction) on the multi-metric space eXwith contractive constant k iff f is fuzzy contractive, with contractiveconstant k, on the standard fuzzy multi-metric space eX = ∪mi=1Xi such

that³Xi,Midi

, ∗´standard fuzzy metric space induced by di for all

1 ≤ i ≤ m.

Definition 14. A sequence (xn)n in a multi-metric space eX = ∪mi=1Xi

is said to be contractive if there exists k ∈ (0, 1) such that dj(xn+1, xn+2) ≤kdi(xn, xn+1), for all n ∈ N and 1 ≤ i, j ≤ m.

Definition 15. Let eX = ∪mi=1Xi be a fuzzy multi-metric space. Wewill say that the sequence (xn)n in eX is fuzzy contractive if there existsk ∈ (0, 1) such that

1

Mj(xn+1, xn+2, t)− 1 ≤ k

µ1

Mi(xn, xn+1, t)− 1¶

for all t > 0, n ∈ N and 1 ≤ i, j ≤ m.


Proposition 3. Let eX = ∪mi=1Xi be the standard fuzzy multi-metricspace induced by the metric di on Xi for all 1 ≤ i ≤ m. The sequence(xn)n in eX is contractive in multi-metric space iff (xn)n is fuzzy con-tractive in standard fuzzy multi-metric space.

Next, we extend the Banach fixed point theorem to fuzzy contractivemappings of complete fuzzy multi-metric spaces.

Theorem 9. Let eX = ∪mi=1Xi be a complete fuzzy multi-metric spacein which fuzzy contractive sequences are Cauchy. Let T : eX −→ eXbe a fuzzy contractive mapping being k the contractive constant. Then1 ≤ Φ‡(T ) ≤ m.

Proof. Fix x ∈ eX. Let xn = Tn(x), n ∈ N. We have for t > 01

Mi(T (x), T 2(x), t)− 1 ≤ k

µ1

Mi(x, x1, t)− 1¶

and by induction

1

Mi(xn+1, xn+2, t)− 1 ≤ k

µ1

Mi(xn, xn+1, t)− 1¶,

n ∈ N.Then (xn)n is a fuzzy contractive sequence, so it is a Cauchy sequence

and, hence, (xn)n converges to z∗, for some z∗ ∈ eX. We will see z∗ isa fixed point for T . By Theorem 2, we have

1

Mi(T (y), T (xn), t)− 1 ≤ k

µ1

Mi(y, xn, t)− 1¶−→ 0

as n −→ ∞. Then limnMi(T (y), T (xn), t) = 1 for each t > 0, and,therefore, limn T (xn) = T (z∗), i.e., limn xn+1 = T (z∗) and then T (z∗) =z∗.For other chosen points u0, v0 ∈ X1, we can also define recursively

un+1 = Tun, vn+1 = Tvn and get the limit points, there exists aninteger i0 such that, limn un = limn vn = w∗ ∈ Xi0, Tu

∗ ∈ Xi0 . Thenfor t > 0 we have

1

Mi0(z∗, u∗, t)

− 1 =1

Mi0(T (z∗), T (u∗), t)

− 1

≤ k

µ1

Mi0(z∗, u∗, t)

− 1¶

= k

µ1

Mi0(T (z∗), T (u∗), t)

− 1¶

≤ k2µ

1

Mi0(z∗, u∗, t)

− 1¶

≤ · · · ≤ knµ

1

Mi0(z∗, u∗, t)

− 1¶−→ 0

EFE ET AL374

as n −→∞.Hence, Mi0(z

∗, u∗, t) = 1 and then z∗ = u∗.Similar consider the points in Xi, 2 ≤ i ≤ m, we get 1 ≤ Φ‡(T ) ≤

m. ¤Now suppose

³ eX,Midi, ∗´is a complete standard fuzzy multi-metric

space where (Xi,Midi , ∗) fuzzy metric space induced by the metric diin Xi for all 1 ≤ i, j ≤ m. From Remark 7 (Xi, di) is complete, then if(xn)n is a fuzzy contractive sequence, by Proposition 3 it is contractivein (Xi, di), hence convergent. So, from Theorem 9 we have the follow-ing corollary, which can be considered the fuzzy version of the classicBanach contraction theorem on complete metric spaces.

Corollary 3. Let eX be a complete standard fuzzy multi-metric spaceand let T : eX −→ eX a fuzzy contractive mapping. Then 1 ≤ Φ‡(T ) ≤m.

References

[1] Z. K. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl. 86(1982) 74−95.[2] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69(1979)

205− 230.[3] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets

and Systems 64(1994) 395− 399.[4] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces,

Fuzzy Sets and Systems 90(1997) 365− 368.[5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems

27(1988) 385− 389.[6] V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy

Sets and Systems 125(2002) 245− 252.[7] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems

12(1984) 225− 229.[8] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kyber-

netica 11(1975) 326− 334.[9] L. Mao, On multi-metric spaces, eprint arXiv: math.GM/0510480, 1/2005.[10] F. Smarandache, Mixed noneuclidean geometries, eprint arXiv:

math/0010119, 10/2000.[11] F. Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Prob-

ability, Set, and Logic, American research Press, Rehoboth 1999.[12] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. 10(1960)

314− 334.[13] L. A. Zadeh, Fuzzy sets, Inform and Control 8(1965) 338− 353.


Statistical Convergence of Double Sequences onIntuitionistic Fuzzy Normed Spaces

Sevda Karakus and Kamil Demirci

Department of Mathematics, Faculty of Sciences and Arts Sinop,Ondokuz Mayis University, 57000 Sinop-TURKEYE-mail: [email protected] , [email protected]

Abstract

The concept of statistical convergence was presented by Steinhaus(1951). This concept was extended to the double sequences by Mursaleenand Edely (2003). In this paper, we dene and study statistical analogueof convergence and Cauchy for double sequences on intuitionistic fuzzynormed spaces. Then we give a useful characterization for statisticallyconvergent double sequences. Furthermore, we display an example suchthat our method of convergence is stronger than the usual convergencefor double sequences on intuitionistic fuzzy normed spaces.

KEY WORDS: Natural double density, statistical convergence, con-tinuous tnorm, continuous tconorm, intuitionistic fuzzy normed space.

1 Introduction

In 1965, the concept of fuzzy sets was introduced by Zadeh [29]. Then manyauthors developed the theory of fuzzy set and applications. The fuzzy logichas been used many elds, like metric and topological spaces [9] ; [10] ; [16] ; [19],theory of functions [4] ; [18] ; [28] ; computer programing [17], econometrics andother elds [1] ; [2] ; [3] ; [8] ; [20] ; [22] : Also, recently, the concepts of intuitionisticfuzzy metric space has been studied by Park [23], and intuitionistic fuzzy normedspace have been studied by Saadati and Park [25] :In this paper we give statistical analogues of convergence and Cauchy for

double sequences which studied in Mursaleen and Osama [21] on intuitionis-tic fuzzy normed spaces:Also we display an example such that our method ofconvergence is stronger than the usual convergence for double sequences onintuitionistic fuzzy normed spaces.Now we recall some notations and denitions which we used in the paper.

Denition 1 [26]A binary operation : [0; 1] [0; 1] ! [0; 1] is said to be acontinuous tnorm if it satises the following conditions:

1


(a) is associative and commutative,

(b) is continuous,

(c) a 1 = a for all a 2 [0; 1];

(d) a b c d whenever a c and b d for each a; b; c; d 2 [0; 1]:

Two typical examples of continuous tnorm are a b = ab and a b =min (a; b) for all a; b 2 [0; 1].

Denition 2 [26]A binary operation : [0; 1] [0; 1] ! [0; 1] is said to be acontinuous tconorm if it satises the following conditions:

(a) is associative and commutative,

(b) is continuous,

(c) a 0 = a for all a 2 [0; 1];

(d) a b c d whenever a c and b d for each a; b; c; d 2 [0; 1]:

Two typical examples of continuous tconorm are a b = min (a+ b; 1) anda b = max (a; b) for all a; b 2 [0; 1].Now we give the concept of intuitionistic fuzzy normed space which has

recently introduced by Saadati and Park [25].

Denition 3 [25] The 5tuple (V; ; ; ; ) is said to be an intuitionistic fuzzynormed space (IFNS) if V is a vector space, is a continuous tnorm, is acontinuous tconorm, and ; fuzzy sets on V (0;1) satisfying the followingconditions for every x; y 2 V and s; t > 0 :

(a) (x; t) + (x; t) 1;

(b) (x; t) > 0;

(c) (x; t) = 1 if and only if x = 0;

(d) (x; t) = x; t

jj

for each 6= 0,

(e) (x; t) (y; s) (x+ y; t+ s);

(f) (x; ) : (0;1)! [0; 1] is continuous,

(g) limt!1

(x; t) = 1 and limt!0

(x; t) = 0;

(h) (x; t) < 1;

(i) (x; t) = 0 if and only if x = 0;

(j) (x; t) = x; t

jj

for each 6= 0,

2

KARAKUS-DEMIRCI378

(k) (x; t) (y; s) (x+ y; t+ s);

(l) (x; ) : (0;1)! [0; 1] is continuous,

(m) limt!1

(x; t) = 0 and limt!0

(x; t) = 0:

In this case (; ) is called an intuitionistic fuzzy norm. We can give anexample as follow:Let (V; kk) be a normed space, and let a b = ab and a b = minfa+ b; 1g

for all a; b 2 [0; 1]: If we dene

0(x; t) :=t

t+ kxk and 0(x; t) :=kxk

t+ kxk :

for all x 2 V and every t > 0; then observe that (V; ; ; ; ) is an intuitionisticfuzzy normed space.Before we present the new denitions and the main theorems, we shall recall

some concepts which we need.By the convergence of a double sequence we mean the convergence in Pring-

sheims sense [24]. A double sequence x = (xjk)1jk=0 is called convergent in the

Pringsheims sense if for every " > 0 there exists N 2 N such that jxjk Lj < "whenever j; k N . L is called the Pringsheim limit of x.A double sequence x = (xjk) is said to be Cauchy sequence if for every " > 0

there exists N 2 N such that jxpq xjkj < " for all p j N , q k N .A double sequence x is bounded if there exists a positive number M such

that jxjkj < M for all j and k.So we can give the (; ) analogue of above two denitions as follow:

Denition 4 Let (V; ; ; ; ) be an IFNS. Then, a double sequence x = (xjk)is said to be convergent to L 2 V with respect to the intuitionistic fuzzy norm(; ) if, for every " > 0 and t > 0; there exists N 2 N such that (xjkL; t) >1 " and (xjk L; t) < " for all j; k N: It is denoted by (; v)2 limx = Lor xjk

(;)2! L as j; k !1:

Denition 5 Let (V; ; ; ; ) be an IFNS. Then, a double sequence x = (xjk)is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm(; ) provided that, for every " > 0 and t > 0; there exists N = N (") andM = M (") such that (xjk xpq; t) > 1 " and (xjk xpq; t) < " for allj; p N , k; q M

Now we rst recall statistical convergence and then in new section, we in-troduce basic denitions and properties which we mention above .

2 Statistical Convergence of Double Sequenceon IFNS

Steinhaus [27] introduced the idea of statistical convergence (see also Fast [11]).If K is a subset of N, the set of natural numbers, then the asymptotic density

3

STATISTICAL CONVERGENCE... 379

of K denoted by (K), is given by

(K) := limn

1

njfk n : k 2 Kgj

whenever the limit exists, when jAj denotes the cardinality of the set A. Asequence x = (xk) of numbers is statistically convergent to L if

(fk 2 N : jxk Lj "g) = 0

for every " > 0. In this case we write st limx = L.Statistical convergence has been investigated in a number of paper [6] ; [7] ;

[12] ; [13] ; [14] ; [15].Now we recall the concept of statistical convergence of double sequences.Let K NN be a two-dimensional set of positive integers and let K (n;m)

be the numbers of (i; j) in K such that i n and j m. Then the two-dimensional analogue of natural density can be dened as follows.The lower asymptotic density of a set K N N is dened as

2 (K) = limn;m

infK (n;m)

nm:

In case the sequence (K (n;m)nm) has a limit in Pringsheims sense [24]then we say that K has a double natural density and is dened as

limn;m

K (n;m)

nm= 2 (K) :

If we consider the set of K =i2; j2

: i; j 2 N

, then

2 (K) = limn;m

K (n;m)

nm lim

n;m

pnpm

nm= 0:

Also, if we consider the set of f(i; 2j) : i; j 2 Ng has double natural density 1=2.If we set n = m, we have a two-dimensional natural density considered by

Christopher [5].Now we recall the concepts of statistically convergent and statistically Cauchy

for double sequence as follows:

Denition 6 [21] A real double sequence x = (xjk) is said to be statisticallyconvergent the number ` provided that, for each " > 0, the set

f(j; k) ; j n and k m : jxjk `j "g

has double natural density zero. In this case we write st2 limj;kxjk = `.

Denition 7 [21]A real double sequence x = (xjk) is said to be statisticallyCauchy provided that, for every " > 0 there exist N = N (") and M = M (")such that for all j; p N; k; q M; the set

f(j; k) ; j n and k m : jxjk xpqj "g

has double natural density zero.

4

KARAKUS-DEMIRCI380

Now we give the analogues of these with respect to the intuitionistic fuzzynorm (; ).

Denition 8 Let (V; ; ; ; ) be an IFNS. A real double sequence x = (xjk)is statistically convergent to L 2 V with respect to the intuitionistic fuzzy norm(; ) provided that, for every " > 0 and t > 0;

K = f(j; k) ; j n and k m : (xjkL; t) 1 " or (xjkL; t) "g (1)

has double natural density zero, i.e., if K (n;m) be the numbers of (j; k) in K

limn;m

K (n;m)

nm= 0: (2)

In this case we write st(;)2 limj;k xjk = L; where L is said to be st(;)2limit.Also we denote the set of all statistically convergent double sequences with respectto the intuitionistic fuzzy norm (; ) by st(;)2 .

By using (2) and the well-known properties of the double natural density,we easily get the following lemma.

Lemma 9 Let (V; ; ; ; ) be an IFNS. Then, for every " > 0 and t > 0; thefollowing statements are equivalent:

(i) st(;)2 limj;k xjk = L

(ii) 2f(j; k) ; j n and k m : (xjk L; t) 1 "g = 2f(j; k) ; j n and k m : (xjk L; t) "g = 0:

(iii) 2f(j; k) ; j n and k m : (xjk L; t) > 1 " and (xjk L; t) <"g = 1:

(iv) 2f(j; k) ; j n and k m : (xjk L; t) > 1 "g = 2f(j; k) ; j n and k m : (xjk L; t) < "g = 1:

(v) st2 lim(xjk L; t) = 1 and st2 lim (xjk L; t) = 0:

Theorem 10 Let (V; ; ; ; ) be an IFNS. If a double sequence x = (xjk) isstatistically convergent with respect to the intuitionistic fuzzy norms (; ); thenthe st(;)2limit is unique.

Proof. Let x = (xjk) be a double sequence. Suppose that st(;)2 limx = L1and st(;)2 limx = L2: Let t > 0 and " > 0: Choose r 2 (0; 1) such that(1 r) (1 r) 1 " and r r ": Then, we dene the following sets:

K;1(r; t) : = f(j; k) 2 N N : (xjk L1; t) 1 rg ;K;2(r; t) : = f(j; k) 2 N N : (xjk L2; t) 1 rg ;K;1(r; t) : = f(j; k) 2 N N : (xjk L1; t) rg ;K;2(r; t) : = f(j; k) 2 N N : (xjk L2; t) rg :

5

Since st(;)2 limx = L1; we have

2fK;1("; t)g = 2fK;1("; t)g = 0 for all t > 0:

Furthermore, using st(;)2 limx = L2; we get


Now let K;("; t) := fK;1("; t) [K;2("; t)g \ fK;1("; t) [K;2("; t)g : Thenobserve that 2fK;("; t)g = 0 which implies 2fN N=K;("; t)g = 1: If(j; k) 2 N N=K;("; t); then we have two possible cases. The former isthe case of (j; k) 2 N N=fK;1("; t) [ K;2("; t)g; and the letter is (j; k) 2N N= fK;1("; t) [K;2("; t)g : We rst consider that

(j; k) 2 N N= fK;1("; t) [K;2("; t)g :

Then we have

(L1 L2; t) (xjk L1;t

2) (xjk L2;

t

2)

> (1 r) (1 r) 1 ":

Since " > 0 was arbitrary, we get (L1 L2; t) = 1 for all t > 0;which yieldsL1 = L2: On the other hand, if (j; k) 2 N N=fK;1("; t)[K;2("; t)g; then wemay write that

(L1 L2; t) (xjk L1;t

2) (xjk L2;

t

2)

< r r ":

Again, since " > 0 was arbitrary, we have (L1 L2; t) = 0 for all t > 0; whichimplies L1 = L2: Therefore, in all cases, we conclude that the st(;)2limit isunique.

Theorem 11 Let (V; ; ; ; ) be an IFNS. If (; )2 limx = L for a doublesequence x = (xjk) ; then st(;)2 limx = L:

Proof. By hypothesis, for every " > 0 and t > 0; there is a number N 2 N suchthat

(xjk L; t) > 1 " and (xjk L; t) < "for all j N and k N: This guarantees that the set

f(j; k) 2 N N : (xjk L; t) 1 " or (xjk L; t) "g

has at most nitely many terms. Since every nite subset of the natural numbershas double density zero, we immediately see that

2f(j; k) 2 N N : (xjk L; t) 1 " or (xjk L; t) "g = 0;

whence the result.The following example shows that the converse of Theorem 11 is not hold in

general.

6

KARAKUS-DEMIRCI382

Example 12 Let (R,jj) denote the space of real numbers with the usual norm,and let a b = ab and a b = minfa + b; 1g for all a; b 2 [0; 1]: For all x 2 Rand every t > 0; consider

0(x; t) :=t

t+ jxj and 0(x; t) :=jxj

t+ jxj :

In this case observe that (R; ; ; ; ) is an IFNS. Now dene a double sequencex = (xjk) whose terms are given by

xjk :=

1; if j and k are squares0; otherwise.

(3)

Then, for every 0 < " < 1 and for any t > 0; let Kn("; t) := f(j; k) ; j n and k m : 0(xjk; t) 1 " or 0(xjk; t) "g: Since

Kn("; t) =

(j; k) ; j n and k m :

t

t+ jxjkj 1 " or jxjkj

t+ jxjkj "

=

(j; k) ; j n and k m : jxjkj

"t

1 " > 0

= f(j; k) ; j n and k m : xjk = 1g= f(j; k) ; j n and k m : j and k are squaresg

we have

2 (Kn("; t)) limn;m

pnpm

nm= 0:

Hence, we get st(0;0)2limx = 0: However, since the sequence x = (xjk) givenby (3) is not convergent in the space (R,jj); by Lemma 4.10 of [25], we also seethat x is not convergent with respect to the intuitionistic fuzzy norm (0; 0):

Theorem 13 Let (V; ; ; ; ) be an IFNS. Then st(;)2 limx = L if andonly if there exists a subset K = f(j; k)g N N, j; k = 1; 2; :::; such that2 (K) = 1 and (; )2 lim

j;k!1(j;k)2K

xjk = L.

Proof. We rst assume that st(;)2 limx = L: Now, for any t > 0 and j 2 N;let

Kr :=

(j; k) 2 N N : (xjk L; t) 1

1

ror (xjk L; t)

1

r

and

Mr =

(j; k) 2 N N : (xjk L; t) > 1

1

rand (xjk L; t) <

1

r

;

(r = 1; 2; :::) :

Then 2 (Kr) = 0 and

7

(1) M1 M2 ::: Mi Mi+1 :::and

(2) 2 (Mr) = 1; r = 1; 2; :::Now we have to show that for (j; k) 2Mr; (xjk) is convergent to L. Suppose

that (xjk) is not convergent to L: Therefore there is " > 0 such that

f(j; k) 2 N N : (xjk L; t) 1 " or (xjk L; t) "g

for innitely many terms.Let

M" = f(j; k) 2 N N : (xjk L; t) > 1 " and (xjk L; t) < "g

and " > 1r (r = 1; 2; :::) :

Then(3) 2 (M") = 0,

and by (1), Mr M". Hence 2 (Mr) = 0 which contradicts (2). Therefore(xjk) is convergent to L.

Conversely, suppose that there exists a subset K = f(j; k)g N Nsuch that 2 (K) = 1 and (; )2 lim

j;kxjk = L, i.e. there exists N 2 N such

that for every " > 0 and t > 0

(xjk L; t) > 1 " and (xjk L; t) < "; 8j; k N:

Now

K" = f(j; k) 2 N N : (xjk L; t) 1 " or (xjk L; t) "g N N f(jN+1; kN+1) ; (jN+2; kN+2) ; :::g :

Therefore 2 (K") 1 1 = 0. Hence x is statistically convergent to L:

Denition 14 Let (V; ; ; ; ) be an IFNS. We say that a double sequencex = (xjk) is statistically Cauchy with respect to the intuitionistic fuzzy norm(; ) provided that, for every " > 0 and t > 0; there exist N = N (") andM =M (") such that for all j; p N , k; q M , the set

f(j; k) ; j n; k m : (xjk xpq; t) 1 " or (xjk xpq; t) "g

has double natural density zero.

Now using a similar technique in the proof of Theorem 13 one can get thefollowing result at once.

Theorem 15 Let (V; ; ; ; ) be an IFNS, and let x = (xjk) be a double se-quence whose terms are in the vector space V . Then, the following conditionsare equivalent:

(i) x is a statistically Cauchy sequence with respect to the intuitionistic fuzzynorm (; ).

8

KARAKUS-DEMIRCI384

(ii) There exists an increasing index sequence K = f(j; k)g N N, j; k =1; 2; ::: such that 2 (K) = 1 and the subsequence fxjkg(j;k)2K is a Cauchysequence with respect to the intuitionistic fuzzy norm (; ):

Now we show that statistically convergence of double sequences on IFNS hassome arithmetical properties similar to properties of the usual convergence onR.

Lemma 16 Let (V; ; ; ; ) be an IFNS. If st(;)2 limxjk = and st(;)2lim yjk = then st(;)2 lim (xjk + yjk) = + :

Proof. Let st(;)2 limxjk = , st(;)2 lim yjk = , t > 0 and " 2 (0; 1).Choose r 2 (0; 1) such that (1 r) (1 r) 1 " and r r ". Then wedene the following sets:

K;1(r; t) : = f(j; k) 2 N N : (xjk ; t) 1 rg ;K;2(r; t) : = f(j; k) 2 N N : (yjk ; t) 1 rg ;K;1(r; t) : = f(j; k) 2 N N : (xjk ; t) rg ;K;2(r; t) : = f(j; k) 2 N N : (yjk ; t) rg :

Since st(;)2 limxjk = ; we have


Similarly, since st(;)2 lim yjk = ; we get


Now let K;("; t) := fK;1("; t) [K;2("; t)g \ fK;1("; t) [K;2("; t)g : Thenobserve that 2fK;("; t)g = 0 which implies 2fN N=K;("; t)g = 1: If(j; k) 2 N N=K;("; t); then we have two possible cases. The former isthe case of (j; k) 2 N N=fK;1("; t) [ K;2("; t)g; and the letter is (j; k) 2N N= fK;1("; t) [K;2("; t)g : We rst consider that

(j; k) 2 N N= fK;1("; t) [K;2("; t)g :

Then we have

((xjk ) + (yjk ) ; t) (xjk ;t

2) (yjk ;

t

2)

> (1 r) (1 r) 1 ":

On the other hand, if (j; k) 2 N N=fK;1("; t)[K;2("; t)g; then we can writethat

((xjk ) + (yjk ) ; t) (xjk ;t

2) (xjk ;

t

2)

< r r ":

9

This show that

2

(j; k) 2 N N : ((xjk ) + (yjk ) ; t) 1 "

or ((xjk ) + (yjk ) ; t) "

= 0

so st(;)2 lim (xjk + yjk) = + .

Lemma 17 Let (V; ; ; ; ) be an IFNS. If st(;)2 limxjk = and 2 Rthen st(;)2 limxjk = :

Proof. Let st(;)2 limxjk = , " 2 (0; 1) and t > 0. First of all we considerthe case of = 0. In this case

(0xjk 0; t) = (0; t) = 1 > 1 ":

Similarly we observe that

(0xjk 0; t) = (0; t) = 0 < "

for = 0: So we obtain (; )2 lim 0x = 0. Then from Theorem 11 we havest(;)2 lim 0xjk = 0:Now we consider the case of 2 R ( 6= 0) : From denition we can write

2 (f(j; k) 2 N N : (xjk ; t) 1 " or (xjk ; t) "g) = 0:

So, if we dene the sets:

K;1("; t) : = f(j; k) 2 N N : (xjk ; t) 1 "gK;1("; t) : = f(j; k) 2 N N : (xjk ; t) "g

then we can say 2 fK;1 ("; t)g = 2 fK;1("; t)g = 0 for all t > 0. Nowlet K;("; t) = K;1("; t) [ K;1("; t) then 2 fK;("; t)g = 0 which implies2 fN NK;("; t)g = 1. If (j; k) 2 N NK;("; t) then for 2 R ( 6= 0)

(xjk ; t) = (xjk ;t

jj )

(xjk ; t) (0;t

jj t)

= (xjk ; t) 1= (xjk ; t) > 1 ":

Similarly, we observe that for 2 R ( 6= 0)

(xjk ; t) = (xjk ;t

jj )

(xjk ; t) (0;t

jj t)

= (xjk ; t) 0= (xjk ; t) < ":

10

KARAKUS-DEMIRCI386

This show that

2 (f(j; k) 2 N N : (xjk ; t) 1 " or (xjk ; t) "g) = 0

so st(;)2 limxjk = .

Lemma 18 Let (V; ; ; ; ) be an IFNS. If st(;)2 limxjk = and st(;)2lim yjk = then st(;)2 lim (xjk yjk) = .

Proof. The proof is clear from Lemma 16 and Lemma 17.

Denition 19 Let (V; ; ; ; ) be an IFNS . We say that a double sequencex = (xjk) is IF-bounded if there exist t > 0 and 0 < r < 1 such that (xjk; t) >1 r and (xjk; t) < r for every (j; k) 2 N N.

Denition 20 Let (V; ; ; ; ) be an IFNS . For t > 0, we dene open ballB (x; r; t) with center x 2 V and radius 0 < r < 1, as

B (x; r; t) = fy 2 V : (x y; t) > 1 r ; (x y; t) < rg :

It follows from Lemma 16 Lemma 17 and Lemma 18, that the set of allIF-bounded statistically convergent double sequences on IFNS is a linear sub-space of the linear normed space `(;)21 (V ) of all IF-bounded sequences onIFNS.

Theorem 21 Let (V; ; ; ; ) be an IFNS and the set st(;)2 (V )\ `(;)21 (V )

is closed linear subspace of the set `(;)21 (V ).

Proof. It is clear that st(;)2 (V ) \ `(;)21 (V ) st(;)2 (V ) \ `

(;)21 (V ) :

Now we show that st(;)2 (V ) \ `(;)21 (V ) st(;)2 (V ) \ `

(;)21 (V ). Let

y 2 st(;)2 (V ) \ `(;)21 (V ) . Since B (y; r; t)\

st(;)2 (V ) \ `

(;)21 (V )6= ?

, there is a x 2 B (y; r; t) \st(;)2 (V ) \ `

(;)21 (V ).

Let t > 0 and " 2 (0; 1). Choose r 2 (0; 1) such that (1 r) (1 r) 1"and r r ". Since x 2 B (y; r; t) \

st(;)2 (V ) \ `

(;)21 (V ), there is a set

K N N with (K) = 1 such that

yjk xjk;

t

2

> 1 r and

yjk xjk;

t

2

< r

and

xjk;

t

2

> 1 r and

xjk;

t

2

< r

for all (j; k) 2 K. Then we have

(yjk; t) = (yjk xjk + xjk; t)

yjk xjk;

t

2

xjk;

t

2

> (1 r) (1 r) 1 "

11

and

(yjk; t) = (yjk xjk + xjk; t)

yjk xjk;

t

2

xjk;

t

2

< r r "

for all (j; k) 2 K. Hence

2 (f(j; k) 2 N N : (yjk; t) > 1 " and (yjk; t) < "g) = 1

and thus y 2 st(;)2 (V ) \ `(;)21 (V ) :

Conclusion 22 In this paper, we obtained results on statistical convergence inintuitionistic fuzzy normed spaces. As every ordinary norm induces a intuition-istic fuzzy norm, the results obtained here are more general than the correspond-ing of normed spaces.

3 REFERENCES

[1] C. D. Aliprantis and K. C. Border. Innite dimensional analysis. SpringerVerlag, Heidelberg (1994).

[2] C. D. Aliprantis, D. J. Brown and O. Burkinshaw. Edgeworth equilibria,Econometrica, 5, 110937 (1997) .

[3] A. Billot. Economic theory of fuzzy equilibria, Springer - Verlag, Berlin(1992).

[4] M. Burgin. Theory of fuzzy limits, Fuzzy Sets Syst.,115, 433-43 (2000).

[5] J. Christoper. The asymptotic density of some k-dimensional sets, Amer.Math. Monthly, 63, 399401 (1956).

[6] J. Connor. The statistical and strong p-Cesàro convergence of sequences,Analysis, 8, 4763 (1988).

[7] J. Connor. A topological and functional analytic approach to statisticalconvergence. Analysis of Divergence. Birkhauser-Verlag, Boston (1999)40313.

[8] G. Debreu. Mathematical Economics: Twenty papers of Gerard Debreu,Cambridge University Press, Cambridge (1983).

[9] Z. Deng. Fuzzy pseudo-metric space, J. Math. Anal. Appl., 86, 7495(1982).

[10] M. A. Erceg. Metric spaces in fuzzy set theory, J. Math. Anal. Appl.,69,20530 (1979).

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[11] H. Fast. Sur la convergence statistique, Colloq. Math., 2, 241-44 (1951).

[12] J. A. Fridy. On statistical convergence, Analysis, 5, 30113 (1985).

[13] J. A. Fridy and C. Orhan. Lacunary statistical convergence, Pacic J.Math.,160, 4351 (1993).

[14] J. A. Fridy and C. Orhan. Lacunary statistical summability, J. Math.Anal.Appl.,173, 497-504 (1993).

[15] J. A. Fridy and M. K. Khan. Tauberian theorems via statistical conver-gence, J. Math. Anal. Appl., 228, 7395 (1998).

[16] A. George, and P. Veeramani. On some result in fuzzy metric space, FuzzySets Syst., 64, 395-9 (1994).

[17] R. Giles. A computer program for fuzzy reasoning, Fuzzy Sets Syst.,4,221-34 (1980).

[18] G. Jäger. Fuzzy uniform convergence and equicontinuity, Fuzzy Sets Syst.,109, 187-98 (2000).

[19] O. Kaleva and S. Seikkala. On fuzzy metric spaces, Fuzzy Sets and Sys-tems,12, 21529 (1984).

[20] J. Madore. Fuzzy physics, Annals of Physics, 219, 187-98 (1992).

[21] Mursaleen and O. H. H. Edely. Statistical convergence of double se-quences, J. Math. Anal. Appl., 288, 22331 (2003).

[22] S. A. Orlovsky. Decision making with a fuzzy preference relation, FuzzySets and Systems, 1, 15567 (1978).

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[25] R. Saadati and J. H. Park. On the intuitionistic fuzzy topological spaces,Chaos, Solitons & Fractals, 27, 33144 (2006).

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[29] L. A. Zadeh. Fuzzy sets, Inform. Control, 8, 338-53 (1965).

13


Common Fixed Points and Compatible Maps of Type (P-1)and Type (P-2) in Menger spaces

Servet Kutukcua, Sushil Sharmab and Cemil YildizcaDepartment of MathematicsFaculty of Science and ArtsOndokuz Mayis University

55139 Kurupelit, Samsun, TurkeyE-mail: [email protected] of MathematicsMadhav Science College,Ujjain, M.P. 456 001, India

E-mail: [email protected] of MathematicsFaculty of Science and Arts

Gazi University06500 Teknikokullar, Ankara, Turkey


April 19, 2007

Abstract

In the present work, we introduce the concepts of compatible maps of type (P-1)and type (P-2), and prove common fixed point theorems for such maps without appealto continuity in Menger spaces.Keywords: Menger space; t-norm; Common fixed point; Compatible maps.AMS Subject Classifications: 54H25, 54E70

1 INTRODUCTIONThere have been a number of generalizations of metric space. One such generalization isMenger space introduced in 1942 by Menger [11] who was use distribution functions insteadof nonnegative real numbers as values of the metric. Schweizer and Sklar [21] studied thisconcept and then the important development of Menger space theory was due to Sehgaland Bharucha-Reid [15]. Sessa [16] introduced weakly commuting maps in metric spaces.Jungck [8] enlarged this concept to compatible maps. The notion of compatible maps inMenger spaces has been introduced by Mishra [12]. Cho et al. [2] and Sharma [17,19] gavefuzzy version of compatible maps and proved common fixed point theorems for compatiblemaps.

In this paper, we introduce the concepts of compatible maps of type (P-1) and type (P-2) in Menger spaces, and show that they are equivalent to compatible maps under certainconditions. In the sequel, we prove common fixed point theorems for compatible mapsof type (P-1) (or type (P-2)) and weak compatible maps without continuity in Menger


spaces illustrating with an example which generalize, extend and fuzzify several well knownfixed point theorems for contractive type maps on metric spaces, Menger spaces, uniformspaces and fuzzy metric spaces. As an application, we have applied one of our result andprobabilistic version of Banach contraction theorem to obtain a fixed point theorem in theproduct of Menger spaces.

2 PreliminariesIn this section, we recall some definitions and known results in Menger space. For moredetails we refer the readers to [1,3-7,8-15,18,20-22].

Definition 2.1. A triangular norm ∗ (shorty t-norm) is a binary operation on the unitinterval [0, 1] such that for all a, b, c, d ∈ [0, 1] the following conditions are satisfied:

(a) a ∗ 1 = a;(b) a ∗ b = b ∗ a;(c) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d;(d) a ∗ (b ∗ c) = (a ∗ b) ∗ c.Some important examples of t-norm are a∗b = max a+ b− 1, 0 and a∗b = min a, b.Definition 2.2. A distribution function is a function F : [−∞,∞] → [0, 1] which is

left continuous on R, non-decreasing and F (−∞) = 0, F (∞) = 1.We will denote by ∆ the family of all distribution functions on [−∞,∞]. H is a special

element of ∆ defined by

H(t) =

½0, t ≤ 01, t > 0.

.

If X is a nonempty set, F : X ×X → ∆ is called a probabilistic distance on X and F (x, y)is usually detoned by Fx,y.

Definition 2.3 (21). The ordered pair (X,F ) is called a probabilistic semimetric space(shortly PSM-space) if X is a nonempty set and F is a probabilistic distance satisfying thefollowing conditions: for all x, y, z ∈ X and t, s > 0,

(PM-1) Fx,y(t) = H(t)⇐⇒ x = y;(PM-2) Fx,y = Fy,x.If, in addition, the following inequality takes place:(PM-3) Fx,z(t) = 1, Fz,y(s) = 1⇒ Fx,y(t+ s) = 1,then (X,F ) is called a probabilistic metric space (shortly PM-space).The ordered triple (X,F, ∗) is called Menger probabilistic metric space (shortly Menger

space) if (X,F ) is a PM-space, ∗ is a t-norm and the following condition is also satisfies:for all x, y, z ∈ X and t, s > 0,

(PM-4) Fx,y(t+ s) ≥ Fx,z(t) ∗ Fz,y(s).For every PSM-space (X,F ), we can consider the sets of the form

Uε,λ = (x, y) ∈ X ×X : Fx,y(ε) > 1− λ .

The family Uε,λε>0,λ∈(0,1) generates a semi-uniformity denoted by UF and a topology τFcalled the F-topology or the strong topology. Namely,

A ∈ τF iff ∀x ∈ A ∃ε > 0 and λ ∈ (0, 1) such that Uε,λ(x) ⊂ A.

UF is also generated by the family Vδδ>0 where Vδ := Uδ,δ.In [22], it is proved that if supt<1(t∗t) = 1, then UF is a uniformity, called F-uniformity,

which is metrizable.

KUTUKCU ET AL392

The F-topology is generated by the F-uniformity and is determined by the F-convergence:

xn → x⇔ Fxn,x(t)→ 1, ∀t > 0.Proposition 2.1 (15). Let (X, d) be a metric space. Then the metric d induces a

distribution function F defined by Fx,y(t) = H(t − d(x, y)) for all x, y ∈ X and t > 0. Ift-norm ∗ is a ∗ b = min a, b for all a, b ∈ [0, 1] then (X,F, ∗) is a Menger space. Further,(X,F, ∗) is a complete Menger space if (X, d) is complete.

Definition 2.4 (12). Let (X,F, ∗) be a Menger space and ∗ be a continuous t-norm.(a) A sequence xn in X is said to be converge to a point x in X (written xn → x) if for

every > 0 and λ ∈ (0, 1), there exists an integer n0 = n0(ε, λ) such that Fxn,x(ε) > 1− λfor all n ≥ n0.

(b) A sequence xn in X is said to be Cauchy if for every > 0 and λ ∈ (0, 1), thereexists an integer n0 = n0(ε, λ) such that Fxn,xm(ε) > 1− λ for all n ≥ n0 and p > 0.

(c) A Menger space in which every Cauchy sequence is convergent is said to be complete.Definition 2.5 (12). Self maps A and B of a Menger space (X,F, ∗) are said to be

compatible if FABxn,BAxn(t)→ 1 for all t > 0, whenever xn is a sequence in X such thatAxn, Bxn → z for some z in X as n→∞.

Lemma 2.1 (20). Let (X,F, ∗) be a Menger space. If there exists k ∈ (0, 1) such that

Fxy(kt) ≥ Fxy(t)

for all x, y ∈ X and t > 0, then x = y.

3 Compatible Maps of Type (P-1) and Type (P-2)In this section, we introduce the concept of compatible mappings of type (P-1) and type(P-2) in Menger spaces and show that they are equivalent to compatible mappings undercertain conditions.

Definition 3.1. Self maps A and B of a Menger space (X,F, ∗) are said to becompatible of type (P) if FABxn,BBxn(t)→ 1 and FBAxn,AAxn(t)→ 1 for all t > 0, wheneverxn is a sequence in X such that Axn, Bxn → z for some z in X as n→∞.

Definition 3.2. Self maps A and B of a Menger space (X,F, ∗) are said to becompatible of type (P-1) if FABxn,BBxn(t) → 1 for all t > 0, whenever xn is a sequencein X such that Axn, Bxn → z for some z in X as n→∞.

Definition 3.3. Self maps A and B of a Menger space (X,F, ∗) are said to becompatible of type (P-2) if FBAxn,AAxn(t) → 1 for all t > 0, whenever xn is a sequencein X such that Axn, Bxn → z for some z in X as n→∞.

Remark 3.1. Clearly, if a pair of mappings (A,B) is compatible of type (P-1) then thepair (B,A) is compatible of type (P-2). Further, if A and B compatible mappings of type(P) then the pair (A,B) is compatible of type (P-1) as well as type (P-2).

The following is an example of pair of self maps in a Menger space which are compatibleof type (P-1) and type (P-2) but not compatible.

Example. Let (X, d) be a metric space with the usual metric d where X = [0, 2] and(X,F, ∗) be the induced Menger space with Fx,y(t) = H(t− d(x, y)), ∀x, y ∈ X and ∀t > 0.Define self maps A and B as follows:

Ax =

½2− x, if 0 ≤ x < 1,2, if 1 ≤ x ≤ 2, and Bx =

½x, if 0 ≤ x < 1,2, if 1 ≤ x ≤ 2.

...MENGER SPACES 393

Take xn = 1−1/n. Then FAxn,1(t) = H(t−(1/n)) and limn→∞ FAxn,1(t) = H(t) = 1. HenceAxn → 1 as n→∞. Similarly, Bxn → 1 as n→∞. Also FABxn,BAxn(t) = H(t−(1−1/n))and limn→∞ FABxn,BAxn(t) = H(t−1) 6= 1, ∀t > 0. Hence the pair (A,B) is not compatible.But FABxn,BBxn(t) = H(t− (2/n)) and limn→∞ FABxn,BBxn(t) = H(t) = 1, ∀t > 0. Hencethe pair (A,B) is compatible of type (P-1). Similarly, the pair (A,B) is compatible of type(P-2).

Next, we cite the following propositions which gives the condition under which theDefinitions 2.5, 3.2 and 3.3 becomes equivalent.

Proposition 3.1. Let A and B be self maps of a Menger space (X,F, ∗) with continuoust-norm ∗ and t ∗ t ≥ t for all t ∈ [0, 1].

(i) If B is continuous then the pair (A,B) is compatible of type (P-1) iff A and B arecompatible.

(ii) If A is continuous then the pair (A,B) is compatible of type (P-2) iff A and B arecompatible.

Proof. (i) Let xn be a sequence in X such that Axn, Bxn → z for some z in X asn→∞ and let the pair (A,B) be compatible of type (P-1). Since B is continuous, we haveBAxn → Bz and BBxn → Bz. Therefore, by (PM-4), we have

FABxn,BAxn(t) ≥ FABxn,BBxn(t/2) ∗ FBBxn,BAxn(t/2)→ 1 ∗ 1 ≥ 1

as n→∞. Hence the mappings A and B are compatible.Now, let A and B be compatible. Therefore, using the continuity of B, we have

FABxn,BBxn(t) ≥ FABxn,BAxn(t/2) ∗ FBAxn,BBxn(t/2)→ 1 ∗ 1 ≥ 1

as n→∞. Hence the mappings A and B are compatible of type (P-1).(ii) Let xn be a sequence in X such that Axn, Bxn → z for some z inX as n→∞ and

let the pair (A,B) be compatible of type (P-2). Since A is continuous, we have AAxn → Azand ABxn → Az. Therefore, by (PM-4), we have

FABxn,BAxn(t) ≥ FABxn,AAxn(t/2) ∗ FAAxn,BAxn(t/2)→ 1 ∗ 1 ≥ 1

as n→∞. Hence the mappings A and B are compatible.Now, let A and B be compatible. Therefore, using the continuity of A, we have

FBAxn,AAxn(t) ≥ FBAxn,ABxn(t/2) ∗ FABxn,AAxn(t/2)→ 1 ∗ 1 ≥ 1

as n→∞. Hence the mappings A and B are compatible of type (P-2).Next, we give some properties of compatible mappings of type (P-1) and type (P-2)

which will be used in our results.Proposition 3.2. Let A and B be self maps of a Menger space (X,F, ∗). If the pair

(A,B) is compatible of type (P-1) and Az = Bz for some z in X then ABz = BBz.Proof. Let xn be a sequence in X defined by xn = x for n ∈ N and let Az = Bz.

Then we have Axn → Az and Bxn → Bz. Since the pair (A,B) is compatible of type(P-1), we have FABz,BBz(t) = FABxn,BBxn(t)→ 1 as n→∞. Hence ABz = BBz.

Proposition 3.3. Let A and B be self maps of a Menger space (X,F, ∗). If the pair(A,B) is compatible of type (P-2) and Az = Bz for some z in X then BAz = AAz.

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Proof. Let xn be a sequence in X defined by xn = x for n ∈ N and let Az = Bz.Then we have Axn → Az and Bxn → Bz. Since the pair (A,B) is compatible of type(P-2), we have FBAz,AAz(t) = FBAxn,AAxn(t)→ 1 as n→∞. Hence BAz = AAz.

Proposition 3.4. Let A and B be self maps of a Menger space (X,F, ∗) with continuoust-norm ∗ and t∗t ≥ t for all t ∈ [0, 1]. If the pair (A,B) is compatible of type (P-1) and xnis a sequence in X such that Axn, Bxn → z for some z in X as n→∞ then BBxn → Azif A is continuous at z.

Proof. Since A is continuous at z and the pair (A,B) is compatible of type (P-1), wehave ABxn → Az and FABxn,BBxn(t)→ 1 as n→∞. Therefore

FAz,BBxn(t) ≥ FAz,ABxn(t/2) ∗ FABxn,BBxn(t/2)→ 1 ∗ 1 ≥ 1

as n→∞. Hence BBxn → Az as n→∞.Proposition 3.5. Let A and B be self maps of a Menger space (X,F, ∗) with continuous

t-norm ∗ and t∗t ≥ t for all t ∈ [0, 1]. If the pair (A,B) is compatible of type (P-2) and xnis a sequence in X such that Axn, Bxn → z for some z in X as n→∞ then AAxn → Bzif B is continuous at z.

Proof. Since B is continuous at z and the pair (A,B) is compatible of type (P-2), wehave BAxn → Bz and FBAxn,AAxn(t)→ 1 as n→∞. Therefore

FBz,AAxn(t) ≥ FBz,BAxn(t/2) ∗ FBAxn,AAxn(t/2)→ 1 ∗ 1 ≥ 1

as n→∞. Hence AAxn → Bz as n→∞.

4 Main ResultsLet A,B,P,Q, S and T be self maps on a Menger space (X,F, ∗) with continuous t-norm∗ and t ∗ t ≥ t, for all t ∈ [0, 1], satisfying:

(a) P (X) ⊆ ST (X), Q(X) ⊆ AB(X),

(b) there exists a constant k ∈ (0, 1) such that

FPx,Qy(kt) ≥ FABx,STy(t) ∗ FPx,ABx(t) ∗ FQy,STy(t)∗FPx,STy(αt) ∗ FQy,ABx((2− α)t)

for all x, y ∈ X,α ∈ (0, 2) and t > 0.

For some arbitrary x0 in X, by (a), we choose x1 in X such that Px0 = STx1, andfor this x1 there exists x2 such that Qx1 = ABx2. Continuing this process we define thesequence yn in X such that

(1) Px2n = STx2n+1 = y2n and Qx2n+1 = ABx2n+2 = y2n+1

for n = 0, 1, 2, ....We need the following lemma proved by Singh and Jain [20] for our result.Lemma 4.1. Let A,B, P,Q, S and T be self maps on a Menger space (X,F, ∗) with

continuous t-norm ∗ and t ∗ t ≥ t for all t ∈ [0, 1], satisfying the conditions (a) and (b).Then the sequence defined by the condition (1) is a Cauchy sequence in X.

Theorem 4.1. Let A,B,P,Q, S and T be self maps on a Menger space (X,F, ∗) withcontinuous t-norm ∗ and t ∗ t ≥ t, for all t ∈ [0, 1], satisfying the conditions (a), (b) and


(c) one of P (X), Q(X), AB(X) and ST (X) is a complete subspace of X,

(d) (P,AB) is compatible of type (P-1) or type (P-2), and (Q,ST ) is weakly compatible.

Then A,B,P,Q, S and T have a unique common fixed point in X.Proof. Let (P,AB) be compatible of type (P-1) and (Q,ST ) be weakly compatible. By

Lemma 4.1, the sequence yn defined by (1) is a Cauchy sequence in X and if AB(X) iscomplete then y2n+1 has a limit z in AB(X). Let w ∈ (AB)−1z, then ABw = z. Weshall use the fact that the subsequence y2n also converges to z. If we take x = w andy = x2n+1 in (b), we have

FPw,Qx2n+1(kt) ≥ FABw,STx2n+1(t) ∗ FPw,ABw(t) ∗ FQx2n+1,STx2n+1(t)∗FPw,STx2n+1(αt) ∗ FQx2n+1,ABw((2− α)t).

Taking n→∞ and α→ 1, we have

FPw,z(kt) ≥ FABw,z(t) ∗ FPw,ABw(t) ∗ Fz,z(t) ∗ FPw,z(t) ∗ Fz,ABw(t)≥ FPw,z(t)

which means that z = Pw. Hence z = ABw = Pw, i.e., w is a coincide point of P and AB.Since the pair (P,AB) is compatible of type (P-1) and ABw = Pw, by Proposition 3.2, wehave P (AB)w = AB(AB)w or Pz = ABz.

Since P (X) ⊆ ST (X), Pw = z implies that z ∈ ST (X). Let u ∈ (ST )−1z, thenSTu = z. It can easily verified by using similar arguments of the previous part of the proofthat z = Qu, hence z = STu = Qu. Since the pair (Q,ST ) is weakly compatible, we haveQ(ST )u = (ST )Qu or Qz = STz.

If we assume that ST (X) is complete, then the argument analogous to the previouscompleteness argument establishes w is a coincide point of P and AB, and u is a coincidepoint of Q and ST . Thus Pz = ABz and Qz = STz.

The remaining two cases pertain essentially to the previous cases. Indeed, if Q(X) iscomplete, then by the condition (a), z ∈ Q(X) ⊂ AB(X). Similarly if P (X) is complete,then z ∈ P (X) ⊂ ST (X). Thus Pz = ABz and Qz = STz.

Now, we prove that z = Pz. If we take x = z and y = x2n+1 in (b), we have

FPz,Qx2n+1(kt) ≥ FABz,STx2n+1(t) ∗ FPz,ABz(t) ∗ FQx2n+1,STx2n+1(t)∗FPz,STx2n+1(αt) ∗ FQx2n+1,ABz((2− α)t).


FPz,z(kt) ≥ FPz,z(t) ∗ FPz,Pz(t) ∗ Fz,z(t) ∗ FPz,z(t) ∗ Fz,Pz(t)≥ FPz,z(t)

which means that z = Pz. Hence z = Pz = ABz. Similarly, we also have z = Qz = STz.Now, we prove that z = Bz. Since z = Pz = ABz, P (Bz) = Bz and AB(Bz) = Bz. If

we take x = Bz and y = x2n+1 in (b), we have

FP (Bz),Qx2n+1(kt) ≥ FAB(Bz),STx2n+1(t) ∗ FP (Bz),AB(Bz)(t) ∗ FQx2n+1,STx2n+1(t)∗FP (Bz),STx2n+1(αt) ∗ FQx2n+1,AB(Bz)((2− α)t).

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FBz,z(kt) ≥ FBz,z(t) ∗ FBz,Bz(t) ∗ Fz,z(t) ∗ FBz,z(t) ∗ Fz,Bz(t)≥ FBz,z(t)

which means that z = Bz. Since z = ABz, z = Bz implies that z = Az. Hencez = Pz = Az = Bz. Similarly we also have z = Qz = Sz = Tz. Therefore z is acommon fixed point of A,B,P,Q, S and T . It is also easy to prove that if (P,AB) iscompatible of type (P-2) and (Q,ST ) is weakly compatible then z is a common fixed pointof A,B,P,Q, S and T .

For uniqueness of common fixed point, let v 6= z be another common fixed point ofA,B,P,Q, S and T . Then, by condition (b) and taking α→ 1, we have

Fz,v(kt) = FPz,Qv(kt) ≥ FABz,STv(t) ∗ FPz,ABz(t) ∗ FQv,STv(t)∗FPz,STv(t) ∗ FQv,ABz(t)

≥ Fz,v(t)

which means that z = v. This completes the proof.If we take A = B = S = T = IX in Theorem 4.1, we have the following:Corollary 4.1. Let P and Q be self maps on a Menger space (X,F, ∗) with continuous

t-norm ∗ and t ∗ t ≥ t for all t ∈ [0, 1]. If there exists a constant k ∈ (0, 1) such thatFPx,Qy(kt) ≥ Fx,y(t) ∗ Fx,Px(t) ∗ Fy,Qy(t)

∗Fy,Px(αt) ∗ Fx,Qy((2− α)t)

for all x, y ∈ X,α ∈ (0, 2) and t > 0, then P and Q have a unique common fixed point.Remark 4.1. Theorem 4.1 is a generalization of the results of Singh and Jain [20], and

Mishra [12] in the sense of that the condition of compatibility of the first pair of self mapshas been restricted to (P-1) type (or (P-2) type) compatibility and no one of the self mapsneed to be continuous in non-complete Menger space.

Example. Let (X, d) be a metric space with the usual metric d where X = [0, 1] and(X,F, ∗) be the induced Menger space with Fx,y(t) = H(t− d(x, y)) for all x, y ∈ X, t > 0.Clearly (X,F, ∗) is a Menger space where t-norm ∗ is defined by a ∗ b = min a, b for alla, b ∈ [0, 1]. Let A,B,P,Q, S and T be maps from X into itself defined as

Ax = x/5, Bx = x/3, Px = x/6, Qx = 0, Sx = x, Tx = x/2

for all x ∈ X. Then

P (X) =

·0,1

6

¸⊂·0,1

2

¸= ST (X)

and

Q(X) = 0 ⊂·0,1

15

¸= AB(X).

If we take k = 1/2, t = 1 and α = 1, we see that the condition (b) of the main Theorem issatisfied. Clearly, conditions (c) and (d) of the main Theorem are also satisfied. Moreover,the pair (P,AB) is compatible of type (P-1) and also type (P-2). In fact, if limn→∞ xn = 0,where xn is a sequence in X such that limn→∞ Pxn = limn→∞ABxn = 0 for some 0 ∈ X,then

limn→∞FP (AB)xn,AB(AB)xn(t) = H(t) = 1.


Similarly, (P,AB) is also compatible of type (P-2). The pair (Q,ST ) is weakly compatiblesince they commute at their coincidence point 0. Thus, all conditions of the main Theoremare satisfied and 0 is the unique common fixed point of A,B, P,Q, S and T .

Following Bylka [1], we consider the family G of functions g : [0,∞)→ [0,∞) such thatg is non-decreasing in [0,∞) and limn→∞ gn(t) =∞ for every t > 0 where gn denotes then-th iteration of g. It is proved in [14] that g(t) ≥ t for all t ≥ 0 and if Fxy(t) ≥ Fxy(g(t))for some t > 0 then x = y.

Theorem 4.2. Let A,B,P,Q, S and T be self maps on a Menger space (X,F, ∗) withcontinuous t-norm ∗ and t ∗ t ≥ t for all t ∈ [0, 1], satisfying:

(a) P (X) ⊆ ST (X), Q(X) ⊆ AB(X),

(b) there exists a function g ∈ G such that

FPx,Qy(t) ≥ FABx,STy(g(t))

for all x, y ∈ X and t > 0,

(c) one of P (X), Q(X), AB(X) and ST (X) is a complete subspace of X,

(d) (P,AB) is compatible of type (P-1) or type (P-2), and (Q,ST ) is weakly compatible.

Then A,B,P,Q, S and T have a unique common fixed point in X.Proof. Using the condition (a), we can construct a sequence yn in X defined by (1).

Then, for all t > 0 and n = 1, 2, ..., we have

Fy2n+1,y2n+2(t) = FPx2n+1,Qx2n+2(t)

≥ FABx2n+1,STx2n+2(g(t))

= Fy2n,y2n+1(g(t)).

Similarly, we have

Fy2n,y2n+1(t) = FPx2n,Qx2n+1(t)

≥ FABx2n,STx2n+1(g(t))

= Fy2n−1,y2n(g(t)).

Therefore

(2) Fy2n,y2n+1(t) ≥ Fy2n−1,y2n(g(t)) ≥ ... ≥ Fy0,y1(gn(t)).

Now, we show that the sequence yn is a Cauchy sequence in X. Let ε and λ be positivereals. Then, for m > n, p = m− n and by using (1), we have

Fyn,ym(ε) ≥ Fyn,yn+1

µε

p

¶∗ Fyn+1,ym

µε(p− 1)

p

¶≥ Fy0,y1

µgnµε

p

¶¶∗ Fyn+1,ym

µε(p− 1)

p

¶≥ Fy0,y1

µgnµε

p

¶¶∗ Fyn+1,yn+2

µε

p

¶∗ Fyn+2,ym

µε(p− 2)

p

¶≥ Fy0,y1

µgnµε

p

¶¶∗ Fy0,y1

µgn+1

µε

p

¶¶∗ Fyn+2,ym

µε(p− 2)

p

¶.

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Since limn→∞ gn(t) = ∞, we have gn³εp

´≤ gn+1

³εp

´and by the hypothesis t ∗ t ≥ t.

Therefore, we have

(3) Fyn,ym(ε) ≥ Fy0,y1

µgnµε

p

¶¶∗ Fyn+2,ym

µε(p− 2)

p

¶.

Using the induction argument, we obtain from (3) that

Fyn,ym(ε) ≥ Fy0,y1

µgnµε

p

¶¶∗ Fyn+k−2,yn+k−1

µε

p

¶∗ Fym−1,ym

µε

p

¶≥ Fy0,y1

µgnµε

p

¶¶∗ Fy0,y1

µgn+k−2

µε

p

¶¶∗ Fy0,y1

µgm−1

µε

p

¶¶≥ Fy0,y1

µgnµε

p

¶¶.

Hence, we can choose n0 ≤ n such that

Fy0,y1

µgnµε

p

¶¶> 1− λ

and then Fyn,ym(ε) > 1−λ for all m > n ≥ n0. This means that yn is a Cauchy sequencein X. Let (P,AB) be compatible of type (P-1) and (Q,ST ) be weakly compatible. Nowsuppose that AB(X) is complete then y2n+1 has a limit z in AB(X). Let w ∈ (AB)−1z,then ABw = z. We shall use the fact that the subsequence y2n also converges to z. Ifwe take x = w and y = x2n+1 in (b), we have

FPw,Qx2n+1(t) ≥ FABw,STx2n+1(g(t))

Taking n→∞, we haveFPw,z(t) ≥ Fz,z(g(t)) = 1

which implies that z = Pw. Hence z = ABw = Pw, i.e., w is a coincide point of P andAB. Since the pair (P,AB) is compatible of type (P-1) and ABw = Pw, by Proposition3.2, we have P (AB)w = AB(AB)w or Pz = ABz.

Since P (X) ⊆ ST (X), Pw = z implies that z ∈ ST (X). Let u ∈ (ST )−1z, thenSTu = z. It can easily verified by using similar arguments of the previous part of the proofthat z = Qu, hence z = STu = Qu. Since the pair (Q,ST ) is weakly compatible, we haveQ(ST )u = (ST )Qu or Qz = STz.

If we assume that ST (X) is complete, then the argument analogous to the previouscompleteness argument establishes w is a coincide point of P and AB, and u is a coincidepoint of Q and ST . Thus Pz = ABz and Qz = STz.

The remaining two cases pertain essentially to the previous cases. Indeed, if Q(X) iscomplete, then by the condition (a), z ∈ Q(X) ⊂ AB(X). Similarly if P (X) is complete,then z ∈ P (X) ⊂ ST (X). Thus Pz = ABz and Qz = STz.

Now, we prove that z = Pz. If we take x = z and y = x2n+1 in (b), we have

FPz,Qx2n+1(t) ≥ FABz,STx2n+1(g(t))

Taking n→∞, we haveFPz,z(t) ≥ FPz,z(g(t))


which means that z = Pz. Hence z = Pz = ABz. Similarly we also have z = Qz = STz.Now, we prove that z = Bz. Since z = Pz = ABz, P (Bz) = Bz and AB(Bz) = Bz. If

we take x = Bz and y = x2n+1 in (b), we have

FP (Bz),Qx2n+1(t) ≥ FAB(Bz),STx2n+1(g(t))

Taking n→∞, we haveFBz,z(t) ≥ FBz,z(g(t))

which means that z = Bz. Since z = ABz, z = Bz implies that z = Az. Hencez = Pz = Az = Bz. Similarly we also have z = Qz = Sz = Tz. Therefore z is acommon fixed point of A,B,P,Q, S and T . It is also easy to prove that if (P,AB) iscompatible of type (P-2) and (Q,ST ) is weakly compatible then z is a common fixed pointof A,B,P,Q, S and T .

It is easy to see that z is unique common fixed point of A,B,P,Q, S and T . Thiscompletes the proof.

If we take g(t) = t/k for k ∈ (0, 1), P = Q and A = B = S = T = IX in Theorem 4.2,we have the following:

Corollary 4.2 (15). Let P be a self map on a Menger space (X,F, ∗) with continuoust-norm ∗ and t ∗ t ≥ t for all t ∈ [0, 1]. If P (X) is complete and there exists a constantk ∈ (0, 1) such that

FPx,Py(kt) ≥ Fx,y(t)

for all x, y ∈ X and t > 0, then P has a unique common fixed point in X.Note that the proof of Corollary 4.2 also follows from Corollary 4.1 since P = Q and

Fx,y(t) = minFx,y(t), Fx,Px(t), Fy,Qy(t), Fy,Px(t), Fx,Qy(t).Remark 4.2. Theorem 4.2 is a generalization of the results of Bylka [1], and Sehgal

and Bharucha-Reid [15] in the sense of that the condition of compatibility of the first pairof self maps has been restricted to (P-1) type (or (P-2) type) compatibility and no one ofthe self maps need to be continuous in non-complete Menger space.

Remark 4.3. In Theorems 4.1 and 4.2, and Corollaries 4.1 and 4.2, the condition "thet-norm ∗ is continuous and t ∗ t ≥ t for all t ∈ [0, 1]" can be replaced by the condition"s ∗ t = mins, t for all s, t ∈ [0, 1]".

5 An ApplicationIn this section, we apply Corollaries 4.1 and 4.2 to establish the following result on theproduct space.

Theorem 5.1. Let (X,F, ∗) be a Menger space with continuous t-norm ∗ and t ∗ t ≥ tfor all t ∈ [0, 1] and P and Q be self maps on the product X ×X with values in X. If thereexists a constant k ∈ (0, 1) such that

FP (x,y)Q(u,v)(kt) ≥ Fxu(t) ∗ Fyv(t) ∗ FxP (x,y)(t) ∗ FuQ(u,v)(t)∗FuP (x,y)(αt) ∗ FxQ(u,v)((2− α)t)(4)

for all x, y ∈ X,α ∈ (0, 2) and t > 0, then P and Q have a unique common fixed point.Proof. From (4), we have

FP (x,y)Q(u,y)(kt) ≥ Fxu(t) ∗ FxP (x,y)(t) ∗ FuQ(u,y)(t)∗FuP (x,y)(αt) ∗ FxQ(u,y)((2− α)t)

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for all x, y, u in X. Therefore, by Corollary 4.1, for each y in X, there exists exactly onepoint z(y) in X such that

(5) P (z(y), y) = z(y) = Q(z(y), y).

Now, for any y, yp in X, by (4) with α = 1, we have

FP (z(y),y)Q(z(yp ),yp )(kt) ≥ Fz(y)z(yp )(t) ∗ Fyyp (t) ∗ Fz(y)P (z(y),y)(t)∗Fz(yp )Q(z(yp ),yp )(t) ∗ Fz(yp )P (z(y),y)(t) ∗ Fz(y)Q(z(yp ),yp )(t)

that is,

Fz(y)z(yp )(kt) ≥ Fz(y)z(yp )(t) ∗ Fyyp (t) ∗ 1 ∗ 1 ∗ Fz(yp )z(y)(t) ∗ Fz(y)z(yp )(t)≥ Fz(y)z(yp )(t) ∗ Fyyp (t)≥ Fz(y)z(yp )(t/k

n) ∗ Fyyp (t)→ Fyyp (t).

Therefore, Corollary 4.2 yields that the map z(.) of X into itself has exactly one fixed pointw in X, i.e. z(w) = w. Hence, by (5), w = z(w) = P (w,w) = Q(w,w). This completes theproof.

References

[1] C. Bylka, Fixed point theorems of Matkowski on probabilistic metric spaces, DemonstratioMath. 29 (1996), 158-164.

[2] Y. J. Cho, H. K. Pathak, S. M. Kang, J. S. Jung, Common fixed points of compatible maps oftype (β) on fuzzy metric spaces, Fuzzy Sets and Systems 93 (1998), 99-111.

[3] G. Constantin, I. Istratescu, Elements of probabilistic analysis, Ed. Acad. Bucuresti and KluwerAcad. Publ., 1989.

[4] O. Hadzic, Common fixed point theorems for families of mapping in complete metric space,Math. Japon. 29 (1984), 127-134.

[5] T. L. Hicks, Fixed point theory in probabilistic metric spaces, Rev. Res. Novi Sad 13 (1983),63-72.

[6] I. Istratescu, On some fixed point theorems in generalized Menger spaces, Boll. Un. Mat. Ital.5(13-A) (1976), 95-100.

[7] I. Istratescu, On generalized complete probabilistic metric spaces, Rev. Roum. Math. PuresAppl. XXV (1980), 1243-1247.

[8] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Sci. 9 (1986),771-779.

[9] S. Kutukcu, A fixed point theorem for contraction type mappings in Menger spaces, AmericanJ. Appl. Sci. 4(6) (2007), 371-373.

[10] S. Kutukcu, C. Yildiz, A. Tuna, On common fixed points in Menger probabilistic metric spaces,Int. J. Contemp. Math. Sci. 2(8) (2007), 383-391.

[11] K. Menger, Statistical metric, Proc. Nat. Acad. 28 (1942), 535-537.[12] S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon. 36

(1991), 283-289.[13] E. Pap, O. Hadzic, R. Mesiar, A fixed point theorem in probabilistic metric spaces and an

application, J. Math. Anal. Appl. 202 (1996), 433-449.[14] R.A. Rashwan, A. Hedar, On common fixed point theorems of compatible mappings in Menger

spaces, Demonstratio Math. 31 (1998), 537-546.


[15] V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping on PM spaces, Math.Systems Theory 6 (1972), 97-100.

[16] S. Sessa On weak commutativity condition of mappings in fixed point considerations, Publ.Inst. Math. Beograd 32 (1982), 149-153.

[17] S. Sharma, Common fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 127(2002), 345-352.

[18] S. Sharma, B. Deshpande, Common fixed point theorems for weakly compatible mappingswithout continuity in Menger spaces, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math.10(2) (2003), 133-144.

[19] S. Sharma, B. Deshpande, Discontinuity and weak compatibility in fixed point considerationon non-complete fuzzy metric spaces, J. Fuzzy Math. 11(2) (2003), 671-686.

[20] B. Singh, S. Jain, A fixed point theorem in Menger Space through weak compatibility, J. Math.Anal. Appl. 301 (2005), 439-448.

[21] B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland, Amsterdam, 1983.[22] B. Schweizer, A. Sklar, E. Thorp, The metrization of SM-spaces, Pasific J. Math. 10 (1960),

673-675.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.10,NO.3,2008 A NEW INTERPOLATORY TYPE QUADRATURE RULE FOR WEIGHTED CAUCHY PRINCIPAL VALUE INTEGRALS,B-G.JANG,H.LEE,K.H.ROH,271 SOME CHARACTERISTICS OF LOGISTIC AND BESSEL RANDOM VARIABLES, S.NADARAJAH,……………………………………………………………….,283 ON THE MOMENTS OF THE B DISTRIBUTION,S.NADARAJAH,……….,301 THE BETA-LAPLACE DISTRIBUTION,T.KOZUBOWSKI,S.NADARAJAH,305 ON THE STABILITY AND ASYMPTOTIC BEHAVIOR OF GENERALIZED QUADRATIC MAPPINGS,H-M.KIM,…………………………………………,319 ON STATISTICAL FUZZY TRIGONOMETRIC KOROVKIN THEORY, O.DUMAN,G.A.ANASTASSIOU,………………………………………………,333 THE LANGRANGE-POINCARE EQUATIONS FOR A REFINEMENT OF A PRINCIPAL GP(n;R)-BUNDLE,G.IVAN,M.IVAN,D.OPRIS,…………………,345 EXPLICIT p-ADIC q-EXPANSION FOR ALTERNATING SUMS OF POWERS, L.JANG,T.KIM,…………………………………………………………………..,355 FUZZY MULTI-METRIC SPACES,H.EFE,C.ALACA,C.YILDIZ,……………,367 STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES ON INTUITIONISTIC FUZZY NORMED SPACES,S.KARAKUS,K.DEMIRCI,………………………,377 COMMON FIXED POINTS AND COMPATIBLE MAPS OF TYPE (P-1) AND TYPE (P-2) IN MENGER SPACES,S.KUTUKCU,S.SHARMA,C.YILDIZ,…………..,391

Volume 10,Number 4 October 2008 ISSN:1521-1398 PRINT,1572-9206 ONLINE







10) Ding-Xuan Zhou





28) T. E. Simos











33) Xin-long Zhou

Nonlinear Random Multi-valued Variational Inclusion Systemsinvolving (A, η)-accretive Mappings in Banach Spaces1

Heng-you Lan

Department of Mathematics, Sichuan University of Science & EngineeringZigong, Sichuan 643000, P. R. China

Email: [email protected]

Abstract In this paper, by using the new random resolvent operator tech-nique associated with (A, η)-accretive mappings, we analyze and establishexistence theorem for a new nonlinear random multi-valued variational in-clusion systems involving (A, η)-accretive mappings in Banach spaces. Ourresults generalize some results of other recent works on strongly monotonequasi-variational inclusions, nonlinear implicit quasi-variational inclusionsand nonlinear mixed quasi-variational inclusion systems.

Key words and phrases: Relaxed random cocoercive mapping, nonlinearrandom multi-valued variational inclusion system, (A, η)-accretive mapping,random resolvent operator technique, random iterative algorithm and con-vergence.

AMS Subject classification: 49J40, 47H05, 47H19

1. Introduction

Very recently, by using the iterative technique and Nadlers theorem, Wu et al. [36] con-struct a new iterative algorithm for solving the following system of nonlinear inclusionsin Banach spaces and prove some new existence results of solutions for the system ofnonlinear inclusions and discuss the convergence of the sequences generated by the al-gorithm. As an application, authors also show the existence of solution for a system offunctional equations arising in dynamic programming of multistage decision processes.

In this paper, for any given elements f : Λ → E and g : Ω → E, and any real-valuedrandom variables λ1(s), λ2(t) > 0, we shall consider the following nonlinear randommulti-valued variational inclusion systems:

Find x : Λ → E, y : Ω → E such that Range(p)⋂

domM1(t, ·) 6= ∅ and

y(t)− x(s)− λ1(s)(N1(s, F (t, y(t)), v(t))− f(s)) ∈ λ1(s)M1(s, p(s, x(s))),∀v(t) ∈ T (t, y(t)),

x(s)− y(t)− λ2(t)(N2(t, G(s, x(s)), u(s))− g(t)) ∈ λ2(t)M2(t, y(t)),∀u(s) ∈ S(s, x(s)),

(1.1)

1This work was supported by the Scientific Research Fund of Sichuan Provincial Education Depart-ment (2006A106).


2 H.Y. Lan

where E and E are two separable real Banach spaces, (Λ,A, µ) and (Ω, C, ν) are twocomplete σ-finite measure spaces, S : Λ × E → 2E and T : Ω × E → 2E are multi-valued mappings, M1 : Λ×E → 2E and M2 : Ω×E → 2E are any nonlinear mappingssuch that for all (s, t) ∈ Λ × Ω, M1(s, ·) : E → 2E is an (A1, η1)-accretive mappingand M2(t, ·) : E → 2E is an (A2, η2)-accretive mapping, N1 : Λ × E × E → E, N2 :Ω×E×E → E, F : Ω×E → E, G, p : Λ×E → E, A1 : Λ×E → E, A2 : Ω×E → E,η1 : Λ×E ×E → E and η2 : Ω×E ×E → E are single-valued mappings, 2Ei denotesthe family of all the nonempty subsets of Ei for i = 1, 2.

The study of such types of problems is motivated by an increasing interest in therandom equations involving the random operators in view of their need in dealing withprobabilistic models in applied sciences is very important. In recent years, Ahmad andBazan [2], Chang [4], Chang and Huang [5], Cho et al. [8], Ganguly and Wadhwa [12],Huang [14], Huang and Cho [15], Huang et al. [16], Khan et al. [19], Lan [21], Noor andElsanousi [32] introduced and studied the research works in these fascinating areas, therandom variational inequality problems, random quasi-variational inequality problems,random variational inclusion problems and random quasi-complementarity problems,respectively.

On the other hand, it is well known that variational inequality type methods havebeen applied widely to problems arising from model equilibria problems in economics,optimization and control theory, operations research, transportation network modelling,and mathematical programming. Very recently, in order to study extensively varia-tional inequalities and variational inclusions, which are providing mathematical modelsto some problems arising in economics, mechanics, and engineering science, Lan et al.[11] first introduced the concept of (A, η)-accretive mappings, which generalizes the ex-isting η-subdifferential operators, maximal η-monotone operators, generalized monotoneoperators (named H-monotone operators), A-monotone operators, (H, η)-monotone op-erators, (A, η)-monotone operators in Hilbert spaces, H-accretive operators, generalizedm-accretive mappings and (H, η)-accretive operators in Banach spaces. The authorsalso studied some properties of (A, η)-accretive mappings and defined resolvent oper-ators associated with (A, η)-accretive mappings, which improved and generalized thecorresponding results of recent works in [1, 6, 9, 10, 11, 20, 22, 24, 26, 27, 28, 31, 33, 38].

Furthermore, the determinate form of the problem (1.1) was studied by Lan et al. [26]when M1(s, ·) and M2(t, ·) are generalized m-accretive mappings for a given determinateelement (s, t) ∈ Λ × Ω, and for appropriate and suitable choices of Ni, Mi, Ai, ηi, λi

(i = 1, 2), S, T, F, G, f, g, p and E, it is easy to see that the problem (1.1) includes anumber (systems) of (random) quasi-variational inclusions, generalized (random) quasi-variational inclusions, (random) quasi-variational inqualities, (random) implicit quasi-variational inequalities studied by many authors as special cases, see, for example:

Example 1.1. Let S : Λ× E → E and T : Ω× E → E be single-valued mappings.Then for each fixed elements (f(s), g(t)) ∈ E ×E and any real-valued random variablesλ1(t), λ2(t) > 0, the problem (1.1) reduces to finding x : Λ → E, y : Ω → E such thatRange(p)

⋂domM1(t, ·) 6= ∅ and

y(t)− x(s)− λ1(s)(N1(s, F (t, y(t)), T (t, y(t)))− f(s)) ∈ λ1(s)M1(s, p(s, x(s))),x(s)− y(t)− λ2(t)(N2(t, G(s, x(s)), S(s, x(s)))− g(t)) ∈ λ2(t)M2(t, y(t)).

(1.2)

Example 1.2. Suppose that p ≡ I, the identity mapping, f(s) = 0 for all s ∈ Λ,g(t) = 0 for all t ∈ Ω, N1(·, x, y) = x + y for all x, y ∈ E and N2(·, z, w) = z + w for

416

Nonlinear Random Multi-valued Variational Inclusion Systems 3

all z, w ∈ E. Then the problem (1.2) is equivalent to the following system of generalnonlinear mixed random quasi-variational inclusions in Banach spaces: find (x(s), y(t)) ∈E × E such that

0 ∈ x(s)− y(t) + λ1(s)(F (t, y(t)) + T (t, y(t))) + λ1(s)M1(s, x(s)),0 ∈ y(t)− x(s)− λ2(t)(G(s, x(s)) + S(s, x(s))) + λ2(t)M2(t, y(t)).

(1.3)

The parametric form of the problem (1.3) was studied by Jeong [17] when M, N arem-accretive mappings and λ1(s) and λ2(t) are constants. Further, the determinate formof the problem (1.3) was introduced and studied by Agarwal et al. [1] when E = H is aHilbert spaces, M1,M2 are two maximal monotone mappings.

Now, for each fixed (s, t) ∈ Λ × Ω, the solution set Q(s, t) of the problem (1.1) isdenoted as

Q(s, t)= (x(s), y(t)) ∈ E × E : ∃u(s) ∈ S(s, x(s)) and v(t) ∈ T (t, y(t)), such that

y(t)− x(s)− λ1(s)(N1(s, F (t, y(t)), v(t))− f(s)) ∈ λ1(s)M1(s, p(s, x(s)))andx(s)− y(t)− λ2(t)(N2(t, G(s, x(s)), u(s))− g(t)) ∈ λ2(t)M2(t, y(t)).

In this paper, by using the new random resolvent operator technique associated with(A, η)-accretive mappings, our main aim is to study the behavior of the solution setQ(ω, λ), and the conditions on these mappings Ni,Mi, S, T, F,G, f, g, p, Ai, ηi, λi and Ei

for i = 1, 2 under which the function Q(s, t) is nonempty and the generalized randomiterative procedures with errors for the element of this solution set Q(s, t) in q-uniformlysmooth Banach spaces is convergence. Our results generalize some results of other re-cent works on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions and nonlinear mixed quasi-variational inclusion systems.

2. Preliminaries

Throughout this paper, we suppose that (Ω,A, µ) is a complete σ-finite measurespace and E is a separable real Banach space endowed with dual space E∗, the norm‖ · ‖ and the dual pair 〈·, ·〉 between E and E∗. We denote by B(E) the class of Borelσ-fields in E. Let 2E and CB(E) denote the family of all the nonempty subsets of E,the family of all the nonempty bounded closed sets of E, respectively. The generalizedduality mapping Jq : E → 2E∗ is defined by

Jq(x) = f∗ ∈ E∗ : 〈x, f∗〉 = ‖x‖q and ‖f∗‖ = ‖x‖q−1, ∀x ∈ E,

where q > 1 is a constant. In particular, J2 is the usual normalized duality mapping. Itis well known that, in general, Jq(x) = ‖x‖q−2J2(x) for all x 6= 0 and Jq is single-valuedif E∗ is strictly convex (see, for example, [37]). If E = H is a Hilbert space, then J2

becomes the identity mapping of H. In what follows we shall denote the single-valuedgeneralized duality mapping by jq.

The modules of smoothness of E is the function ρE : [0,∞) → [0,∞) defined by

ρE(t) = sup12‖x + y‖+ ‖x− y‖ − 1 : ‖x‖ ≤ 1, ‖y‖ ≤ t.

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4 H.Y. Lan

A Banach space E is called uniformly smooth if limt→0

ρE(t)t = 0 and E is called q-uniformly

smooth if there exists a constant c > 0 such that ρE ≤ ctq, where q > 1 is a real number.It is well known that Hilbert spaces, Lp(or lp) spaces, 1 < p < ∞, and the Sobolev

spaces Wm,p, 1 1 be a given real number and E be a real uniformly smoothBanach space. Then E is q-uniformly smooth if and only if there exists a constant cq > 0such that for all x, y ∈ E, jq(x) ∈ Jq(x), there holds the following inequality

‖x + y‖q ≤ ‖x‖q + q〈y, jq(x)〉+ cq‖y‖q.

In this paper, we will use the following definitions and lemmas.Definition 2.1. A operator x : Ω → E is said to be measurable if for any E ∈ B(E),

t ∈ Ω : x(t) ∈ E ∈ A.Definition 2.2. A operator F : Ω × E → E is called a random operator if for any

x ∈ E, F (t, x) = y(t) is measurable. A random operator F is said to be continuous(resp. linear, bounded) if for any t ∈ Ω, the operator F (t, ·) : E → E is continuous (resp.linear, bounded).

Similarly, we can define a random operator b : Ω × E × E → E. We shall writeFt(x) = F (t, x(t)) and bt(x, y) = b(t, x(t), y(t)) for all t ∈ Ω and x(t), y(t) ∈ E.

It is well known that a measurable operator is necessarily a random operator.Definition 2.3. A multi-valued operator G : Ω → 2E is said to be measurable if for

any E ∈ B(E), G−1(E) = t ∈ Ω : G(t) ∩ E 6= ∅ ∈ A.Definition 2.4. A operator u : Ω → E is called a measurable selection of a multi-

valued measurable operator Γ : Ω → 2E if u is measurable and for any t ∈ Ω, u(t) ∈ Γ(t).Definition 2.5. A multi-valued operator F : Ω×E → 2E is called a random multi-

valued operator if, for any x ∈ E, F (·, x) is measurable. A random multi-valued operatorF : Ω×E → CB(E) is said to be H-continuous, if for any t ∈ Ω, F (t, ·) is continuous inH(·, ·), where H(·, ·) is the Hausdorff metric on CB(E) defined as follows: for any givenD, K ∈ CB(E),

H(D, K) = max

supx∈D

infy∈K

d(x, y), supy∈K

infx∈D

d(x, y).

Definition 2.6. Let E be a q-uniformly smooth Banach space. Then a randomoperator g : Ω× E → E is said to be

(i) m-relaxed accretive in the second argument, if

〈gt(x)− gt(y), jq(x(t)− y(t))〉 ≥ −m(t)‖x(t)− y(t)‖q, ∀x(t), y(t) ∈ E, t ∈ Ω,

where m(t) is a real-valued random variable;

(ii) s-cocoercive in the second argument, if there exists a real-valued random variables(t) > 0 such that

〈gt(x)− gt(y), jq(x(t)− y(t))〉 ≥ s(t)‖gt(x)− gt(y)‖q, ∀x(t), y(t) ∈ E, t ∈ Ω;

418


(iii) γ-relaxed cocoercive in the second argument, if there exists a positive real-valuedrandom variable γ(t) such that

〈gt(x)− gt(y), jq(x(t)− y(t))〉 ≥ −γ(t)‖gt(x)− gt(y)‖q, ∀x(t), y(t) ∈ E, t ∈ Ω;

(iv) (α, ε)-relaxed cocoercive in the second argument, if there exist positive real-valuedrandom variables α(t) and ε(t) such that

〈gt(x)− gt(y), jq(x(t)− y(t))〉 ≥ −α(t)‖gt(x)− gt(y)‖q + ε(t)‖x(t)− y(t)‖q,

for all x(t), y(t) ∈ E, t ∈ Ω;

(v) µ-Lipschitz continuous if there exists a real-valued random variable µ(t) > 0 suchthat

‖gt(x)− gt(y)‖ ≤ µ(t)‖x(t)− y(t)‖, ∀x(t), y(t) ∈ E, t ∈ Ω.

Remark 2.1. Clearly, every m-cocoercive mapping is m-relaxed cocoercive, whileeach r-strongly monotone mapping is (r + r2, 1)-relaxed cocoercive with respect to I.Further, we can find some operators which are cocoercive and relaxed cocoercive. See,for example, [23, 33, 35].

Definition 2.7. Let E be a q-uniformly smooth Banach space, η : Ω×E×E → E andA,H : Ω× E → E be random single-valued operators. Then a multi-valued measurableoperator M : Ω× E → 2E is said to be

(1) σ-H-Lipschitz continuous, if there exists a measurable function σ : Ω → (0,+∞)such that for any t ∈ Ω,

H(Mt(x),Mt(y)) ≤ σ(t)‖x(t)− y(t)‖, ∀x(t), y(t) ∈ E;

(2) η-accretive if for all x(t), y(t) ∈ E, u(t) ∈ Mt(x), v(t) ∈ Mt(y), t ∈ Ω,

〈u(t)− v(t), jq(ηt(x, y))〉 ≥ 0;

(3) strictly η-accretive if for all x(t), y(t) ∈ E, u(t) ∈ Mt(x), v(t) ∈ Mt(y), t ∈ Ω,

〈u(t)− v(t), jq(ηt(x, y))〉 ≥ 0,

and equality holds if and only if u(t) = v(t) for all t ∈ Ω;(4) r-strongly η-accretive, if there exists a real-valued random variable r(t) > 0 such

that

〈u(t)− v(t), jq(ηt(x, y))〉 ≥ r(t)‖x(t)− y(t)‖2,

for all x(t), y(t) ∈ E, u(t) ∈ Mt(x), v(t) ∈ Mt(y), t ∈ Ω;(5) α-relaxed η-accretive, if there exists a real-valued random variable α(t) > 0 such

that for all x(t), y(t) ∈ E, u(t) ∈ Mt(x), v(t) ∈ Mt(y), t ∈ Ω,

〈u(t)− v(t), jq(ηt(x, y))〉 ≥ −α(t)‖x(t)− y(t)‖q;

(6) m-accretive, if M is accretive and (I+ρ(t)Mt)(E) = E for all t ∈ Ω and real-valuedrandom variable ρ(t) > 0, where I denotes the identity operator on E and Mt(·) = M(t, ·)for all t ∈ Ω;

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6 H.Y. Lan

(7) generalized m-accretive, if M is η-accretive and (I+ρ(t)Mt))(E) = E for all t ∈ Ωand (equivalently, for some) ρ(t) > 0;

(8) H-accretive if M is accretive and (Ht+ρ(t)Mt)(E) = E for all t ∈ Ω and ρ(t) > 0,where Ht(·) = H(t, ·) for all t ∈ Ω;

(9) (H, η)-accretive, if M is η-accretive and (Ht + ρ(t)Mt)(E) = E for all t ∈ Ω andρ(t) > 0;

(10) (A, η)-accretive with real-valued random variable m(t) if(i) M is m-relaxed η-accretive, (ii) (At + ρ(t)Mt)(E) = E for every t ∈ Ω and

ρ(t) > 0, where At(·) = A(t, ·) for all t ∈ Ω.In a similar way, we can define strictly η-accretivity and strongly η-accretivity of the

single-valued mapping A.Remark 2.1. For appropriate and suitable choices of m,A, η and E, it is easy to see

that Definition 2.7 includes a number of definitions of monotone operators and accretivemappings (see [25]).

Definition 2.8. The operator η : Ω×E×E → E is said to be τ -Lipschitz continuousif there exists a real-valued random variable τ(t) > 0 such that

‖ηt(x, y)‖ ≤ τ(t)‖x(t)− y(t)‖, ∀x(t), y(t) ∈ E, t ∈ Ω.

Definition 2.9. Let A : Ω × E → E be a strictly η-accretive mapping and M :Ω×E → 2E be an (A, η)-accretive mapping. For any given measurable function ρ : Ω →(0,∞), the resolvent operator J

ρ(t),At

ηt,Mt: E → E is defined by:

Jρ(t),At

ηt,Mt(u) = (At + ρ(t)Mt)−1(u), ∀u ∈ E, t ∈ Ω.

Lemma 2.2. ([25]) Let E be a q-uniformly smooth Banach space and η : Ω×E×E →E be τ -Lipschitz continuous, A : Ω × E → E be a r-strongly η-accretive mappingand M : Ω × E → 2E be an (A, η)-accretive mapping. Then the resolvent operatorJ

ρ(t),At

ηt,Mt: E → E is τq−1(t)

r(t)−ρ(t)m(t) -Lipschitz continuous, i.e.,

‖Jρ(t),At

ηt,Mt(x)− J

ρ(t),At

ηt,Mt(y)‖ ≤ τ q−1(t)

r(t)− ρ(t)m(t)‖x− y‖, ∀x, y ∈ E, t ∈ Ω,

where ρ(t) ∈ (0, r(t)m(t)) is a measurable function for all t ∈ Ω.

3. Random Iterative Algorithms

In this section, we suggest and analyze a new class of iterative methods and constructsome new random iterative algorithms with errors for solving problem (1.1).

Lemma 3.1. ([3]) Let M : Ω × E → CB(E) be a H-continuous random multi-valued mapping. Then for any measurable mapping x : Ω → E, the multi-valuedmapping M(·, x(·)) : Ω → CB(E) is measurable.

Lemma 3.2. ([3]) Let M, V : Ω × E → CB(E) be two measurable multi-valuedmappings, ε > 0 be a constant and x : Ω → E be a measurable selection of M . Thenthere exists a measurable selection y : Ω → E of V such that for any t ∈ Ω,

‖x(t)− y(t)‖ ≤ (1 + ε)H(M(t), V (t)).

420


Lemma 3.3. For each fixed (s, t) ∈ Λ×Ω, an element (x(s), y(t)) ∈ Q(s, t) is a solutionto the problem (1.1) if and only if there are (x(s), y(t)) ∈ E × E such that

ps(x) ∈ Jλ1(s),A1s

η1s,M1s[A1s(ps(x)) + y(t)− x(s)−λ1(s)(N1s(Ft(y), Tt(y))− f(s))],

y(t) ∈ Jλ2(t),A2t

η2t,M2t[A2t(y) + x(s)− y(t)−λ2(t)(N2t(Gs(x), Ss(x))− g(t))],

where Jλ1(s),A1s

η1s,M1s= (A1s +λ1(s)M1s)−1 and J

λ2(t),A2t

η2t,M2t= (A2t +λ2(t)M2t)−1 are the corre-

sponding resolvent operator of an (A1, η1)-accretive mapping M1(s, ·), (A2, η2)-accretivemapping M2(t, ·), respectively, Ai is an ri-strongly monotone mapping for i = 1, 2.

Proof. The proof directly follows from the Definition 2.9 and some simple arguments.Based on Lemma 3.3, we can develop a new iterative algorithm for solving general

nonlinear random equation (1.1) as follows:Let S : Λ×E → CB(E) and T : Ω×E → CB(E) be a multi-valued mappings and

αn : Λ → (0, 1] and βn : Ω → (0, 1] be two measurable step size functions for all n ∈ N .For any given (z0(·), w0(·)) ∈ E × E, we choose (x0(·), w0(·)) ∈ E × E such that

ps(x0) = Jλ1(s),A1s

η1s,M1s(z0), y1(t) = J

λ2(t),A2t

η2t,M2t(w0),

then it is easy to know that x0 : Λ → E is measurable. Further, by Lemma 3.1 andHimmelberg [13], we know that for the chosen x0(·) and w0(·), the multi-valued mappingS(·, x0(·)) T (·, w0(·)) are measurable. Let

z1(s) ∈ (1− α0)z0(s) + α0[A1s(ps(x0)) + y0(t)− x0(s)−λ1(s)(N1s(Ft(y0), Tt(y0))− f(s))] + α0d0(s) + e0(s),

w1(t) ∈ (1− α0(t))w0(t)+α0[A2t(y0) + x0(s)− y0(t)− λ2(t)(N2t(Gs(x0), Ss(x0))− g(t))] + h0(t),

where λ1(s), λ2(t) and A1s, A2t, ps, N1s, N2t, Gs, Ft are the same as in (1.1). Then itis easy to know that z1 : Λ → E and w1 : Ω → E are measurable.

For z1(·) ∈ E and w1(·) ∈ E, we take x1(·) ∈ E and y1(·) ∈ E such that

ps(x1) = Jλ1(s),A1s

η1s,M1s(z1), y1(t) = J

λ2(t),A2t

η2t,M2t(w1),

then it is easy to know that x1 : Λ → E and y1 : Ω → E are measurable. Let

z2(s) ∈ (1− α1)z1(s) + α1[A1s(ps(x1)) + y1(t)− x1(s)−λ1(s)(N1s(Ft(y1), Tt(y1))− f(s))] + α1d1(s) + e1(s),

w2(t) ∈ (1− α1)w1(t)+α1[A2t(y1) + x1(s)− y1(t)− λ2(t)(N2t(Gs(x1), Ss(x1))− g(t))] + h1(t).

By induction, we can get an iterative algorithm for solving the nonlinear operatorequation problem (1.1) as follows:

Algorithm 3.1.STEP 1. For any given (z0(·), w0(·)) ∈ E × E, choose (x0(·), y0(·)) ∈ E × E.

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8 H.Y. Lan

STEP 2. Let

ps(xn) = Jλ1(s),A1s

η1s,M1s(zn),

yn(t) = Jλ2(t),A2t

η2t,M2t(wn),

zn+1(s) ∈ (1− αn)zn(s) + αn[A1s(ps(xn)) + y1(t)− xn(s)−λ1(s)(N1s(Ft(yn), Tt(yn))− f(s))] + αndn(s) + en(s),

wn+1(t) ∈ (1− αn)wn(t)+βn[A2t(yn) + xn(s)− yn(t)− λ2(t)(N2t(Gs(xn), Ss(xn))− g(t))] + hn(t),

n = 0, 1, 2, · · · .

(3.1)

STEP 3. Choose sequence αn and dn(s), en(s), h(t) such that for n ≥ 0, αnis a sequence in (0, 1] with

∑∞n=0 αn = ∞, and dn(s), en(s), h(t) ∈ E (n ≥ 0) are real-

valued random errors to take into account a possible inexact computation of the resolventoperator point satisfying the following conditions:

(i) dn(s) = d′n(s) + d′′n(s);(ii) limn→∞ ‖d′n(s)‖ = 0;(iii)

∑∞n=0 ‖e′′n(s)‖ < ∞,

∑∞n=0 ‖en(s)‖ < ∞,

∑∞n=0 ‖hn(t)‖ < ∞.

STEP 4. If zn+1, wn+1, xn, yn, dn, en, h(t) and αn satisfy (3.1) to sufficient accuracy,stop; otherwise, set n := n + 1 and return to STEP 2..

Remark 3.1. From Algorithm 3.1, we can get corresponding algorithms for solvingExamples 1.1, 1.2 and other special cases. See, for example, [7, 10, 11, 18, 26, 28, 34,36, 38] and the references therein.

4. Main Results

In this section, we will prove the existence of solution of the problem (1.1) and theconvergence of the iterative sequences generated by the algorithm introduced in Section3.

Lemma 4.1. Let an, bn, cn be three nonnegative real sequences satisfying thefollowing condition: there exists a natural number n0 such that

an+1 ≤ (1− tn)an + bntn + cn, ∀n ≥ n0,

where tn ∈ [0, 1] with∞∑

n=0tn = ∞, lim

n→∞ bn = 0, and∞∑

n=0cn < ∞. Then an → 0(n →∞).

Proof. The proof directly follows from the proof of Lemma 2 in Liu [29].Theorem 4.1. Let E be a q-uniformly smooth Banach space, Ai : Bi → Bi be

ri-strongly monotonefor all i = 1, 2, A1 be π-Lipschitz continuous, S : Λ× E → CB(E)be κ1-H-Lipschitz continuous in the second variable, T : Ω × E → CB(E) be κ2-H-Lipschitz continuous in the second variable, M1 : Λ×E ×E → 2E be (A1, η1)-accretivewith real-valued random variable m1(s) in the second variable and M2 : E × E × Λ →2E be (A2, η2)-accretive with real-valued random variable m2(t) in the second variable.Let η1 : Λ × E × E → E be τ1-Lipschitz continuous, η2 : Ω × E × E → E be τ2-Lipschitz continuous, N1 : Λ×E×E → E be (γ1, α1)-relaxed cocoercive and µ1-Lipschitzcontinuous in the second variable, N2 : Ω×E×E → E be (γ2, α2)-relaxed cocoercive andµ2-Lipschitz continuous in the second variable, and let N1 be β1-Lipschitz continuousin the third variable, and N2 be β2-Lipschitz continuous in the third variable. LetG : Λ × E → E be ξ1-Lipschitz continuous in the second variable, F : Ω × E → E be

422

ξ2-Lipschitz continuous in the second variable, p : Λ × E → E be δ-strongly monotoneand σ-Lipschitz continuous in the second variable and p1 : Λ × E → E defined byp1(s, x) = A1(s, p(s, x)) = A1s(ps(x)) for all (s, x) ∈ Λ×E be $-strongly monotone andς-Lipschitz continuous in the second variable. If there exist real-valued random variablesλ1(s) ∈ (0, r1(s)

m1(s)), λ2(t) ∈ (0, r2(t)m2(t)) such that

τq−11 (s)

r1(s)−λ1(s)m1(s) = τq−12 (t)

r2(t)−λ2(t)m2(t) , k = q

√1− qδ(s) + cqσq(s) < 1,

q

√1− qλ2(t)α2(s) + cqλ

q2(t)µ

q2(t)ξ

q1(s) + qλ2(t)γ

q2(t)ξ

q1(s)

< τ1−q2 (t)(r2(t)− λ2(t)m2(t))[1− k − τq−1

1 (s) q√

1−q$(s)+cqςq(s)

r1(s)−λ1(s)m1(s) ]− λ2(t)κ1(s)β2(t),q

√1− qλ1(s)α1(t) + cqλ

q1(s)µ

q1(s)ξ

q2(t) + qλ1(s)γ

q1(s)ξ

q2(t)

< τ1−q1 (s)(r1(s)− λ1(s)m1(s))[1− τq−1

2 (t) q√

1−qr2(t)+cqπq(t)

r2(t)−λ2(t)m2(t) ]− λ1(s)β1(s)κ2(t),

(4.1)

where cq1 , cq2 are the constants as in Lemma 2.1, then for each (s, t) ∈ Λ × Ω, thefollowing results follows.

(1) There exist x∗(s), y∗(t) ∈ E such that (x∗(s), y∗(t)) is a solution of the problem(1.1), i.e., the solution set Q(s, t) of the problem (1.1) is nonempty.

(2) xn(s) → x∗(s) and yn(t) → y∗(t) as n → ∞, where xn(s) and yn(t) are theiterative sequences generated by Algorithm 3.1.

Proof. In the sequel, from Lemma 3.3, we first define mappings Φ : Λ×Ω×E×E → Eand Ψ : Λ× Ω× E × E → E as follows

Φ(s, t, x, v) = x(s)− ps(x)+J

λ1(s),A1s

η1s,M1s[A1s(ps(x)) + y(t)− x(s)− λ1(s)(N1s(Ft(y), v)− f(s))],

Ψ(s, t, u, y) = Jλ2(t),A2t

η2t,M2t[A2t(y) + x(s)− y(t)− λ2(t)(N2t(Gs(x), u)− g(t))]

(4.2)

for all (s, t, x, y) ∈ Λ× Ω× E × E.Now define ‖ · ‖∗ on E × E by

‖(x, y)‖∗ = ‖x‖+ ‖y‖, ∀(x, y) ∈ E × E.

It is easy to see that (E × E, ‖ · ‖∗) is a Banach space (see [10]). By (4.2), for anygiven λ1(s) > 0 and λ2(t) > 0, define P : Λ× Ω× E × E → 2E × 2E by

P (s, t, x, y) = (Φ(s, t, x, v),Ψ(s, t, u, y)) : ∀u ∈ Ss(x), v ∈ Tt(y) and(s, t, x, y) ∈ Λ× Ω× E × E.

For any (s, t, x, y) ∈ Λ × Ω × E × E, since Ss(x) ∈ CB(E), Tt(y) ∈ CB(E),p, A1, A2, η1, η2, N1, N2, J

λ1(s),A1s

η1s,M1s, J

λ2(t),A2t

η2t,M2tare continuous, we have P (s, t, x, y) ∈

CB(E × E). Now for each fixed (s, t, x, y) ∈ Λ× Ω× E × E, we prove that P (s, t, x, y)is a multi-valued contractive mapping.

In fact, for any (s, t, x, y), (s, t, x, y) ∈ Λ×Ω×E ×E and any (a1, a2) ∈ P (s, t, x, y),there exist u ∈ Ss(x), v ∈ Tt(y) such that

a1 = x(s)− ps(x)

+Jλ1(s),A1s


a2 = Jλ2(t),A2t

η2t,M2t[A2t(y) + x(s)− y(t)− λ2(t)(N2t(Gs(x), u)− g(t))].

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10 H.Y. Lan

Note that Ss(x) ∈ CB(E), Tt(y) ∈ CB(E), it follows from Nadler’s result [30] that thereexist u ∈ Ss(x) and v ∈ Tt(y) such that

‖u− u‖ ≤ H(Ss(x), Ss(x)), ‖v − v‖ ≤ H(Tt(y), Tt(y)). (4.3)

Setting

b1 = x(s)− ps(x)

+Jλ1(s),A1s


b2 = Jλ2(t),A2t

η2t,M2t[A2t(y) + x(s)− y(t)− λ2(t)(N2t(Gs(x), u)− g(t))],

we have (b1, b2) ∈ P (s, t, x, y). It follows from Lemma 2.2 that

‖a1 − b1‖≤ ‖x(s)− x(s)− (ps(x)− ps(x))‖

+‖Jλ1(s),A1s

η1s,M1s[A1s(ps(x)) + y(t)− x(s)− λ1(s)(N1s(Ft(y), v)− f(s))]

−Jλ1(s),A1s

η1s,M1s[A1s(ps(x)) + y(t)− x(s)− λ1(s)(N1s(Ft(y), v)− f(s))]‖

≤ ‖x(s)− x(s)− (ps(x)− ps(x))‖

+τ q−11 (s)

r1(s)− λ1(s)m1(s)‖x(s)− x(s)− [A1s(ps(x))−A1s(ps(x))]‖

+‖y(t)− y(t)− λ1(s)[N1s(Ft(y), v)−N1s(Ft(y), v)]‖+λ1(s)‖N1s(Ft(y), v)−N1s(Ft(y), v)‖. (4.4)

By the assumptions on p, N1, A1, F, T and (4.3), we have

‖x(s)− x(s)− (ps(x)− ps(x))‖q ≤ (1− qδ(s) + cqσq(s))‖x(s)− x(s)‖q, (4.5)

‖x(s)− x(s)− [A1s(ps(x))−A1s(ps(x))]‖q

≤ (1− q$(s) + cqςq(s))‖x(s)− x(s)‖q, (4.6)

‖y(t)− y(t)− λ1(s)[N1s(Ft(y), v)−N1s(Ft(y), v)]‖q

≤ ‖y(t)− y(t)‖q + cqλq1(s)‖N1s(Ft(y), v)−N1s(Ft(y), v)‖q

−qλ1(s)〈N1s(Ft(y), v)−N1s(Ft(y), v), jq(y(t)− y(t))〉≤ (1− qλ1(s)α1(t) + cqλ

q1(s)µ

q1(s)ξ

q2(t) + qλ1(s)γ

q1(s)ξ

q2(t))‖y(t)− y(t)‖q, (4.7)

‖N1s(Ft(y), v)−N1s(Ft(y), v)‖≤ β2(s)‖v − v‖ ≤ β2(s)H(Tt(y), Tt(y)) ≤ β1(s)κ2(t)‖y(t)− y(t)‖, (4.8)

where cq is the constants as in Lemma 2.1. Combining (4.5)-(4.8) with (4.4), we infer

‖a1 − b1‖ ≤ θ1‖x(s)− x(s)‖+ ϑ1‖y(t)− y(t)‖, (4.9)

where

θ1 = q

√1− qδ(s) + cqσq(s) +

τ q−11 (s) q

√1− q$(s) + cqςq(s)

r1(s)− λ1(s)m1(s),

ϑ1 =τ q−11 (s)

r1(s)− λ1(s)m1(s)[λ1(s)β1(s)κ2(t)

+ q

√1− qλ1(s)α1(t) + cqλ

q1(s)µ

q1(s)ξ

q2(t) + qλ1(s)γ

q1(s)ξ

q2(t) ].

424

On the other hand, by the assumptions of S,A2, N2, G and (3.6), we can obtain

‖y(t)− y(t)− [A2t(y)−A2t(y)]‖q ≤ (1− qr2(t) + cqπq(t))‖y(t)− y(t)‖q,

‖x(s)− x(s)− λ2(t)(N2t(Gs(x), u)−N2t(Gs(x), u))‖q

≤ (1− qλ2(t)α2(s) + cqλq2(t)µ

q2(t)ξ

q1(s) + qλ2(t)γ

q2(t)ξ

q1(s))‖x(t)− x(t)‖q,

‖N2t(Gs(x), u)−N2t(Gs(x), u)‖ ≤ β2(t)κ1(s)‖x(t)− x(t)‖,

and

‖a2 − b2‖ ≤ τ q−12 (t)

r2(t)− λ2(t)m2(t)‖y(t)− y(t)− [A2t(y)−A2t(y)]‖

+‖x(s)− x(s)− λ2(t)(N2t(Gs(x), u)−N2t(Gs(x), u))‖+λ2(t)‖N2t(Gs(x), u)−N2t(Gs(x), u)‖

≤ θ2‖x(s)− x(s)‖+ ϑ2‖y(t)− y(t)‖, (4.10)

where

θ2 =τ q−12 (t)

r2(t)− λ2(t)m2(t)[λ2(t)κ1(s)β2(t)

+ q

√1− qλ2(t)α2(s) + cqλ

q2(t)µ

q2(t)ξ

q1(s) + qλ2(t)γ

q2(t)ξ

q1(s) ],

ϑ2 =τ q−12 (t) q

√1− qr2(t) + cqπq(t)

r2(t)− λ2(t)m2(t).

It follows from (4.9) and (4.10) that

‖a1 − b1‖+ ‖a2 − b2‖ ≤ υ(‖x(s)− x(s)‖+ ‖y(t)− y(t)‖), (4.11)

whereυ = maxθ1 + θ2, ϑ1 + ϑ2.

It follows from condition (4.1) that υ < 1. Hence, from (4.11), we get

d((a1, a2), P (s, t, x, y)) = inf(b1,b2)∈P (s,t,x,y)

(‖a1 − b1‖+ ‖a2 − b2‖)≤ υ‖(x(s), y(t))− (x(s)− y(t))‖∗.

Since (a1, a2) ∈ P (s, t, x, y) is arbitrary, we obtain

sup(a1,a2)∈P (s,t,x,y)

d((a1, a2), P (s, t, x, y)) ≤ υ‖(x(s), y(t))− (x(s)− y(t))‖∗.

By using the same argument, we can prove

sup(b1,b2)∈P (s,t,x,y)

d(P (s, t, x, y), (b1, b2)) ≤ υ‖(x(s), y(t))− (x(s)− y(t))‖∗.

It follows from the definition of the Hausdorff metric H on CB(E × E) that

H(P (s, t, x, y), P (s, t, x, y)) ≤ υ‖(x(s), y(t))− (x(s)− y(t))‖∗

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12 H.Y. Lan

for all (s, x, x) ∈ Λ×E×E, (t, y, y) ∈ Ω×E×E, i.e., P (s, t, x, y) is a multi-valued contrac-tive mapping, which is uniform with respect to (s, t) ∈ Λ×Ω. By a fixed point theoremof Nadler [30], for each (s, t) ∈ Λ×Ω, P (s, t, x, y) has a fixed point (x∗(s), y∗(t)) ∈ E×E,i.e., (x∗(s), y∗(t)) ∈ P (s, t, x∗(s), y∗(t)). By the definition of P , we know that there existu∗(s) ∈ Ss(x∗) and v∗(t) ∈ Tt(y∗) such that (4.2) holds. Thus, it follows from Lemma3.3 that (x∗(s), y∗(t)) ∈ Q(s, t) is a solution of the problem (1.1) and so Q(s, t) 6= ∅ forall (s, t) ∈ Λ× Ω.

Next, we prove the conclusion (2). Let (x∗(s), y∗(t)) ∈ Q(s, t) is a solution of theproblem (1.1). It follows from Lemma 3.3 that

ps(x∗) ∈ Jλ1(s),A1s

η1s,M1s[A1s(ps(x∗)) + y∗(t)− x∗(s)− λ1(s)(N1s(Ft(y∗), Tt(y∗))− f(s))],

y∗(t) ∈ Jλ2(t),A2t

η2t,M2t[A2t(y∗) + x∗(s)− y∗(t)− λ2(t)(N2t(Gs(x∗), Ss(x∗))− g(t))],

i.e.,

ps(x∗) = Jλ1(s),A1s

η1s,M1s(z∗), y∗(t) = J

λ2(t),A2t

η2t,M2t(w∗),

z∗(s) = A1s(ps(x∗)) + y∗(t)− x∗(s)− λ1(s)(N1s(Ft(y∗), v∗)− f(s)),∀v∗ ∈ Tt(y∗),

w∗(t) = A2t(y∗) + x∗(s)− y∗(t)− λ2(t)(N2t(Gs(x∗), u∗)− g(t)),∀u∗ ∈ Ss(x∗).

(4.12)

Since Ss(x∗), Ss(xn), Tt(y∗), Tn(yn) ∈ CB(E) for all n ≥ 0, for any given n ≥ 0 andε > 0, it follows from Nadler [30] that there exist un ∈ Ss(xn), vn ∈ Tt(yn) such that

‖un − u∗‖ ≤ (1 + ε)H(Ss(xn), Ss(x∗)), ‖vn − v∗‖ ≤ (1 + ε)H(Tt(yn), Tt(y∗)).

Thus, from (3.1), (4.12) and the proofs of (4.9) and (4.10), for all vn ∈ Tt(yn) andv∗ ∈ Tt(y∗), we have

‖zn+1(s)− z∗(s)‖≤ (1− αn)‖zn(s)− z∗(s)‖+ αn(‖d′n(s)‖+ ‖d′′n(s)‖) + ‖en(s)‖

+αn‖xn(s)− x∗(s)− (A1s(ps(xn))−A1s(ps(x∗)))‖+αn‖yn(t)− y∗(t)− λ1(s)(N1s(Ft(yn), vn)−N1s(Ft(y∗), vn))‖+αnλ1(s)‖N1s(Ft(y∗), vn)−N1s(Ft(y∗), v∗)‖

≤ (1− αn)‖zn(s)− z∗(s)‖+ αn‖d′n(s)‖+ (‖d′′n(s)‖+ ‖en(s)‖)+αn

q

√1− q$(s) + cqςq(s)‖xn(s)− x∗(s)‖

+αn[ q

√1− qλ1(s)α1(t) + cqλ

q1(s)µ

q1(s)ξ

q2(t) + qλ1(s)µ

q1(s)ξ

q2(t)

+λ1(s)β1(s)κ2(t)(1 + ε)]‖yn(t)− y∗(t)‖ (4.13)

and

‖wn+1(t)− w∗(t)‖≤ (1− αn)‖wn(t)− w∗(t)‖

+αn‖yn(t)− y∗(t)− (A2t(yn)−A2t(y∗))‖+αn‖xn(s)− x∗(s)− λ2(t)(N2t(Gs(xn), un)−N2t(Gs(x∗), un))‖

426


+αnλ2(t)‖N2t(Gs(x∗), un)−N2t(Gs(x∗), u∗)‖+ ‖h(t)‖≤ (1− αn)‖wn(t)− w∗(t)‖

+αnq

√1− qr2(t) + cqπq(t)‖yn(t)− y∗(t)‖

+αn[ q

√1− qλ2(t)α2(s) + cqλ

q2(t)µ

q2(t)ξ

q1(s) + qλ2(t)µ

q2(t)ξ

q1(s)

+λ2(t)β2(t)κ1(s)(1 + ε)]‖xn(s)− x∗(s)‖+ ‖h(t)‖. (4.14)

On the other hand, by Lemma 2.2 and (4.5) we know that

‖yn(t)− y∗(t)‖ ≤ ‖Jλ2(t),A2t

η2t,M2t(wn)− J

λ2(t),A2t

η2t,M2t(w∗)‖

≤ τ q−12 (t)

r2(t)− λ2(t)m2(t)‖wn(t)− w∗(t)‖ (4.15)

and

‖xn(s)− x∗(s)‖≤ ‖xn(s)− x∗(s)− (ps(xn)− ps(x∗))‖+ ‖Jλ1(s),A1s

η1s,M1s(zn)− J

λ1(s),A1s

η1s,M1s(z∗)‖

≤ q

√1− qδ(s) + cqσq(s)‖xn(s)− x∗(s)‖+

τ q−11 (s)

r1(s)− λ1(s)m1(s)‖zn(s)− z∗(s)‖,

which implies that

‖xn(s)− x∗(s)‖ ≤τq−11 (s)

r1(s)−λ1(s)m1(s)

1− q

√1− qδ(s) + cqσq(s)

‖zn(s)− z∗(s)‖. (4.16)

Combining (4.13), (4.14) with (4.15) and (4.16), we get

‖zn+1(s)− z∗(s)‖+ ‖wn+1(t)− w∗(t)‖≤ [1− αn + αnι(ε)](‖zn(s)− z∗(s)‖+ ‖wn(t)− w∗(t)‖)

+αn‖d′n(s)‖+ (‖d′′n(s)‖+ ‖en(s)‖+ ‖h(t)‖), (4.17)

where ι(ε) = maxθ(ε), ϑ(ε),

θ(ε) =τq−11 (s)

r1(s)−λ1(s)m1(s)

1− q


q

√1− q$(s) + cqςq(s) + λ2(t)β2(t)κ1(s)(1 + ε)

+ q

√1− qλ2(t)α2(s) + cqλ

q2(t)µ

q2(t)ξ

q1(s) + qλ2(t)γ

q2(t)ξ

q1(s) ,

ϑ(ε) =τ q−12 (t)

r2(t)− λ2(t)m2(t) q

√1− qr2(t) + cqπq(t) + λ1(s)β1(s)κ2(t)(1 + ε)

+ q

√1− qλ1(s)α1(t) + cqλ

q1(s)µ

q1(s)ξ

q2(t) + qλ1(s)γ

q1(s)ξ

q2(t)

427

14 H.Y. Lan

Let ε → 0. Then we have θ(ε) → θ, ϑ(ε) → ϑ and ι(ε) → ι, where ι = maxθ, ϑ and

θ =τq−11 (s)

r1(s)−λ1(s)m1(s)

1− q


q

√1− q$(s) + cqςq(s) + λ2(t)β2(t)κ1(s)

+ q

√1− qλ2(t)α2(s) + cqλ

q2(t)µ

q2(t)ξ

q1(s) + qλ2(t)γ

q2(t)ξ

q1(s) ,

ϑ =τ q−12 (t)

r2(t)− λ2(t)m2(t) q

√1− qr2(t) + cqπq(t) + λ1(s)β1(s)κ2(t)

+ q

√1− qλ1(s)α1(t) + cqλ

q1(s)µ

q1(s)ξ

q2(t) + qλ1(s)γ

q1(s)ξ

q2(t)

Since 0 < ι < 1, 1− ι > 0, it follows from (4.17) that

‖zn+1(s)− z∗(s)‖+ ‖wn+1(t)− w∗(t)‖≤ [1− αn(1− ι)](‖zn(s)− z∗(s)‖+ ‖wn(t)− w∗(t)‖)

+αn(1− ι) · 1(1− ι)

‖d′n(s)‖+ (‖d′′n(s)‖+ ‖en(s)‖+ ‖h(t)‖). (4.18)

Since∞∑

n=0αn = ∞, it follows from Lemma 4.1 and (4.18) that

‖(zn(s), wn(t))− (z∗(s), w∗(t))‖∗ → 0 as n →∞,

i.e., zn(s) → z∗(s) for all s ∈ Λ and wn(t) → w∗(t) for all w ∈ Ω. Hence, by (4.15)and (4.16), we know that the sequence xn(s) converges to x∗(s) for all s ∈ Λ and thesequence yn(t) converges to y∗(t) for all t ∈ Ω. This completes the proof.

Remark 4.1. We note that Hilbert space and Lp (or lp) (2 ≤ p < ∞) spaces are2-uniformly smooth Banach spaces. Further, in Theorem 4.1, if Ni is strongly accretivein the second variable, i.e., when γi = 0 (i = 1, 2) in Theorem 4.1, then we can obtainthe corresponding results. Our results improve and generale the known results in [7, 10,11, 18, 26, 28, 34, 36, 38]

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430

Topological Dynamic Classification of Antitriangular Maps1

Zhiming Luoa,b, Xianhua Tanga and Gengrong Zhangc

a Department of Mathematics, Central South University, Changsha, Hunan, 410083, P.R.C.;b Department of Information, Hunan Business College, Changsha, Hunan, 410205, P.R.C.;

c Department of Mathematics, Guangxi University, Nanning, Guangxi, 530004, P.R.C.

[Abstract]:We classify antitriangular maps on In by using topological dynamics. More pre-

cisely, we prove that the following properties are equivalent: (1) zero topological entropy; (2)

UR(F ) = R(F ); (3) type less than or equal to 2∞; and (4) AP (F ) = (x1, x2, · · · , xn) ∈ In :

lim F 2n(x1, x2, · · · , xn) = (x1, x2, · · · , xn). These result allow us to decide when the behavior

of these n-dimensional dynamical systems is complicated.

Keywords: zero topological entropy; uniformly recurrent points; recurrent points; almost pe-

riodic points

AMS(2000) Subject Classification: 58F10, 54H20

§1 Introduction

we consider n-dimensional maps F : In → In with

F (x1, x2, · · · , xn) = (fn(xn), fn−1(xn−1), · · · , f1(x1))

where (x1, x2, · · · , xn) ∈ In = [0, 1]×[0, 1]×· · ·×[0, 1] and fi : I → I, i = 1, 2, · · · , n are continuous

maps. When n = 2, these maps have been proposed to give a mathematical description of a

competitive production process called Cournot duopoly. And then, we say that F is a Cournot

map on I2, f1 and f2 are called reaction maps. The Cournot duopoly has been studied in

literatures [7], [8], [10], [13], [15] and [16]. In these papers, when the behavior of these two-

dimensional dynamical systems is complicated is studied. In our paper, we studied when the

behavior of these n-dimensional dynamical systems is complicated.

It is well known that positive topological entropy implies a complicated dynamical behavior

for continuous maps defined on I. More precisely, when f : I → I is continuous, the topological

entropy of f is positive if and only if fn has a horseshoe for some n ∈ N . Roughly speaking,

this implies the existence of a closed invariant subset Y ⊂ I holding that fn | Y is conjugated

to a shift of k symbols for some k ∈ N(see[17]or[2]). Additionally, h(f) = 0 implies thatf has a

restricted type of periodic orbits and simple topological dynamics. Let UR(f), R(f)and AP (f)

denote the sets of all uniformly recurrent points, all recurrent points and all almost periodic

points, resp.1This work supported by the National Science Foundation of China (10661001), the Postdoctor Sci-

entific Foundation of Central South University and the Foundation of GuangXi University (X061022);

Email:zhi ming [email protected]

1


In [17], the authors proved the following Theorem 1.1:

Theorem 1.1 Let f : I → I be continuous , the following properties are equivalent:

(a) h(f) = 0,

(b) the period of any periodic point is a power of two ,

(c) UR(f) = R(f),

(d) AP (f) = x ∈ I : limn→∞ f2n(x) = x.

Usually, Theorem 1.1 is used as a criterion to determine whether an interval map f has a

complicated dynamical behavior. Theorem 1.1 could be expected to be extended to more general

setting, for instance, to continuous maps defined on In. However, Theorem 1.1 fails in general in

case of a type of n-dimensional maps, called triangular maps with the form T (x1, x2, · · · , xn) =

(f1(x), f2(x1, x2), · · · , fn(x1, x2, · · · , xn)). For example, two-dimensional triangular maps have

the same periodic structure as one-dimensional maps [12]. But there is a T of type 2∞ such

that h(T ) > 0 (see[13] or [4]). Examples of triangular maps holding h(T)=0, type 2∞ but

UR(f) 6= R(f) can be found in [15].

Peiodic structure of antitriangular maps and interval maps are quite similar but not exactly

the same [5]. However, the dynamics of n-dimensional antitriangular maps are more closely re-

lated to one-dimensional dynamics than triangular maps. In this paper, we proved the following

theorem 4.1, which generalizes Theorem 1.1 to n-dimensional antitriangular maps.

Theorem 4.1 Let

F (x1, x2, · · · , xn) = (fn(xn), fn−1(xn−1), · · · f1(x1))

an antiargular ,Then the following properties are equivalent :

(a)h(F)=0,

(b)The period of any periodic point of F is a power of 2 ,

(c)UR(F)=R(F),

(d)AP (F ) = (x1, x2, · · · , xn) ∈ In : lim F 2s(x1, x2, · · · , xn) = (x1, x2, · · · , xn).

§2 Preliminaries

Let ϕ : X → X be a continuous map on a compact metric space X. ϕ0 means the

identity map , ϕ1 = ϕ and ϕn+1 = ϕ ϕn, n ≥ 0, where denotes the composition of maps,

Orbϕ(z) = ϕn(z)∞n=0 will denotes the orbit of z ∈ X. z ∈ X is said to be periodic if ϕn(z) = z

for some n ∈ N . The smallest positive integer satisfying this condition is called the order or

period of z. Let Per(ϕ) denote the set of periods of ϕ.

In the case of an interval map ϕ, we say that it has type 2∞ if Per(ϕ) = 2n : n ∈N. A antitriangular map F (x1, x2, · · · , xn) = (fn(xn), fn−1(xn−1), · · · , f1(x1) has type 2∞ if

f1 fn, f2 fn−1, · · · , fn f1 have type 2∞.

2

LUO ET AL432

The set of periodic points is denoted by P (ϕ). AP (ϕ) denotes the set of almost periodic

points, that is, those point x ∈ X such that for any neighborhood V = V (x) of x, there exists

an N = N(V ) ∈ N such that ϕkN (x) ∈ V , for every k ≥ 0. UR(ϕ) denotes the set of uniformly

recurrent points, that is, those points x ∈ X such that for any neighborhood V = V (x) of x, there

exists N = N(V ) ∈ N such that ϕr(x), ϕr+1(x), · · · , ϕr+N−1(x)∩V 6= ∅ for all r > 0. Finally,

the ω-limit set of ϕ at the point x is denoted by w(x, ϕ), that is, those points y ∈ X such that

there exists a sequence (ni)∞i=1 such that ϕni(x) → y as ni →∞. R(ϕ) = x ∈ X : x ∈ w(x, ϕ)is the set of recurrent points. It is well known that (see [17])

P (ϕ) ⊆ AP (ϕ) ⊆ UR(ϕ) ⊆ R(ϕ)

The definition of topological entropy of ϕ, h(ϕ) can be seen in [1,2] or [19]. For more

information on these topics see for instance [17].

Let f : I → I be a continuous map of zero topological entropy. In order to generalize The-

orem 1.1 to n-dimensional antitriangular maps, we need some additional information concerning

the structure of infinite ω-limit sets of continuous interval maps with zero topological entropy.

In [18], we can see that for any infinite ω(x, f) there is a sequence of compact intervals

J0 ⊃ J1 ⊃ J2 ⊃ · · · Ji ⊃ Ji+1 ⊃ · · ·

such that for each k ∈ N the following statements hold :

(a) f i(Jk)2k

i=1 are pairwise disjoint and f2k(Jk) = Jk.

(b) Jk+1⋃

f2k(Jk+1) ⊂ Jk

(c) w(x, f) ⊂⋃2k−1

i=0 f i(Jk). In particular , w(x, f) ⊂⋂∞

k=0

⋃2k−1i=0 f i(Jk)

(d) For each i ∈ N,w(x, f)⋂

f i(Jk) 6= ∅.In order to write easily the sets f j(Ji) for i ∈ N and 0 ≤ j < 2j , we consider the sets

Xi = 0, 1i whose elements are finite sequences of i elements composed of 0’s and 1’s . On

each Xi, the adding machine transformation A : Xi → Xi is defined by A(θ1, θ2, · · · , θi) :=

(θ1, θ2, · · · , θi) + (1, 0, · · · , 0) for all (θ1, θ2, · · · , θi) ∈ Xi, where the addition is modulo 2 from

the left to the right. For example, A(1, 1, 0, 0, 1, · · · , 1) = (0, 0, 1, 0, 1, · · · , 1). The map A can

extend to infinite sequences as follows . Let X∞ = 0, 1∞, for any α = (αi)∞i=1 ∈ X∞,define

A(α) = α + (1,0),where 0= (0, 0, · · ·). Notice that if 1= (1, 1, · · · , 1), then A(1) = 0. For

α ∈ X∞ and i ∈ N , define α |i= (α1, α2, · · · , αi) ∈ Xi by taking the first i elements of the

sequence α. For θ ∈ Xi and υ ∈ Xj , let

θ ∗ υ = (θ1, θ2, · · · , θi, υ1, υ2, · · · , υj) ∈ Xi+j .

If α ∈ X∞ and θ ∈ Xi, we similarly define θ ∗α ∈ X∞. Let 0 = 1 and 1 = 0 and in a similar

way α ∈ X∞ and θ ∈ Xi are defined . For a chosen i ∈ N , let K0|i = Ji, and for 0 ≤ j ≤ 2i.

KAj(0|i) = f j(K0|i) = f j(Ji). Notice that for all α ∈ X∞, α|i = Aj(0|i) for each i ∈ N

and some 0 ≤ j ≤ 2i. Also ,A2i(0 |i) = 0 |i . Hence the sequence of compact intervals (Kα|i)

∞i=1

3

...ANTITRIANGULAR MAPS 433

decreases to a compact (possibly degenerate) interval Kα with all the the previous notation,

conditions (a)-(d)can be rewritten as follows ([8]).

Theorem 2.1 Let f : I → I be continuous with h(f) = 0 and ω(x, f) be an infinite

ω−limit set for some x ∈ I, then there exists a sequence of compact intervals

J0 ⊃ K(0) ⊃ K(0,0) ⊃ · · · ⊃ K0|i ⊃ · · ·

such that for each k ∈ N :

(a) f j(K0|k)2k

j=1 are pairwise disjoint and f2k(K0|k) = K0|k .

(b)For each θ ∈ Xk,Kθ∗0⋃

Kθ∗1 ⊂ Kθ.

(c)w(x, f) ⊂⋃

θ∈XkKθ. In particular, w(x, f) ⊂

⋂∞k=0

⋃θ∈Xk

=⋃

α∈X∞ Kα.

(d)For each θ ∈ Xk, w(x, f)⋂

Kθ 6= ∅.

(e)If Kα is non-degenerate (| Kα |> 0) for some α ∈ X∞, then Kα is a wandering interval

(that is, f j(Kα)⋂

f i(Kα) = ∅, for all 0 ≤ i < j) .

For any pair of disjoint interval J,K ⊆ I, Write dist(J,K) = infd(x, y) : x ∈ J, y ∈ K.Int(J) denotes the interior of J.

§3 Auxiliary results

Let f : I → I be continuous interval maps with h(f) = 0. Let

ω(x, f) ⊂∞⋂

k=0

⋃θ∈X∞

Kθ =⋃

α∈X∞

Kα

be an infinite w-limit set of f . For all α ∈ X∞ and k ∈ N , let K+α|k and K−

α|k be the right and

left-side components of Kα|k \Kα. Then

Proposition 3.1[8] Under the above assumption , let x ∈ R(f) be such that

x ∈ w(x, f) ⊂∞⋂

k=0

⋃θ∈Xk

=⋃

α∈X∞

Kα,

then there exists an increasing sequence of positive integers (ni)∞i=1, with

limi→∞

ni = ∞,

such that for all k ∈ N and i ≥ 1 one of the following possibilities holds:

(a)If x ∈ Kα with | Kα |= 0, then fk2ni (x) ∈ Kα|ni,

(b)If x ∈ Kα with | Kα |> 0, then either f2ni(1+2k)(x) ∈ K+

α|nior f2ni(1+2k)

(x) ∈ K−α|ni

.

Now, we consider an arbitrary n-dimensional antitriangular map

F (x1, x2, · · · , xn) = (fn(xn), fn−1(xn−1), · · · , f1(x1)),

4

LUO ET AL434

where fi : I → I, i = 1, 2, · · · , n are continuous interval maps. Noticing that for all k ∈ N , it

holds that

F 2k(x1, x2, · · · , xn) = ((fn f1)k(x1), (fn−1 f2)k(x2), · · · , (f1 fn)k(xn)), (3.1)

and

F 2k+1(x1, x2, · · · , xn) = (fn(f1fn)k(xn), fn−1(f2fn−1)k(xn−1), · · · , f1(fnf1)k(x1)), (3.2)

So, the dynamical behavior of F must be closely related to the dynamical behavior of fn f1, fn−1 f2, · · · , f1 fn.

From the above, we have the following Proposition:

Proposition 3.2 Under the above conditions we have that

(a)P (F ) = P (fn f1)× P (fn−1 f2)× · · · × P (f1 fn) = P (F 2)

(b)AP (F ) = AP (fn f1)×AP (fn−1 f2)× · · · ×AP (f1 fn) = AP (F 2)

(c)UR(F ) = UR(F 2) ⊂ UR(fn f1)× UR(fn−1 f2)× · · · × UR(f1 fn)

(d)R(F ) = R(F 2) ⊂ R(fn f1)×R(fn−1 f2)× · · · ×R(f1 fn)

(c)h(F ) = h(fn f1) = h(fn−1 f2) = · · · = h(f1 fn)

Proof. The proof is similar to [14,10,3] and is omitted.

In order to prove our main result, we must investigate the structure of the sets R(F)

and UR(F) for antitriangular maps of zero topological entropy, that is, when fn f1, fn−1 f2, · · · , f1 fn have zero topological entropy . we study in this section the relationship among

R(fn f1), R(fn−1 f2), · · · , R(f1 fn) and R(F ).

We start with the following results in which we maintain the notation of Theorem 2.1

and write Kα1(fn f1),Kα2(fn−1 f2), · · · ,Kαn(f1 fn) to indicate that the system of intervals

depends on the compositions fn f1, fn−1 f2, · · · , f1 fn, respectively.

Proposition 3.3 Let xi ∈ R(fn−i+1fi), i = 1, 2, · · · , n. Assume that w(xi, fn−i+1fi), i =

1, 2, · · · , n are infinite. If xi = Kαi(fn−i+1 fi) for some αi ∈ X∞, i = 1, 2, · · · , n, then

(x1, x2, · · · , xn) ∈ R(F )⋂

UR(F ).

Proof Let ε > 0 be arbitary , by Theorem 2.1, there must exist an m ∈ N such that

max1≤i≤n

|Kαi|m(fn−i+1 fi)| < ε,

then , for any open neighborhood U of (x1, x2, · · · , xn) there is an m ∈ N such that

Kα1|m(fn f1)×Kα2|m(fn−1 f2)× · · · ×Kαn|m(f1 fn) ⊆ U

Applying (3.1) and Theorem 2.1, we obtain

F k2m+1(x1, x2, · · · , xn) ∈ Kα1|m(fn f1)×Kα2|m(fn−1 f2)× · · · ×Kαn|m(f1 fn) ⊆ U

5

for all k ∈ N . It follows that (x1, x2, · · · , xn) ∈ AP (F ). By (2.1) and Proposition 3.2,

(x1, x2, · · · , xn) ∈ AP (F ) ⊂ UR(F ) ⊂ R(F ), which ends the proof.

Let α ∈ X∞ and Kα = Kα(ϕ) = [z, w] be as in Theorem 2.1, where z, w ∈ I is a continuous

interval map of zero topological entropy. Define

z(ϕ) = n ∈ N : ϕ2n(Kα) ⊂ Kα|n∗αn+1

⊂ K−α|n,

and

w(ϕ) = n ∈ N : ϕ2n(Kα) ⊂ Kα|n∗αn+1

⊂ K+α|n.

Proposition 3.4 Let xi ∈ Kαi(fn−i+1fi) be such that | Kαi(fn−i+1fi) |> 0, for every i ∈1, 2, · · · , n. Assume that xi ∈ R(fn−i+1 fi), for every i ∈ 1, 2, · · · , n. If

⋂ni=1 xi(fn−i+1 fi)

is infinite,then (x1, x2, · · · , xn) ∈ R(F )⋂

UR(F ).

Proof. Without loss of generality, we suppose that Kαi(fn−i+1 fi) = [xi, xi0 ] for some

xi0 ∈ I, i ∈ 1, 2, · · · , n.

Since that⋂n

i=1 xi(fn−i+1 fi) is infinite, then there exists a sequence of positive integers

(nj)∞j=1, with limi→∞ nj = ∞ such that nj ∈⋂n

i=1 xi(fn−i+1 fi) for all j ∈ N. Hence (fn−i+1 fi)2

nj (xi) ∈ K−αi|nj

(fn−i+1 fi), for i = 1, 2, · · · , n. On the other hand , applying Theorem 2.1 ,

for any open neighborhood U of (x1, x2, · · · , xn) there exists an n ∈ N such that∏ni=1 K−

αi|m(fn−i+1 fi) ⊆ U for m ≥ n , In particular, we obtain an j ∈ N such that∏ni=1 K−

αi|nj(fn−i+1 fi) ⊆ U . By using (3.7), we have

F 2nj+1

(x1, x2, · · · , xn) = ((fn f1)2nj (x1), (fn−1 f2)2

nj (x2), · · · , (f1 fn)2nj (xn))

∈∏n

i=1 K−αi|nj

(fn−i+1 fi) ⊂ U ;

By Proposition 3.1(b) and (3.1), we have

F 2nj+1(1+2k)(x1, x2, · · · , xn) ∈ ((fnf1)2nj (1+2k)(Kα1(fnf1)), (fn−1f2)2

nj (1+2k)(Kα2(fn−1f2)), · · · , (f1 fn)2

nj (1+2k)(Kαn(f1 fn)))

⊂ ((fn f1)2nj+1k(Kα1|nj ∗α1,nj+1

(fn f1)), (fn−1 f2)2nj+1k(Kα2|nj ∗α2,nj+1

(fn−1 f2)), · · · , (f1

fn)2nj+1k(Kαn|nj ∗αn,nj+1

(f1 fn)))

= Kα1|nj ∗α1,nj+1(fn f1)×Kα2|nj ∗α2,nj+1(fn−1 f2)× · · · ×Kαn|nj ∗αn,nj+1(f1 fn)

⊂ K−α1|nj

(fn f1)×K−α2|nj

(fn−1 f2)× · · · ×K−αn|nj

(f1 fn)) ⊂ U for all k ∈ N .

Then, according to the definition of uniformly recurrent point, we obtain

(x1, x2, · · · , xn) ∈ UR(F ) ⊂ R(F ).

Proposition 3.5 Let xi ∈ Kαi(fn−i+1 fi) be such that | Kαi(fn−i+1 fi) |> 0, for every

i ∈ 1, 2, · · · , n. Assume that (x1, x2, · · · , xn) ∈ R(F ), then⋂n

i=1 xi(fn−i+1 fi) is infinite.

Proof. Suppose Kαi(fn−i+1 fi) = [xi, xi0 ] for some xi0 ∈ I, i ∈ 1, 2, · · · , n( the other

cases are similar to this .) Now let (x1, x2, · · · , xn) ∈ R(F ) and⋂n

i=1 xi(fn−i+1 fi) finite. Let

m = max⋂n

i=1 xi(fn−i+1 fi). By Proposition 3.1(b), we have that Kαi|k(fn−i+1 fi)) 6= ∅ for

each i ∈ 1, 2, · · · , n and anyk ∈ N. Moreover , according to Theorem 2.1 (e), we known that

Int(Kαi(fn−i+1 fi)))⋂

Orbfn−i+1fi)(xi) = ∅, i = 1, 2, · · · , n.

6

LUO ET AL436

On the other hand, for each i ∈ 1, 2, · · · , n, we have either

((fn−i+1 fi)2j(Kαj (fn−i+1 fi)) ⊂ K

αj |j∗(αj ,j+1)(fn−i+1 fi)) ⊂ K+

αj |j (fn−i+1 fi)

or

((fn−i+1 fi)2j(Kαj (fn−i+1 fi)) ⊂ K

αj |j∗(αj ,j+1)(fn−i+1 fi)) fi)) ⊂ K−

αj |j (fn−i+1 fi).

Let U = Int(K−α1|l(fn f1) × K−

α2|l(fn−1 f2)× · · · × K−αn|l(f1 fn)) for some l > m. Being

(x1, x2, · · · , xn) ∈ R(F ) = R(F 2), let s ∈ N satisfy that

F 2s(x1, x2, · · · , xn) ∈ K−α1|k(fn f1))×K−

α2|k(fn−1 f2)× · · · ×K−αn|k(f1 fn))).

By Theorem 2.1 and (3.7),s = 2lq for some q ∈ N( notice that we can assume that 2l ≤ s). If q

is odd ,q = 1 + 2k, then by Proposition 3.1(b) and (3.1) we would have that

F 2l+1(Kα1(fn f1)×Kα2(fn−1 f2)× · · · ×Kαn(f1 fn))

⊂ K−α1|k(fn f1)×Kα2|k(fn−1 f2)× · · · ×Kαn|k(f1 fn)

and so l ∈⋂n

i=1 xi(fn−i+1 fi), a contradiction . If q is even, q = l.2l, l odd , r > 0, by (3.1), we

would have

F 2l+r+1(Kα1(fnf1)×Kα2(fn−1f2)×· · ·×Kαn(f1fn)) ⊂ K−

α1|l(fnf1)×K−α2|l(fn−1f2)×· · ·×K−

αn|l(f1fn))

By(3.4) and since

K−α1|l+r

(fnf1) ⊂ K−α1|l(fnf1),K−

α2|l+r(fn−1f2) ⊂ K−

α2|l(fn−1f2), · · · ,K−αn|l+r

(f1fn) ⊂ K−αn|l(f1fn)

we have that

F 2l+r+1(Kα1(fn f1)×Kα2(fn−1 f2)× · · · ×Kαn(f1 fn))

⊂ K−α1|l+r

(fn f1)×K−α2|l+r

(fn−1 f2)× · · · ×K−αn|l+r

(f1 fn))

and then we can similarly conclude that l + r ∈⋂n

i=1 xi(fn−i+1 fi) which is a contradiction.

Proposition 3.6 Let xi ∈ R(fn−i+1 fi), i = 1, 2, · · · , n. Assume that w(xi, fn−i+1 fi), (i =

1, 2, · · · , n) are infinite. Let I0 ⊂ 1, 2, · · · , n.If xi ∈ Kαi(fn−i+1 fi)with | Kαi(fn−i+1 fi) |> 0

for some αi ∈ X∞, i ∈ Io;xi = Kαi(fn−i+1 fi) for some αi ∈ X∞, i ∈ 1, 2, · · · , n/I0 then

(x1, x2, · · · , xn) ∈ R(F )⋂

UR(F ).

Proof Without lose generality,we suppose that I0 = 1, 2, · · · , p . Let ε > 0, Assume , for

instance , that Kαi(fn−i+1 fi) = [xi, xi0 ] for some xi0 ∈ I, i ∈ 1, 2, · · · , p ( the other cases

are analogous ). By the proof of Theorem 3.3 and Theorem 3.5, we would have that⋂i∈Io

xi(fn−i+1 fi)

is infinite.

By Theorem 2.1, we have that

limm→∞

max| K−α1|m(fnf1) |, · · · , | K−

αp|m(fn−p+1fp) |, | Kαp+1|m(fn−pfp+1) |, · · · , | Kαn|m(f1fn) | = 0,

7


then , for any open neighborhood U of (x1, x2, · · · , xn), there is a k ∈ N such that

K−α1|m(fn f1)× · · · × K−

αp|m(fn−p+1 fp)× Kαp+1|m(fn−p fp+1)× · · · × Kαn|m(f1 fn) ⊆ U

for all mi ≥ k, i = 1, 2, · · · , n. On the other hand ,by Proposition 3.1 (b) there is an increasing

sequence of positive integer (ni)∞i=1 such that

(fn f1)2ni(1+2l)(x1) ∈ K−α1|ni

(fn f1), · · · , (fn−p+1 fp)2ni(1+2l)(xp) ∈ K−αp|ni

(fn−p+1 fp)

for all l ∈ N , By Proposition 3.1 (a) (fj fn−j+1)2nl(xj) ∈ Kαj |n(fn−j+1fj) for all n, l ∈ N, j =

p + 1, p + 2, · · · , n. Then there is an ni ∈ N such that, applying (3.1) F 2ni+1(1+2l)(x1, x2, . . . , xn)

∈ K−α1|ni

(fn f1)× · · · K−αp|ni

(fn−p+1 fp)× Kαp+1|ni(fn−p fp+1)× · · · ×Kαn|ni

(f1 fn) ⊆ U

for all l ∈ N . Therefore, (x1, x2, · · · , xn) ∈ UR(F ) and by (2.1) (x1, x2, · · · , xn) ∈ R(F ).

The following result concludes our study of the set R(F )and UR(F ) for antiargular maps

of zero entropy .

Theorem 3.7 Let F (x1, x2, · · · , xn) = ((fn f1)(x1), (fn−1 f2)(x2), · · · , (f1 fn)(xn)) an

antitriangular map having zero topological entropy. Then (x1, x2, · · · , xn) ∈ UR(F ) if and only

if (x1, x2, · · · , xn) ∈ R(F ).

Proof According to (2.1) and Proposition 3.2 , we must prove that R(F 2) ⊆ UR(F 2). So

let (x1, x2, · · · , xn) ∈ R(F 2). Noting that, by Proposition 3.2 (d) and Theorem 1.1, it holds

that xi ∈ R(fn−i+1 fi) = UR(fn−i+1 fi), i = 1, 2, · · · , n. We distinguish two cases: (a)

w(xi, fn−i+1 fi) is finite if i = 1, 2, · · · , n; (b) w(xi, fn−i+1 fi) is finite if i = 1, 2, · · · ,m, and

w(xi, fn−i+1 fi) is infinite if i = m + 1,m + 2, · · · , n.

(a) If xi is a periodic point of fn−i+1 fi, i = 1, 2, · · · , n then according to (2.1) and

Proposition 3.2 (x1, x2, · · · , xn) ∈ P (F ) ⊂ UR(F ).

(b) Now let xi be a periodic of fn−i+1fi of period 2si , i = 1, 2, · · · ,m. and w(xi, fn−i+1fi),

infinite,i = m + 1,m + 2, · · · , n.. Let s = maxsi|i = 1, 2, . . . ,m, U be an open neighaborhood

of (x1, x2, . . . , xn).Following the proof of Propositions 3.3-3.6,for any neighborhood Viof xi, i =

m + 1,m + 2, · · · , n. there exists a positive integer t ≥ s such that (fn−i+1 fi)2t(1+2k)(xi) ∈ Vi

for all k ∈ N, (i = m + 1,m + 2, · · · , n.) And then

F 2t+1(1+2k)(x1, x2, · · · , xn) = (x1, x2, · · · , xm, (fn−m fm+1)2t(1+2k)(xm+1), · · · ,

(f1 fn)2t(1+2k)(xn) ∈ V1 × V2 × · · · × Vn ⊂ U and therefore (x1, x2, · · · , xn) ∈ UR(F 2).

§4 Main theorem

In this section, we will prove our main result

Theorem 4.1 Let

F (x1, x2, · · · , xn) = (fn(xn), fn−1(xn−1), · · · f1(x1))

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LUO ET AL438

an antiargular ,Then the following properties are equivalent :

(a)h(F)=0,

(b)The period of any periodic point of F is a power of 2 ,

(c)UR(F)=R(F),

(d)AP (F ) = (x1, x2, · · · , xn) ∈ In : lim F 2s(x1, x2, · · · , xn) = (x1, x2, · · · , xn).

Proof (a) ⇒ (b) By Proposition 3.2 (e),h(F ) = h(fn−i+1 fi), i = 1, 2, · · · , n. By Theorem

1.1 fn−i+1 fi(i = 1, 2, · · · , n) have only periodic points with period a power of two . By (3.1)

and Proposition 3.2(a) F must have only periodic pionts with period a power of two .

(b) ⇒ (a) By (3.1) and Proposition 3.2 (a),fn−i+1 fi, i = 1, 2, · · · , n have only periodic

points with period a power of two . By Theorem 1.1 and Proposition 3.2 (e),h(F ) = h(fn−i+1 fi) = 0, (i = 1, 2, · · · , n).

(a) ⇒ (c) By Proposition 3.2 ,h(F ) = h(fn−i+1 fi) = 0, (i = 1, 2, · · · , n). By Theorem 3.7

it holds that UR(F ) = R(F ).

(c) ⇒ (a) If UR(F ) = R(F ), then by proposition 3.2 ,UR(F 2) = R(F 2). Applying again

Proposition 3.2 , UR(fn−i+1 fi) = R(fn−i+1 fi). Then , by Theorem 1.1 and 3.2(e), we

conclude that h(F ) = h(fn−i+1 fi) = 0, i = 1, 2, · · · , n.

(a) ⇐⇒ (d) If h(F ) = 0, then h(fn−i+1 fi) = 0, i = 1, 2, · · · , n. and by Theorem 1.1,

we obtain AP (fn−i+1 fi) = xi ∈ I : lim(fn−i+1 fi)2s(xi) = xi.Notice that AP (F ) =

AP (F 2) = AP (fn f1)×AP (fn−1 fn−1)× · · · ×AP (f1 fn) (Proposition 3.2 (b)). Then ,By

(3.1) AP (F ) = (x1, x2, · · · , xn) ∈ In : lim F 2s(x1, x2, · · · , xn) = (x1, x2, · · · , xn).

The converse implication is analogous and so the proof concludes.

§5 Conclusion

References

[1] Alseda Ll,Llibre J,Misirurewicz M. Combinatorial dynamics and entropy in dimension one.

Singapore:World Scientific;1993.

[2] Balibrea F,Canovas Pena JS,Jimenez Lopez V. Some results on entropy and sequence en-

tropy. Int J Bifurc Chaos Appl Sci Eng,1999,9:1731-1742.

[3] Balibrea F,Esquembre F, Linero A. Smooth triangular maps of type 2∞ with positive topo-

logical entropy. Int J Bifurc Chaos Appl Sci Eng,1995,5:1319-1324.

[4] Balibrea F, Linero A. On the periodic structure of the σ−permutation maps on the unit

square.Ann Math Sil,1999,13:39-49.

[5] Balibrea F, Linero A. Some results on topological dynamics of antitriangular maps.Acta

Math Hun, 2000, 88(1-2) 169-178.

[6] Bischi GI,Gardini L,Mammana C. Multistability and cyclic attractors in duopoly games.

Chaos, Solitons and Fractals,2000,11:543-564.

[7] Canovas J.S., Linero A., Topological dynamic classification of duopoly games.Chaos Solitons

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and Fractals 2001,12:1259-1266.

[8] Balibrea F., Canovas J.S., Linero A., New results on topological dynamics of antitriangular

maps. Applied general Topology, 2001, 2(1) 51-61.

[9] Kolyada SF. On dynamics of triangular maps of the square. Ergodic Theory Dynamical

Systems,1990,20:171-181.

[10] Kopel M. Simple and complex adjustement dynamics in Courtnot duopoly games. Chaos,Solitons

and Fractals,1996,7:2031-2048.

[11] Puu T. Chaos in duopoly pricing.Chaos,Solitons and Fractals,1991,1:573-581.

[12] Smital J. and Tefankova M., Distributional chaos for triangular maps, Chaos, Solitons Frac-

tals, 2004,21(5), 1125-1128

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Some Results in Intuitionistic Fuzzy Metric

Spaces

Jin Han Park, Yong Beom ParkDivision of Mathematical Sciences, Pukyong National University,599-1 Daeyeon 3-Dong, Nam-Gu, Pusan 608-737, South [email protected](J.H. Park), [email protected](Y.B. Park)

Reza SaadatiDepartment of Mathematics, Amol Islamic Azad University,

Amol 46176-54553, [email protected]

June 1, 2007

Abstract

In this paper we prove some results of metric spaces including Uniformcontinuity theorem and Ascoli-Arzela theorem for intuitionistic fuzzy met-ric spaces. We also prove that every intuitionistic fuzzy metric space hasa countably locally finite basis and use this result to conclude that everyintuitionistic fuzzy metric space is metrizable.

Key Words: Intuitionistic fuzzy metric space, Cauchy sequence, com-pleteness, uniform continuity, equicontinuity.

2000 AMS Subject Classifications: 54A40, 54E70

1 Introduction

One of the most important problems in fuzzy topology, which may have veryimportant applications in quantum particle physics particularly in connectionswith both string and ε(∞) theory which were given and studied by Elnaschie[9, 10], is to obtain an appropriate concept of fuzzy metric space. This problemhas been investigated by many authors [3, 11, 13] from different points of views.In particular, George and Veeramani [7] have introduced and studied a notionof fuzzy metric space with the help of continuous t-norms, which constitutes aslight but appealing modification of the one due to Kramosil and Michalek [13].Using the idea of intuitionistic fuzzy sets [1, 2], Park [17] introduced the notionof intuitionistic fuzzy metric spaces with the help of continuous t-norms andcontinuous t-conorms as a generalization of fuzzy metric space due to George

1


and Veeramani [7]. In this paper, we introduce the notion of uniform continuityand equicontinuity in an intuitionistic fuzzy metric space and prove Uniformcontinuity theorem for intuitionistic fuzzy metric space. We prove that if anequicontinuous sequence of functions from an intuitionistic fuzzy metric spaceX to a complete intuitionistic fuzzy metric space Y converges for each point of adense subset of X, then it converges at each point of X and the limit function iscontinuous. Using this result we prove Ascoli-Arzela theorem for intuitionisticfuzzy metric space. Finally, we prove that in an intuitionistic fuzzy metric spaceX every open cover admits a countably locally finite refinement which coversX. We use this result to prove that every intuitionistic fuzzy metric space hasa countably locally finite base.

2 Intuitionistic fuzzy metric spaces

Definition 2.1 [18] A binary operation ∗ : [0.1]× [0, 1] → [0, 1] is a continuoust-norm if ∗ is satisfying the following conditions:

(a) ∗ is commutative and associative;(b) ∗ is continuous;(c) a ∗ 1 = a for all a ∈ [0, 1];(d) a ∗ b ≤ c ∗ d whenever a ≤ c, b ≤ d and a, b, c, d ∈ [0, 1].

Definition 2.2 [18] A binary operation : [0.1]× [0, 1] → [0, 1] is a continuoust-conorm if is satisfying the following conditions:

(a) is commutative and associative;(b) is continuous;(c) a 0 = a for all a ∈ [0, 1];(d) a b ≤ c d whenever a ≤ c, b ≤ d and a, b, c, d ∈ [0, 1].

Note 2.3 The notions of t-norms and t-conorms are known as the axiomaticskeletons that we use for characterizing fuzzy intersections and unions, respec-tively. These concepts were originally introduced by Menger [15] in his study ofstatistical metric spaces. Several examples for these notions were proposed bymany authors (see [4, 5, 6, 8, 12, 18, 19] ).

Because there are three redundant conditions in definition of intuitionisticfuzzy metric space in [17], we modify the definition as follows:

Definition 2.4 A 5-tuple (X, M, N, ∗, ) is said to be an intuitionistic fuzzymetric space if X is an arbitrary set, ∗ is a continuous t-norm, is a continuoust-conorm and M , N are fuzzy sets on X2 × (0,∞) satisfying the followingconditions: for all x, y, z ∈ X, s, t > 0,

(a) M(x, y, t) + N(x, y, t) ≤ 1;(b) M(x, y, t) > 0;(c) M(x, y, t) = 1 if and only if x = y;(d) M(x, y, t) = M(y, x, t);(e) M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s);

2

PARK ET AL442

(f) M(x, y, ·) : (0,∞) → [0, 1] is continuous;(g) N(x, y, t) = N(y, x, t);(h) N(x, y, t) N(y, z, s) ≥ N(x, z, t + s);(i) N(x, y, ·) : (0,∞) → [0, 1] is continuous.

Then (M, N) is called an intuitionistic fuzzy metric on X. The functionsM(x, y, t) and N(x, y, t) denote the degree of nearness and the degree of non-nearness between x and y with respect to t, respectively.

Remark 2.5 Every fuzzy metric space (X, M, ∗) is an intuitionistic fuzzy met-ric space of the form (X, M, 1−M, ∗, ) such that t-norm ∗ and t-conorm areassociated [14], i.e. x y = 1 − ((1 − x) ∗ (1 − y)) for any x, y ∈ X.

Example 2.6 [17] Let (X, d) be a metric space. Denote a ∗ b = ab and a b =min1, a+ b for all a, b ∈ [0, 1] and let Md and Nd be fuzzy sets on X2 × (0,∞)defined as follows:

Md(x, y, t) =htn

htn + m d(x, y), Nd(x, y, t) =

d(x, y)ktn + m d(x, y)

,

for all h, k, m, n ∈ R+. Then (X, Md, Nd, ∗, ) is an intuitionistic fuzzy metricspace. We call this (Md, Nd) as the intuitionistic fuzzy metric induced by d.

Example 2.7 [17] Let X = N. Define a ∗ b = max0, a + b − 1 and a b =a + b − ab for all a, b ∈ [0, 1] and let M and N be fuzzy sets on X2 × (0,∞) asfollows:

M(x, y, t) =

xy

if x ≤ y,yx if y < x,

N(x, y, t) = y−x

yif x ≤ y,

x−yx if y < x,

for all x, y ∈ X and t > 0. Then (X, M, N, ∗, ) is an intuitionistic fuzzy metricspace.

Remark 2.8 Note that, in the above example, t-norm ∗ and t-conorm arenot associated. And there exists no metric d on X satisfying

M(x, y, t) =t

t + d(x, y), N(x, y, t) =

d(x, y)t + d(x, y)

,

where M(x, y, t) and N(x, y, t) are as defined in above example. Also note thatthe above functions (M, N) is not an intuitionistic fuzzy metric with the t-normand t-conorm defined as a ∗ b = mina, b and a b = maxa, b.Definition 2.9 [17] Let (X, M, N, ∗, ) be an intuitionistic fuzzy metric space,and let r ∈ (0, 1), t > 0 and x ∈ X. The set B(x, r, t) = y ∈ X : M(x, y, t) >1 − r is called the open ball with center x and radius r with respect tot. Define τ(M,N) = A ⊂ X : for each x ∈ A, there exist t > 0 and r ∈(0, 1) such that B(x, r, t) ⊂ A. Then τ(M,N) is a topology on X. Clearly,this topology is Hausdorff and first countable.

3

INTUITIONISTIC FUZZY METRIC SPACES 443

Theorem 2.10 [17] Let (X, M, N, ∗, ) be a intuitionistic fuzzy metric spaceand τ(M,N) be the topology on X induced by the intuitionistic fuzzy metric.Then for a sequence xn in X, xn → x if and only if M(xn, x, t) → 1 andN(xn, x, t) → 0 as n → ∞.

Definition 2.11 [17] Let (X, M, N, ∗, ) be an intuitionistic fuzzy metric space.Then

(a) a sequence xn in X is said to be Cauchy if for each ε > 0 and eacht > 0, there exists n0 ∈ N such that M(xn, xm, t) > 1 − ε for all n, m ≥ n0.

(b) (X, M, N, ∗, ) is called complete if every Cauchy sequence is convergentwith respect to τ(M,N).

Theorem 2.12 Every intuitionistic fuzzy metric space is normal.

Proof Let (X, M, N, ∗, ) be given intuitionistic fuzzy metric space and F, Gbe two disjoint closed subsets of X. Let x ∈ F . Then x ∈ Gc. Since Gc

is open, there exist tx > 0 and rx ∈ (0, 1) such that B(x, rx, tx) ∩ G = ∅.Similarly, there exist ty > 0 and ry ∈ (0, 1) such that B(y, ry, ty) ∩ F = ∅ forall y ∈ G. Let s = minrx, tx, ry, ty. Then we choose a s0 ∈ (0, s) such that(1−s0)∗ (1−s0) > 1−s. Put U =

⋃x∈F B(x, s0,

s2) and V =

⋃y∈G B(y, s0 ,

s2).

Then U and V are open sets such that F ⊂ U and G ⊂ V . Now we claim thatU ∩ V = ∅. Let z ∈ U ∩ V . Then there exist x ∈ F and y ∈ G such thatz ∈ B(x, s0,

s2) and z ∈ B(y, s0 ,

s2 ). Now, we have

M(x, y, s) ≥ M(x, z,s

2) ∗ M(y, z,

s

2)

≥ (1 − s0) ∗ (1 − s0)> 1 − s.

Hence y ∈ B(x, s, s). But since s < rx, tx, B(x, s, s) ⊂ B(x, rx, tx) and thusB(x, rx, tx) ∩G = ∅, which is a contradiction. Hence X is normal.

Remark 2.13 From above theorem we can easily deduce that every metrizablespace is normal. Since every intuitionistic fuzzy metric space is normal, Urysohnlemma and Tietze extension theorem are true in the case of intuitionistic fuzzymetric space.

3 Some theorems in intuitionistic fuzzy metric spaces

Definition 3.1 A function f from an intuitionistic fuzzy metric space X toan intuitionistic fuzzy metric space Y is said to be uniformly continuous iffor given r ∈ (0, 1) and t > 0, there exist r0 ∈ (0, 1) and t0 > 0 such thatM(x, y, t0) > 1 − r0 implies M(f(x), f(y), t) > 1− r for all x, y ∈ X.

Theorem 3.2 (Uniform continuity theorem) If f is continuous functionfrom a compact intuitionistic fuzzy metric space X to an intuitionistic fuzzymetric space Y , then f is uniformly continuous.

4

PARK ET AL444

Proof Let s ∈ (0, 1) and t > 0 be given. Then we can find r ∈ (0, 1) suchthat (1 − r) ∗ (1 − r) > 1 − s. Since f : X → Y is continuous, for each x ∈ X,we can find rx ∈ (0, 1) and tx > 0 such that M(x, y, tx) > 1 − rx impliesM

(f(x), f(y), t

2

)> 1 − r. But rx ∈ (0, 1) and then we can find sx ∈ (0, rx)

such that (1 − sx) ∗ (1 − sx) > 1 − rx. Since X is compact and B(x, sx, tx

2 ) :x ∈ X is an open covering of X, there exist x1, x2, . . . , xk in X such thatX =

⋃ki=1 B(xi, sxi ,

txi

2 ). Put s0 = min sxi and t0 = min txi

2 , i = 1, 2, . . . , k.For any x, y ∈ X, if M(x, y, t0) > 1 − s0, then M(x, y,

txi

2 ) > 1 − sxi . Sincex ∈ X, there exists a xi such that M(x, xi,

txi

2 ) > 1 − sxi . Hence we haveM(f(x), f(xi), t

2) > 1 − r. Now

M(y, xi, txi) ≥ M

(x, y,

txi

2

)∗ M

(x, xi,

txi

2

)≥ (1 − sxi) ∗ (1 − sxi ) > 1 − rxi.

Therefore, M(f(y), f(xi), t

2

)> 1 − r. Now we have

M(f(x), f(y), t) ≥ M

(f(x), f(xi),

t

2

)∗ M

(f(y), f(xi),

t

2

)≥ (1 − r) ∗ (1 − r) > 1 − s.

Hence f is uniformly continuous.

Remark 3.3 Let f be an uniformly continuous function from the intuitionisticfuzzy metric space X into the intuitionistic fuzzy metric space Y . If xn is aCauchy sequence in X, then f(xn) is also a Cauchy sequence in Y .

Theorem 3.4 Every compact intuitionistic fuzzy metric space is separable.

Proof Let (X, M, N, ∗, ) be the given compact intuitionistic fuzzy metricspace. Let r ∈ (0, 1) and t > 0. Since X is compact, there exist x1, x2, . . . , xn inX such that X =

⋃ni=1 B(xi, r, t). In particular, for each n ∈ N, we can choose

a finite subset An such that X =⋃

a∈AnB(a, 1

n , 1n ). Let A =

⋃n∈N An. Then

A is countable. We claim that X ⊂ A. Let x ∈ X. Then for each n ∈ N, thereexists an ∈ An such that x ∈ B(an, 1

n, 1

n). Thus an converges to x. But since

an ∈ A for all n, x ∈ A. Hence A is dense in X and thus X is separable.

Definition 3.5 Let X be any nonempty set and (Y, M, N, ∗, ) be an intuition-istic fuzzy metric space. Then a sequence fn of functions from X to Y is saidto be converge uniformly to a function f from X to Y if for given r ∈ (0, 1) andt > 0, there exists n0 ∈ N such that M(fn(x), f(x), t) > 1 − r for all n ≥ n0

and for all x ∈ X.

Definition 3.6 A family F of functions from an intuitionistic fuzzy metricspace X to a complete intuitionistic fuzzy metric space Y is said to be equicon-tinuous if for given r ∈ (0, 1) and t > 0, there exist r0 ∈ (0, 1) and t0 > 0 suchthat M(x, y, t0) > 1 − r0 implies M(f(x), f(y), t) > 1 − r for all f ∈ F .

5


Lemma 3.7 Let fn be an equicontinuous sequence of functions from an intu-itionistic fuzzy metric space X to a complete intuitionistic fuzzy metric space Y .If fn converges for each point of a dense subset D of X, then fn convergesfor each point of X and the limit function is continuous.

Proof Let s ∈ (0, 1) and t > 0 be given. Then we can find r ∈ (0, 1) such that(1 − r) ∗ (1 − r) ∗ (1 − r) > 1 − s. Since F = fn is an equicontinuous family,for given r ∈ (0, 1) and t > 0, there exist r1 ∈ (0, 1) and t1 > 0 such that foreach x, y ∈ X, M(x, y, t1) > 1− r1 ⇒ M(fn(x), fn(y), t

3) > 1− r for all fn ∈ F .Since D is dense in X, there exists y ∈ B(x, r1, t1) ∩ D and fn(y) convergesfor that y. Since fn(y) is a Cauchy sequence, for given r ∈ (0, 1) and t > 0,there exists n0 ∈ N such that M(fn(y), fm(y), t

3) > 1 − r for all m, n ≥ n0.Now for any x ∈ X, we have

M(fn(x), fm(x), t)

≥ M(fn(x), fn(y),t

3) ∗ M(fn(y), fm(y),

t

3) ∗ M(fm(x), fm(y),

t

3)

≥ (1 − r) ∗ (1 − r) ∗ (1 − r)> 1 − s.

for all m, n ≥ n0. Hence fn(x) is a Cauchy sequence in Y . Since Y is complete,fn(x) converges. Let f(x) = lim fn(x). We claim that f is continuous. Letso ∈ (0, 1) and t0 > 0 be given. Then we can find r0 ∈ (0, 1) such that(1−r0)∗(1−r0)∗(1−r0) > 1−s0. Since F is equicontinuous, for given r0 ∈ (0, 1)and t0 > 0, there exist r2 ∈ (0, 1) and t2 > 0 such that M(x, y, t2) > 1 − r2 ⇒M(fn(x), fn(y), t0

3 ) > 1 − r0 for all fn ∈ F . Since fn(x) converges to f(x), forgiven r0 ∈ (0, 1) and t0 > 0, there exists n1 ∈ N such that M(fn(x), f(x), t0

3 ) >1 − r0 for all n ≥ n1. Also since fn(y) converges to f(y), for given r0 ∈ (0, 1)and t0 > 0, there exists n2 ∈ N such that M(fn(y), f(y), t0

3) > 1 − r0 for all

n ≥ n2. Now for all n ≥ maxn1, n2, we have

M(f(x), f(y), t0)

≥ M

(f(x), fn(x),

t03

)∗ M

(fn(x), fn(y),

t03

)∗ M

(fn(y), f(y),

t03

)≥ (1 − r0) ∗ (1 − r0) ∗ (1 − r0)> 1 − s0.

Hence f is continuous.

Theorem 3.8 (Ascoli-Arzela theorem) Let X be a compact intuitionisticfuzzy metric space and Y be a complete intuitionistic fuzzy metric space. Let Fbe an equicontinuous family of functions from X to Y . If fnn∈N is a sequencein F such that fn(x) : n ∈ N is a compact subset of Y for each x ∈ X, thenthere exists a continuous function f from X to Y and a subsequence gn offn such that gn converges uniformly to f on X.

6

PARK ET AL446

Proof Since X is a compact intuitionistic fuzzy metric space, by Theorem 3.4,X is separable. Let D = xi : i = 1, 2, . . . be a countable dense subset of X.By hypothesis, for each i, fn(xi) : n ∈ N is compact subset of Y . Since everyintuitionistic fuzzy metric space is first countable space [17], every compactsubset of Y is sequentially compact. Thus by standard argument, we have asubsequence gn of fn such that gn(xi) converges for each i = 1, 2, . . .. ByLemma 3.7, there exists a continuous function f from X to Y such that gn(x)converges to f(x) for all x ∈ X. Now we claim that gn converges uniformly tof on X. Let s ∈ (0, 1) and t > 0 be given. Then we can find r ∈ (0, 1) suchthat (1 − r) ∗ (1 − r) ∗ (1 − r) > 1 − s. Since F is equicontinuous, there existr1 ∈ (0, 1) and t1 > 0 such that M(x, y, t1) > 1−r1 ⇒ M(gn(x), gn(y), t

3) > 1−rfor all n. Since X is compact, by Theorem 3.2, f is uniformly continuous.Hence for given r ∈ (0, 1) and t > 0, there exist r2 ∈ (0, 1) and t2 > 0 suchthat M(x, y, t2) > 1 − r2 ⇒ M(f(x), f(y), t

3 ) > 1 − r for all x, y ∈ X. Letr0 = minr1, r2 and t0 = mint1, t2. Since X is compact and D is dense in X,X =

⋃ki=1 B(xi, r0, t0) for some k. Thus for each x ∈ X, there exists i, i ≤ k,

such that M(x, xi, t0) > 1−r0. But since r0 = minr1, r2 and t0 = mint1, t2,we have, by the equicontinuity of F , M(gn(x), gn(xi), t

3) > 1 − r and we alsohave, by the uniform continuity of f , M(f(x), f(xi), t

3) > 1− r. Since gn(xj)converges to f(xj), for r ∈ (0, 1) and t > 0, there exists n0 ∈ N such thatM(gn(xj), f(xj), t

3 ) > 1 − r for all n ≥ n0, and for all j = 1, 2, · · · , n. Now foreach x ∈ X, we have

M(gn(x), f(x), t)

≥ M(gn(x), gn(xi),t

3) ∗ M(gn(xi), f(xi),

t

3) ∗ M(f(xi), f(x),

t

3)

≥ (1 − r) ∗ (1 − r) ∗ (1 − r)> 1 − s.

Hence gn converges uniformly to f on X.

Lemma 3.9 Let X be any nonempty set and (Y, d) be a metric space. Let(Y, M, N, ∗, ) be the induced intuitionistic fuzzy metric space. Then a sequencefn of functions from X to Y converges uniformly to a function f from X toY with respect to d if and only if fn converges uniformly to f with respect tothe intuitionistic fuzzy metric (M, N).

Proof We prove only the sufficiency part since the necessity part is similar.Let ε > 0 and t > 0. Put r = ε

t+ε . Since fn converges uniformly to f withrespect to the intuitionistic fuzzy metric (M, N), there exists k ∈ N such thatM(fn(x), f(x), t) > 1− r for all n ≥ k and for all x ∈ X and hence by Theorem2.10, d(fn(x), f(x)) < ε for all n ≥ k and for all x ∈ X. Hence fn convergesuniformly to f with respect to the metric d.

Lemma 3.10 Let (X, d) and (Y, d′) be metric spaces. Let (X, M, N, ∗, ) and(Y, M ′, N ′, ∗′, ′) be the corresponding induced intuitionistic fuzzy metric spaces.

7

Then a family F of functions from X to Y is equicontinuous with respect to themetric if and only if F is equicontinuous with respect to the intuitionistic fuzzymetric.

Proof We prove only the necessity part since the sufficiency part is similar.Let r ∈ (0, 1) and t > 0. Put ε = rt

1−r. Since F is equicontinuous with respect

to the metric, there exists δ > 0 such that d(x, y) < δ implies d(f(x), f(y)) < εfor all f ∈ F . Put t0 = t and r0 = δ

t0+δ . Then M(x, y, t0) > 1 − r0 ⇒M(f(x), f(y), t) > 1 − r for all f ∈ F . Hence F is equicontinuous with respectto the intuitionistic fuzzy metric.

Corollary 3.11 Let X be a compact metric space and Y be a complete metricspace. Let F be an equicontinuous family of functions from X to Y . If fnn∈N

is a sequence in F such that fn(x) : n ∈ N is a compact subset of Y for eachx ∈ X, then there exists a continuous function f from X to Y and a subsequencegn of fn such that gn converges uniformly to f on X.

Proof Let (Y, d) be the given metric space and (Y, M, N, ∗, ) be the inducedintuitionistic fuzzy metric space. Then (Y, M, N, ∗, ) is complete if and only if(Y, d) is complete. Hence by Lemmas 3.9 and 3.10 and Theorem 3.8, we get therequired result.

Now, we shall prove that every intuitionistic fuzzy metric space is metrizable.

Lemma 3.12 Let (X, M, N, ∗, ) be an intuitionistic fuzzy metric space. If Ais an open covering of X, then there is an open covering B of X such that B isa countably locally finite refinement of A.

Proof Since A is open covering of X, by well-ordering theorem we can choosea well ordering < for A. For each n ∈ N and U ∈ A, define Sn(U) = x ∈ X :B(x, 1

n , 1n ) ⊂ U and Rn(U) = Sn(U) − ⋃

V <U V . If V, W ∈ A with V < Wand if x ∈ Rn(V ) and y ∈ Rn(W ), we show that M(x, y, 1

n ) ≤ 1 − 1n . Since

x ∈ Rn(V ), we have x ∈ Sn(V ). Since y ∈ Rn(W ) and V < W , y /∈ V andhence M(x, y, 1

n) ≤ 1 − 1n . For given n ∈ N, we can find s ∈ (0, 1

n) such that(1 − s) ∗ (1 − s) ∗ (1 − s) > 1 − 1

n. Let En(U) =

⋃B(x, s, 13n

) : x ∈ Rn(U).Then clearly En(U)’s are open [17]. We claim that En(U)’s are disjoint. LetV, W ∈ A with V < W and let x ∈ En(V ) and y ∈ En(W ). Then we show thatM(x, y, 1

3n) ≤ 1 − s. If M(x, y, 13n) > 1 − s, since x ∈ En(V ) and y ∈ En(W ),

there exist x0 ∈ Rn(V ) and y0 ∈ Rn(W ) such that x ∈ B(x0, s,13n) and y ∈

B(y0 , s,13n

). Also since V < W , we have M(x0, y0,1n) ≤ 1 − 1

n. But

1 − 1n

≥ M(x0, y0,1n

)

≥ M(x, x0,13n

) ∗ M(x, y,13n

) ∗ M(y, y0,13n

)

≥ (1 − s) ∗ (1 − s) ∗ (1 − s) > 1 − 1n

,

8

PARK ET AL448

which is a contradiction and hence M(x, y, 13n) ≤ 1 − s.

Let En = En(U) : U ∈ A. We claim that En refines A. If y ∈ En(U),then there exists x ∈ Rn(U) such that y ∈ B(x, s, 1

3n). Since s < 1n , we have

y ∈ B(x, s, 13n) ⊂ B(x, 1

n , 1n ) ⊂ U . Since En(U) ⊂ U for all U ∈ A, En refines A.

We claim that En is locally finite. Since s ∈ (0, 1), we can find r0 ∈ (0, 1) suchthat (1− r0) ∗ (1 − r0) > 1− s. For each x ∈ X, B(x, r0,

16n) intersects at most

one element of En. For, if B(x, r0,16n

) intersect En(U) and En(V ) with U < V ,then there exist y ∈ En(U) and z ∈ En(V ) such that M(x, y, 1

6n) > 1 − r0 and

M(x, z, 16n) > 1 − r0. Since U < V , we have M(y, z, 1

3n) ≤ 1 − s. But

M(y, z,13n

) ≥ M(x, y,16n

) ∗ M(x, z,16n

)

≥ (1 − r0) ∗ (1 − r0) > 1 − s,

which is a contradiction. Hence En is locally finite. Now, we consider the familyB =

⋃n∈N En. Let x ∈ X. Since A is cover of X, there exists a U ∈ A such that

U is the first element of A that contains x. Since U is open, there exists n ∈ Nsuch that B(x, 1

n , 1n ) ⊂ U . Then x ∈ Sn(U) and since U is the first element of A

that contains x, x ∈ Rn(U) and thus x ∈ En(U). Hence B is an open coveringof X such that B is countably locally finite refinement of A.

Theorem 3.13 Every intuitionistic fuzzy metric space has a countably locallyfinite basis.

Proof For each n ∈ N, let An = B(x, 1n , 1

n) : x ∈ X. Then An covers Xfor each n ∈ N. By Lemma 3.12, there exists an open covering Bn of X whichis a countably locally finite refinement of An. Let B =

⋃n∈N Bn. Then B is

countably locally finite. We claim that B is a basis for X. Let x ∈ X. Givenr ∈ (0, 1) and t > 0, we can find n0 ∈ N such that (1 − 1

n0) ∗ (1 − 1

n0) > 1 − r

and t > 2n0

. Let B ∈ Bn0 with x ∈ B. Since Bn0 refines An0 , there exists ax0 ∈ X such that B ⊂ B(x0,

1n0

, 1n0

). For any y ∈ B, we have

M(x, y, t) > M(x, y,2n0

) ≥ M(x, x0,1n0

) ∗ M(y, x0,1n0

)

≥ (1 − 1n0

) ∗ (1 − 1n0

) > 1 − r.

Then y ∈ B(x, r, t) and thus B ⊂ B(x, r, t).

Corollary 3.14 Every intuitionistic fuzzy metric space is metrizable.

Proof By Theorem 2.12 and 3.13, every intuitionistic fuzzy metric space isregular and has a basis that is countably locally finite. Hence by Nagata-Smirnovmetrization theorem ([16], Theorem 40.3, p. 250), it is metrizable.

Remark 3.15 The topologies induced by a metric and the corresponding in-tuitionistic fuzzy metric are same. Hence by Theorem 3.13, we can deduce thatevery metric space has a basis that is countably locally finite. Since intuitionis-tic fuzzy metric is a generalization of fuzzy metric, the results for fuzzy metricspace are particular cases of the above theorem.

9

4 Conclusions

As a generalization of fuzzy metric, the notion of intuitionistic fuzzy metric wasintroduced by Park [17]. Because, in intuitionistic fuzzy metric space, the degreeof nearness and the degree of non-nearness between two points with respect tosome value are allowed, there is room for more flexibility. In this paper, we provesome results of metric spaces including Uniform continuity theorem and Ascoli-Arzela theorem for intuitionistic fuzzy metric spaces and also prove that everyintuitionistic fuzzy metric space is metrizable. Applications of intuitionisticfuzzy metric may have been done in quantum particle physics particularly inconnections with both string and ε(∞) theory [9, 10].

Acknowledgments

This work was supported by the Korea Research Foundation Grant funded bythe Korean Government(MOEHRD) (KRF-2006- 521-C00017).

References

[1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96(1986).

[2] K.T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems,33, 37-46 (1989).

[3] Deng Zi-Ke, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl., 86, 74-95(1982).

[4] J. Dombi, A general class of fuzzy operators, the De Morgan class of fuzzyoperators and fuzziness measures induced by fuzzy operators, 8 (1982),49-62.

[5] D. Dubois and H. Prade, New results about properties and semantics offuzzy set-theoretic operators, In Fuzzy sets: Theory and Applications toPolicy Analysis and Information Systems (P.P. Wang and S.K. Chang,eds.), Plenum Press, New York, 1980.

[6] M.J. Frank, On the simultaneous associativity of F (x, y) and x+y−F (x, y),Aequationes Math., 19, 194-226 (1979).

[7] A. George and P. Veeramani, On some results in fuzzy metric spaces, FuzzySets and Systems, 64, 395-399 (1994).

[8] H. Hamacher, Uber logische Verknupfungen unscharfer Aussagen und derenZugehorige Bewertungsfunktionen, In Progress in Cybernetics and SystemsResearch Vol. 3 (R. Trappl, G.J. Klir and L. Ricciardi, eds.), Hemisphere,Washington DC, 1978.

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[9] M.S. Elnaschie, On the uncertainty of Cantorian geometry and two-slitexperiment, Chaos, Soliton and Fractals, 9, 517–529 (1998).

[10] M.S. Elnaschie, On the verifications of heterotic strings theory and ε(∞)

theory, Chaos, Soliton and Fractals, 11, 397-407 (2000).

[11] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems,12, 215-229 (1984).

[12] E.P. Klement, Operations on fuzzy sets: an axiomatic approach, Informa-tion Sciences, 27, 221-232 (1984).

[13] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces,Kybernetica, 11, 326-234 (1975).

[14] R. Lowen, Fuzzy Set Theory, Kluwer Academic Publishers, Dordrecht,1996.

[15] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci., 28, 535-537 (1942).

[16] J.R. Munkres, Topology (2nd ed.), Prentice-Hall, New Jersey, 2000.

[17] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals,22, 1039-1046 (2004).

[18] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math., 10,314-334 (1960).

[19] R.R. Yager, On a general class of fuzzy connectives, Fuzzy Sets and Systems,4, 235-242 (1980).

[20] L.A. Zadeh, Fuzzy sets, Inform and Control, 8, 338-353 (1965).

11


1

A Preconditioned Linear Sampling Method in In-verse Acoustic Obstacle Scattering

G. Pelekanos∗, K. H. Leem∗, and V. Sevroglou†

Abstract — The problem of determining the shape of an obstacle from far-fieldmeasurements is considered. It is well known that Kirsch’s (F ?F )1/4-methodhas been widely used for shape reconstructions obtained via the singular systemof the ill conditioned discretized far-field operator F . In this work we presenta preconditioned version of this method. In particular, the singular system ofan appropriate preconditioner constructed via the Algebraic Multigrid Method(AMG) is used for the reconstructions since it exploits stable eigenvalues, incontrast to the ones of the disretized far field operator.

Keywords:AMG Preconditioning, Inverse Scattering, Linear Sampling Method.

1. INTRODUCTION

For inverse acoustic scattering the original linear sampling method was introducedby Colton and Kirsch [7], and mathematically clarified in [8]. Kirsch [9] improvedthe original version of the linear sampling method, leading to the so-called (F ?F )1/4

-method. Recently, Arens [2] presented a proof of convergence for this method forthe case of acoustic scattering by sound soft obstacles. Linear sampling methodsinvolve the solution of a linear Fredholm equation of the first kind, the far-fieldequation, which is written for each point x0 inside the scatterer and whose integralkernel is the far-field pattern, i.e. far-field data that are usually contaminated withsignificant noise. Its right-hand side is an exactly known analytic function. Conse-quently, in the second version of the linear sampling method, Kirsch, characterizedthe obstacle in terms of the spectral data of the far field operator F . Two of the at-tractive futures of this method is its computational speed, and the very low amountof a priori information on the scatterers. In other words, it is not necessary to apriori know the number of scatterers or the kind of the boundary condition thatis satisfied by the total field. One of its disadvantages though is that it only givesan explicit characterization of the scattering obstacle (i.e. it only determines thesupport of the scatterer).

It is well known that every numerical implementation of an inverse scatteringmethod requires at some point regularization in order to cope with the ill-posednessof the problem, and the linear sampling method is not an exception. In most numer-ical applications of the linear sampling method, Tikhonov regularization has been

∗Department of Mathematics and Statistics, Southern Illinois University, Edwardsville,IL 62026, USA. E-mails: [email protected], [email protected]

†Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece . E-mail:[email protected]


2

employed and the regularization constant was computed via Morozov’s discrep-ancy principle [8], which involved the computation of the zeros of the discrepancyfunction at each point of the grid, a process that is time-consuming. In addition,the noise level in the data should be known a priori, something that in real lifeapplications is not the case in general.

In this work we propose a characterization of the object via a preconditioned(F ?F )1/4 -method. This paper attempts to alleviate the ill-posedness of this prob-lem via the construction of an appropriate preconditioner. The preconditioner iseffectively constructed via the Algebraic Multigrid Method (AMG) from the de-scretized form of the operator (F ?F )1/2 and is symmetric and positive definite. Itis very important to point out that in our approach we are not dealing with theeigenvalues of the ill possed matrix that corresponds to the discretized form of theoperator F , but we rather dealing with the more stable eigenvalues of the matrixthat is constructed via the AMG.

We organize our paper as follows. Section 2 will be devoted to the formulationof the problem. Consequently, section 3 will deal with the idea behind the selectionof our preconditioner as well as with its implementation within the framework ofthe linear sampling method. In section 4 we will describe the construction of thepreconditioner via the Algebraic Multigrid Method (AMG). Finally, in section 5we will show the effectiveness of the preconditioner via numerical examples for thecase of inpenetrable scatterers. For the reconstructions we will use simulated dataobtained by means of the Nystrom method [6].

2. FORMULATION OF THE PROBLEM

It is well known that the propagation of time-harmonic acoustic fields in a homo-geneous medium, in the presence of a sound soft obstacle D, is modeled by theexterior boundary value problem (direct obstacle scattering problem)

42u(x) + k2u(x) = 0, x ∈ <2 \ D (2.1)u(x) + ui(x) = 0, x ∈ ∂D (2.2)

where k is a real positive wavenumber and ui is a given incident field, that in thepresence of D will generate the scattered field u.In addition, the scattered field u will satisfy the Sommerfeld radiation condition

limr→∞

√r(

∂u

∂r− iku) = 0 (2.3)

|r| = |x|, x ∈ <2 \ D, and the limit is taken uniformly for all directions x = x/|x|.The Green formula implies that the solution u of the direct obstacle scattering

problem above has the asymptotic behaviour

u(x) = u∞(x)eikr

√r

+ O(r−3/2) (2.4)

PELEKANOS ET AL454

3

for some analytic function u∞, called the far field-pattern of u, and defined on theunit sphere Ω. In the case of the inverse problem, it represents the measured data.In particular, the inverse problem that will be considered here, is the problem offinding the shape of D from a complete knowledge of the far-field pattern.

We now define the far-field equation

(Fgz)(x) =eiπ/4

√8πk

e−ikx·z (2.5)

where the right hand side is the far-field pattern of the fundamental solution of theHelmholtz equation, z ∈ <2 and F : L2(Ω) → L2(Ω) is given by

(Fg)(x) =∫

Ω

u∞(x; d) g(d) ds(d), d ∈ Ω (2.6)

It is well known that the first version of the linear sampling method [7] solves thelinear operator equation (2.5) based on the numerical observation that its solutionwill have a large norm outside and close to ∂D. Hence, reconstructions are obtainedby plotting the norm of the solution. However, the problem is that the right-handside does not in general belong to the range of the operator F . Kirsch [9] wasable to overcome this difficulty with the introduction of a new version of the linearsampling method based on appropriate factorization of the far-field operator F . Inthis method, Kirsch is elegantly using the spectral properties of the operator F tocharacterize the obstacle. In particular, the following far-field equation is now usedin place of equation (2.5)

(F ?F )1/4gz =eiπ/4

√8πk

e−ikx·z (2.7)

and the spectral properties of F are used for the reconstructions. However, dueto noisy data, the discretized version of the far field operator F is characterizedby numerical instability which may result to false information about its singularsystem. In the next section, we will show that we can overcome this difficulty via theconstruction of an appropriate preconditioner with eigenvalues far away from zero(in contrast to the eigenvalues of F that cluster around zero) and whose singularsystem will be used for the reconstructions. That way, Tikhonov regularization canbe avoided and the noise in the data does not need to be known in advance.

3. THE PRECONDITIONED LINEAR SAMPLING METHOD

The far field operator F maps a density g onto the far-field pattern of thescattered field generated from the incident field ui = vg via the Herglotz waveoperator H : L2(Ω) → L2(∂D) given by

Hg(x) =∫

Ω

eikx·d g(d) ds(d), x ∈ ∂D (3.1)

...ACOUSTIC OBSTACLE SCATTERING 455

4

As indicated in [9], closely related to F , is the operator G : L2(∂D) → L2(Ω)defined by Gh = u∞, where u∞ ∈ L2(Ω) is the far field pattern of the solution uof the exterior Dirichlet problem with boundary data h ∈ L2(∂D).

The far-field operators H and G are related through the single layer boundaryoperator S : L2(∂D) → L2(∂D), defined by

Sφ(x) =∫

∂D

Φ(x, y)φ(y) ds(y), φ ∈ L2(∂D) (3.2)

where Φ denotes the fundamental solution of the Helmholtz equation in two dimen-sions and defined by

Φ(x, y) =i

4H

(1)0 (k|x− y|) x 6= y (3.3)

It is not difficult to show that

H = ζ S? G? on L2(∂D) (3.4)

andF = −GH (3.5)

with ζ =eiπ/4

√8πk

in two dimensions [9]. Using (3.4) and (3.5), we arrive to the

factorization [9]

F = −1ζGS?G? (3.6)

We now consider a singular system (µj , ψj , ψj) of the operator F in L2(Ω).Since F is normal [9], we have µj = |λj | and ψ = sign(λj)ψ, where λj are theeigenvalues of F . Using that system, Kirsch showed that the sequence defined by

φj =1

i√

λj

G?ψj (3.7)

forms a Riesz basis in H−1/2(∂D) [9].The above result combined with Picard’s theorem yields a characterization of

the range of G [9]:

Range(G) = ∞∑

j=1

ρjψj :∞∑

j=1

|ρj |µj

< ∞ = Range((F ?F )1/4) (3.8)

Our approach from the other hand is based on the operator F = (F ?F )−1/2.Since Range((F ?F )1/2)⊆ Range((F ?F )1/4) we define the operator G1 : M →Range(G1) as a restriction of G i.e.

G1 = G |M with M = φ ∈ L2(∂D) | Gφ ∈ Range((F ?F )1/2)

PELEKANOS ET AL456

5

Applying the operator F on both sides of (2.7) we obtain

(F?F)1/4gz = Fζe−ikx·z, (3.9)

with the understanding that the right hand side of the equation (2.7) is an elementof Range((F ?F )1/2). Comparing now equations (2.7) and (3.9), it is easy to seethat solving the latter is a matter of using the singular system (µj , χj , χj) of theoperator F instead of F . Furthermore, all theoretical results presented above, andwith details in [9], remain valid with just minor modifications. In particular, it iseasy to show that equation (3.8) can now take the form

Range((F?F)1/4) = ∞∑

j=1

ρjχj :∞∑

j=1

|ρj |µj

< ∞ = Range((FG1) (3.10)

where the µj ’s denote the eigenvalues of F .The discretized form of F is constructed using the AMG, and numerical ex-

periments presented in section 5 show that it is better conditioned than F , henceits use will provide reliable results even without the use of Tikhonov regulariza-tion. The following section describes the way the authors constructed an effectivepreconditioner for the solution of the discretized form of equation (2.7).

4. CONSTRUCTION OF THE AMG PRECONDITIONER

Frequently, the solution of the n× n system Ax = b requires multiplication ofboth of its sides by a nonsingular n× n matrix M−1, called preconditioner, i.e.

M−1Ax = M−1b (4.1)

The role of the preconditioner is to make the matrix M−1A better conditionedthan the original matrix A and hence to accelerate the convergence of the iterativemethod [13, 14] used. Developing efficient preconditioners has been one of themajor research interests in computational electromagnetics [10, 12].

AMG methods have been developed for solving linear systems posed on large,unstructured grids because they do not require geometric grid information. Classi-cal AMG was originally used for the solution of linear systems with symmetric andpositive definite M -matrices. Recently, many new AMG approaches are developedto solve more general linear systems [3, 5].

As mentioned in section 3, we are trying to solve the discretized form of the pre-conditioned problem (3.9). Notice that the preconditioner M−1 introduced in (4.1)corresponds to the discretized form of F . This preconditioner can be constructedby solving the following systems:

Mx = e, (4.2)

where the right-hand side vector e = e(k), k = 1, · · · , n, and with e(k) being thekth canonical vector. Hence, the solution x corresponds to the kth column of M−1.

6

In this paper, we suggest to solve (4.2) via the AMG. We begin our discussionby introducing the basic framework of AMG. AMG does not require to access thephysical grids of problems. With “grids” we mean sets of indices of the unknownvariables. Hence the grid set for (4.2) is Ω = 1, 2, · · · , n, since the unknownvector x in (4.2) has components x1, x2, ..., xn. The main idea of AMG is to removethe smooth error by coarse grid correction, where smooth error is the error noteliminated by relaxation on the fine grid, which also can be characterized by smallresiduals [4]. In order to develop the multi-grid algorithm, we consider the setsof grids in each level. The number 0 stands for the finest-grid level. Then thenumbers 1, 2, · · · , lmax represent the corresponding coarse-grid levels. A directsolver was used at the coarsest-grid level lmax. Hence, equation (4.2) can be writtenas M0x0 = e0 and the set of finest grid set is Ω0 = Ω. Assume now that we havedefined a relaxation scheme, a set of coarse-grid points Ωl, a coarse-grid operatorAl, where l = 1, 2, · · · , lmax, and intergrid transfer operators I l+1

l (restriction) andI ll+1 (interpolation), where l = 0, 1, · · · , lmax − 1. For the relaxation scheme we

employ Gauss-Seidel iterations.We are now ready to perform an AMG V-cycle as follows:

Algorithm 4.1 AMG V-Cycle

xl ← AMGV(Ml,xl, el)

if Ωl = coarsest grid, then

xl ← DIRECTSOLVE (Mlxl = el)

else

xl ← Relax ν1 times on Mlxl = el on Ωl with initial guess xl

el+1 ← I l+1l (el −Mlxl)

xl+1 ← 0,

xl+1 ← AMGV(Ml+1,xl+1, el+1).

Correct xl = xl + I ll+1xl+1.

xl ← Relax ν1 times on Mlxl = el on Ωl with xl.

endif

In general, the restriction operator I l+1l is defined by the transpose of the

interpolation operator I ll+1, i.e., I l+1

l = (I ll+1)

T and the coarse grid operator Ml+1

is constructed from the fine grid operator Ml by the Galerkin approach:

Ml+1 = I l+1l Ml I l

l+1, (4.3)

so that AMG satisfies the principle that the coarse-grid problem needs to providea good approximation to fine-grid error in the range of interpolation [4]. Hence, to

PELEKANOS ET AL458

7

set up an AMG preconditioner, we need to find a suitable coarsening strategy andan effective interpolation operator. The creation of the coarse-grid sets Ωl, wherel = 1, 2, · · · , lmax is based on a combinatorial clustering algorithm developed byVanek, Mandel and Brezina in [15] with normalized edge weights:

ωij = |Mij |/√|Mii| · |Mjj |. (4.4)

The first step of their coarsening algorithm iterates through the nodes Ωl =Cl

1, Cl2, · · · , Cl

nl creating clusters j | ωij ≥ η1 for a given tolerance η1 > 0,

provided no node in j | ωij ≥ η1 is already a cluster. Two nodes i and j aresaid to be strongly connected if ωij ≥ η1. In the second step, unassigned nodesare assigned to a cluster from step one to which the node is strongly connected,if any. In the last step, the remaining nodes are assigned to clusters consisting ofstrong neighborhoods intersecting with the set of remaining nodes. As explainedearlier, the next important phase in AMG is to construct an interpolation opera-tor. In smoothed aggregation AMG [11], we solve a local linear system to obtainan interpolation vector that interpolates a value for a coarse-grid cluster onto itsneighborhood. In other words, we consider one cluster Cl+1

k , called Ck, from theset of coarse-grid points Ωl+1. For each cluster Ck, we define a neighborhood Nk

asNk = j 6∈ Ck | ωij ≥ η2, i ∈ Ck,

and also have an associated interpolation vector pk, which is the kth column of theinterpolation operator I l

l+1 = [p1, p2, · · · , pnl+1 ]. Denote MIJ = [ aij | i ∈ I, j ∈ J ]to be an |I|× |J | matrix where I and J are sets of nodes. For a cluster Ck, its localmatrix LCk

is given by

LCk=

[MCkCk

MCkNk

MNkCkMNkNk

]. (4.5)

Then its corresponding interpolation vector pk, the kth column of I ll+1, can be

obtained by solving the local linear system below:

LCkpk = εCk

, (4.6)

where εCkis the vector given by

(εCk)i =

1 if i ∈ Ck,

0 otherwise.(4.7)

Using I l+1l = (I l

l+1)T and Ml+1 = I l+1

l MlIll+1, we have defined all components of

the AMG method.After the discussion above it becomes apparent that solving the system (4.2)

via an AMG V-cycle (Algorithm 4.1) for k = 1, · · · , n, we obtain the desired pre-conditioner M−1.


8

−10 −5 0 5 10−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

10

20

30

40

50

60

Figure 1: Visualization of a circle (left). Profile reconstruction via the AMGprecondtitioner (right).

−50 0 50 100 150 200−4

−2

0

2

4

6

8

10

12

14

Real(λ)

Imag

(λ)

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Real(λ)

Imag

(λ)

Figure 2: Eigenvalue distribution of F and F (left). Eigenvalue distributionof F and F around zero (right).

5. NUMERICAL APPLICATIONS

In this section we present reconstructions of impenetrable objects from a finitenumber of u∞ measurements. In particular, synthetic data produced using Nystrommethod are used. Visualizations follow from equation (3.10) in a very naturalway. The key fact in our approach is the utilization of the singular system ofthe numerically stable discretized form of the operator F , in place of the singularsystem of F . In the sequel, along with domain reconstructions, we provide someeigenvalue plots for the discretized operators F and F respectively. In the numericalexperiments below, a pointwise random noise is imposed on F in a similar way asin [1]. In particular, let NR and NI be random matrices such that the noisy far-fieldmatrix Fδ is defined by

Fδ = F + ε (NR + iNI)F,

where ε is given, and the corresponding noise level is computed in the matrixspectral norm, ‖Fδ − F‖2 ≤ δ.

PELEKANOS ET AL460

9

−10 −5 0 5 10−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

10

20

30

40

50

60

Figure 3: Visualization of a circle and a ellipse (left). Profile reconstructionvia the AMG precondtitioner (right).

−20 0 20 40 60 80 100−5

0

5

10

15

20

Real(λ)

Imag

(λ)

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Real(λ)

Imag

(λ)


In the figures of eigenvalue distribustions, ′×′ and ′′ represent the eigenvaluesof the discretized forms of the operators F and F respectively. It is observed thateven in the presence of noisy far-field measurements the eigenvalues of F are locatedfar away from zero, in contrast to the eigenvalues of F that are clustered around it.

In our first numerical experiment we are reconstructing the profile of a circleof radius 1. The far-field pattern has been synthetically produced via Nystrommethod in the case of 21 incident and observed directions with k = 1. The objectis located in a grid of 63× 63 points. Furthermore, a 1% noise is pointwise addedto each element of the far-field matrix as described above. The left image in figure1 shows the obstacle that we are about to reconstruct, whereas the right one showsthe profile reconstructed via the AMG. Furthermore, the left image in figure 2shows the eigenvalue distributions for the operators F and F respectively, and theright one captures their eigenvalues around zero. Notice that very few eigenvaluesof F appear within a ”small window” around zero, in contrast to the ones of F .This fact suggests that costly identification of the optimal value of the parameter


10

−10 −5 0 5 10−10

−8

−6

−4

−2

0

2

4

6

8

10

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

10

20

30

40

50

60

Figure 5: Visualization of a kite and a ellipse (left). Profile reconstructionvia the AMG precondtitioner (right).

−100 0 100 200 300 400 500 600 700−10

0

10

20

30

40

50

60

Real(λ)

Imag

(λ)

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Real(λ)

Imag

(λ)


in Tikhonov’s regularization can be avoided. In addition, the amount of noise inthe data doesn’t need to be known a priori.

In our second numerical experiment D is given by two disjoint obstacles D =D1 ∪D2, where D1 is the vertical ellipse with axes 1 and 2 and D2 is the circle ofradius 1 and center at the point (4, 5). The objects are located in a grid of 40× 40points. We took k = 2 and computed the far field pattern for 40 incident andobserved directions. In addition 2% noise has been added to the far-field matrix.Figure 3 shows the original(left) and the reconstructed profile (right). The lastimage of figure 4 shows the eigenvalue distribution for the the operators F and F ,respectively. Similarly as in the previous example the eigenvalues of F are movingfar away from zero and the method yielded reliable reconstructions.

In our last numerical experiment we are dealing with an example consideredin [9], were the profiles of a kite and an ellipse are considered. The ellipse hasaxes 0.8, 0.4 and center (0, 0), while the kite is parametrized by x(t) = (cos t +0.65 cos(2t) − 0.65, 1.5 sin t), t ∈ [0, π], then rotated by 70 degrees and shifted to

PELEKANOS ET AL462

11

center (3, 5). In this case the objects are located in a grid of 80× 80 points, k = 1,and the far field pattern is computed for 80 incident and observed directions. Inthis example 5% noise has been added to the data. Original profiles as well asreconstructions and eigenvalue distibutions are shown in figures 5 and 6. It turnsout that our reconstructions are comparable to the ones of [9], that obtained viaTikhonov regularization.

In all three numerical experiments the construction of the preconditioner tookjust a few seconds on a Pentium 4 at 3.4GHz CPU with 3 Gb of memory.

References

[1] K. Anagnostopoulos and A. Charalambopoulos. The linear sampling methodfor the transmission problem in 2D anisotropic elasticity Inverse Problems(2006) 22, 553-577.

[2] T. Arens, Why linear sampling works. Inverse Problems (2002) 20, 163-173.

[3] R. E. Bank and R. K. Smith, An algebraic multilevel multigraph algorithm.SIAM J. Sci. Comput. (2002) 23, 1572-1592.

[4] A. Brandt, S. McCormick and J. Ruge, Sparse and its applications (Cam-bridge:Cambridge Univ. Press), 1985.

[5] M. Brezina, A. J. Cleary, R. D. Falgout, V. E. Henson et al. Algebraic Multigridbased on element interpolation (AMGe) SIAM J. Sci. Comput. (2000) 22,1570-1592.

[6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering The-ory. (New York:Springer-Verlag), 1992.

[7] D. Colton and A. Kirsch, A simple method for solving the inverse scatteringproblems in the resonance region. Inverse Problems (1996) 12, 383-393.

[8] D. Colton, M. Piana and R. Potthast, A simple method using Morozov’s dis-crepancy principle for solving inverse scattering problems, Inverse Problems(1999) 13, 1477-93

[9] A. Kirsch, Characterization of the shape of a scattering obstacle using thespectral data of the far field operator. Inverse Problems (1998) 14, 1489-1512.

[10] J. Lee, J. Zhang and C.C. Lu, Sparse inverse preconditioning of multilevel fastmultipole algorithm for hybrid integral equations in electromagnetics. IEEE,Trans. Antennas Propagat. (2004), 52:9 2277-2287.

[11] K. H. Leem, S. Oliveira and D. E. Stewart, Algebraic Multigrid (AMG) for sad-dle point systems from meshfree discretizations. Num. Lin. Alg. Appl. (2004)11, 293-308.

[12] K. H. Leem and G. Pelekanos, Algebraic multigrid preconditioner for homo-geneous scatterers in electromagnetics. IEEE Trans. Antennas Propag. (2006)AP-23, 2081-2087.


12

[13] Y. Saad and H. A. van der Vorst, Iterative solution of linear systems in the20th century. J. Comp. Appl. Math. (2000) 123, 1-33.

[14] L. N. Trefethen and D. Bau, Numerical Linear Algebra (Siam), 1997.

[15] P. Vanek, J. Mandel, and M. Brezina, Algebraic multigrid by smoothed ag-gregation for second and forth order elliptic problems. Computing (1996) 56,179-196.

PELEKANOS ET AL464

Asymptotic distribution of the sample averagevalue-at-risk

Stoyan V. StoyanovChief Financial Researcher,

FinAnalytica Inc., Seattle, USA

e-mail: [email protected]

Svetlozar T. Rachev∗

Department of Econometrics and Statistics,

University of Karlsruhe, D-76128 Karlsruhe, Germany and

Department of Statistics and Applied Probability,

University of California Santa Barbara, CA 93106, USA


October 2, 2007

∗Rachev gratefully acknowledges research support by grants from Division of Mathe-matical, Life and Physical Sciences, College of Letters and Science, University of California,Santa Barbara, the Deutschen Forschungsgemeinschaft and the Deutscher AkademischerAustausch Dienst.


Abstract

In this paper, we prove a result for the asymptotic distributionof the sample average value-at-risk (AVaR) under certain regularityassumptions. The asymptotic distribution can be used to derive as-ymptotic confidence intervals when AV aRε(X) is calculated by theMonte Carlo method which is adopted in many risk management sys-tems. We study the effect of the tail behavior of the random variableX on the convergence rate and the improvement of a tail truncationmethod.

Keywords average value-at-risk, risk measures, heavy-tails, asymptoticdistribution, Monte Carlo

1 Introduction

The average value-at-risk (AVaR) risk measure has been proposed in theliterature as a coherent alternative to the industry standard Value-at-Risk(VaR), see Artzner et al. (1998) and Pflug (2000). It has been demonstratedthat it has better properties than VaR for the purposes of risk managementand, being a downside risk-measure, it is superior to the classical standarddeviation and can be adopted in a portfolio optimization framework, seeRachev et al. (2006), Stoyanov et al. (2007), Biglova et al. (2004), and Rachevet al. (2008).

The AVaR of a random variable X at tail probability ε is defined as

AV aRε(X) = −1

ε

∫ ε

0

F−1(p)dp.

where F−1(x) is the inverse of the cumulative distribution function (c.d.f.)of the random variable X. The random variable may describe the return ofstock, for example. A practical problem of computing portfolio AVaR is thatusually we do not know explicitly the c.d.f. of portfolio returns. In orderto solve this practical problem, the Monte Carlo method is employed. Thereturns of the portfolio constituents are simulated and then the returns of theportfolio are calculated. In effect, we have a sample from the portfolio returndistribution which we can use to estimate AVaR. The larger the sample, thecloser the estimate to the true value. However, with any finite sample, thesample AVaR will fluctuate about the true value and, having only a sampleestimate, we have to know the probability distribution of the sample AVaRin order to build a confidence interval for the true value. The sample AVaRequals,

1

STOYANOV-RACHEV466

AV aRε(X) = −1

ε

∫ ε

0

F−1n (p)dp.

where F−1n (p) denotes the inverse of the sample c.d.f. Fn(x) = 1

n

∑ni=1 IXi ≤

x in which IA denotes the indicator function of the event A.The problem of computing the distribution of the sample AVaR is a com-

plicated one even if we know the distribution of X. From a practical view-point, X describes portfolio returns which can be a complicated function ofthe joint distribution of the risk drivers. Therefore, we can only rely on largesample theory to gain insight into the fluctuations of sample AVaR. Thatis, for a large n, we can use a limiting distribution to calculate a confidenceinterval.

In this paper, first we prove a limit theorem for the sample AVaR inSection 2 . The limit theorem does not give answers to the question of howmany simulations are necessary in order for the limiting distribution to beacceptable as a model for practical purposes. This number depends alsoon the distribution of X. A major factor is the tail behavior of X and,more precisely, how heavy the left tail of the distribution is. We study thisproblem in Section 3.1 assuming that X has Student’s t distribution. Finally,we illustrate the impact of a tail truncation method in a finite and infinitevariance case.

2 A limit theorem

In this section, we prove the following limit theorem.

Theorem 1. Suppose that X is random variable with finite second mo-ment EX2 < ∞. Furthermore, suppose that the c.d.f. of X is differentiableat x = qε, where qε is the ε-quantile of X. Then, as n →∞,

√n

σε

(AV aRε(X)− AV aRε(X)

)w→ N(0, 1) (1)

wherew→ denotes weak limit and

σ2ε =

1

ε2D(max(qε −X, 0)). (2)

Proof. We apply the following more general result,

φ(Fn)− φ(F )w→ N(0, λ2)

2

ASYMPTOTIC DISTRIBUTION... 467

where φ is a differentiable functional, Fn is the empirical c.d.f., F is the c.d.f.of X, and

λ2 = D(φ′(δXi− F )) =

∫

R

(IFφ(x))2dF (x) < ∞

in which IFφ stands for the influence function of the functional φ, δXiis the

cdf of the observation Xi. By the definition of the influence function,

φ′(δXi− F ) =

d

dt(φ((1− t)F + tδXi

))|t=0 =d

dt(φ(Ft))|t=0.

The proof of the main result reduces to calculating the influence functionof φ(F ) and then calculating the variance λ2. We need the assumed propertiesof the c.d.f. for the calculation of the influence function. In our case, fromthe definition of AVaR,

φ(F ) = −1

ε

∫ ε

0

F−1(p)dp

= −F−1(ε) +1

ε

∫ F−1(ε)

−∞F (p)dp.

(3)

The influence function can be directly calculated,

IFφ(x) =d

dt(φ(Ft))|t=0

= − d

dt(F−1

t (ε))|t=0 +1

ε

d

dt

(∫ F−1t (ε)

−∞Ft(p)dp

)∣∣∣∣∣t=0

The second term is differentiated separately below

d

dt

(∫ F−1t (ε)

−∞Ft(p)dp

)∣∣∣∣∣t=0

= εd

dt(F−1

t (ε))|t=0 + max(qε − x, 0)−∫ qε

−∞F (y)dy

where where qε stands for the ε-quantile of X and we take advantage of thechain rule

d

dt

(∫ f(t)

a

G(t, y)dy

)= G(t, f(t))f ′(t) +

∫ f(t)

a

Gt(t, y)dy

3

STOYANOV-RACHEV468

in which f(x) is a monotonically increasing function. In computing the deriv-ative we used that F (x) is differentiable at x = qε. Finally, for the influencefunction we obtain

IFφ(x) =1

εmax(qε − x, 0)− 1

ε

∫ qε

−∞F (y)dy

Now we can calculate the variance,

λ2 = D(IFφ(X)) =1

ε2D(max(qε −X, 0)).

It is also straightforward to check that E(IFφ(X)) = 0,

E(IFφ(X)) =1

εE max(qε −X, 0)− 1

ε

∫ ε

0

pdF−1(p) = 0

follows after integration by parts.

The variance of the asymptotic normal distribution is not possible tocalculate if we do not know the cdf F (x) of X. Therefore, if we have onlya sample of i.i.d. observations, the variance σ2 has to be estimated. Tothis end, expressing the variance in terms of conditional moments may bemore useful. The variance of the asymptotic normal distribution given in (2)equals

σ2ε =

q2ε

ε− 2qε

εE(X|X ≤ qε) +

1

εE(X2|X ≤ qε)− (qε − E(X|X ≤ qε))

2 (4)

An estimate of σ2ε can be obtained by estimating the conditional moments

and the corresponding quantile from the sample.Furthermore, we would like to remark on a consistency with the classical

theory behind constructing confidence intervals for the mean of a randomvariable. Suppose that the tail probability approaches one. In this case, theAVaR turns into the mean of X,

limε→1

AV aRε(X) = EX,

the sample AVaR turns into the sample average,

limε→1

AV aRε(X) =1

n

n∑i=1

Xi,

where X1, . . . , Xn is a sample if i.i.d. observations, and the variance of theasymptotic normal distribution becomes the variance of X,

4


limε→1

σε = DX.

Therefore, we obtain as a special case the classical CLT

√n√

DX

(1

n

n∑i=1

Xi − EX

)w→ N(0, 1).

3 Monte Carlo experiments

In this section, our goal is to investigate the effect of the tail behavior onthe rate of convergence in (1). We are also interested in the question if tailtruncation improves the convergence and by how much. Generally, the tailtruncation method consists in “replacing” the tails of X with the tails ofa thin-tailed distribution “far away” from the center of the distribution ofX, for example beyond the 0.1% and 99.9% quantiles. The tail truncationmethod has applications in finance for modeling the distribution of stockreturns, a practical reason being that stock exchanges close if a severe marketcrash occurs. This method also has application in derivatives pricing with aheavy-tailed distributional assumption for the return of the underlying, seeRachev et al. (2005) and the references therein.

In the following sections, we start with Student’s t distribution and in-vestigate the convergence rate in the limit relation (1) as degrees of freedomincrease. We address the same questions with a truncated Student’s t distri-bution in which the truncation is done in the simplest possible way — we setthe the values of the random variable which are beyond the 0.1% and 99.9%quantiles to be equal to the corresponding quantiles. As a result, small pointmasses appear at the 0.1% and 99.9% quantiles. We also focus on the classof stable distributions and truncated stable distributions in which the sametruncation technique is adopted as in the case of Student’s t distribution.

3.1 The effect of tail thickness

The impact of the tail behavior on the rate of convergence in Theorem 1 isfirst studied when X has Student’s t distribution, X ∈ t(ν), with ν ≥ 3. Weneed the condition on the degrees of freedom in order for the random variableto have finite variance. Taking advantage of the expression for the density,

fν(x) =Γ

(ν+12

)

Γ(

ν2

) 1√νπ

(1 +

x2

ν

)− ν+12

, x ∈ R,

5

STOYANOV-RACHEV470

it is possible to compute explicitly the variance in equation (2). In fact, forthis purpose the expression in (4) is more appropriate. As a first step, wecalculate the two conditional expectations.

E(X|X ≤ qε) =1

ε

∫ qε

−∞xfX(x)dx

=1

ε

∫ qε

−∞x

Γ(

ν+12

)

Γ(

ν2

) 1√νπ

(1 +

x2

ν

)− ν+12

dx

=1

ε

Γ(

ν+12

)

Γ(

ν2

) 1√νπ

ν

2

∫ qε

−∞

(1 +

x2

ν

)− ν+12

d

(1 +

x2

ν

)

= −1

ε

Γ(

ν+12

)

Γ(

ν2

)√

ν

(ν − 1)√

π

(1 +

q2ε

ν

) 1−ν2

, if ν > 1.

(5)

E(X2|X ≤ qε) =1

ε

∫ qε

−∞x2fX(x)dx

=1

ε

∫ qε

−∞x2 Γ

(ν+12

)

Γ(

ν2

) 1√νπ

(1 +

x2

ν

)− ν+12

dx

=1

ε

Γ(

ν+12

)

Γ(

ν2

) 1√νπ

ν

1− ν

∫ qε

−∞xd

(1 +

x2

ν

)− ν+12

+1

= qεE(X|X ≤ qε) +ν

ε(ν − 2)Fν−2

(qε

√ν − 2

ν

), if ν > 2.

(6)

where the last equality follows by integration by parts and Fν(x) is the c.d.f.of Student’s t distribution with ν degrees of freedom. Plugging these expres-sions in (4), we obtain the expression for the variance σ2

ε .Note that, besides an equation for σ2

ε , we can explicitly calculate theAVaR of X since in the case of Student’s t distribution we can express AVaRas a conditional expectation,

AV aRε(X) = −E(X|X ≤ qε).

Having an expression for the variance allows us to use the test of Kol-mogorov and address the question oh how many simulations are needed inorder to accept the hypothesis that the distribution of the random variablein the left-hand side of the limit relation (1),

6


ν ε = 0.01 ε = 0.053 70000 170004 60000 90005 50000 70006 23000 45007 14000 42008 13000 41009 12000 4000

10 12000 390015 11000 385025 10000 380050 10000 3750∞ 10000 3300

Table I: The number of observations sufficient to accept the normal distrib-ution as an approximate model for different values of ν and ε.

√n

σε

(AV aRε(X)− AV aRε(X)

), (7)

is standard normal. If we accept the null hypothesis for a given value of n,then the standard normal distribution can be used as an approximate modeland we can calculate not only confidence intervals but also other character-istics based on it.

Table I shows the values of n sufficient to accept the null hypothesis in thetest of Kolmogorov for different degrees of freedom and tail probabilities. Wechose ε = 0.01 and ε = 0.05 since these values are frequently used in financialindustry in value-at-risk estimation. The numbers in the table are calculatedby generating independently 2000 samples of a given size and then from eachsample (7) is estimated. In effect, we obtain 2000 observations from thedistribution of (7).

In line with intuition, the numbers Table I indicate that when the tailis heavier, we need a larger sample in order for the asymptotic law to besufficiently close to the distribution of (7) in terms of the Kolmogorov metric.Another expected conclusion is that as the tail probability increases, a smallersample turns out to be sufficient.

In Table II, we calculated the 95% confidence interval for AVaR when thesample size changes from 250 to 10000 observations. We generated 2000 inde-pendent samples and then computed the quantity in equation (7). Thus, the95% confidence intervals are obtained from 2000 observations of the randomvariable in (7). As n increases, the two quantiles approach the correspond-ing quantiles of the standard normal distribution. Note that the largest

7

STOYANOV-RACHEV472

n = 250 n = 500 n = 1000 n = 5000 n = 10000ν q2.5% q97.5% q2.5% q97.5% q2.5% q97.5% q2.5% q97.5% q2.5% q97.5%

3 -1.110 2.011 -1.257 2.173 -1.352 2.202 -1.633 2.037 -1.664 2.0074 -1.337 2.144 -1.442 2.229 -1.543 2.082 -1.744 2.230 -1.756 2.1765 -1.441 2.153 -1.529 2.224 -1.728 2.190 -1.843 2.060 -1.807 2.0096 -1.522 2.134 -1.618 2.033 -1.701 2.115 -1.848 1.987 -1.955 1.9827 -1.627 2.050 -1.668 1.975 -1.827 2.043 -1.841 2.048 -1.913 2.0148 -1.655 2.028 -1.760 2.145 -1.836 2.032 -1.898 2.034 -1.866 1.9399 -1.720 1.938 -1.753 2.146 -1.798 2.075 -1.866 2.005 -1.905 2.007

10 -1.747 1.925 -1.809 1.980 -1.762 2.078 -1.822 1.950 -1.962 2.00015 -1.813 1.751 -1.848 1.896 -1.891 1.956 -1.969 1.941 -1.968 1.87325 -1.848 1.760 -1.933 2.028 -1.897 1.950 -1.939 1.957 -1.899 1.92350 -1.898 1.948 -1.962 1.900 -1.971 1.973 -1.961 1.914 -1.895 1.948∞ -1.921 1.761 -1.976 1.920 -1.964 1.822 -1.869 1.907 -2.004 1.937

Table II: The 95% confidence bounds generated from 2000 simulations fromthe distribution of (7) with ε = 0.01. The corresponding quantiles of N(0, 1)are q2.5% = −1.96 and q2.5% = 1.96.

n = 10000 is generally below the sample sizes for ε = 0.01 given in TableI. Nevertheless, the relative discrepancies between the quantiles given inTable II and the corresponding standard normal distribution quantiles areless than 5% for ν ≥ 6.1 The relative discrepancies between the quantilesgiven in Table III the corresponding standard normal distribution quantilesfor n = 10000 have the same magnitude. However, in this case n = 10000 iswell above the sample sizes given in Table I for ε = 0.05. As a result, we canconclude that even smaller samples than the ones given in Table I can lead to95% confidence intervals obtained via resampling from (7) being close to thecorresponding 95% confidence interval obtained from the limit distributioneven though the Kolmogorov test fails for such samples. For instance, therelative deviation between the quantiles given in Table II for n = 5000 andthe corresponding standard normal distribution quantiles are below 7% forn ≥ 6, which is a small deviation for all practical purposes.

As a result of this analysis, we can conclude that for the purposes ofbuilding confidence intervals for AV aRε(X) when X ∈ t(ν), with ν ≥ 6 andε = 0.01, 0.05, we can safely employ the asymptotic law when the samplesize we use for AVaR estimation contains more than 5000 observations. IfStudent’s t distribution is fitted on daily stock-returns time series, such values

1If we generate a sample of 2000 observations from the standard normal distribution, arelative deviation below 6% between the estimated quantile q2.5% and the correspondingstandard normal quantile happens with about 95% probability, and below 7.7% with about99% probability.

8


3 -1.422 2.110 -1.543 2.016 -1.549 1.981 -1.725 1.947 -1.883 1.9874 -1.647 2.169 -1.737 2.235 -1.787 2.171 -1.900 2.226 -1.849 2.1155 -1.749 2.081 -1.811 2.096 -1.757 2.148 -1.868 2.015 -1.937 2.1006 -1.810 2.071 -1.896 2.030 -1.921 1.941 -1.958 1.998 -1.886 2.0327 -1.786 2.215 -1.824 1.990 -1.809 2.086 -1.986 2.030 -1.916 2.0158 -1.932 2.131 -1.870 2.058 -1.755 2.090 -1.937 2.014 -1.915 1.9529 -1.848 2.139 -1.884 2.081 -1.930 2.023 -1.995 1.964 -1.863 2.048

10 -1.906 2.103 -2.021 1.966 -1.839 2.087 -2.009 1.930 -1.989 1.99515 -1.797 1.905 -1.929 2.056 -1.944 1.952 -1.924 1.973 -1.947 1.97925 -1.958 1.950 -1.994 1.956 -1.939 1.968 -2.085 1.993 -1.894 1.94450 -1.986 1.927 -1.980 1.823 -1.962 1.883 -1.911 1.969 -2.002 1.935∞ -2.013 1.828 -1.953 1.869 -1.975 1.893 -2.034 1.958 -1.903 1.944

Table III: The 95% confidence bounds generated from 2000 simulations fromthe distribution of (7) with ε = 0.05. The corresponding quantiles of N(0, 1)are q2.5% = −1.96 and q2.5% = 1.96.

for ν are very common.Figure 1 illustrates the differences in the convergence rate when X has

Student’s t distribution with ν = 3, which corresponds to heavier tails, andν = 10. Since high degrees of freedom imply more light tails, smaller samplesare sufficient for the density of (7) to be closer to the standard normal density.

3.2 The effect of tail truncation

The stochastic stability of sample AVaR increases dramatically after tailtruncation. In this section, we repeat the calculations from the previoussection but when X has Student’s t distribution with the tails truncated atq0.1% and q99.9% quantiles. The random variable Y is said to have a truncateddistribution at these quantiles if it has the representation

Y = XIq0.1% ≤ X ≤ q99.9%+ q0.1%IX < q0.1%+ q99.9%IX > q99.9%

in which X ∈ t(ν), IA denotes the indicator of the event A, and q0.1%,q99.9% are the corresponding quantiles of X. The tail truncation introducessmall point masses at the two quantile levels.

The two conditional expectations in (4) can be related to the correspond-ing conditional expectations of X. In the following, we assume that the tailprobability ε is larger from the tail probability of the left truncation point,

9

STOYANOV-RACHEV474

−5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

n = 250n = 1000n = 10000n = 70000N(0,1)

−5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

n = 250n = 1000n = 10000n = 12000N(0,1)

Figure 1: The density of (7) approaching the N(0, 1) density as the samplesize increases with ν = 3 (top) and ν = 10 (bottom).

10


ν ε = 0.01 ε = 0.053 12000 40004 11500 36005 11000 33006 11000 32007 10500 31008 10000 30009 10000 3000

10 10000 300015 10000 295025 10000 290050 10000 2900∞ 10000 2900

Table IV: The number of observations sufficient to accept the normal distri-bution as an approximate model for different values of ν and ε.

ε > 0.001. Under this assumption, the ε-quantile of X is the same as theε-quantile of Y .

E(Y |Y ≤ qε) = E(X|X ≤ qε)− 0.001

εE(X|X ≤ q0.1%) +

0.001qε

ε

E(Y 2|Y ≤ qε) = E(X2|X ≤ qε)− 0.001

εE(X2|X ≤ q0.1%) +

0.001q2ε

ε

in which the conditional expectations of X can be computed according to for-mulae (5) and (6). Plugging the expressions for the conditional expectationsof Y in the expression for σ2

ε , we obtain the variance of the asymptotic dis-tribution. Furthermore, the tail truncation does not break the link betweenAVaR and the conditional expectation, therefore

AV aRε(Y ) = −E(Y |Y ≤ qε).

In the following, we investigate the convergence rate of

√n

σε

(AV aRε(Y )− AV aRε(Y )

), (8)

for different degrees of freedom to the standard normal distribution and wecompare the results to the ones in the previous section.

Table IV is the counterpart of Table I for the truncated distribution. It isimpressive how the sample size sufficient to accept the null hypothesis in theKolmogorov test decreases after tail truncation. The most dramatic change

11

STOYANOV-RACHEV476

3 -1.723 1.699 -1.847 1.932 -1.850 1.958 -1.966 1.921 -1.860 1.9364 -1.759 1.694 -1.863 1.819 -1.903 1.860 -1.989 1.942 -1.964 1.8865 -1.808 1.536 -1.884 1.871 -1.926 1.932 -1.961 1.964 -1.782 2.0666 -1.947 1.565 -1.937 1.759 -2.002 1.734 -2.057 1.946 -1.981 1.9587 -1.960 1.524 -1.960 1.666 -1.965 1.844 -2.101 1.932 -1.981 1.9278 -2.002 1.567 -2.015 1.693 -1.903 1.802 -1.952 1.856 -1.917 1.9289 -1.963 1.552 -2.030 1.748 -2.106 1.779 -1.965 2.026 -1.932 1.938

10 -2.003 1.596 -2.119 1.709 -2.034 1.850 -1.925 1.813 -1.990 1.95215 -2.090 1.485 -2.159 1.650 -2.065 1.786 -1.983 1.847 -2.035 1.85525 -2.183 1.502 -2.084 1.578 -2.093 1.747 -2.016 1.806 -1.954 1.87750 -2.272 1.509 -2.089 1.632 -2.042 1.726 -1.938 1.914 -2.056 1.970

Table V: The 95% confidence bounds generated from 2000 simulations fromthe distribution of (8) with ε = 0.01. The corresponding quantiles of N(0, 1)are q2.5% = −1.96 and q2.5% = 1.96.

is in the case ν = 3. Now we need only 12000 observations compared to70000 in the non-truncated case.

Tables V and VI are the counterparts of Tables II and III. The relativedeviation of the quantiles q2.5% and q97.5% of the random variable in (8) fromthose of the standard normal distribution are below 7% for all degrees offreedom and n = 10000, and, with a few exceptions, for n = 5000. CompareFigure 2 and the top plot in Figure 1 for an illustration of the improvement inthe convergence rate. These results indicate that the asymptotic distributioncan be used to obtain a 95% confidence bound for the sample AVaR for alldegrees of freedom if the sample size contains more than 5000 observations.

3.3 Infinite variance distributions

A critical assumption behind the limit result in Theorem 1 is the finite vari-ance of X. To be more precise, the condition of finite variance can be loosenedto finite downside semi-variance,

D max(−X, 0) < ∞,

because it is the behavior of the left tail which is important. As a conse-quence, the sample AVaR of distributions with infinite variance, but finitedownside semi-variance, may still follow Theorem 1.

However, there are infinite variance distributions for which

D max(−X, 0) = ∞

12


3 -1.815 2.116 -1.866 2.041 -1.939 2.018 -1.944 1.975 -2.045 1.8744 -1.756 2.150 -1.811 2.073 -2.052 2.060 -1.923 1.973 -1.922 1.8545 -1.820 1.954 -1.971 2.032 -1.916 2.036 -1.826 1.960 -1.941 1.8836 -1.899 2.089 -1.981 2.036 -2.012 2.012 -1.955 1.933 -1.921 2.0117 -2.001 2.032 -1.921 1.997 -1.949 1.980 -1.980 1.936 -2.016 1.9158 -1.888 1.995 -1.922 2.050 -1.907 1.917 -1.942 1.911 -1.910 1.9039 -2.017 2.003 -1.892 1.918 -1.899 2.017 -1.931 2.001 -2.009 1.967

10 -1.928 1.814 -1.992 1.960 -1.870 1.949 -1.845 2.076 -1.992 1.89815 -2.059 1.983 -2.020 2.007 -1.961 1.922 -1.953 1.870 -1.936 1.87425 -1.999 1.854 -2.038 1.945 -1.889 2.028 -2.031 1.916 -1.975 1.89050 -1.960 1.898 -2.028 1.898 -1.947 1.906 -2.015 2.002 -1.959 1.911

Table VI: The 95% confidence bounds generated from 2000 simulations fromthe distribution of (8) with ε = 0.05. The corresponding quantiles of N(0, 1)are q2.5% = −1.96 and q2.5% = 1.96.

−5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

n = 250n = 1000n = 5000n = 10000N(0,1)

Figure 2: The density of (8) approaching the N(0, 1) density as the samplesize increases with ν = 3 and ε = 0.01.

13

STOYANOV-RACHEV478

−5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

n = 250n = 1000n = 5000n = 10000N(0,1)

Figure 3: Lack of convergence, X has a stable distribution with X ∈S1.5(1, 0, 0) and ε = 0.05.

and, therefore, the limit result in Theorem 1 does not hold for them. Suchis the class of stable distributions which arises from generalizations of theCentral Limit Theorem and has been proposed as a model for stock returndistributions, see Rachev and Mittnik (2000).

Stable distributions are introduced by their characteristic functions. Xis said to have a stable distribution if its characteristic function is

ϕ(t) = EeitX =

exp−σα|t|α(1− iβ t

|t| tan(πα2

)) + iµt, α 6= 1

exp−σ|t|(1 + iβ 2π

t|t| ln(|t|)) + iµt, α = 1

Except for a couple of representatives, generally no closed-form expressionsfor their densities and c.d.f.s are known. If α < 2, then X has infinitevariance. If 1 < α ≤ 2, then X has finite mean and the AVaR of X canbe calculated. In our calculations, we will use the semi-analytic formula inStoyanov et al. (2006).

Even though we know that Theorem 1 does not hold for a stable dis-tribution with α < 2, we simulate 2000 draws from the random variable

14

−5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

n = 250n = 1000n = 5000n = 10000N(0,1)

Figure 4: After tail truncation at q0.1% and q99.9%, there is a fast convergenceto N(0, 1), α = 1.5 and ε = 0.05.

in equation (7) in which σε is estimated from a generated sample by esti-mating the corresponding conditional moments. In theory these the secondconditional moment explodes but for any finite sample its estimate is a finitenumber. Our goal is to see what happens when Theorem 1 does not hold.Figure 3 illustrates such a divergent case in which α = 1.5 and ε = 0.05. Thelack of convergence is quite obvious.

Stable distributions with α < 2 in combination with a tail truncationmethod have been proposed as a model for the returns of the underlying inderivatives pricing. It is interesting to see how much the simple truncationtechnique we applied in the previous section can change Figure 3. Withits tails truncated according to our simple method, the random variable be-comes with a bounded support and, therefore, it has finite variance. As aconsequence, Theorem 1 holds. Figure 4 illustrates this change. We observe aquick convergence rate, similar to the one illustrated in Figure 2 for Student’st distribution.

15

STOYANOV-RACHEV480

4 Conclusion

In this paper, we study the asymptotic distribution of sample AVaR. Undercertain regularity conditions, we prove a limit theorem in which the limitingdistribution is the normal distribution. We study how the convergence ratein the limit theorem is influenced by the tail behavior of the random vari-able. An expected result is that, other things equal, more observations areneeded when the tail is heavier. We find out that a simple tail truncationmethod improves dramatically the convergence rate. As a consequence, theasymptotic distribution is reliable for confidence interval calculations whenthe number of simulations is more than 5000 if the random variable has atruncated Student’s t distribution.

We also consider an infinite variance case in which the random variable asa stable distribution with finite mean. We illustrate the lack of convergenceand demonstrate the improvement due to tail truncation at high quantiles.

16


References

Artzner, P., F. Delbaen, J.-M. Eber and D. Heath (1998), ‘Coherent measuresof risk’, Math. Fin. 6, 203–228.

Biglova, A., S. Ortobelli, S. Rachev and S. Stoyanov (2004), ‘Different ap-proaches to risk estimation in portfolio theory’, The Journal of PortfolioManagement Fall 2004, 103–112.

Pflug, G. (2000), ‘Some remarks on the value-at-risk and the conditionalvalue-at-risk’, In: Uryasev, S. (Ed.), Probabilistic Constrained Optimiza-tion: Methodology and Applications. Kluwer Academic Publishers, Dor-drecht .

Rachev, S., D. Martin, B. Racheva-Iotova and S. Stoyanov (2006), ‘Stableetl optimal portfolios and extreme risk management’, forthcoming in De-cisions in Banking and Finance, Springer/Physika, 2007 .

Rachev, S., F. Fabozzi and C. Menn (2005), Fat-tails and skewed asset returndistributions, Wiley, Finance.

Rachev, S. T., Stoyan V. Stoyanov and F. J. Fabozzi (2008), Advanced sto-chastic models, risk assessment, and portfolio optimization: The ideal risk,uncertainty, and performance measures, Wiley, Finance.

Rachev, S.T. and S. Mittnik (2000), Stable Paretian Models in Finance, JohnWiley & Sons, Series in Financial Economics.

Stoyanov, S., G. Samorodnitsky, S. Rachev and S. Ortobelli (2006), ‘Comput-ing the portfolio conditional value-at-risk in the α-stable case’, Probabilityand Mathematical Statistics 26, 1–22.

Stoyanov, S., S. Rachev and F. Fabozzi (2007), ‘Optimal financial portfolios’,forthcoming in Applied Mathematical Finance .

17

STOYANOV-RACHEV482

A note on generalized twisted q-Euler numbers

and polynomials

C. S. Ryoo†∗, J.J. Seo†, T. Kim‡

†Department of Mathematics,Hannam University, Daejeon 306-791, Korea


‡ EECS,Kyungpook National University, Taegu, 702-701, Korea


Abstract In this paper we construct a new generalized twisted q-Euler polynomials and

generalized twisted q-Euler numbers attached to χ. We investigate some properties which

are related to the generalized twisted q-Euler Polynomials. We also derive the existence of a

specific interpolation function which interpolate the generalized twisted q- Euler polynomials

at negative integer.

2000 Mathematics Subject Classification - 11B68, 11S40

Key words- Euler numbers, Euler polynomials, Generalized Euler numbers, Generalized

Euler polynomials, Euler numbers, Euler polynomials, Generalized twisted q-Euler numbers,

Generalized twisted q-Euler polynomials

1. Introduction

Throughout this paper we use the following notations. By Zp we denotethe ring of p-adic rational integers, Q denotes the field of rational numbers, Qp

denotes the field of p-adic rational numbers, C denotes the complex numberfield, and Cp denotes the completion of algebraic closure of Qp. Let νp be thenormalized exponential valuation of Cp with |p|p = p−νp(p) = p−1. When onetalks of q-extension, q is considered in many ways such as an indeterminate, acomplex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C one normally assumethat |q| < 1. If q ∈ Cp, we normally assume that |q − 1|p < p−

1p−1 so that

qx = exp(x log q) for |x|p ≤ 1.

[x]q = [x : q] =1− qx

1− q, cf. [1, 2, 3, 4, 5, 9, 11] .

∗This work was supported by Hannam University Research Fund, 2007

1

Hence, limq→1[x] = x for any x with |x|p ≤ 1 in the present p-adic case. Forg ∈ UD(Zp,Cp) = g|g : Zp → Cp is uniformly differentiable function, thep-adic q-integral (or q-Volkenborn integration ) was defined by [3]

Iq(g) =∫

Zp

g(x)dµq(x) = limN→∞

1[pN ]q

pN−1∑x=0

qxg(x).

Let d be a fixed integer and let p be a fixed prime number. For any positiveinteger N , we set

X = Xd = lim←−N

(Z/dpNZ), X1 = Zp,

X∗ =⋃

0<a<dp(a,p)=1

(a + dpZp),

a + dpNZp = x ∈ X | x ≡ a (mod dpN ),

where a ∈ Z satisfies the condition 0 ≤ a < dpN . For any positive integer N ,

µq(a + dpNZp) =qa

[dpN ]q

is known to be a distribution on X, cf.[1, 2, 3, 4, 6, 7, 8, 9]. For g ∈ UD(Zp,Cp),∫

Zp

g(x)dµ1(x) =∫

X

g(x)dµ1(x), cf. [4, 5].

The Euler numbers En are usually defined by means of the following generatingfunction:

eEt =2

et + 1=

∞∑n=0

Entn

n!, cf. [8, 9, 10]

where the symbol En is interpreted to mean that En must be replaced by En

when we expand the one on the left. These numbers are classical and importantin mathematics and in various places like analysis, number theory. Frobeniusextended such numbers as En to the so-called Frobenius-Euler numbers Hn(u)belonging to an algebraic number u with |u| > 1. Let u be an algebraic number.For u ∈ C with |u| > 1, the Frobenius-Euler numbers Hn(u) belonging to u aredefined by the generating function

eH(u)t =1− u

et − u=

∞∑n=0

Hn(u)tn

n!, cf. [1,4,9,10],

with the usual convention of symbolically replacing Hn by Hn. The Eulerpolynomials En(x) are defined by

eE(x)t =2

et + 1ext =

∞∑n=0

En(x)tn

n!, cf. [3,4,5,8,9,10].

2

RYOO ET AL484

For u ∈ C with |u| > 1, the Frobenius-Euler polynomials Hn(u, x) belonging tou are defined by

eH(u,x)t =1− u

et − uext =

∞∑n=0

Hn(u, x)tn

n!, cf. [8, 9, 10].

T. Kim[2] gave relation between Bn,w and Hn(u), nth Euler numbers as follows:

Bn,w =n

w − 1Hn−1(w−1) if w 6= 1.

In [2], Kim defined the locally constant function as follows: Let

Tp = ∪m≥1Cpm = limm→∞

Cpm ,

where Cpm = w|wpm

= 1 is the cyclic group of order pm. For w ∈ Tp, wedenote by φw : Zp → Cp the locally constant function x 7−→ wx. If we takeg(x) = φw(x)etx, then we easily see that

∫

Zp

φw(x)etxdµ1(x) =t

wet − 1.

T.Kim [2] treated analogue of Bernoulli numbers, which is called twisted Bernoullinumbers. We define the twisted Bernoulli polynomials Bn,w(x)

ext t

wet − 1=

∞∑n=0

Bn,w(x)tn

n!.

By using Taylor series of etx in the above equation, we obtain∫

Zp

xnφw(x)dµ1(x) = Bn,w,

where Bn,w = Bn,w(0).Now, we consider the case q ∈ (−1, 0) corresponding to q-deformed fermionic

certain and annihilation operators and the literature given therein [3,4,5,7,8].The expression for the Iq(g) remains same, so it is tempting to consider thelimit q → −1. That is,

I−1(g) = limq→−1

Iq(g) =∫

Zp

g(x)dµ−1(x) = limN→∞

∑

0≤x<pN

g(x)(−1)x. (1.1)

If we take g1(x) = g(x + 1) in (1.1), then we easily see that

I−1(g1) + I−1(g) = 2g(0). (1.2)

From (1.2), we obtain

I−1(gn) = (−1)nI−1(g) + 2n−1∑

l=0

(−1)n−1−lg(l),

3

GENERALIZED TWISTED Q-EULER NUMBERS... 485

where gn(x) = g(x + n). Ryoo and Kim[9] treated analogue of Euler numbers,which is called twisted Euler numbers. We can consider twisted Euler numbers.If we take g(z) = φw(z)etz, (w ∈ Tp) in (1.2), then we have

I−1(φw(z)etz) =2

wet + 1=

∞∑n=0

En,wtn

n!.

We define twisted Euler numbers En,w as follows:

Fw(t) =2

wet + 1=

∞∑n=0

En,wtn

n!.

Twisted Euler polynomials En,w(z) are defined by means of the generating func-tion

Fw(t, z) =2

wet + 1ezt = I−1(φw(x)et(z+x)) =

∞∑n=0

En,w(z)tn

n!,

where En,w(0) = En,w. Let χ be the Dirichlet character with conductor f( =odd) ∈N. Ryoo, Kim and Jang[8] studied on the generalized Euler numbers and poly-nomials. The generalized Euler numbers associated with χ, En,χ, was definedby means of the generating function

Fχ(t) =2

∑f−1a=0 χ(a)(−1)aeat

eft + 1=

∞∑n=0

En,χtn

n!.

Generalized Euler polynomials En,χ(x), was also defined by means of the gen-erating function

Fχ(t, z) =2

∑f−1a=0 χ(a)(−1)aeat

eft + 1ezt =

∞∑n=0

En,χ(z)tn

n!.

Substituting g(x) = χ(x)φw(x)etx into (1.2), then the generalized twisted Eulernumbers En,χ,w are defined by means of the generating functions

Fχ,w(t) =∫

X

φw(x)etxχ(x)dµ−1(x)

=2

∑f−1a=0 eta(−1)aχ(a)φw(a)

φw(f)eft + 1=

∞∑n=0

En,χ,wtn

n!

By using the above equation, En,χ,w are defined by

En,χ,w =∫

X

φw(x)xnχ(x)dµ−1(x).

Generalized twisted Euler polynomials En,χ,w(z) are defined by

Fχ,w(t, z) = Fχ,w(t)ezt =∫

X

φw(x)etxχ(x)dµ−1(x)etz

=

(2

∑f−1a=0 eta(−1)aχ(a)φw(a)

φw(f)eft + 1

)ezt =

∞∑n=0

En,χ,w(z)tn

n!.

4

RYOO ET AL486

We set

Fχ,w(t, z) =2

∑f−1a=0(−1)aχ(a)φw(a)e(a+z)t

φw(f)eft + 1.

Observe that if w → 1, then Fχ,w(t, z) → Fχ(t, z) and En,χ,w(z) → En,χ(z).Ryoo and Kim [9]studied on the twisted Euler zeta function and twisted Hur-witz Euler zeta function. We gave the relation between twisted Euler numbersand twisted l-functions at non-positive integers. Observe that if χ = χ0, thenFχ,w(t, z) → Fw(t, z) and En,χ,w(z) → En,w(z). T. Kim defined the generalizedq-Euler numbers as follows: The generalized q-Euler numbers associated withχ, En,χ,q, are defined by

En,χ,q =∫

X

χ(x)[x]nq dµ−q(x).

We give the generating function of q-Euler numbers as follows:

Fχ,q(t) = [2]q∞∑

l=0

χ(l)ql(−1)le[l]qt =∞∑

n=0

En,χ,qtn

n!. (1.5)

C.S.Ryoo [10] introduced the generalized q-Euler polynomials associated with χ.The generalized q-Euler polynomials associated with χ, En,χ,q(z), are definedby

En,χ,q(z) =∫

X

χ(x)[z + x]nq dµ−q(x).

Hence we give the generating function of q-Euler polynomials as follows:

Fχ,q(t, z) = [2]q∞∑

l=0

χ(l)ql(−1)le[z+l]qt =∞∑

n=0

En,χ,q(z)tn

n!.

Observe that if q → 1, then Fχ,q(t, z) → Fχ(t, z) and En,χ,q(z) → En,χ(z). T.Kim and C.S. Ryoo studied the generalized q-Euler numbers and q-Euler poly-nomials and derived a Dirichlet’s type l-series which interpolates the generalizedq-Euler polynomials En,χ,q(z).

The purpose of this paper is to construct the generalized twisted q-Eulerpolynomials En,χ,w,q(z) attached to χ and derive a new Dirichlet’s type l-serieswhich interpolates the generalized twisted q-Euler polynomials En,χ,w,q(z).

2. Generalized twisted q-Euler numbers andpolynomials

In this section, we introduce the generalized twisted q-Euler numbers andq-Euler polynomials. These numbers will be used to prove the analytic con-tinuation of the q-l-series. Let χ be the Dirichlet character with conductorf( =odd) ∈ N. Then the generalized twisted q-Euler numbers associated withχ, En,χ,w,q, are defined by

En,χ,w,q =∫

X

χ(x)φw(x)[x]nq dµ−q(x). (2.1)

5


By using p-adic q-integral, we obtain,

En,χ,w,q =[f ]nq[f ]−q

f−1∑a=0

χ(a)(−1)aqaφw(a)En,wf ,qf

(a

f

).

Note that∫

X

χ(x)φw(x)[x]nq dµ−q(x)

= [2]qf−1∑a=0

χ(a)(−1)aφw(a)qa

(1

1− q

)n n∑

l=0

(n

l

)(−1)lqal 1

1 + φw(f)qf(1+l).

Hence we have

En,χ,w,q = [2]qf−1∑a=0


(1

1− q

)n n∑

l=0

(n

l

)(−1)lqal 1

1 + φw(f)qf(1+l).

By simple calculation, we obtain

[2]qf−1∑a=0


(1

1− q

)n n∑

l=0

(n

l

)(−1)lqal 1

1 + φw(f)qf(1+l)

= [2]q∞∑

l=0

χ(l)qlφw(l)(−1)l[l]nq .

We have the following theorem.

Theorem 2. Let χ be a primitive Dirichlet character of conductor f(=odd) ∈ N, we have

En,χ,w,q = [2]q∞∑

l=1

χ(l)qlφw(l)(−1)l[l]nq .

We set

Fχ,w,q(t) =∞∑

n=0

En,χ,w,qtn

n!.

By using Theorem 2, we obtain

Fχ,w,q(t) =∞∑

n=0

En,χ,w,qtn

n!=

∞∑n=0

[2]q∞∑

l=0

χ(l)qlφw(l)(−1)l[l]nqtn

n!

= [2]q∞∑

l=0

χ(l)qlφw(l)(−1)l∞∑

n=0

[l]nqtn

n!

= [2]q∞∑

l=0

χ(l)qlφw(l)(−1)le[l]qt.

6

RYOO ET AL488

Hence we give the generating function of q-Euler numbers as follows:

Fχ,w,q(t) = [2]q∞∑

l=0

χ(l)qlφw(l)(−1)le[l]qt =∞∑

n=0

En,χ,w,qtn

n!.

The generalized twisted q-Euler polynomials associated with χ, En,χ,w,q(z), aredefined by

En,χ,w,q(z) =∫

X

χ(x)φw(x)[z + x]nq dµ−q(x). (2.2)

By using p-adic q-integral, we obtain,

En,χ,w,q(z) =[f ]nq[f ]−q

f−1∑a=0

χ(a)(−1)aφw(a)qaEn,wf ,qf

(a + z

f

).

Since∫

X

χ(x)φw(x)[z + x]nq dµ−q(x)

= [2]qf−1∑a=0


(1

1− q

)n n∑

l=0

(n

l

)(−1)lqalqzl 1

1 + φw(f)qf(1+l),

we have

En,χ,w,q(z)

= [2]qf−1∑a=0


(1

1− q

)n n∑

l=0

(n

l

)(−1)lq(a+z)l 1

1 + φw(f)qf(1+l).

By simple calculation, we obtain

[2]qf−1∑a=0


(1

1− q

)n n∑

l=0

(n

l

)(−1)lq(a+z)l 1

1 + φw(f)qf(1+l)

= [2]q∞∑

l=0

χ(l)qlφw(a)(−1)l[z + l]nq .

(2.3)We set

Fχ,w,q(t, z) =∞∑

n=0

En,χ,w,q(z)tn

n!.

By using (2.3), we obtain

Fχ,w,q(t) = [2]q∞∑

l=0

χ(l)qlφw(a)(−1)le[z+l]qt.

7


Hence we give the generating function of the generalized twisted q-Euler poly-nomials as follows:

Fχ,w,q(t, z) = [2]q∞∑

l=0

χ(l)qlφw(a)(−1)le[z+l]qt =∞∑

n=0

En,χ,w,q(z)tn

n!.

Observe thatFχ,w,q(t, z) =

∫

X

χ(x)φw(x)e[x+z]qtdµ−q(x).

By using the above equation, we have

∞∑n=0

En,χ,w,q(z)tn

n!=

∫

X

χ(x)φw(x)e[z+x]qtdµ−q(x)

=∞∑

n=0

(∫

X

χ(x)φw(x)[z + x]nq dµ−q(x))

tn

n!

=∞∑

n=0

(n∑

k=0

(n

k

)qzl[z]n−k

q

∫

X

χ(x)φw(x)[x]kqdµ−q(x)

)tn

n!

By using comparing coefficients tn

n! , we have the following theorem.

Theorem 3. For any positive integer n, we have

En,χ,w,q(z) =n∑

k=0

(n

k

)qzl[z]n−k

q Ek,χ,w,q.

We have the following remark.

Remark 1. Note that

(1) En,χ,w,q(0) = En,χ,w,q,

(2) If χ = χ0, then En,χ,w,q(z) = En,w,q(z), En,χ,w,q = En,w,q,

(3) If q → 1, then En,χ,w,q(z) = En,χ,w(z), En,χ,w,q = En,χ,w,

(4) If q → 1, then Fχ,w,q(t, z) = Fχ,w(t, z),

(5) If χ = χ0, then Fχ,w,q(t) = Fw,q(t), Fχ,w,q(t, z) = Fw,q(t, z),

(6) If χ = χ0, q → 1 then En,χ,w,q(z) = En,w(z), En,χ,w,q = En,w.

(7) If χ = χ0, q → 1, w → 1 then En,χ,w,q(z) = En(z), En,χ,w,q = En.

In complex case, the generating function of generalized twisted q-Euler numbers

is given by

Fχ,w,q(t) = [2]q∞∑

l=0

χ(l)qlwl(−1)le[l]qt =∞∑

n=0

En,χ,w,qtn

n!, (2.4)

8

RYOO ET AL490

where q ∈ C with |q| < 1 and w is the rth root of unity. Generalized twistedq-Euler polynomials En,χ,w,q(z), was also defined by means of the generatingfunction

Fχ,w,q(t, z) = Fχ,w,q(t)ezt = [2]q∞∑

l=0

χ(l)qlwl(−1)le[z+l]qt =∞∑

n=0

En,χ,w,q(z)tn

n!.

(2.5)We define interpolation functions of the generalized twisted q-Euler numbersand polynomials. Thus we need the following relations. By using (2.4), we have

(d

dt

)k

Fχ,w,q(t)∣∣t=0

= [2]q∞∑

l=1

χ(l)qlwl(−1)l[l]kq , (2.6)

and (d

dt

)k( ∞∑

n=0

En,χ,w,qtn

n!

)∣∣t=0

= Ek,χ,w,q, for k ∈ N. (2.7)

By (2.6), (2.7), we have the following theorem.

Theorem 4. For any positive integer k, we have

Ek,χ,w,q = [2]q∞∑

n=1

χ(n)wnqn(−1)n[n]kq .

The above generating function is used to construct a q-Dirichlet series. Wedefine q-analogue Dirichlet’s type l-function as follows:

Definition 1. Let s ∈ C.

lw,q(s, χ) = [2]q∞∑

n=1

(−1)nχ(n)wnqn

[n]sq.

By using Definition 1 and Theorem 4, we have the following theorem.


Ek,χ,w,q = lw,q(−k, χ).

Observe thatlimw→1

lw,q(s, χ) = lq(s, χ),

which is a q-analogue Dirichlet’s type l-function ([10]).By using (2.5), we have

(d

dt

)k

Fχ,w,q(t, z)∣∣t=0

= [2]q∞∑

l=1

χ(l)wlql(−1)l[z + l]kq , (2.8)

9

and (d

dt

)k( ∞∑

n=0

En,χ,w,q(z)tn

n!

)∣∣t=0

= Ek,χ,w,q(z), for k ∈ N. (2.9)

By (2.8), (2.9), we have the following theorem.


Ek,χ,w,q(z) = [2]q∞∑

n=0

χ(n)wnqn(−1)n[n + z]kq .

By using the above theorem, we define the two-variable q-analogue l-series.

Definition 2. Let s ∈ C.

lw,q(s, z|χ) = [2]q∞∑

n=0

(−1)nχ(n)wnqn

[n + z]sq.

Note that lw,q(s, z|χ) is analytic continuation in C with only simple pole ats = 1, and lw,q(s, 1|χ) = lw,q(s|χ). By using Definition 2 and Theorem 6, wehave the following theorem.


Ek,χ,w,q(z) = lw,q(−k, z|χ).

If w → 1, then lw,q(s, z|χ) → lq(s, z|χ), where lq(s, z|χ) is the two-variablel-series cf [10].

In general, how many roots does En,χ,w,q(z) have ? Find the numbers ofcomplex zeros cEn,χ,w,q(z) of the En,χ,w,q(z), Im(z) 6= 0. Using numerical ex-periments, we hope to investigate the structure of the complex roots of thegeneralized q-Euler polynomials En,χ,w,q(z). The authors have no doubt thatinvestigation along this line will lead to a new approach employing numericalmethod in the field of research of the generalized q-Euler polynomials En,χ,w,q(z)to appear in mathematics and physics. For related topics the interested readeris referred to [7, 8].

References

[1] M. Cenkci, M. Can, V. Kurt, q-adic interpolation functions and kummertype congruence for q-twisted and q-generalized twisted Euler numbers, Ad-van. Stud. Contemp. Math., 2004(9), 203-216.

[2] T. Kim, An analogue of Bernoulli numbers and their applications, Rep.Fac. Sci. Engrg. Saga Univ. Math., 1994(22), 21-26.

[3] T. Kim , On the q-extension of Euler and Genocchi numbers , Journal ofMathematical Analysis and Applications, 2007(326), 1458-1465.

10

RYOO ET AL492

[4] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 2002(9), 288-299.

[5] T. Kim, A note on q-Volkenborn integration, Proc. Jangjeon Math. Soc.,2001(2), 45-49.

[6] T. Kim, On p-adic q-l-functions and sums of powers, Journal of Mathe-matical Analysis and Applications, in Press.

[7] C. S. Ryoo, T. Kim, R.P. Agarwal, A numerical investigation of the rootsof q-polynomials, Inter. J. Comput. Math., 2006 83(2), 223-234.

[8] C. S. Ryoo, T. Kim, L. C. Jang, A note on generalized Euler numbers andpolynomials , Inter. J. Comput. Math., in Press.

[9] C. S. Ryoo, T. Kim, Generating functions of the twisted Euler Numbersand polynomials associated with their interpolation functions , Submitted.

[10] C. S. Ryoo, A note on generalized q-Euler numbers and polynomials, Sub-mitted.

[11] Y. Simsek, Theorem on twisted L-function and twisted Bernoulli numbers,Advan. Stud. Contemp. Math., 2006(12), 237-246.

11


Pade Spline Functions

Tian-Xiao He∗

Department of Mathematics and Computer Science

Illinois Wesleyan University

Bloomington, IL 61702-2900, USA

[email protected]

Abstract

We present here the definition of Pade spline functions, their expres-sions, and the estimate of the remainders of pade spline expansions.Some algorithms are also given.

AMS Subject Classification: 41A15, 65D07, 65D15.

Key Words and Phrases: Pade two-point approximation, Padespline, Rational Hermite interpolation, pade spline expansion.

1 Introduction

Pade approximation is derived by expanding a function as a ratio of twopower series and both the numerator and denominator coefficients are thusdetermined (cf. Baker [1-2], Baker and Graves-Morris [3], and Brent, Gus-tavson, and Yun [4]). In this paper, we shall use two points Pade approx-imation to construct Pade spline functions. The main idea initially camefrom the author’s talk at the Joint U.S.- China Workshop on ApproximationTheory that took place in April, 1985, Hangzhou, China ([5]).

Let

∗The research of this author was partially supported by ASD Grant and sabbaticalleave of the Illinois Wesleyan University.

1


4 : a = x0 < x1 < · · · < xn = b

be an arbitrary partition on the interval [a, b], and let f be a kth differentiablefunction defined on [a, b] with function value and dervatives at each node xi

(i = 0, 1, · · · , n)

y(m)i = f (m)(xi), m = 0, 1, · · · , k − 1; i = 0, 1, · · · , n.

Denote by πk the collection of all polynomials of degree less than or equal tok. We now give the definition of Pade spline functions.

Definition 1.1 We call R(k)r,` (4) the set of Pade spline function of order

k with nodes xi (i = 0, 1, . . . , n), if any function R(x) ∈ R(k)r,` satisfies the

following coditions for all x ∈ [xi−1, xi], i = 1, 2, . . . , n,

(i) R(x) = Pi(x)Qi(x)

, Pi(x) ∈ πr, Qi(x) ∈ π`,

(ii)∑k−1

m=0 y(m)i−1

(x−xi−1)m

m!− Pi(x)

Qi(x)= O

((x− xi−1)

k),

(iii)∑k−1

m=0 y(m)i

(x−xi)m

m!− Pi(x)

Qi(x)= O

((x− xi)

k),

(iv) r + ` = 2k − 1.

From Definition 1.1, we immediately know R(k)r,` (4) ∈ Ck. In addition,

the rational Hermite interpolation and rational contact interpolation can beeasily obtained by using the pade spline functions.

Although Definition 1.1 only gives the piecewise expression of Pade splinefunctions, we might discuss its globe expression as follows.

Suppose Qi(xi) 6= 0. Denote

Gi(x) =k−1∑m=0

R(m)(xi)(x− xi)

m

m!.

Thus

R(x)−Gi(x) =Pi(x)−Gi(x)Qi(x)

Qi(x)

2

T-X HE496

has multiple roots at xi of order k; i.e.,

Pi(x)−Gi(x)Qi(x) = (x− xi)kFi(x),

where deg Fi(x) ≤ maxr, k + `− k = maxr− k, `. Thus, the expressionsof R(x) on [xi−1, xi] and [xi, xi+1] are respectively

Pi(x)

Qi(x)=

k−1∑m=0

R(m)(xi)(x− xi)

m

m!+ (x− xi)

k Fi(x)

Qi(x)(1.1)

and

Pi+1(x)

Qi+1(x)=

k−1∑m=0

R(m)(xi)(x− xi)

m

m!+ (x− xi)

k Fi+1(x)

Qi+1(x). (1.2)

Consequently,

Pi+1(x)

Qi+1(x)− Pi(x)

Qi(x)=

[Fi+1(x)

Qi+1(x)− Fi(x)

Qi(x)

](x− xi)

k

=Mi(x)

Qi(x)Qi+1(x)(x− xi)

k,

where Mi(x) = Qi(x)Fi+1(x) − Fi(x)Qi+1(x) is in πr+`−k. Therefore, forx ∈ [xi, xi+1], we have

Pi+1(x)

Qi+1(x)− P1(x)

Q1(x)=

i∑j=1

(Pj+1(x)

Qj+1(x)− Pj(x)

Qj(x)

)

=i∑

j=1

Mj(x)

Qj(x)Qj+1(x)(x− xj)

k+, (1.3)

where Mj(x) ∈ πr+`−k and

(x− xj)+ :=

x− xj if x ≥ xj

0 if x < xj.

From Eq. (1.3) we obtain the globe expression of Pade spline function R(x)as follows.

3

PADE SPLINE FUNCTIONS 497

R(x) =P1(x)

Q1(x)+

n−1∑j=1

Mj(x)

Qj(x)Qj+1(x)(x− xj)

k+, (1.4)

where Mj(x) ∈ πr+`−k is completely determined by P1(x), Qj(x) (j = 1, 2, . . . , n)as well as the values and the first k derivatives of R(x) at xj (j = 1, 2, . . . , n).

We can also show that if any real-valued function R(x) defined on [a, b]

can be written as in Eq. (1.4) with Mj(x) ∈ πr+`−k, then R(x) ∈ R(k)r,` (4);

i.e., R(x) is a Pade spline function defined as in Definition 1.1. Indeed, as-sume that R(x) shown as in (1.4) is given, where Mj(x) ∈ πr+`−k, P1(x) ∈ πr,Qi(x) ∈ π` (i = 1, 2, . . . , n), and the greatest common divisor (P1(x), Q1(x)) =1, there exist p(x) and q(x) such that

p(x)Q1(x) + P1(x)q(x) ≡ 1.

By multiplying φ1(x) = (x−x1)kM1(x) on the both sides of the last equation,

we obtain

p(x)φ(x)Q1(x) + P1(x)q(x)φ(x) = φ(x). (1.5)

Since we can write

p(x)φ(x) = P1(x)r1(x) + s1(x)

and

q(x)φ(x) = Q1(x)r2(x) + s2(x),

Eq. (1.5) can be changed to

[P1(x)r1(x) + s1(x)] Q1(x) + P1(x) [Q1(x)r2(x) + s2(x)] = φ(x). (1.6)

If (x− x1) 6 |s2(x), we set Q2(x) = −s2(x) and

P2(x) = P1(x) [r1(x) + r2(x)] + s1(x).

If (x − x1)|s2(x), then (x − x1) 6 |Q1(x) because of (P1(x), Q1(x)) = 1. Wethus denote Q1(x) = −s2(x)−Q1(x) and

P2(x) = P1(x) [r1(x) + r2(x)− 1] + s1(x).

4

T-X HE498

Therefore in either case, we can write Eq. (1.6) as

P2(x)Q1(x)− P1(x)Q2(x) = (x− x1)kM1(x),

where (x− x1) 6 |Q2(x).Similarly, we cane decompose (x− xj)

kMj(x) into

(x− xj)kMj(x) = Pj+1(x)Qj(x)− Pj(x)Qj+1(x), (1.7)

where Qj(xj) 6= 0, Qj+1(xj) 6= 0, and j = 1, 2, . . . , n − 1. Since Mj(x) ∈πr+`−k, Pj(x) ∈ πr for all j = 1, 2, . . . , n− 1.

For x ∈ [xi, xi+1] (j = 0, 1, . . . , n− 1), from Eqs. (1.4) and (1.7), we have

R(x) =P1(x)

Q1(x)+

i∑j=1

(Pj+1(x)

Qj+1(x)− Pj(x)

Qj(x)

)=

Pi+1(x)

Qi+1(x).

In addition, since

[Mi(x)(x− xi)

k](m)

∣∣∣∣x=xi

= 0

for i = 1, 2, . . . , n− 1 and m = 0, 1, . . . , k− 1, by using the Lemma shown asin [6], we obtain

[(x− xi)

kMi(x)

Qi(x)Qi+1(x)

](m)∣∣∣∣∣∣x=xi

= 0.

Consquently,

[Pi+1(x)

Qi+1(x)

](m)∣∣∣∣∣∣x=xi

=

[Pi(x)

Qi(x)

](m)∣∣∣∣∣∣x=xi

.

It follows that R(x) ∈ R(k)r,` . Thus we have established the following result.

Theorem 1.2 Function R(x) defined on [a, b] and shown as in Eq. (1.4) is

in R(k)r,` (4) if and only if Mj(x) ∈ πr+`−k.

5


2 Algorithm

In this section, we will give two algorithms for constructing Pade spline func-tions. Our first algorithm is to construct the functions piece by piece by usingcontinued fractions. The second algorithm is based on the general expres-sion of Pade spline functions shown as in (1.4). To describe the algorithms

clearly, we only consider the Pade spline function set R(k)k−1,k, which is the

most important set in the Pade approximation. The first algorithms is alsoan improvement of [7-8].

For x ∈ [xi, xi+1], i = 0, 1, . . . , n− 1, write R(x) = Pi(x)Qi(x)

as its continuedfraction form:

Pi(x)

Qi(x)= ai,0 +

x− xi||ai,1

+ · · ·+ x− xi||ai,k−1

+x− xi||ai+1,0

+x− xi+1||ai+1,1

+ · · ·+ x− xi+1||ai+1,k−1

. (2.1)

Denote

Si,0(x)

Ti,0(x)= ai,0 +

x− xi||ai,1

+ · · ·+ x− xi||ai,k−1

. (2.2)

It is easy to find that[Si,0(x)

Ti,0(x)

](m)∣∣∣∣∣∣x=xi

=

[Pi(x)

Qi(x)

](m)∣∣∣∣∣∣x=xi

= y(m)i

for m = 0, 1, . . . , k − 1, which implies by the Lemma in [6]

S(m)i,0 (xi) = [f(x)Ti,0(x)](m)

∣∣∣x=xi

, (2.3)

where m = 0, 1, . . . , k − 1. From Eq. (2.3) we can find the coefficients ofSi,0(x) and Ti,0(x). Then, by using the following relations, (2.4) and (2.5),we can determine the coefficient set ai,0, ai,1, . . . , ai,k−1 of the continuedfraction (2.1).

Si,0 = Πk−1j=0ai,j

1 +k−2∑j=0

x− xi

ai,jai,j+1

+∑

0≤j<`≤k−3

(x− xi)2

ai,jai,j+1ai,`+1ai,`+2

6

T-X HE500

+∑

0≤j<`<m≤k−4

(x− xi)3

ai,jai,j+1ai,`+1ai,`+2ai,m+2ai,m+3

+ · · ·

, (2.4)

Ti,0(x) = Πk−1j=1ai,j

1 +k−2∑j=1

x− xi

ai,jai,j+1

+∑

1≤j<`≤k−3

(x− xi)2

ai,jai,j+1ai,`+1ai,`+2

+ · · ·

. (2.5)

Denote

Si+1,0(x)

Ti+1,0(x)= ai+1,0 +

x− xi+1||ai+1,1

+ · · ·+ x− xi+1||ai+1,k−1

(2.6)

and

Si,−1(x)

Ti,−1(x)= ai,0 +

x− xi||ai,1

+ · · ·+ x− xi||ai,k−2

. (2.7)

Then,

Pi(x)

Qi(x)=

Si,0Si+1,0 + (x− xi)Si,−1Ti+1,0

Ti,0Si+1,0 + (x− xi)Ti,−1Ti+1,0

.

Similar to Eq. (2.3), from [6] we have

[Si,0Si+1,0 + (x− xi)Si,−1Ti+1,0](m)∣∣∣x=xi+1

= f(x) [Ti,0Si+1,0 + (x− xi)Ti,−1Ti+1,0](m)∣∣∣x=xi+1

. (2.8)

In Eq. (2.8), since Si,−1 and Ti,−1 have been determined from (2.3), we thusfind Si+1,0(x) and Ti+1,0(x). We can also establish the relations betweenthe functions Si+1,0(x) and Ti+1,0(x) and the coefficient set ai+1,j : j =0, 1, . . . , k − 1, which is as the same as Eqs. (2.4) and (2.5) except anindex change of i → i + 1. From the relations we finally determine the setai+1,j : j = 0, 1, . . . , k − 1.Example 2.1. As an example, we now consider the case of k = 2. Obviously,we have

7

Si,0(x) = ai,0ai,1 + x− xi, Ti,0(x) = ai,1,

Si+1,0(x) = ai+1,0ai+1,1 + x− xi+1, Ti+1,0 = ai+1,1,

Si,−1(x) = ai,0, Ti,−1(x) = 1.

Thus, (2.3) is reduced to

[ai,0ai,1 + x− xi](m)∣∣∣x=xi

= ai,1y(m)i , m = 0, 1.

Assume that y′i 6= 0, we solve ai,0 = yi and ai,1 = 1/y′i.From Eq. (2.8) we have

[(yi

y′i+ x− xi

)(ai+1,0ai+1,1 + x− xi+1) + yiai+1,1(x− xi)

](m)∣∣∣∣∣∣x=xi+1

=

f(x)

[1

y′i(ai+1,0ai+1,1 + x− xi+1) + ai+1,1(x− xi)

](m)∣∣∣∣∣∣x=xi+1

for m = 0, 1. From the last equation it can be found that

ai+1,0 =y′i(xi+1 − xi)(yi+1 − yi)

y′i(xi+1 − xi)− (yi+1 − yi),

ai+1,1 =[yi+1 − yi − y′i(xi+1 − xi)]

2

y′i [y′iy′i+1(xi+1 − xi)2 − (yi+1 − yi)2]

.

Substituting the obtained coefficient set ai,0, ai,1, ai+1,0, ai+1,1 into the ex-

pression of the Pade spline function R(x) ∈ R(2)2,1

R(x) = ai,0 +x− xi||ai,1

+x− xi||ai+1,0

+x− xi+1||ai+1,1

yields

R(x) = [(x− xi)(x− xi+1) + ai,0ai,1(x− xi+1) + ai+1,0ai+1,1(x− xi)

+ai,0ai+1,1(x− xi) + ai,0ai,1ai+1,0ai+1,1] / [ai,1(x− xi+1)

+ai+1,1(x− xi) + ai,1ai+1,0ai+1,1]

8

T-X HE502

for i = 0, 1, . . . , n− 1.We now discuss the second algorithm. Denote a(i)

m = y(m)i /m! and ì =

x− xi (i = 0, 1, . . . , n), and write

Pi(x) = α(i)0 + α

(i)1 ì + · · ·+ α

(i)k−1`

k−1i

= α(i)0 + α

(i)1 ì+1 + · · ·+ α

(i)k−1`

k−1i+1 (2.9)

Qi(x) = β(i)0 + β

(i)1 ì + · · ·+ β

(i)k `k

i

= β(i)0 + β

(i)1 ì+1 + · · ·+ β

(i)k `k

i+1 (2.10)

From conditions (ii) and (iii) in Definition 1.1, we have

k−1∑m=0

y(m)i

(x− xi)m

m!− Pi(x)

Qi(x)= (x− xi)

k∞∑

j=0

cj(x− xi)j

k−1∑m=0

y(m)i+1

(x− xi+1)m

m!− Pi(x)

Qi(x)= (x− xi+1)

k∞∑

j=0

dj(x− xi+1)j

Substituting expressions (2.9) and (2.10) into the last two equtions yields

k−1∑m=0

k∑j=0

a(i)m β

(i)j `m+j

i −k−1∑j=0

α(i)j `j

i =2k−1∑j=k

r(i)j `j

i (2.11)

k−1∑m=0

k∑j=0

a(i+1)m β

(i)j `m+j

i+1 −k−1∑j=0

α(i)j `j

i+1 =2k−1∑j=k

r(i)j `j

i+1. (2.12)

Therefore we obtain

α(i)j =

j∑µ=0

a(i)j−µβ

(i)µ (2.13)

and

α(i)j =

j∑µ=0

a(i+1)j−µ β(i)

µ (2.14)

for j = 0, 1, . . . , k − 1.Denote hi = xi+1 − xi. From Eq. (2.9) we have

9


α(i)j =

∂jPi(x)

j!∂xj

∣∣∣∣∣x=xi+1

=1

j!

(j!α

(i)j +

(j + 1)!

1!α

(i)j+1`i

∣∣∣x=xi+1

+(j + 2)!

2!α

(i)j+2`

2i

∣∣∣x=xi+1

+ · · ·+ (k − 1)!

(k − j − 1)!α

(i)k−1`

k−j−1i

∣∣∣x=xi+1

)

=k−j−1∑ν=0

(j + ν

ν

)α

(i)j+νh

νi (2.15)

for j = 0, 1, . . . , k − 1. Similarly, from Eq. (2.10) we obtain

β(i)j =

k−j∑ν=0

(j + ν

ν

)β

(i)j+νh

νi (2.16)

for j = 0, 1, . . . , k. Substituting (2.15), (2.16), and (2.13) into (2.14) yields

k−j−1∑µ=0

(j + µ

µ

) j+µ∑ν=0

α(i)j+µ−νβ

(i)ν hµ

i =j∑

µ=0

a(i+1)j−µ

k−ν∑ν=0

(µ + ν

ν

)β

(i)µ+νh

νi , (2.17)

where j = 0, 1 . . . , k − 1. We separate the left-hand side of Eq. (2.17) intotwo parts and write them as

j∑ν=0

β(i)ν

k−j−1∑µ=0

(j + µ

µ

)a

(i)j+µ−νh

µi

+k−j−1∑ν=1

q(i)ν+j

k−j−1∑µ=ν

(j + µ

µ

)a

(i)µ−νh

µi

=

j∑ν=0

β(i)ν

k−j−1∑µ=0

(j + µ

µ

)a

(i)j+µ−νh

µi

+

k−1∑ν=j+1

q(i)ν

k−j−1∑µ=ν−j

(j + µ

µ

)a

(i)j+µ−νh

µi

. (2.18)

Similarly, we can change the right-hand side of Eq. (2.17) to

j∑ν=0

β(i)ν

ν∑µ=0

(ν

ν − µ

)a

(i+1)j−µ hν−µ

i

+k∑

ν=j+1

q(i)ν

j∑µ=0

(ν

ν − µ

)a


i

.

(2.19)

10

T-X HE504

Substituting expressions (2.18) and (2.19) into (2.17) yields the followingequations for j = 0, 1, . . . , k − 1:

j∑ν=0

β(i)ν

k−j−1∑µ=0

(j + µ

µ

)a

(i)j+µ−νh

µi −

ν∑µ=0

(ν

ν − µ

)a


i

+

k−1∑ν=j+1

q(i)ν

k−j−1∑µ=ν−j

(j + µ

µ

)a

(i)j+µ−νh

µi −

j∑µ=0

(ν

ν − µ

)a


i

−β

(i)k

j∑µ=0

(k

k − µ

)a

(i+1)j−µ hk−µ

i

= 0. (2.20)

Eqs. (2.20) is a homogeneous system of k + 1 unknowns, β(i)0 , β

(i)1 , . . ., β

(i)k ,

consisting of k equations. Hence, it has nontrivial solution. To simplyfy theexpression of (2.20), we denote

b(i)j,ν :=

∑k−j−1µ=0

(j+µµ

)a

(i)j+µ−νh

µi

−∑νµ=0

(ν

ν−µ

)a


i if 0 ≤ ν ≤ j,∑k−j−1µ=ν−j

(j+µµ

)a

(i)j+µ−νh

µi

−∑jµ=0

(ν

ν−µ

)a


i if j + 1 ≤ ν ≤ k − 1,

−∑jµ=0

(k

k−µ

)a

(i+1)j−µ hk−µ

i if ν = k

(2.21)

and rewrite (2.20) as

k∑ν=0

b(i)j,νβ

(i)ν = 0, j = 0, 1, . . . , k − 1. (2.22)

After finding

Qi(x) =k∑

j=0

β(i)j (x− xi)

j

=

∣∣∣∣∣∣∣∣∣∣∣

1 x− xi (x− xi)2 · · · (x− xi)

k

b(i)0,0 b

(i)0,1 b

(i)0,2 · · · b

(i)0,k

......

......

b(i)k−1,0 b

(i)k−1,1 b

(i)k−1,2 · · · b

(i)k−1,k

∣∣∣∣∣∣∣∣∣∣∣, (2.23)

11


by (2.13) we have

Pi(x) =k−1∑ν=0

β(i)ν

k−1∑j=ν

a(i)j−ν(x− xi)

j

+ β(i)k · 0 =

∣∣∣∣∣∣∣∣∣∣∣

∑k−1j=0 a

(i)j ti(x)j ∑k−1

j=1 a(i)j−1ti(x)j · · · ∑k−1

j=k−1 a(i)j−k+1ti(x)j 0

b(i)0,0 b

(i)0,1 · · · b

(i)0,k−1 b

(i)0,k

......

......

...

b(i)k−1,0 b

(i)k−1,1 · · · b

(i)k−1,k−1 b

(i)k−1,k

∣∣∣∣∣∣∣∣∣∣∣,

where ti(x) = x−xi. We now calculate r(i)j and r

(i)j in Eqs. (2.11) and (2.12).

First, from (2.11) we obtain

r(i)µ =

k∑j=0

a(i)µ−jβ

(i)j , (2.24)

where µ = k, k + 1, . . . , 2k − 1, and a(i)ν = 0 for all ν ≥ k. Comparing the

last equation with (2.23) yields

r(i)µ =

∣∣∣∣∣∣∣∣∣∣∣

a(i)µ a

(i)µ−1 a

(i)µ−2 · · · a

(i)µ−k

b(i)0,0 b

(i)0,1 b

(i)0,2 · · · b

(i)0,k

......

......

b(i)k−1,0 b

(i)k−1,1 b

(i)k−1,2 · · · b

(i)k−1,k

∣∣∣∣∣∣∣∣∣∣∣. (2.25)

Secondly, from (2.12) we have

r(i)µ =

k∑j=0

a(i+1)µ−j β

(i)j , (2.26)

where µ = k, k +1, . . . , 2k− 1, and a(i)ν = 0 for all ν ≥ k. Substituting (2.16)

into (2.26) yields

r(i)µ =

k∑j=0

k−j∑ν=0

(j + ν

ν

)β

(i)j+νh

νi

a(i+1)µ−j

=k∑

ν=0

ν∑j=0

(ν

ν − j

)a

(i+1)µ−j hν−j

i

β(i)ν . (2.27)

12

T-X HE506

Denoting c(i+1)µ,ν =

∑νj=0

(ν

ν−j

)a

(i+1)µ−j hν−j

i in (2.27) and using (2.23), we obtain

r(i)µ =

k∑ν=0

c(i+1)µ,ν β(i)

µ =

∣∣∣∣∣∣∣∣∣∣∣

c(i+1)µ,0 c

(i+1)µ,1 c

(i+1)µ,2 · · · c

(i+1)µ,k

b(i)0,0 b

(i)0,1 b

(i)0,2 · · · b

(i)0,k

......

......

b(i)k−1,0 b

(i)k−1,1 b

(i)k−1,2 · · · b

(i)k−1,k

∣∣∣∣∣∣∣∣∣∣∣. (2.28)

Therefore, Eqs. (2.11) and (2.11) are evantually obtained as

k−1∑m=0

y(m)i

(x− xi)m

m!− Pi

Qi

=2k−1∑j=k

r(i)j

Qi

(x− xi)j,

k−1∑m=0

y(m)i+1

(x− xi+1)m

m!− Pi

Qi

=2k−1∑j=k

r(i)j

Qi

(x− xi+1)j,

from which we have the Pade spline function defined on [a, b] with the form

R(x) =P0

Q0

+n−2∑i=0

2k−1∑µ=k

(r(i)µ

Qi

−r(i)µ

Qi+1

) (x− xi+1)µ+, (2.29)

where r(i+1)µ , r(i)

µ , and Qi are given by Eqs. (2.25), (2.28), and (2.23), respec-tively.

References

[1] G. A. Baker, Jr., The Theory and Application of The Pade Approxi-mant Method, In Advances in Theoretical Physics, Vol. 1 (Ed. K. A.Brueckner). New York: Academic Press, pp. 1-58, 1965.

[2] G. A. Baker, Jr., Essentials of Pad Approximants in Theoretical Physics,New York: Academic Press, pp. 27-38, 1975.

[3] G. A. Baker, Jr. and P. Graves-Morris, Pad Approximants, Second Edi-tion, New York: Cambridge University Press, 1996.

13


[4] R. P. Brent, F. G. Gustavson, and D. Y. Y. Yun, Fast Solution ofToeplitz Systems of Equations and Computation of Pad Approximants,J. Algorithms, 1 (1980), 259-295.

[5] T. X. He, Pade Spline Functions, Conference talk in the JointU.S.- China Workshop on Approximation Theory that took place inHangzhou, China, April, 1985.

[6] H. E. Salzer, Equally-weighted formulas for numerical differentiation,Numer. Math., 4 (1962/1963), 381–392.

[7] G. Q. Zhu and T. X. He, A note on rational spline functions, Numer.Math. J. Chinese Univ., 3 (1981), no. 3, 276–278.

[8] G. Q. Zhu and T. X. He, A method of calculation for the n-point sec-tional Pad approximant, Math. Numer. Sinica (China), 3 (1981), no. 2,179–182.

14

T-X HE508

1

Null Controllability Of Nonlinear Infinite Neutral Systems

With Multiple Delays In Control

R. A. Umana

Department of Mathematics and Computer Science

Federal University of Technology, Owerri, Imo State, Nigeria.


Abstract

Sufficient conditions are developed for the null controllability of nonlinear infinite neutral

systems with time varying multiple delays in control. It is shown that if the uncontrolled system

is uniformly asymptotically stable, and if the linear system is controllable, then the nonlinear

infinite neutral system is null controllable. An example is provided to illustrate the obtained

results.

Keywords

Controllability, infinite neutral systems, multiple delays, uniform asymptotic stability.

1. Introduction

Neutral functional differential equations are characterized by a delay in the derivative. Equations

of this type have applications in the study of electrical networks containing lossless transmission

lines. It is well known that the mixed initial-boundary hyperbolic partial differential equation

which arises in the study of lossless transmission lines can be replaced by an associated neutral

functional differential equation [1]. The aim of this paper is to study the null controllability of

such systems by introducing multiple delays in control. For motivation of time varying multiple

delays in control variables refer to the book by Klamka [13].


2

The problem of controllability and null controllability of functional differential equations in finite

dimensional space has been studied by several authors. These include Chukwu [7] Balachandran

et al [2], Umana [16], Umana and Nse [17], Iheagwam and Onwuatu [11], and Fu [8]. Several

authors have extended the controllability and null controllability concepts to infinite dimensional

systems. These include Balachandran and Dauer [3], Sinha [15], Onwuatu [14], Balachandran

and Anandhi [5], Iyai [12], and Balachandran and Leelamani [6].

It is well known [10] that if the linear control system

( ) ( ) ( ) ( ) ( )x t A t x t B t u t= +& (1.1)

is proper and if the free system

( ) ( ) ( )x t A t x t=& (1.2)

is uniformly asymptotically stable, then (1.1) is null controllable with constraints. Chukwu [7]

obtained an analogous result for the delay system

( ) ( , ) ( ) ( ) ( , , ( ))t tx t L t x B t u t f t x u t= + +& (1.3)

where

0

0( , ) ( ) ( ) ( , ) ( )k k

kL t A t t A t s s ds

γφ φ φ

∞

−=

= − +∑ ∫ . (1.4)

Sinha [15] studied the nonlinear infinite delay system

0( ) ( , ) ( ) ( ) ( ) ( ) ( , , ( ))t tx t L t x B t u t A x t d f t x u tθ θ θ

−∞= + + + +∫& (1.5)

and showed that (1.5) is Euclidean null controllable if the linear base system

( ) ( , ) ( ) ( )tx t L t x B t u t= +& (1.6)

is proper and if the free system

0( ) ( , ) ( ) ( )tx t L t x A x t dθ θ θ

−∞= + +∫& (1.7)

UMANA510

3

is uniformly asymptotically stable, provided that f satisfies some growth conditions.

Balachandran and Dauer [3] studied this problem for system (1.5) with distributed delays in the

control. This problem for system (1.5) with time varying multiple delays has also been studied by

the same authors [4]. Onwuatu [14] extended Sinha’s results to neutral systems.

In this paper, nonlinear infinite neutral control systems with time varying multiple delays of the

following form are considered:

0

0( , ) ( , ) ( ) ( ) ( ) ( )

p

t t i ii

d D t x L t x B t u t h A x t ddt

θ θ θ−∞

== + − + +∑ ∫ ( , , ( ))tf t x u t+ (1.8)

( ) ( ), ( ,0]x t t tφ= ∈ −∞

where ( , )L t φ is as defined in (1.4), ( )A θ is an n n× continuous matrix and ( , , ( ))tf t x u t is a

nonlinear continuous matrix function.

Our results extend those of [3,15,4] to neutral systems and those of [14,12] to neural systems

with multiple delays in the control.

2. Preliminaries

Let n and m be positive integers, R the real line ( , )−∞ ∞ . Denote by nR , the space of real n -

tuples with the Euclidean norm defined by ⋅ . If 0 1[ , ]J t t= is any interval of R , the usual

Lebesgue space of square integrable functions from J to mR will be denoted by 2 ( , )mL J R .

Let 0hγ ≥ > be a given real number and let ([ ,0], )nB B Rγ= − be the Banach space of functions

which are continuous on [ ,0]γ− with 0sup ( ) , ([ ,0], )ns s B Rγφ φ φ γ− ≤ <= ∈ − . If x is a function

from 0[ , )t γ− ∞ to nR , let tx , [0, )t ∈ ∞ , be a function from [ ,0]γ− to nR , defined by

( ) ( )tx s x t s= + , [ ,0]s γ∈ − . Similarly, if u is a function from 0[ , )t γ− ∞ to mR , let tu , [0, )t ∈ ∞

NULL CONTROLLABILITY... 511

4

be a function from [ ,0]γ− to mR , defined by ( ) ( )tu s u t s= + , [ ,0]s γ∈ − . In system (1.8), assume

that ( , ) : nD R B R⋅ ⋅ × → is defined by

( , ) ( ) ( , )t tD t x x t g t x= −

where

0

1( , ) ( ) ( ( )) ( , ) ( )n n

ng t A t w t A t s s ds

γφ φ φ

∞

−=

= − +∑ ∫

and where 0 ( )nw t γ< ≤ and ( )nA t and ( , )A t s are n n× matrix functions.

A nonautonomous linear homogeneous neutral differential equation is defined to be

( , ) ( , )t td D t x L t xdt

= (2.1)

Our objective is to study the null controllability of system (1.8) through its linear control base

system

0( , ) ( , ) ( ) ( )

p

t t i ii

d D t x L t x B t u t hdt =

= + −∑ (2.2)

and its free system

0( , ) ( , ) ( ) ( )t t

d D t x L t x A x t ddt

θ θ θ−∞

= + +∫ . (2.3)

Here iB are continuous n m× matrix functions, each kA is a continuous n n× matrix function for

0 kh γ≤ ≤ , ( )A θ is an n n× matrix whose elements are square integrable on ( ,0]−∞ . , ,D L f

satisfy enough smoothness conditions to ensure that a solution of (1.8) exists through each 0( , )t φ

in J B× , is unique, depends continuously on 0( , )t φ and can be continued to the right as the

trajectory remains in a bounded set in J B× .

UMANA512

5

If 0( , ) :T t t B B→ , 0t t≥ is defined by 0 0( , ) ( , )tT t t x tφ φ= , where 0( , )x t φ is the solution of (2.1),

then there exists an n n× matrix function ( , )X t s defined for 0 s γ≤ ≤ , [0, )t ∈ ∞ , continuous in

s from the right, of bounded variation in s , ( , ) 0,X t s t s t γ= < ≤ + , such that the solution

0( , )x t φ of (2.2) is given by

00 0

0( , , , ) ( , ) (0) ( , ) ( ) ( )

pt

i iti

x t t u T t t X t s B s u s h dsφ φ=

= + −∑∫ (2.4)

The corresponding solution of (1.8) is given by

00 0

0( , , , , ) ( , ) (0) ( , ) ( ) ( )

pt

i iti

x t t u f T t t X t s B s u s h dsφ φ=

= + −∑∫

( )0

0( , ) ( ) ( )

t

tX t s A x t d ds

γθ θ θ

−+ +∫ ∫

0

( , ) ( , , ( ))t

stX t s f s x u s ds+∫ . (2.5)

Observe that the uniqueness of solution of (2.1) implies that

2 1 1 0 2 0( , ) ( , ) ( , )T t t T t t T t t= , 1 2 0, 0t t t≥ ≥ .

If we let

0

0, 0( )

, 0s

X sI s

γ− ≤ <⎧= ⎨ =⎩

then 0 0 0 0( , ) ( ) ( , ) ( , )( )tT t t X s X t s t X t s= + = ⋅ . Therefore, 0 0( , ) ( , )T t t I X t t= .

Now, let us assume that the functions 0 1( ) :[ , ] , 0,1, 2,...,ih t t t R i p→ = are twice continuously

differentiable and strictly increasing functions in the time interval 0 1[ , ]t t . Moreover, ( )ih t t≤ for

0 1[ , ]t t t∈ , and 0,1,2,...,i p= . Let us introduce the time-lead functions

0 1 0 1( ) :[ ( ), ( )] [ , ], 0,1, 2,...,i i ir t h t h t t t i p→ = ,

such that ( ( ))i ir h t t= for 0 1[ , ]t t t∈ .

6

Furthermore, only for simplicity and compactness of notation let us assume that 0 ( )h t t= and for

1t t= the functions ( )ih t satisfy the following inequalities

1 1 1 1 1 0( ) ( ) ... ( )p p mh h t h t h t t− += ≤ ≤ ≤ ≤ =

1 1 1 1 1 0 1 1( ) ( ) ... ( ) ( )m mh t h t h t h t t−≤ ≤ ≤ ≤ = (2.6)

Using the time-lead function and the inequalities (2.6) we have

0

01 0 1 0 1

0( , , , , ) ( , ) (0) ( , ) ( )( ) ( )

i

m t

i i i it hi

x t t u f T t t X t s r B s r s r s dsφ φ η−

== + + + +∑∫ &

1

01

1( , ) ( )( ) ( )i

i

p t h

i i i it hi m

X t s r B s r s r s dsη−

−= +

+ + + +∑ ∫ &

1

00( , ) ( )( ) ( )

m t

t i i i iti

X t s r B s r s r u s ds=

+ + + +∑∫ &

( )1

0

0

1( , ) ( ) ( )t

t rX t s A x s d dsθ θ θ

−+ +∫ ∫

( )1

01( , ) , , ( )

t

stX t s f s x u s ds+∫ (2.7)

where ( ) ( )u s sη= for 0 0[ , ]s t r t∈ − .

For brevity, we introduce as in [4], the following notations:

0

01 1

0( , ) ( , ) ( )( ) ( )

i

m t

i i i it hi

H t X t s r B s r s r s dsη η−

== + + +∑∫ &

1

01

1( , ) ( )( ) ( )i

i

p t h

i i i it hi m

X t s r B s r s r s dsη−

−= +

+ + + +∑ ∫ &

( )1

01 1 0 1 1( , ) ( , ) (0) ( , ) ( , ) , , ( )

t

stq t T t t H t X t s f s x u s dsη φ η= + + ∫

( )1

0

0

1( , ) ( ) ( )t

t rX t s A x s d dsθ θ θ

−+ +∫ ∫ ,

UMANA514

7

0

( , ) ( , ) ( )( )i

i j j j jj

G t s X t s r B s r s r=

= + + +∑ & .

Define the controllability matrix of (2.2) at time t by

00( , ) ( , ) ( , )

t

m mtW t t G t s G t s dsΤ= ∫

where • denotes matrix transpose.

Definition 2.1: The system (1.8) is said to be null controllable on 0 1[ , ]t t if for each

( )[ ,0], nB Rφ γ∈ − , there exists a 1 0t t≥ , ( )2 0 1[ , ],u L t t IU∈ , IU a compact convex subset of mR ,

such that the solution 0( , , , , )x t t u fφ of (1.8) satisfies 0 0( , , , )tx t u fφ φ= and 1 0( , , , , ) 0x t t u fφ = .

3. Main Results

Theorem 3.1: Assume for system (1.8) that

(i) the constraint set IU is an arbitrary compact subset of mR ,

(ii) the system (2.3) is uniformly asymptotically stable so that the solution of (2.3) satisfies

0( )0( , , ,0, 0) t tx t t Me αφ φ− −≤ for some 0α > , 0M > ,

(iii) the linear control system (2.2) is controllable in nR ,

(iv) the continuous function f satisfies ( ) ( ), ( ), ( ) exp( ) ( ), ( )f t x u t x uβ π⋅ ⋅ ≤ − ⋅ ⋅ , for all

( ) ( ) ( )0 2 0 1, ( ), ( ) [ , ) [ ,0], [ , ],nt x u t B R L t t IUγ⋅ ⋅ ∈ ∞ × − × , where

( )0

( ), ( )t

x u ds Kπ∞

⋅ ⋅ ≤ < ∞∫ and 0β α− ≥ ,

then system (1.8) is Euclidean null controllable.

Proof: By (iii), 10 1( , )W t t− exists for each 1 0t t> . Suppose the pair of functions x , u form a

solution pair to the set of integral equations

11 0 1 1( ) ( , ) ( , ) ( , )mu t G t t W t t q t ηΤ −= − (3.1)

8

for some suitably chosen 1 0t t t≥ ≥ , ( ) ( )u t tη= , 0 0[ , ]t t tγ∈ − and

00( ) ( , ) (0) ( , ) ( , ) ( )

t

mtx t T t t H t G t s u s dsφ η= + + ∫

( ) ( )0 0

0( , ) ( ) ( ) ( , ) , , ( )

t t

st tX t s A x t d ds X t s f s x u s ds

γθ θ θ

−+ + +∫ ∫ ∫ (3.2)

( ) ( )x t tφ= , 0 0[ , ]t t tγ∈ − .

Then u is square integrable on 0 1[ , ]t tγ− and x is a solution of (1.8) corresponding to u with

initial state ( )0 0( ) ( ), ,z t x t φ η= .

Also

1

0

11 1 0 1 1 0 1 1 0 1( ) ( , ) (0) ( , ) ( , ) ( , )[ ( , ) (0) ( , )

t Tm mt

x t T t t G t s G t s W t t T t t H tφ φ η−= − +∫

( )( )1

0

0

1( , ) , , ( ) ( ) ( ) ]t

stX t s f s x u s A x t d ds ds

γθ θ θ

−+ + +∫ ∫

( )( )1

0

0( , ) , , ( ) ( ) ( ) 0

t

stX t s f s x u s A x t d ds

γθ θ θ

−+ + + =∫ ∫ . (3.3)

We now show that 0 1:[ , ]u t t IU→ is a compact constraint subset of mR , that is u a≤ for some

constant 0a > . Since (2.3) is uniformly asymptotically stable and iB are continuous in t , it

follows that

11 0 0 1 1( , ) ( , )mG t t W t t kΤ − ≤ , for some 1 0k > ,

( )1 0 2 1 0( , ) (0) exp ( )T t t k t tφ α≤ − − , for some 2 0k > ,

( )3 1 0( , ) exp ( )H t k t tη α≤ − − , for some 3 0k > .

Hence,

( ) ( ) ( )1

01 2 1 0 3 1( ) exp ( ) exp ( ) exp( ) ( ), ( )

t

tu t k k t t k M t s s x u dsα α β π⎡ ⎤≤ − − + + − − − ⋅ ⋅⎢ ⎥⎣ ⎦∫ ,

UMANA516

9

and therefore

( )1 2 3 1 0 1( ) ( )exp ( ) exp( )u t k k k t t KM tα α≤ + − − + −⎡ ⎤⎣ ⎦ (3.4)

since 0β α− ≥ and 0 0s t≥ ≥ . Hence, by taking 1t sufficiently large, we have ( )u t a≤ ,

0 1[ , ]t t t∈ which proves that u is an admissible control.

We now prove the existence of a solution pair of the integral equations (3.1) and (3.2). Let B be

the Banach space of all functions

0 1 0 1( , ) :[ , ] [ , ] n mx u t h t t h t R R− × − → ×

where ( )0 1[ , ], nx B t h t R∈ − and ( )2 0 1[ , ], mu L t h t R∈ − with the norm defined by

2 2( , )x u x u≤ + ,

where 121

0

2

2 ( )t

t hx x s ds

−= ∫ , and

121

0

2

2 ( )t

t hu u s ds

−= ∫ .

Define the operator :T B B→ by ( , ) ( , )T x u y v= ,

where

11 0 1 1( ) ( , ) ( , ) ( , )mv t G t t W t t q t ηΤ −= − for 0 1[ , ]t t t∈ (3.5)

and

( ) ( )v t tη= for 0 0[ , ]t t tγ∈ − ;

00( ) ( , ) (0) ( , ) ( , ) ( )

t

mty t T t t H t G t s v s dsφ η= + + ∫ ( )

0

0( , ) ( ) ( )

t

tX t s A x t d ds

γθ θ θ

−+ +∫ ∫

( )0

( , ) , , ( )t

stX t s f s x u s ds+∫ for t J∈ (3.6)

and ( ) ( )y t tφ= for 0 0[ , ]t t tγ∈ − . From equation (3.4), we have shown that ( )v t a≤ , t J∈ and

also 0 0:[ , ]v t h t IU− → , so ( )v t a≤ . Hence,


10

12

1 0 02( ) ( )v t a t h t β≤ + − = .

Also

( )0

2 3 0 4 1( ) exp ( ) ( ) exp( )t

ty t k k t t k v s ds KM tα α≤ + − − + + −∫

where 4 sup ( , )mk G t s= . Since 0α > , 0 0t t≥ ≥ , it follows that

2 3 4 1 0( ) ( )y t k k k a t t KM β≤ + + − + ≡ , t J∈

and

( ) sup ( )y t tφ δ≤ ≡ , 0 0[ , ]t t h t∈ − .

Hence, if max ,λ β δ= , then

12

1 0 12 ( )y t h tλ β≤ + − ≡ < ∞ .

Let 0 1max ,r β β= . Then letting

2 2( ) ( , ) : ,Q r x u B x r u r= ∈ ≤ ≤ ,

it follows that : ( ) ( )T Q r Q r→ . Since ( )Q r is closed, bounded and convex, by Riesz’s theorem,

it is relatively compact under the transformation T . The Schauder fixed point theorem implies

that T has a fixed point ( , ) ( )x u Q r∈ . This fixed point ( , )x u of T is a solution pair of the

integral equations (3.5), (3.6). Hence, the system (1.8) is Euclidean null controllable.

4. Applications

If we now specialize to the constant systems with multiple delays in the control defined by

( )1 0 1 0 1( ) ( ) ( ) ( ) ( ) ( )d x t A x t h A x t A x t h B u t B u t hdt −− − = + − + + − (4.1)

( )1 0 1 0 1( ) ( ) ( ) ( ) ( ) ( )d x t A x t h A x t A x t h B u t B t hdt −− − = + − + + −

0

0 exp( ) ( )C x t dηθ θ θ−∞

+ +∫ (4.2)

UMANA518

11

( )1 0 1 0 1( ) ( ) ( ) ( ) ( ) ( )d x t A x t h A x t A x t h B u t B u t hdt −− − = + − + + −

( )0

0 exp( ) ( ) , ( ), ( ), ( )C x t d f t x t x t h u tηθ θ θ−∞

+ + + −∫ (4.3)

then the following results follow:

Theorem 4.1: If rank 0 0 0[ , ]B A B n= , then system (4.1) is completely controllable on 0 1[ , ]t t .

Proof: This is equivalent to Theorem 2 of Gahl [9].

Theorem 4.2: In system (4.2), assume that

(i) (4.2) with 0u = is uniformly asymptotically stable,

(ii) rank 0 0 0[ , ]B A B n= ,

then system (4.2) is null controllable with constraints.

Proof: By (ii), (4.1) is completely controllable. Hence (i) and (ii) satisfy the requirements of

Theorem 3.1 and the proof is complete.

Theorem 4.3: For system (4.3), assume that

(i) f satisfies all smoothness conditions for the existence and uniqueness of solutions,

(ii) the zero solution of (4.2) with 0u = is uniformly asymptotically stable,

(iii) rank 0 0 0[ , ]B A B n= ,

(iv) ( ,0, 0,0) 0f t = ,

then system (4.3) is null controllable with constraints.

Proof: Immediate from Theorems 3.1 and 4.1.

Example

Consider the system

( )1 0 1 0 1( ) ( ) ( ) ( ) ( ) ( )d x t A x t h A x t A x t h B u t B u t hdt −− − = + − + + −


12

( )0

0 exp( ) ( ) , ( ), ( ), ( )C x t d f t x t x t h u tηθ θ θ−∞

+ + + −∫ (4.4)

where

1

0 11 0

A−⎛ ⎞

= ⎜ ⎟⎝ ⎠

, 0

1 11 2

A−⎛ ⎞

= ⎜ ⎟−⎝ ⎠, 1

0 30 1

A ⎛ ⎞= ⎜ ⎟−⎝ ⎠

, 0

01

B ⎛ ⎞= ⎜ ⎟⎝ ⎠

, 1

10

B−⎛ ⎞

= ⎜ ⎟⎝ ⎠

, 0

0 00 1

C ⎛ ⎞= ⎜ ⎟−⎝ ⎠

with

( ) ( )0

, ( ), ( ), ( )sin ( ) ( ) cos ( )tf t x t x t h u t

e x t x t h u t−

⎛ ⎞− = ⎜ ⎟+ −⎝ ⎠

.

The characteristic roots of the homogeneous equation

( ) 0

1 0 1( ) ( ) ( ) ( ) exp( ) ( )d x t A x t h A x t A x t h x t ddt

ηθ θ θ− −∞− − = + − + +∫ (4.5)

is

02 2 23 1 (3 ) (2 3 ) ( 1) exp[( ) ] 0e e dλ λλ λ λ λ λ λ λ η θ θ− −

−∞+ + + − + − + + + =∫ (4.6)

Every root of (4.6) has negative real part. Hence, by Theorem 1 of Sinha [15], system (4.5) is

uniformly asymptotically stable.

We now show that the linear base system

( )1 0 1 0 1( ) ( ) ( ) ( ) ( ) ( )d x t A x t h A x t A x t h B u t B u t hdt −− − = + − + + − (4.7)

is controllable on any interval [0, ]t , 0t > . By Theorem 4.1, we show that rank 0 0 0[ , ]B A B n= .

But

rank 0 0 0[ , ]B A B = rank0 11 2

⎛ ⎞=⎜ ⎟−⎝ ⎠

rank0 1

21 0

n⎛ ⎞= =⎜ ⎟

⎝ ⎠.

Since rank 0 0 0[ , ]B A B = 2 for each 0t > , the system (4.7) is controllable on each [0, ]t , 0t > on

nR . We conclude that system (4.4) is null controllable, by Theorem 3.1, since

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13

( ) ( ), ( ), ( ),0 sin ( ) ( ) ( )t tf t x t x t h e x t x t h e tπ− −− ≤ + − ≤ ≡ .

Conclusion

We have derived sufficient conditions for the null controllability of nonlinear infinite neutral

systems with time varying multiple delays in control. These conditions are given with respect to

the uniform asymptotic stability of the free linear base system and the controllability of the linear

controllable base system, with the assumption that the perturbation function f satisfies some

smoothness and growth conditions.

References

[1] R. Brayton, Nonlinear oscillations in a distributed network, Quart. Appl. Math. 24, 239-301

(1976).

[2] K. Balachandran, J. P. Dauer and P. Balasubramanian, Local null controllability of nonlinear

functional differential systems in Banach spaces, J. Optim. Theory Appl. 88, 61-75 (1995).

[3] K. Balachandran and J. P. Dauer, Null controllability of nonlinear infinite delay systems with

distributed delays in control, J. Math. Anal. Appl. 145, 274-281 (1990).

[4] K. Balachandran and J. P. Dauer, Null controllability of nonlinear infinite delay systems with

time varying multiple delays in control, Appl. Math. Letters 9, 115-121 (1996).

[5] K. Balachandran and E. R. Anandhi, Controllability of neutral functional integrodifferential

infinite delay systems in Banach spaces, Taiwanese J. Math. 8, 689-702 (2004).

[6] K. Balachandran and A. Leelamani, Null controllability of neutral evolution

integrodifferential systems with infinite delay, Math. Prob. Engineering 2006, 1-18 (2006).

[7] E. N. Chukwu, On the null controllability of nonlinear delay systems with restrained controls,

J. Math. Anal. Appl. 76, 283-296 (1980).


14

[8] X. Fu, Controllability of neutral functional differential systems in abstract space, Applied

Math. Comp. 141, 281-296 (2003).

[9] R. D. Gahl, Controllability of nonlinear systems of neutral type, J. Math. Anal. Appl. 66, 33-

42 (1978).

[10] H. Hermes and J. P. Laselle, Functional Analysis and Time Optimal Control, Academic

Press, New York, 1969.

[11] V. A. Iheagwam and J. U. Onwuatu, Relative controllability and null controllability of linear

delay systems with distributed delays in the state and control, J. Nigerian Asso. Math. Physics

9, 221-238 (2005).

[12] D. Iyai, Euclidean null controllability of infinite neutral differential systems, ANZIAM J. 48,

285-293 (2006).

[13] J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrect,

1991.

[14] J. U. Onwuatu, Null controllability of nonlinear infinite neutral system, Kybernetika 29,

325-336 (1993).

[15] A. S. C. Sinha, Null controllability of nonlinear infinite delay systems with restrained

controls, Int. J. Control 42, 735-741 (1985).

[16] R. A. Umana, Relative null controllability of linear systems with multiple delays in state and

control, J. Nigerian Asso. Math. Physics 10, 517-522 (2006).

[17] R. A. Umana and C. A. Nse, Null controllability of nonlinear integrodifferential systems

with delays, Journal of Advances in Modelling 61, 73-84 (2006).

UMANA522

Moments of the Scaled

Burr Type X Distributionby

M. Zhou1, D. Yang2, Y. Wang3 & S. Nadarajah4

Abstract: A recent paper by Surles and Padgett proposed the scaled Burr type X distribution anddiscussed various properties, including moments. The paper claimed that closed-form expressionsfor E(Xk) are possible only for certain special cases: when the parameter of the distribution isassumed to be an integer or when k = 2 (the latter represented as an infinite sum). In this note, weshow that one can derive simple expressions for E(Xk) for all even k ≥ 2 without any restrictionon the parameter of the distribution. The expressions only involve the gamma function and itsderivatives.

1 Introduction

Surles and Padgett (2005) defined the scaled Burr X distribution with shape parameter θ and scaleparameter σ by the cdf

F (x) =[1− exp

−

(x

σ

)2]θ

(1)

for x > 0, θ > 0 and σ > 0. Surles and Padgett (2005) discussed various properties of thisdistribution, including moments and their approximations, maximum likelihood estimators andtheir asymptotic properties as well as types I and II censoring. This distribution is a particular caseof the exponentiated Weibull distribution introduced by Mudholkar et al. (1995); see Mudholkarand Hutson (1996), Nassar and Eissa (2003) and Nadarajah and Gupta (2005) for more recentdevelopments.

This note concerns the moment properties of a random variable X having the cdf (1). Surles andPadgett (2005) claim that ‘closed-form expressions for the moments only exist for certain specialcases . . ..” In particular, two closed-form expressions are given:

E(Xk

)= σkθΓ

(k

2+ 1

) θ−1∑j=0

(−1)j

(θ − 1

j

)1

(j + 1)k/2+1

applicable when θ ≥ 1 is an integer; and,

E(X2

)= θσ2

∞∑i=0

1i(θ + i)

1Author’s address: Department of Statistics, University of Nebraska, Lincoln, Nebraska 68583, USA2Author’s address: Department of Statistics, University of Nebraska, Lincoln, Nebraska 68583, USA3Author’s address: Department of Statistics, University of Nebraska, Lincoln, Nebraska 68583, USA4Corresponding author’s address: School of Mathematics, University of Manchester, Manchester M60 1QD, UK,


1


applicable for any θ > 0. In this note, we show that one can derive simple expressions for E(Xk)for any even k ≥ 2 and for any θ > 0. The expressions only involve the gamma function and itsderivatives.

2 Moments

Theorem 1 derives the expression for the kth moment for any even k ≥ 2 and for any θ > 0.

Theorem 1 If X is a random variable with the cdf (1) then

E(Xk

)= θΓ(θ)σk(−1)k/2 ∂k/2

∂βk/2

Γ(β + 1)

Γ(θ + β + 1)

∣∣∣∣∣β=0

(2)

for any even k ≥ 2 and for any θ > 0.

Proof: The pdf corresponding to (1) is:

f(x) =2θx

σ2 exp−

(x

σ

)2 [

1− exp−

(x

σ

)2]θ−1

and so one can express E(Xk) as

E(Xk

)=

2θ

σ2

∫ ∞

0xk+1 exp

−

(x

σ

)2 [

1− exp−

(x

σ

)2]θ−1

dx

= (−1)k/2σkθ

∫ 1

0yθ−1 log(1− y)k/2 dy, (3)

which follows by substituting y = 1−exp−(x/θ)2. The result in (2) follows by applying equation(2.6.9.5) in Prudnikov et al (1986, volume 1) to calculate the integral in (3).

The result in (2) can be used to derive simple expressions for moments of even-order. Corollary1 illustrates this for the first five even-order moments.

Corollary 1 If X is a random variable with the cdf (1) then the first five even-order moments aregiven by

E(X2

)= σ2

[γ + Ψ(θ + 1)

],

E(X4

)= (1/6)σ4

[π2 − 6Ψ

′(θ + 1) + 6γ2 + 12γΨ(θ + 1) + 6Ψ2 (θ + 1)

],

E(X6

)= (1/2)σ6

[4ζ (3) + 2Ψ

′′(θ + 1) + π2γ + π2Ψ(θ + 1)− 6Ψ

′(θ + 1) γ

−6Ψ′(θ + 1) Ψ (θ + 1) + 2γ3 + 6γ2Ψ(θ + 1) + 6γΨ2 (θ + 1) + 2Ψ3 (θ + 1)

],

E(X8

)= (1/20)σ8

[3π4 − 20Ψ (3θ + 1) + 160ζ (3) γ + 160ζ (3)Ψ (θ + 1) + 80Ψ

′′(θ + 1) γ

2

ZHOU ET AL524

+80Ψ′′(θ + 1) Ψ (θ + 1)− 20π2Ψ

′(θ + 1) + 60

Ψ

′(θ + 1)

2+ 20π2γ2

+40π2γΨ(θ + 1) + 20π2Ψ2 (θ + 1)− 120Ψ′(θ + 1) γ2

−240Ψ′(θ + 1) γΨ(θ + 1)− 120Ψ

′(θ + 1) Ψ2 (θ + 1) + 20γ4 + 80γ3Ψ(θ + 1)

+120γ2Ψ2 (θ + 1) + 80γΨ3 (θ + 1) + 20Ψ4 (θ + 1)

],

E(X10

)= (1/12)σ10

[60π2γ2Ψ(θ + 1)− 120Ψ

′′(θ + 1) Ψ

′(θ + 1) + 240ζ (3) γ2

−60π2Ψ′(θ + 1) Ψ (θ + 1) + 60π2γΨ2 (θ + 1) + 180

Ψ

′(θ + 1)

2Ψ(θ + 1)

+20π2γ3 − 60π2Ψ′(θ + 1) γ + 120Ψ

′′(θ + 1) Ψ2 (θ + 1) + 180

Ψ

′(θ + 1)

2γ

+480ζ (3) γΨ(θ + 1) + 60γ4Ψ(θ + 1) + 120γ3Ψ2 (θ + 1) + 120γ2Ψ3 (θ + 1)+60γΨ4 (θ + 1) + 9π4γ + 9π4Ψ(θ + 1) + 40ζ (3)π2 − 240ζ (3)Ψ

′(θ + 1)

−360Ψ′(θ + 1) γ2Ψ(θ + 1) + 240ζ (3)Ψ2 (θ + 1) + 120Ψ

′′(θ + 1) γ2

+240Ψ′′(θ + 1) γΨ(θ + 1) + 20π2Ψ3 (θ + 1)− 120Ψ

′(θ + 1) γ3

−120Ψ′(θ + 1) Ψ3 (θ + 1) + 12Ψ5 (θ + 1) + 12γ5 − 60Ψ

′′′(θ + 1) γ

−60Ψ′′′

(θ + 1) Ψ (θ + 1) + 20Ψ′′(θ + 1) π2 − 360Ψ

′(θ + 1) γΨ2 (θ + 1)

+12Ψ′′′′

(θ + 1) + 288ζ (5)

],

where γ = 0.5772156649 · · · is Euler’s constant, Ψ(x) = d log Γ(x)/dx is the digamma function andζ(x) =

∑∞k=1 k−x is the zeta function.

References

Mudholkar, G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: some propertiesand a flood data application. Communications in Statistics—Theory and Methods, 25, 3059-3083.

Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family.Technometrics, 37, 436–445.

Nadarajah, S. and Gupta, A. K. (2005). On the moments of the exponentiated Weibull distribution.Communications in Statistics—Theory and Methods, 34.

Nassar, M. M. and Eissa, F. H. (2003). On the exponentiated Weibull distribution. Communica-tions in Statistics—Theory and Methods, 32, 1317–1336.

Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series (volumes 1, 2and 3). Amsterdam: Gordon and Breach Science Publishers.

Surles, J. G. and Padgett, W. J. (2005). Some properties of a scaled Burr type X distribution.Journal of Statistical Planning and Inference, 128, 271–280.

3

X DISTRIBUTION 525

On the Rate of Approximation of Meyer-Konig andZeller Operators

Xiao-Ming Zeng

Department of Mathematics, Xiamen University, Xiamen 361005, ChinaE-mail: [email protected]

Abstract

In this paper the asymptotic property of Meyer-Konig and Zeller operators Mn forbounded functions on [0, 1] is studied. An asymptotic convergence theorem of this typeapproximation is established by means of some probabilistic methods and results and ac-curate estimate technique to the basis functions of the operators Mn. The main resultof this paper subsumes the approximation of the operators Mn for functions of boundedvariation as a special case.

2000 Mathematics Subject Classification: 41A10, 41A36, 41A25.

Keywords: Rate of approximation, Meyer-Konig and Zeller operators, Probability dis-tribution, Basis functions.

1 INTRODUCTION

For a function f defined on [0, 1], Meyer-Konig and Zeller operator Mn [10] isdefined by

Mn(f, x) =∞∑

k=0

f(k

n + k)mn,k(x), 0 ≤ x < 1,

Mn(f, 1) = f(1), mn,k(x) =

(n + k − 1

k

)xk(1− x)n. (1)

If replacing the basis function mn,k(x) in definition (1) with the new basis

function mn,k(x) =

(n + k

k

)xk(1 − x)n+1, one get a modified version of Mn,

which belongs to Cheney and Sharma [4]. The asymptotic convergence propertiesof Bernstein type operators for bounded functions have been studied in [11]. In

1

this paper we study the asymptotic convergence property of the operators Mn

for bounded functions on [0, 1]. By means of some probabilistic methods andresults and accurate estimate technique to the basis functions mn,k, we establishan asymptotic convergence theorem of this type approximation. Our investigationsubsumes the approximation of operators Mn for functions of bounded variationas a special case.

The following three quantities were first introduced in [11]. For their basicproperties, one refers to [11].

Ωx−(f, δ1) = supt∈[x−δ1,x]

|f(t)− f(x)|, Ωx+(f, δ2) = supt∈[x,x+δ2]

|f(t)− f(x)|,

Ω(x, f, λ) = supt∈[x−x/λ,x+(1−x)/λ]

|f(t)− f(x)|,

where f ∈ I, x ∈ [0, 1] is fixed, 0 ≤ δ1 ≤ x, 0 ≤ δ2 ≤ 1− x, and λ ≥ 1.The following example shows that in the case of approximation of functions of

bounded variation, the above quantities may give better asymptotic estimate thanusing the total variation of function of bounded variation.

Example 1. Consider the function f0(x) =

x2sin(π/x), x ∈ (0, 1]

0, x = 0.

f0(x) is bounded variation on [0, 1] by the boundedness of f′0(x). On the interval

[0, n−1] taking points:1n

>1

n + 1/2>

1n + 1

>1

(n + 1) + 1/2> ... >

1n + n

>1

(n + n) + 1/2> 0.

It is easy to observe that1/n∨

0

(f0) ≥(

1n + 1/2

)2

+(

1(n + 1) + 1/2

)2

+ ... +(

1(n + n) + 1/2

)2

> (n + 1)(

12n + 1/2

)2

> (4n)−1,

and obviously, Ω0+(f0, n−1) ≤ n−2.

The main result of this paper is as follows:

Theorem 1. Let f be bounded on [0, 1], f(x+) and f(x−) exsit at a fixedpoint x ∈ (0, 1) and r = x/(1− x). Then for n ≥ 2 we have∣∣∣∣Mn(f, x)− f(x+) + f(x−)

2− Af,n,x

3√

2πxn

∣∣∣∣ ≤4

nx(1− x)

n∑

k=1

Ω(x, gx,√

k) + O(n−1),

(2)

2

X-M ZENG528

where gx(t) is defined by

gx(t) =

f(t)− f(x+), x < t ≤ 1;0, t = x;

f(t)− f(x−), 0 ≤ t < x.(3)

and

Af,n,x =

3(1− x)f(x) + (2x− 1)f(x+) + (x− 2)f(x−), nr = [nr](2x− 1 + 3(1− x)(nr − [nr]))(f(x+)− f(x−)), nr 6= [nr]

, (4)

in (4), [nr] denotes the greatest integer not exceeding nr.

From Theorem 1 we get an interesting asymptotic formula as follows.

Corollary 1. Under the conditions of Theorem 1, if Ω(x, gx, λ) = o(λ−1), thenwe have the following asymptotic formula

Mn(f, x) =f(x+) + f(x−)

2+

Af,n,x

3√

2πxn+ o(n−1/2). (5)

We point out that approximation of functions of bounded variation is the spa-cial case of Theorem 1. From Theorem 1 we get immediately

Corollary 2. Let f be a function of bounded variation on [0, 1], and let∨b

a(f)denote the total variation of f on [a, b], x ∈ (0, 1) and r = x/(1 − x). Then forn ≥ 2 we have∣∣∣∣Mn(f, x)− f(x+) + f(x−)

2− Af,n,x

3√

2πxn

∣∣∣∣ ≤4

nx(1− x)

n∑

k=1

Ω(x, gx,√

k) + O(n−1)

≤ 4nx(1− x)

n∑

k=1

x+x/√

k∨

x−x/√

k

(gx) + O(n−1), (6)

where Af,n,x and gx(t) are defined as in Theorem 1.

2 A SET OF LEMMAS

Each of the following four lemmas will be required in the proof of Theorem 1.

3

RATE OF APPROXIMATION... 529

Lemma 1. For n ≥ 2, x ∈ [0, 1] there holds

Mn((t− x)2, x) ≤ 2x(1− x)n

. (7)

Proof . By [2, Lemma 2.1] and simple calculation, for n ≥ 2 there holds

∆ =∞∑

k=0

(k

n + k− x

)2 (n + k)!k!n!

xk(1− x)n+1 ≤(

1 +2x

n− 1

)x(1− x)2

n + 1.

ThusMn((t− x)2, x) ≤ ∆

1− x≤ 2x(1− x)

n.

Using Bojanic-Cheng-Khan’s method [3, 5, 9] and Lemma 1 we obtain

Lemma 2. For gx(t) defined in (3) we have

|Mn(gx, x)| ≤ 4nx(1− x)

n∑

k=1

Ω(x, gx,√

k). (8)

Because the method of proof of Lemma 2 is well known (cf. [3, 5, 7, 9, 12]), wehere omit the details of the proof.

Lemma 3. Let ξk∞k=1 be a sequence of independent and identically dis-tributed random variables with the expectation E(ξ1) = a1, the variance E(ξ1 −a1)2 = σ2 > 0, E(ξ1 − a1)4 < ∞, and let Fn stand for the distribution function ofn∑

k=1(ξk − a1)/σ

√n. If Fn is a lattice distribution and F ∗

n is a polygonal approxima-

tion of Fn (see the following Definition 1), then the following equation holds for allt ∈ (−∞,+∞)

F ∗n(t)− 1√

2π

∫ t

−∞e−u2/2du− E(ξ1 − a1)3

6σ3√

n(1− t2)

1√2π

e−t2/2 = O(n−1). (9)

The proof of Lemma 3 can be found in [6, p. 540-542].

Definition 1 ([6, p. 540, Definition]). Let F be concentrated on the lattice ofpoints b±nh, but on no sublattice (that is, h is the span of F). A polygonal approx-imation F ∗ to F is a distribution function with a polygonal graph with vertices atthe midpoints b± (n + 1/2)h lying on the graph of F. Thus

F ∗(t) = F (t) if t = b± (n + 1/2)h; (10)

4

X-M ZENG530

F ∗(t) = 1/2[F (t) + F (t−)] if t = b± nh. (11)

The following Lamma 4 is an accurate estimate technique to the basis func-tions of the operators Mn, which is a key auxiliary result in the proof of Theorem 1.

Lemma 4. For x ∈ (0, 1), r = x/(1− x), we have

mn,[nr](x) =1− x√2πxn

+ O((nx)−3/2

), (12)

andmn,[nr]+1(x) =

1− x√2πxn

+ O((nx)−3/2

). (13)

Proof . We first show that

(nr

[nr]

)[nr]+1/2 (n + [nr]n + nr

)n+[nr]−1/2

= 1 + O([nr]−1). (14)

Set W1(n, r) =(

nr[nr]

)[nr]+1/2 (n+[nr]n+nr

)n+[nr]−1/2, and write nr = [nr] + ε

(0 ≤ ε < 1), then

W1(n, r) =(

1 +ε

[nr]

)[nr]+1/2 (1 +

ε

n + [nr]

)−(n+[nr]−1/2)

.

Thus

logW1(n, r) = ([nr] + 1/2)log(

1 +ε

[nr]

)− ([n + [nr]− 1/2)log

(1 +

ε

n + [nr]

)

= ([nr]+1/2)

(ε

[nr]+ O

(ε

[nr]

)2)−([n+[nr]−1/2)

(ε

n + [nr]+ O

(ε

n + [nr]

)2)

= O([nr]−1),

which implies thatW1(n, r) = 1 + O([nr]−1).

Using Stirling’s formula:

n! = (n/e)n√

2πneθn/12n, 0 < θn < 1,

5

and by direct calculations we find that√

2πnx

1− xmn,[nr](x) =

√2πnx

1− x

(n + [nr]− 1)![nr]!(n− 1)!

x[nr](1− x)n

=√

2πnx

1− x

n

n + [nr](n + [nr])!

[nr]!n!x[nr](1− x)n

=(n + [nr])n+[nr]−1/2

[nr][nr]+1/2nn−1x[nr]+1/2(1− x)n−1ec(x,n),

=(

nr

[nr]

)[nr]+1/2 (n + [nr]n + nr

)n+[nr]−1/2

ec(x,n),

where− 1

12n− 1

12[nr]≤ c(x, n) ≤ 1

12n.

Thus, it follows from Eq. (12) that√

2πnx

1− xmn,[nr](x) = 1 + O([nr]−1),

which derives the estimation (14). Furthermore, note that

mn,[nr]+1(x)−mn,[nr](x) = mn,[nr](x)(

n + [nr][nr] + 1

x− 1)

,

and since nr = [nr] + ε (0 ≤ ε < 1), then

n + [nr][nr] + 1

x− 1 =(1− x)([nr] + ε) + [nr]x− [nr]− 1

[nr] + 1=

ε− εx− 1[nr] + 1

,

that is

mn,[nr]+1(x) = mn,[nr](x)(

ε− εx− 1[nr] + 1

+ 1)

.

Thus, we get (13) directly from (12). The proof of Lemma 4 is completed.

3 PROOF OF MAIN RESULT

Proof of Theorem 1. For any f ∈ IB, if f(x+) and f(x−) exist at x, by Bojanic-Cheng decomposition it follows that

Mn(f, x)− f(x+) + f(x−)2

=f(x+)− f(x−)

2Mn(sgnx, x)

6

X-M ZENG532

+2f(x)− f(x+)− f(x−)

2Mn(δx, x) + Mn(gx, x). (15)

where gx(t) is defined in (3), sgnx(t) =

1, t > x0, t = x−1, t < x

, and δx(t) =

1, t = x0, t 6= x

.

We need to estimate every term in the right side of (15). The term Mn(gx, x)has been estimated in Lemma 2. Let r = x/(1− x). Direct calculation gives

Mn(δx, x) =

mn,[nr](x), nr = [nr]

0, nr 6= [nr]. (16)

Below we estimate Mn(sgnx, x).Let ξi∞i=1 be a sequence of independent random variables with the same ge-

ometric distribution P (ξi = k) = xk(1 − x), k = 0, 1, 2, · · ·, and x ∈ (0, 1) is aparameter. Direct computations give

Eξ1 =x

1− x, E(ξ1 − Eξ1)2 =

x

(1− x)2,

E(ξ1 − Eξ1)3 =x2 + x

(1− x)3, E(ξ1 − Eξ1)4 =

x3 + 7x2 + x

(1− x)4< ∞.

Let ηn =n∑

i=1ξi and Fn stand for the distribution function of

n∑i=1

(ξi − Eξ1)/σ√

n.

Then the probability distribution of the random variable ηn is

P (ηn = k) =

(n + k − 1

k

)xk(1− x)n = mn,k(x).

Thus

Mn(sgnx, x) = −∑

k<nx

mn,k(x) +∑

k>nx

mn,k(x)

= 1−∑

k<nx

mn,k(x)−∑

k≤nx

mn,k(x)

= 1− P (ηn < nx)− P (ηn ≤ nx)

= 1− Fn(0−)− Fn(0) (17)

On the other hand, for F ∗n(t), the polygonal approximation of Fn(t), from (10),

(11), we obtain

F ∗n(0) =

1√2π

∫ 0

−∞e−u2/2du +

E(ξ1 − a1)3

6σ3√

n

1√2π

+ O(n−1).

7

=12

+1 + x

6√

2πxn+ O(n−1). (18)

We need to estimate the terms F ∗n(0) − Fn(0) and F ∗

n(0) − Fn(0−), write r =x/(1− x). If nr just is a natural number, i.e nr = [nr], then 0 is a lattice point ofFn. From (11) we get

2F ∗n(0)− Fn(0−)− Fn(0) = 0. (19)

If nr 6= [nr], thenFn(0) = Fn(0−) =

∑

k≤[nr]

mn,k(x),

which implies that Fn(t) =∑

k≤[nr]mn,k(x) on the interval

[−nr−[nr]

σ√

n, 1+[nr]−nr

σ√

n

),

since Fn(t) is a step function.We need to compute F ∗

n(0). If 0 < nr − [nr] ≤ 1/2, from (10) and (11) it isknown that

F ∗n

(−nr − [nr]

σ√

n

)=

12

∑

k≤[nr]−1

mnk(x) +∑

k≤[nr]

mnk(x)

,

andF ∗

n

([nr]− nr + 1/2

σ√

n

)=

∑

k≤[nr]

mnk(x).

Since F ∗n(t) is a polygonal approximation of Fn(t), by a simple calculation we get

F ∗n(0) =

∑

k≤[nr]

mnk(x) + (nr − [nr]− 1/2)mn,[nr](x).

Thus for 0 < nr − [nr] < 1/2

F ∗n(0)− Fn(0) = F ∗

n(0)− Fn(0−) = (nr − [nr]− 1/2)mn,[nr](x). (20)

Similarly, for 1/2 < nr − [nr] < 1

F ∗n(0)− Fn(0) = F ∗

n(0)− Fn(0−) = (nr − [nr]− 1/2)mn,[nr]+1(x). (21)

Collecting the above estimates (17)–(21) and by means of Lemma 4 and somesimple computations we obtain

Mn(sgnx, x) =

− 1 + x

3√

2πxn+ O(n−1), nr = [nr]

(6nr − 6[nr]− 2)(1− x)− 23√

2πxn+ O(n−1), nr 6= [nr].

(22)

8

X-M ZENG534

Theorem 1 now follows from (15), (16), (22) and Lemma 2 with the simple calcu-lation.

Acknowledgement

The present investigation was supported by NSFC under Grant 10571145.

References

[1] J. A. H. Alkemade, The second moments for the Meyer-Konig and Zeller operators,J. Approx. Theory 40 (1984), 261-273.

[2] M. Becker and R. J. Nessel, A global approximation theorem for the Meyer-Konigand Zeller operators, Math. Z. 160 (1978), 195-206.

[3] R. Bojanic and M. Vuilleumier, On the rate of convergence of Fourier-Legendre seriesof functions of bounded variation, J. Approx. Theory 31 (1981), 67-79.

[4] E. W. Cheney and A. Sharma, Bernstein power series, J. Canad. Math. 16 (1964),241-252.

[5] F. Cheng, On the rate of convergence of Benstein polynomials of functions of boundedvariation, J. Approx. Theory 39 (1983), 259-274.

[6] W. Feller, ”An Introduction to Probability Theory and Its Applications”, John Wiley& Sons, Inc. New York, London, Toronto, 1971.

[7] S. Guo, On the rate of convergence of integrated Meyer-Konig and Zeller operatorsfor functions of bounded variation, J. Approx. Theory 59 (1989), 183-192.

[8] V. Gupta, On a new type Meyer-Konig and Zeller operators, J. Inequal. Pure Appl.Math. 3 (2002), 1-10.

[9] M. K. Khan, On the rate of convergence of Bernstein power series for functions ofbounded variation, J. Approx. Theory 57 (1989), 90-103.

[10] W. Meyer-Konig and K. Zeller, Bernsteinsche potenzreihen, Studia Math. 19 (1960),89-104.

[11] X. M. Zeng and F. Cheng, On the rate of approximation of Bernstein type operators,J. Approx. Theory 109 (2001), 242-256.

[12] X. M. Zeng and A. Piriou, On the rate of convergence of two Bernstein-Bezier typeoperators for bounded variation functions, J. Approx. Theory 95 (1998), 369-387.

9



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538


539


540

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.10,NO.4,2008 NONLINEAR RANDOM MULTI-VALUED VARIATIONAL INCLUSION SYSTEMS INVOLVING (A,h)-ACCRETIVE MAPPINGS IN BANACH SPACES,H-Y.LAN,415 TOPOLOGICAL DYNAMIC CLASSIFICATION OF ANTITRIANGULAR MAPS, Z.LUO,X.TANG,G.ZHANG,………………………………………………………..,431 SOME RESULTS IN INTUITIONISTIC FUZZY METRIC SPACES,J.H.PARK,Y.B.PARK,R.SAADATI,……………………………………..,441 A PRECONDITIONED LINEAR SAMPLING METHOD IN INVERSE ACOUSTIC OBSTACLE SCATTERING,G.PELEKANOS,K.LEEM,V.SEVROGLOU,……….,453 ASYMPTOTIC DISTRIBUTION OF THE SAMPLE AVERAGE VALUE-AT-RISK, S.STOYANOV,S.T.RACHEV,………………………………………………………,465 A NOTE ON GENERALIZED TWISTED q-EULER NUMBERS AND POLYNOMIALS,C.S.RYOO,J.J.SEO,T.KIM,………………………………………,483 PADE SPLINE FUNCTIONS,T-X.HE,……………………………………………...,495 NULL CONTROLLABILITY OF NONLINEAR INFINITE NEUTRAL SYSTEMS WITH MULTIPLE DELAYS IN CONTROL,R.UMANA,……………………….....,509 MOMENTS OF THE SCALED BURR TYPE X DISTRIBUTION,M.ZHOU,D.YANG, Y.WANG,S.NADARAJAH,………………………………………………………….,523 ON THE RATE OF APPROXIMATION OF MEYER-KONIG AND ZELLER OPERATORS,X-M.ZENG,…………………………………………………………..,527

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