v01'ume 16 -...
TRANSCRIPT
V01'UME 16
1986
Published by :
The Vijnana Parishad of India DAY ANAND VEDIC POSTGRADUATE COLLEGE
Bu:ndelkhand University
ORAi, U.P., INDIA
Jl\l'ANABHA
EDI.TORS
H .. M .. Srivastava
University of Victoria Victoria, B. C., Canada
AND
R. C .. Singh Chan.de!
D. V. Postgraduate College Orai, U.P., India
Editorial Ad.visoFy Board
Chief Advisor : J. N. Kapur (Delhi)
R. G. Buschman (Laramie, WY) K. L. Chung (Stanford) R. S Mishra (Lucknow) R. C. Mehrotra (Jaipur) C. Prasad (Roorkee) B. E. Rhoades (Bloomington. IN) D. Roux (Milano) H. K. Srivastava (Lucknow)
L. Carlitz (Durham, NC) L. S. Kothari (Delhi) K. M. Saksena (Kanpur) S. P. Singh (St. John's) L. J. Slater (Cambridge, U.K.) K. N. Srivastava (Bhopal) A. B. Tayler (Oxford. UK.) V. K. Verma (Delhi)
The Vijn.ana Pa.rishad Of India
( Society for Applications to Mathematics )
Office : D. V. Postgraduate College, Orai - 285001, U. P., India
President :
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COUNCIL
J. N. Kapur (Delhi)
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s~cretary~ Treasurer ; (D. V. Postgraduate College, Orai) R. C. Singh Chandel (Orai)
Foreign Secretary:
C. M. Joshi (Udaipur) R. N. Pandey (Varanasi)
R. K. Saxena (Jodhpur) V. P. Saxena (Gwalior)
H. M. Srivastava (Victoria, Canada)
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S. N. Singh (Faizabad) R. P, Singh (Bhopal)
Printed at: Navin Printing Press, Gopal Ganj, Orai-285001, U.P., India
.'Jfianabha, Vol. 16, 1986
AN UNIFIED APPROACH FOR THE CONTRACTIVE
MAPPINGS
By
D. DELBOSCO
Dipartmento di Matematica, Universita di· Torino
Via Principe Amedeo 8, 10123 Torino, Italy
( Received: June 15, 1985)
ABSTRACT
In this paper we introduce a new definition of cor~tra~ti ve type
mappings called p-contraction. A self mappi~g T of 'metric ~pace (X, d)
. is called a p-contraction if there exists a map P': R3:t. ~R* w:1ere
R* = {t € R : t ;? O}, safofying some s;U.itable hypothesis, such that,
d(Tx, Ty) ~ p( d(x, y), d(x, Tx), d(y; Ty)).
for all pair x, y € X.
The definition of p-contraction~ which . contains some of previous
generalization of contraction mappings, implies a!! conclusions of
Banach contraction principle.
1. INTRODUCTION
We recall Banach's contraction principle and its basic conseq ue
'nces. A selfrnapping .of a metric spac (X, d) is c~lled a contraction if
there exists h, 0 ~ h < 1. such that for eaeh x, y EX
d(Tx, Ty) ~ h d(x, y).
Any contraction map T of a complete metric space (X, d) has the
following properties :
2 ]
(A) T has a unique fixed point z* EX.
(B) Convergence of iterates: T11 x ___,.. z*, as n ___,. oo.
(C) Uniform convergence : there exists a neighbourhood U of z* such
that Tn (U) = {T"x : x E U} _,.. {z*}, which means that for any
neighbourhood V of z* we can. find an i.nte:ger no such that
T" (U) C i: for all n. greater than: n0 •
(D) Tis continuous.
On the other hand, Meyers· f6J proved that every self mapping T of
complete metric space (X, d) having properties A, B~ C and Il is a
contraction map under suitable topologically equivalent metric.
T!he purpose of th rs paper is to inve~tigate p-contraction mapp
ings: (se·e: precise definition below), mappings having some iterate
which is ap-contraction and common fix.ed points of two mappings.
Applications of previous theorem of Meyers are also given.
2. Results on p-Contraction Mappings.
Let us introduce the set P of continuous p : R3* _,.. R*, where
R* = {t E R : t ;> O} satisf,Ying the following properties :
(i) p( I, I, l) = h < I ;
(ii) Let u > 0. If u ~ p(u, v, v) or u ~p(v. u, v) or u ~ p(v, v, u)
then u ~.hv.
Definition l. A self mapping of a metric space (X, d) is called a
p-contraction if there exists a map p E P, such that, for all x, y E X
d1.Tx, Ty) ~ p( d(x, y), d(x, Tx), d()', 7)))
( 3
Now we indicate 3 examples of contractiv~ type mappings, \Vhich
can be regarded as a p-cor,traction, under suitable function p.
Indeed, If we put p(u, v, z) = au + bv + cz ta+ b + c < 1),
t,hen, we get t~ ~oJl.o,,w,µi,g well know,n :
DefinitioQ. 2. (S. Reich [7], I. Rus f9] ).
There ~xist non negative numbers a, b, c,. a + b + c < 1, such that
for all pair x, y e X
d(Tx, Ty) :::;;; a d(x, y} + b 4(x, Tx) + c d(y, Ty).
Moreover if we put p(u, v, z) = h max (u, v. z) (O ~ h < I),
we have
Definition 3.
There exists a number h, 0 <;;::;; h < I, such that fo,r aJI pair x, y ~ ¥
dtTx, Ty) :::;;; It max {d(x, y), d(x, T1;), d(y, Ty) },
At last if we choose p(u, v, z) = (aur + bv1• + cz•')lfr, r > 0,
(a + b + c < 1) then we find the follo)ViI\~ :
Definition 4. (Delbo,co) [2].
There exist non negative numbers a, b, c aµd thepe exists r > 0, such
that for all pair x, y e X.
d(Tx, Ty)" :::;;; a d(x, y)r + b d(x, Tx)" + c d(y, Ty)'.
Tf1eore~ 2.1 Let T be a p-contraction mapping of a complete metric
space (X, d). Then T has properties (A) and (B).
Proof. Fix z E x. Suppose z # Tz and set Zn = T Zn_J = Tri z. if we
put x = Zn and y = z,,_1 , then inequality (I) becomes :
4 ]
(2.1) d(Zn+V Zn) ~ p( d(Zn, Zn_J), d(z,,, Zn+J), d(z,q, Zn)).
Hypothesis (ii) on function p implies
(2.2) d(in+1 , Zn) ,;;;;; h d(zn , Zn_J).
On the other hand, hypothesis (i) on p, proves that (zn) N is a nE
Cauchy sequence and completeness of X assures that Zn -? z* E X.
To prove that z* is fixed point of T, we take x = z* and y = z,,
in inequality (l) and we obtain
(2.3) d(Tz*, Tzn) ~ p( d(z*, Zn). d(z"", Tz*). d(z,., Tzn) ).
Smee p is continuous in each variable, letting n -7 oo, we get
(2.4) d(Tz*, z,;,) ~ p( 0, d(z*, Tz *), 0).
d(z*, Tz*) must be zero; hence z* = Tz*
Finally we prove that mapping T has a unique fixeJ point.
Indeed, let us suppose by contradiction, that Tx = x, Ty = y and
that x ;:ft y. Erom inequality (I) we get :
d(x, y) = d(Tx, Ty) ~ p( d(x. y), 0, 0).
It follows that d(x, y) = 0 i.e. x = y.
The proof is, thus, complete
Theorem 2.2 Let The a continuous p-contraction selfmarping of a
complete metric space (X, d). Then T has property (CJ.
Proof. For any r > 0, we define Vr ,-= {x E X : d(x, Tx) < r}. We
~how 1hat v,. is an open subset of X, since it contains a. neighbourhoed
for any point. lndeed, if y E V,, d(y, Ty) = a < r, then continuity of
T implies that there exists r such that
[ 5·
(2.5) d(z, y) < r' implies d(z, Tz) < a i r < r.
V, is a neighbourhood.of unique fixed point z*. since z* ~ V, and
V, is an open subset of mearic space X.
Now we prove property (C) about T, i e.
(2.6) {T" V,} ~ {z,J, as n becomes infinite.
Let us consider z = Ty E T(V,) = {Tx: x E V,}. From inequality
( 1) we can write that
(2.7) d(Ty. Tiy) <C,,p( d(y, Ty), d(y, Ty), d(Ty, T2y))
we use hypothe~is (ii) on function to deduce that
(2.8) d(Ty, T2y) <C,, h d(y, Ty).
Therefore it follows that
(2.9) T(V,) C Vi,.
and, by induction,
(2. IO) P'(V.) C Vh"r .
1 hus, inclusion (2.10) proves (2.6) (uniform convergence of
iterates - or - property l C) ).
Theorem 2.3 Let T be a continuous p-contraction self mapping of a
complete metric space (X, d). Then Tis a contraction map for suita
ble topologically equivalent metric.
Proof. It suffices to remark that T has properties (A), (B), (C) from
previous Theorem 2.1 and 2.2 and T has also property (D) for hypo
thesis. Now applying Meyers's result [6J, one proves this theorem.
6 ]
3. MAPPINGS HAVING SOME ITERATE p-CONTRACTION.
Sometimes a self mapping Tis not a p-contraction, but it can .have
some iterate which is a p-contraction mapping, under suitable function
p E P.
From this point of view, we state and prove some results.
Theorem 3.1 Let T be a self mapping of complete metric space (X, d)
satisfying the following condition
(1 *) there exists an integer n such that for all pair x, y EX
d(Tnx, T"y) ~ p( d(x, y), d(x, P•x), d(y, Tny))
for some p E P. Then T has properties (A) and (R).
Proof. If we put S = Tn,. then Theorem 2.1 shows that S has a uni
que fixed point that we call z*.
1\ow v.e prove that z* is also a fixed point of T. Indeed let us
consider these equalities
(3.1) T11(Tz*) = T(T11z*)
Tn\Tz*) = Tz*
S(Tz*) = Tz*
Uniqueness cf fixed point of Sand (3. I) imply
(3.2) Tz* = z*
Now we prove that z* is the unique fixed point of T. Suppose
x -::F y, Tx = x and Ty = y and apply (I*)
(3.3) d(x, y) = d(T11x, Tny) ~ p( d(x, y), 0, 0)
\
d(x, y) .;;;;; p(O, 0, d(x, y))
d(x, y) = 0
Hence x = y and uniqueness of fixed point is proved.
[ 7
To prove property (B) it suffices to remark that we can deduce
the convergence of sequence Pz from the convergence of sequence
Sizo , putting z0 = z, Tz, T2z ... , pi-Iz.
Theorem 3.2 Let Ta continuous self mapping of a complete metric
space (X. d) satisfying (l*) Then Thas property (C).
Proof. For any r > 0 we define v, = {x Ex: d(x, rnx) < r}. where
n is the integer of condition (I*).
We ~hall prove that V,. is an open subset of X containing the fixed
point z*, hence a neighbourhooJ of z*.
Indeed if y E V, , d(y, Tr•y) = a < r, then continuity of T implies
that there exists r such that
(3.4) d(z, y) < r' implies d(z, Tz) < a i r < r.
To prove property ( C), we consider sequence {T''" (Vr )} (k E N) and
deduce the following convergence result
(3.:) {pm (V,.)1,EN ~ {zd, ask becomes infinite.
Suppose z ET" (V,.), then we can write z = T"y, with y EV,.
From condition (I*) it follows
(3.6) d(z, T"z) = d(Tny, T2ny) ,;;;;; p( d(y, T"y), d(y, P•y), d(Tr•y, T2 11y))
Now hypothesis ,i) and (ii) on function p assure that
(3.7) d(z, T"z) ~ h d(y, T 11y) ~hr.
8 J
Therefore we have
(3.8) Tn (Vr) C Viir
and, by induction,
(3.9) pn (V,) C Vh kr.
Thus uniform convergence (3.6) is proved as consequence of inclu
sion (3.9), letting n __,. oo.
4. Common Fixed Points Of Two Mappings.
The purpose o~ this section is to investigate common fixed points
of two mappings f and g satisfying the following condition
(l**) There exists a functionp E P, such that, for all pair x, y EX
d(fx, gy) ~ p( d(x, y), d(x, fx) d(y, gy))
Theorem 4.1 Let f and g be two self mapping of a comp!ete metric
space (X, d) satisfying ( 1 * *). Then f and g have a unique common
fixed point.
Proof. Fix x E X and define the following sequence
( 4.1) X1 = fx, X2 = g X1 , ... , X2n-l = Jx2n-2, X2n = g X2n-l , ...
If we put y = x1 in(!**) we have
(4.2) d(xb x2) = d( f•, gx1) <.,p( d(x, x1), d(x:, x1)· d(x1, Xz))
d(xv x2) ~ h d(x, x1). ·
Similarly we get
(4.3) d(x2, xa) ;;;:; h d(xi. x2) ~ 1z2 d(x, x1)
and, by induction,
/'\
[ 9
(4 4) d(xn, Xn+i) ~ hn d(x, X1).
Inequality (4.4) proves that sequence (4.1) is Cauchy.
Completeness of X assures that limit z* of sequence (4.1) belongs
to X.
To prove that z* is a fixed point off an:i g, we set x = z* and
y = x2n_1 in condition (1 **)and deduce
(4.5) d(fz*' X2n) .::;;; p( d(z*' X2n-1), d(z*, fz*), d(x2n-1, X2n))
Leiting n -+ =, ( 4.5) shwos that z * is a fixed point of g. On the
other hand, z* is the unique common fixed point of two mappings f and g. Tlus has simple and standard proof.
Finally we make use of this Theorem 4.1, to study the class of
selfmapp;ng of a metric space (X, d) which satisfy the following conl
ition
(I***) There exist two integers n, m, such that for all pair x, y EX
d(Tnx, T"'y) ~ p( d(x, y), d(x, Tnx), d(y, Tmy) )
for some function p E P.
Theorem 4.2 Let T be a self mapping of a complete metric space
(X, d) satisfying (l***). Then T has prop<:rties (A) and (B).
Proof. The proof of property (A) follows directly from Theorem 4 l
taking f = T 11 and g = Tm. Indeed there exists a unique fixed' point
such that f'"z* = z* = T"'z*.
But we have
T 11(Tz*) = T(Tnz*) = Tz* = T(Tmz*) = T""(Tz*)
hence, uniqueness of common fixed point implies
10 ]
T(z*) = z*.
To prove property (B), let us suppose that n > m. Theorem (4.1)
implies that fer any fixed z E X the sequence
(4 6) z, Tnz, Tn+mz. T2n+mz, T2n+'d.mz, ..
converges to fixed point z*.
Putting z = x, Tx, T2x, ... , Tn-lx, in (4 6) we ottain n sequences
cor.taini11g all elements of sequence (Tkx)kEN·
The property (B) is thus proved.
5. Comparison with some recent similar results.
Recently, wme authors PJ ts] have established fixed point theor
ems involving !unction q: ](5* --+ R:f. satisfying :
(a) q is continuous;
(b) q is increasing, or nondecreasing; in each variable;
(c) q (t, t, t, t, 1) < t for each t > 0.
Now we pre>e1;t the corresponding contractive defrnition: ,. A self mapping T of a complete metric space is "q" contraction if there
exists a function c; such that for all pair x, y E X
d(Tx, Ty) ~ p( d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), dly, Tx)).
Our purpose is to show that f ,.rnctio.1 p is not the restriction of
some function q to its first three component.
Let us consider the following example :
p(u, v, z) =h max {u exp l(u-1) (v-<) (z-u)]2, v, z}, 0 <; h < I.
l 11
Since function p, as defined above, is not increasing or not decre
asing in its first coordinate variable, p can not satisfy (b), and hence,
can not be the restriction of some function q to its first three variable.
REFERENCES
[l] R. M. T. Bianchini, Su un problema di S. Reich riguardante la
teoria dei punti fissi, Boll. Un. Mat. Ital. 5(1972), 103-108.
[2] D. Delbosco, Un estensione di un teorema sul punto fisrn di S.
Reich, Rend. Sem. Mat. Univ. Polit. Torino, 35 (1976-77),
233-238.
[3] S. A. Hussain-V. M. Sehgal; A fixed point theorem with a func
tional inequality, Pub!. Inst. Math. Beograd. 21 (1977),
89-91.
[4] R. Kannan, Some results on fixed point. II, Amer. Math. Monthly.,
76, 4'.)5-403.
(5] S. Kasahara, On some recent results on fixed point, Math. Semi
nar Nutes, Kohe U., 6 (197~), 373-382.
[6] P. R. Meyers. A converse to Banch's contraction theorem, J. Res.
Nat. Bur. Standards, 71 B (1967), 73-76.
(7] S. Reich, Some remarl<:s concerning contraction mappings, Canad.
Ma1h. Bull. 14 (1971), 121-124.
[8] B. E. Rhoades, A comparison of various definitions of contract
ive mappings, Trans. Amer. Math. Soc 226(1977), 257-290.
[9] I A. Rus, Some fixed point theorems in metric spaces, Rend.
Sem. Mat. Univ. Tri·ste. 3 (1971), 169-172.
Ji'Hinabha, Vol. 16, 1986
FIXED POINTS OF SET-VALUED MAPPINGS OF
CONVEX METRIC SPACES*
By
M. D. GUAY
Department of Mathematics, University of Southern Maine
Portland, Maine 04103, U. S. A,
and
K. L. SINGH
Department of Mathematics Fayetteville State University
Fayetteville, North Carolina 28301
t Received: August 3, 1985)
ABSTRACT
The principal aim of the paper is to prove the e'Cistence of fixed
points for set-valued mappings of convex metric spaces. Such spaces
include Banach spaces and starshaped subsets of Banach spaces. and
our results ge1~eralize those of Assad, Kirk, Markin, Nadler, and
others.
INTRODUCTION
In 1970 Takahashi [IO] introduced the definition of convexity in
metric spc:.ce given in Definiti0n 0. l below, and generalized some
* This research supported in part by a grant from NSERG (Canada).
1980 Mathematics Subject Classification
Primary : 47Hl0, 54H25 Secondary : 46825
Key Words and Phrases
Fixed point, set-valued mapping, convex metric space
14 ]
important fixed-point theorems previously provej for Banach spaces.
Subsequently. Machado [5], Tallman [11], Naimpa1ly and Singh [8],
and Naimpally, Singh and Whitfield [9], among others have obtained
additional results in this setting. This paper is a continuation of these
investigations for set-valued mappings. Our results are motivated by
the results Samanta [ 12] and extends his results to convex metric spaces.
Definition O. 1 Let X be a metric space and I be the closed unit
interval. A mapping W: X x X x I--+X is said to be a convex structure
on X if for all x, y E X, ,\ E I,
d(u, W(x, y, ,\)) < ,\ d(u, x) +(l - A) d(u, y) for all u EX. The space
X together with a convex structure is called a convex metric space.
Clearly any convex subset cf a Banach space is a convex metric space
with W(x, y, ,\) = ,\x +(I- t\)y.
Definition O. 2 Let X be a convex metric space. A nonempty subset
K of X is convex if W(x, y, ,\) E K whenever x, y E K and ,\ s /.
Takahashi has shown that open and closed balls are convex, and that
the arbitrary intersection of convex sets is convex.
Definition 0. 3 A convex metric space X will be said to have Prop
erty (C) if every tounded decreasing net of nonempty closed convex
subsets of X has a nonempty intersection.
By Srriulian's theorem [2, p. 433], every weakly compact convex subset
of a Banach space has Property ( C).
1 Fixed points for set-valued mappings.
Fixed points of set-valued mappings on Banach spaces haw been
studied extensively. In this section we pre5ent results which generalize
results of Assad and Kirk [I], Kir~ (3], Mor:dn L6], Nadler [7], and
others.
j I !
(~
,_J '< fr~
~ 15
A few additional definitions will be needed.
Definition t. 1 Let X be a convex metric space and C be a nonem
pty, closed, convex, bounded set in X. Following Kirk [3], for x" X
we set
rx(C) = sup d(x, y),
J'EC
r (C) = inf {rx (C): XE C}
We then define D = {x E C : rro ( C) = r ( C) } to the center of C.
Definition t. 2 If A, B" 2x, then the Hausdorff distance D (A, B)
is defined as D(A, B) = inf {o: : A C N(B, e) and B C N(A, o:), where
N(C. E) = {x EX: g y" C > d(x, y) < E}.
We denote the diameter of a subset E of X by
<l(E) = jUp {d(x. y): x, y E EJ.
Definition 1. 3 A point x E E is a diametral point of E provided
sup {d(x, y) : y E £} = 13(£), the diameter of E.
Definition 1. 4 A convex metric space is siad to have norm::tl structure
if for each closed bounded convex subset E of X which contains at
least two points, there exists x " E which is not a diametral point of E.
It is clear that any compact convex metric space, and every boun
ded, closed convex subset of a uniformly convex Banach space will
have normal structure.
Theorem 1.5 Let X be a complete convex metric space with Property
(C) and Kbe a closed, convex, bounded subset of X with normal stru-
cture.
16 ] .
If T : K ___,,. 2K is a mapping such that
(i) T(x) n K ==/=- ¢ for x E K,
and
(ii) for every closed convex subset of L of K satisfying
T(:.:) n K ==/=- ¢ for all z E L
D( T(x) n L, T(y) n L):;;;;; d(x, y) whenever x, y EL. x ==/=- y,
then T has a fixed point.
Proof. Let G be the family of all nonempty, closed convex subsets
M of K such that T(x) n M ==/=- ¢ for all x E M. Then G is nonempty
since KE G. Partially order G by ir.c!usion. Let t = {F;}. '" be a dec/E Ll
reasing chain in G. From Property (C) if follows that . n/\ F1 ==!=- ¢. !Eu
Let F0 = i~f, F'-o .a nonempty, closed convex subset of M. Let x E F~.
By hypothesis, T(x) . ~ F, ==/=- ¢.For any F; E ~ T(x)nF. is compact IE Ll
and {T(x) n F1 } ., • is a family of nonempty closed sursets of the com.J 9' l
pact set T(x) n F, having the finite intersection property. Cons~quentiy
. -0 . (T(x) n F1) ==/=- ¢ and therefore . n /\ ( T( 1) n F; ) ==/=- ef>, i.e. j~l . /Eu
T(x) n Fo ==!=- g,. Thus any cham in G has a greatest lower bound and
by Zorn's Lein ma there is a minirnal mem'Jer Nin G. We claim that
N is a. singleton ~et. If not, then as shown by Takahashi ( l OJ. the center
of N, denoted by P is a nonempty proper closed convex subset of N.
It is enough to show that P belongs to G. Let y E P. Then y E /V and
T(y) n N ==/=- ef>. Take z E T(y) n N Let Q = B[z, r(N)J n N (where
Br(x) = {y EX :~d(x, J).;:;;; r} ). Then Q is a nonempty closed convex
subset of N. Let x E Q. Using hypotnesis (ii) we have
r
-~
( 17
D(T (x) n N, T(y) n N) .,;;; d(x, y).
Since z E T(y) n N, there is a u E Tly) n N such that
d(u, y).,;;; D(T(x) n N, T(y) n N) .,;;; d(x, y). Thus u E_ Q. Herce
T(x) n Q =F rp for all x E Q. By the minimality of N we have N c; Q;
that is, N c B[z, r(N)]. Consequently, z E p and we have T(y) n p =F rp
for all y E P. Therefore P E G, a contradiction of the minimality of N.
Hence N consists of a single element which is a fixed point of r. The following example show that condition (ii), can not be replaced
by D(Tx, Ty) .,;;; d(x, y) for ·au x, y E K, x =F y.
Example 1. 6 Let X = R2 with the Euclidean norm and
K = { (x, 0): 0.,;;; x.,;;; l}. Then Kis a bo.mded, closed CJnV.::< s1bset
of a reflexive Banach space X. Define T: K .....,. 2x as folloiis :
7(x, 0) = {(vt (~ - x), v~ 0 - x)), (x + t, O)}, 0.,;;; x.,;;; t
. . x x . . . T(x+f:,0)={(1- vi, ,if
2-),(x,0)},0'<x< t
Then T(x) intersects K for every x in K.
lf (x, 0), (y, 0)< K such that x, y E (0, H 9r x, y E [!, IJ,. then
D( T(x, 0), T(y, 0)) .,;;; 11 (t, 0) -- (y, 0) 11 .
If x, y E [O, n then
D(T(x, 0), T(y + ~" 0) .,;;; [( v~ 0 - x) - y) 2 + ( ifk 0 - x))2 p 12
.,;;; [x2 + y2 + v2xy -:- (x + vt y) + i- )1/2
.,;;; 11 (x, 0) - (y + t, 0) II
whenever ( v2 + 2) XJ-" ~ (1 + vf:) y; /,,e., x :;;;;; f: with y # 0.
18 ]
Thus D( T(x, 0), T(y, 0)) ~ II (x, 0) - (y, 0) II for all
(x, 0), (y, 0), x ::/=-yin K. Now for x, y € [O, !],
D( T(x, 0) n K, T(y + i. 0) n K)
= 11 (x + ~· 0) - (y, 0) 11
=llx+i-yll
== 11 (x, 0) - (y + i, 0) 11
=lly+i-xll
w:tenever x > y. So Tis nonexpansive and satisfies all but condition
(ii) of Theorem I.5. Clearly T has no fixed.point.
Corollary J. 8 LetX be a reflexive Banach space and K be a boun -
ded, closed convex subset of X which has normal. structure. Let
T: K _,.. 2x be a mapping such that
(i) T(x) n K ::/=- cp for all z E K
(ii) for any closed convex subset L of K satisfying
T(z) n L * cp for all z E L
D( T(x) n L, T(y) n L) ~ d(x,y)
whenever x, y E L, x ::/=- y. Then T has a fixed point
The following result of Takahashi (Thm 3 [IO]) follows from Theo
rem I. 5.
Corollary 1. 9 Let X be a convex metric space having Property (C).
Let K be a nonempty bounded closed convex s:lbset of X with normal
structure. lf Tis a nonexpansive mapping of K into itself, then T has
a fixed point in K.
'f, i )
.~
(
[ 19
The following result of Kirk [4) follows from Corollary I. 9.
CoroUary 1. 10 Let K be a nonempty bounded, closed convex subset
of a reflexive Banach space X and suppose that K has normal structure.
If Tis a nonexpansive mapping of K into itself, then T has a fixed
point in K.
To obtain the next theorem we first establish a lemma whose proof
requires that the mapping W: X x X x I -+ X be continuous in ,\ " /.
Lemma 1. 11 Lel H and K be two closed subsets of the con vex
metric space X such that H n K # rf>. If W: X x X x I-+ Xis cont
inuous in ,\ " I and H is convex, then df!(K) = cfo iff H C K. Here
dH(K) denotes the realative boundary of H n Kin H.
Proof. Ii H C. K,then H - K = <P and dH(K) = <fa. Conversely, su
ppose that dH(K)=<fo, while H-K # <fo. Let x ".H n Kandy" H-K.
Since H is convex, Wtx, y, ,\) e H for all x, y e 11 and ,\ e /. Let
y=inf {,\: W(x, y, A) e K} and {Yn} be a sequence in I converging toy
such that W(x, y, Yn) e K Since K is closed; and Wis continuous in A,
W(x, y, y)=lim W(x, y Yn) is in K. Assume that y >0 and let e >0 be n
given. Taken" (0,y) such that y-n < -d( e . Then W(x, y, y) e Se x, y)
( W(x, y, y) n (H - K). It follows that W(x, y, y) e dH(K) and cons-
equently, that dH(K)-;:/= <fo, a contradictiou. The result follows.
Theorem 1. 12 Lei X be a convex metric space having Property (C)
such that the mapping Wis continuous in A e /. Let K be a nonempty
closed convex subset of X. Let L be a r.onempty closed bounded convex
subset of A having normal structure, and T: L --+ 1x be a m·appmg
satisfying T(x) n-K-;:/= cf; for all x e L. If M is a nonempty closed con
vex subset of K such that T(x) n M # cf; whenever x e M n L, ~nd
20]
(i) D( T(x) n M, T(y) n M) < d(x, y)
for all x, y E M n L and
(ii) T(w) n (M n L) ::/= cp, W..E dM(L), then T has a fixed point.
Proof. Let G be the family of all nonempty closed convex subsets M
of K such that Mn L ::/= cfo and T(x)n M ::f= cfo whenever x E M n L. ·
Then G is nonempty since KE G. Partially order G be set inclusion.
Let t = { F;} . 11 be a decreasing chain i"n G. Since each F1 is closed IE Ll
and convex, it follows from property (C) that Fo = n F; is a none-iE 6
mpty closed convex subset of M. Let x E F0 . Then, by hypothesis,
Ttx) n F; ::/= cp for all i E 6. Mon;over' for a.nY Fi E t ' T(x) n F; js
compact and {T(x) n Fj}.-....... is a family of nonempty closed. sub ets ;,z . of the compact set T(x) n Fi having the finite intersection property.
Therefore, n (T(x)n F1) ::/= cp and consequently n (T(x)n F;) ::/= </;; j":;pf fEt0, .
that is, T(x) n Fo ::/= cfo. Thus every chain in G has a greatest lower
bound and by Zorn's Lemma there is a minimal element . Nin G. If
dN(L) = cfo, then by lemma 11. I, N c; L. In this case Theorem 1.5 is
applicable and T has a fixed point. Thus. we a~sume that dN(L) ::/= cp
atn,d show that N n Lis a singleton set. Let 'f'i(N n L) > 0. Since L
has normal structure, there is a point a in L n N such that
0 < a = 0 sup {d(a, b)} < 'f'i(N n L) b E NnL
Consider the ~et H = {x E N: N n L c; B [x, a]},
(I)
a closed convex subset of N. Since a E H, His nonempty and, by (I)
H ::/= N. We show that HE G. Let ZE H n L and y E dN(L). Then
y E N n L and hence d(z, y) ,;;;;; a. Now; sinci NE G, from hypothesis .
(i) we have for y, z E N n L:
~·
:A
[ 21
D( T(y) n N, T(z) n N) ~ d(y, z). (2)
Consequently, D( T(y) n N, T(z) n N) ~ a.
Since y E dN(L), by hypothesis (ii),. T(y) n (N n £)=I= rP· Take
u E T(y) n (N n· £). Then, using (2) we obtain for v E T(z) n N.
d(u, v) ~ D( T(z) n N, T(y) n N) ~ d(z, y). Let W = B[v, a] n N.
Then Wis a closed convex subset of K with W n L =I= rfo (since u e
w n L). lf w E w n L c N n L, then d(z, w) ~a.
Further, D( T(w) n N, T(:) n N) ~ d(z, w) ~a. Since v E Ttz) n N,
there exists an elements in T(w)n N such that d(v, s) ~PC T(w)n N,
T(z) n N) ~ d(w, z) ~- oc. This shows that s e W, or. that
T(1-1) n. w_ =t q, for all w E w n L. Therefore w E G. By the
minimality of N we have N = W. Hence N n L C NB [v, a] and
v E H. We now have T(z) n H =I= rP for all z E H n L and it follows
that He G. But His a prcper subset of N, a contradiction. Hence
N n L is a singleton set. Let N n L = {x}. Since by assumption,
<P =I= dN(L) c N n N, dN(L) = {x}. Now by hypothesis (ii), x E T(x).
This completes the proJf.
Remark 1.13· The example re' ow shows that condition (ii) of Theorem
I 12 can not be replaced by the following condition: (ii)' If x e dK(L)
thenT(x) n L =I= <fa.
Example 1. 14 Let X = R2 with the usual metric and K = { (x, 0):
0 ~ x ~ oo} and· L = { (x, 0): 0 ~ x ~ l}. Define T: L ~ 2x .by
T(x, 0) = {(l + x, 0), (0, 1 - x) }, (x, 0) e L.
1hen K is a closed and convex subset of a reflexive Banach space and ,
L is a closed; bounded convex subset of K. The mapping T satisfies
the condition T(x)n K =I= cf> for all x e L, co:1dition (i) of Theorem 1.12,
and condition (ii)·. By takinJ .\.:f = [!, oo) we sQe that hypothesis (ii)
of Theorem 1 12 is not safisfied. Clearly T do.!3 not have a fixed
point. ,··~J·
22 ]
REFERENCES
[I] N. A. Assad and W.A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific f. Math. 43 (1972). 553-562.
[2] N. Dunford and J. T. Schwartz, Line.ar operators. (Part I: General Theory). Wiley Jnterscience, New York (1964).
[3] W. A. Kirk, A fixed point theorem for. mapings which do not . increase distances, Amer. Math. Monthly 72 (1965), 1004- 1006
[4) W. A. Kirk, Fixed point theorems for nonlinear nonexpansive and generalized contraction mappings, Pacific J. Math. 38 (1971),· 89-94.
(5] . H. V. Machado, A characterization of conve< subsets of normal spaces, Kodai Math. Sem. Rep. 25 (1973), 307-320.
[6] J. T. Markin, A fixed point theorem for set-valued mappings, Bull. Amer. Math. Soc. 74 (1968) 639-640.
[7) S. B. Nadler, Jr., Multi-valued contraction mapping;, Pacific J. Math. 30 (1909)475-488. ,
[8] S. A. Naimpally and K. L. Singh, Fixed points and common fixed points in convex metric space, Preprint.
[9] S. A. Naimpally K. L. Singh and J H. M. Whitfield, Fixed points in convex metric spaces, Preprint.
[IO] W. Takahashi, A convexity in metric spaces and nonexpansive mappings I, Kodai Math. Sem. lJ,.ep. 22 ( 1970). 142-149.
{11] ;L. A. Tallman, fixed points for codensing multifunctions in metric; spaces with convex structures, Koiai Math. Sem. Rep. 29 (1977), 62-70.
[ 11] S K. Samanta, Fixed point theorems for multivalued mappings over a reflexive Banach space, Bull. Ca!~·utta A1ath'. Soc 71 (1977), 45-50.
Jiianabha, Vol. 16, 1986
COMMON FIXED POINT IN METRIC SPACES
By
Duane E. Anderson
Department of Mathematical, Sciences
University of Minnesota, Duluth
Duluth, Minnesota-55812, U. S. A,,
K. L. SINGH
Department of Mathematical Sciences
Fayetteville State University
Fayetteville, North Carolina"28301, U, S. A
and
J. H. M. Whitfield*
Department of Mathematics Lakehead University
Thunder Bay, Ontario, Canada.
t Received: August 3, 1985 )
INTRODUCTION
In 1970 , Takahashi [ 8 ] introduced a notion of convexity in
metric spaces ( see Definition I . 1 ) and generalized some fixed-point
theorems in Banach spaces. Subsequently, Itoh { 4 J , Guay , Singh ,
and Whitfield { 3 J Naimpally, Singh, and whitfield { 7 J, Tallman
{ 9 J , and Machado { 6 ] , among others, have studied fjxed - point
theorems in convex metric spaces. This paper is a continuation of these
investigations.
I. PrelJminari~s and definit:on
Throughout this paper, we will assume that X is a convex metric
*This author's research wc.s supported in part by a grant from NSERC in Canada.
24 ]
space having property (C) .
Let K be a nonempty, closed, bounded convex subset of a convex
metric space X . Let A ( K ) denote the collection of all nonempty
subsets of K, and for A , B E A ( K) , let II( A , B) = sup
{d( a, b): a E A, b EB}, let r,,,(K) = ll(x, k) = sup { b (x, y): y E K },
II( K ) denotes the diameter of K. Let r( K ) - inf (3( x , K ) : x € K),
and Kc - ( x € K : r ( K ) ~ II ( x , K ) } .
It is well known [ 8 ] that if K is a closed, bounded, convex subset
of a convex metric space having property l C) then Kc is a nonempty
closed convex subset of K . Furthermore, if K has normal structure
then II ( Kc ) < o ( K ) , whenever a ( K ) > O ,
In the sequel, we use I for the interval [ 0, I ] . The following
notion wa~ introduced by Takahashi [ 8 ] •
Definition 1. 1 Let X l:e a rr:etric space. A mapping
W: X x X x I ----)> X is said to be a co11vex structure on X if for all
x, y E X, ,\ E /, d(u, W(x. y, ..\)) ~ /..d(u, x) + (I - ..\) d(u, y) for all
~ E X, X together \\ i1l1 a convex structure is called a convex mPtric
space ..
Remark 1. 1 lf Xis a Banach space. then as a metric space with
·d(x, y) = II x ~ 'y II , ihe rrapping W: X x X x I_,. X defined by
JV(x, y, /..) == ltx + (1 - ..\)y is a convex stri.;cture. More generally, if
X is a linear space with a translation invariant metric satisfying
d(/..x + (1 - /..) y, 0) ~lid(.;, 0) +(I - ..\) d(y, 0), it is a convex met
ric space. There are mariy other examples, but we consider these as
pc:.radigmatic.
Definition l. 2 Let X be a convex metric space. A nonempty subset
[ 25
K C Xis.convex if W(x, y, ,\) e K, whenever x, ye Kand A e /.
Remark 1. 2 Takahashi [8] has shown that the open spheres,
E(x, r) = {ye X: d(x, y) < r}, and clo>ed spheres, B[x, r] = {ye X:
d(x, y) ~ r} are convex Also, if {K"' : a e 6} is a family of convex
&ubsets of X, then n{K"': a e 6} is convex.
Definition 1. 3 A convex metrie space Xis saic to have property (C)
if every decreasing net of nonempty, closed c0nvex subsets of X has
nonempty intersection.
Remark 1. 3 It is a theorem of Smulian [2, p. 433) that a convex
subst.t of a Banach space has property (C) if and only if it is weakly
compact.
Definition 1. 4 Let B be a bounded set in a convex metric space X,
and let a(B) be its diameter, i. e., o(B) =sup {d(x, y): x, ye B}. A
convex subset S·of Xis said to have normal structure if every bounded
convex subset S1 of S which contains more than one point has a point
which is a nonJiametral point of Si. i. e., there is a point x of S 1 such
that sup { d(x, y): y e S1} < o(S1)·
It is easily seen that the following definition is equivalent to the
above definition. A bounded, closed conve.{ subset S of a metric space
Xis said to have normal structure if for eac.1 closed convex subset S1
of S which contains more than one point, there exists an x e S1 and
~(S), 0 < cx{S) < 1, such that sup {d(x, y): ye S1} = r,,(S1)
~ a{S1) o(S1).
2. Main Results
Theorem 2. 1 K be a nonempty, closed, bounded, convex subset
of X. Assume Kha-; nor.nal structure. Let T: K -+ A(K) be a mapping
26 J
satisfying for each closed convex subset F of K invariant under T and
with ll(F) > 0, there exists some ix(F), 0 < ix(F) < 1, such that
'8(Tx, Ty) < max { ll{x, F), a(F) '8(F)} for all x, y E F. Then T has a
fixed point.
Proof let G be the family of all nonempty, closed convex
subsets of K, each of which is mapped into itself by T. Partially order
G by set inclusion. Using property (C) and Zorn's lemma, we infer
that G has a minimal element, say M. We show that M consists of a
single point. Suppose M contains more than one element. By the defi
nition of normal structure, there exists an x0 E M and a 1{M),
0 < a 1 (M) < I, such that '8(xo, M) = sup { d(xo, y): y E M}
< ()(.1 (M) '8(M). lf '8(Tx, Ty) ,,:;;; '8(x, M) for all x, ye M, let
Mo= {x € M: '8(x, M) ~ a 1 O(M)}. Otherwise, by hypothesis, there
, exists at(M) > 0 0 < ()(.(M) < 1, such that o(Tx, Ty) ,,:;;; a3(M) for
some x, y E M. Let ~=max{ix, a 1} and H = {x E M: 3(x, M),,:;;; ~0(1H)}.
His nonempty, since xo E H. Clearly, His convex. Since x _,.. o(x, iH)
is continuous, His closed. Let x E H, o(Tx, Ty)~ max {o(x, M), ao(M)}
~ ~ll(M) for y E M. Hence, T(M) is contained in a closed ball of cen
ter Tx and of radius ~o(M). By the minimality of M if m e Tx, then
MC S[m, ~o(M)J (the closed spherical ball of center m and radius
. ~'8(M)), whence m E H and T(H) C H.
But o(H) ~ ~o(M) < o(M), which contradicts the minimality of
M. Thus, Mis a singleton, and this completes the proof.
CoroJlary 2. 1 Let K be a bounded, closed convex subset of X.
Assume K has normal structure. Let T be a mapping of K into itself
which satisfies for each closed conve.:\ subset F of K invariant undM T,
with o(F) > 0, there exists some rJ.(F), 0 < a(F) < I, such that
j. \ ,.
l 27
d(Tx, Ty) ~ max { 8(x, F), a(F) 8(F)} for each x, y EX, then T has a
fixed point.
Corollary 2. 2 Let K be a nonempty, closed, bounded convex
subset of X. Assume K has normal structure. Let T be a self-mapping
of K satisfying: for each closed convex subset F of K invariant under
T, with o(F) > 0, there exists some a(F), 0 < rJ.(F) < l, such that
d(Tx, Ty)~ max { d(x. y), r(F) a(F) o(E)} for x, y E F,
then T bas a fixed point.
The follow;ng example shows that our results generalize those of
Browder [I], Kirk [5], and Takahashi [8].
Example 2. 1 Let X = R with the usual norm and K = [O, 4].
Define T : K _,. K by
{o for x * 4
T(x) = 3 for x = 4.
Each closed convex subset F of K invariant under T with 8(F) > 0 is
of the form [O, r], r ~ 4. For r < 4, T([O, r]) = {O}, and in this case
I Tx - Ty I ~ max { I x - y I , r(F), a 8(F) }, for x, y E F, is satisf
ied for any a > 0. For r = 4, T([O, r]) = {O, 3}, hence for ix = 4/5,
I Tx - Ty l ~ max t I x - y I , r(K), aa{K)} for all x, y E K is sati
sfied with 0 as a fixed point of T. Since I T4 - T3 I = I 3 - 0 I = 3,
I T4 - T3 I ~ I 4 - 3 I = 1 is not satisfied for 3, 4 E K. In fact,
I Tx - Ty j ~ max { I x - y I , r(F)} is also not satisfied for F = K.
Theorem 2. 2 Let K be a closed, bounded, convex sub3et of X.
Let T1 , T2 : K _,. K be two mappings satisfying :
(i) d(T1x, T2Y) ~ max {[d(x, T1x) + d(y, T2y)]/2,
[d(x, T2Y) + d(y, T1x)]/3,
[d(x, y) + d(x, T1x) + d( y, T2y)]/3) for all x, y EK;
28 1
(ii) T1A C A if and only if T2A c A for each closed convex subset
A of K with o(K) > O :
(iii) either sup d(z, T 1z) :;;;;;; o(A)/2, or sup d(z, T~z) :;;;;;; o(A)/2 ZEA ZEA
fer each closed convex subset A of K invariant under T1 and T2 with
~(A) > 0. Then, T1 and T2 have a unique common fixed point.
Proof Let G be the family of all nonempty, closed, convex subsets
of K, each of which is mapped into itself by T1 and T2• Partially order
G by set-inclusion. By property (C) and Zorn's Lemma, it follows
that K has a minimal element, say F. Suppose o(F) > 0. Without loss
of generality, we may assume that sup d(z, T2z) :;;;;;; a(F)/2. Let x E Fe. z E F
Since x E Fe ' o(x, F) = r(F). Also, o(F)/2 :;;;;;; r(F). Therefore, for
y E F, ! [d(x, T1x) + d(y, T2y)] :;;;;;; ~ [o(x, F) + sup d(z, T2z)] z E F
:;;;;;; ~- [r (F) + -o(F)/2.] :;;;;;; r(F) Likewise, 1/3 [d(x, T2y) + d(y, T1x)J
:;;;;;; 1/3 [r(F) + r (F):;;;;;; o r(F), and 1/3 [d(x, y) + d(x, T1x) + d(y, T2y)]
:;;;;;; 1/3 [r(F) + r(F) + o(F)/2] :;;;;;; 1(F). Hence, d(T1x, T2y) :;;;;;; max
{ [ d(x, Tix) + d(y, T2Y)J/2, [d(x, T2y) + d(y, Tix)]/3 + [d(x, y)+
d(x, Tix)+ d(y, T2Y)]/3} :;;;;;; r(F). So, T2(F) CB [Tix, r(F)] and, hence.
T2 (F n B) c F n Bas T2 (F) c F. By hypothesis (ii). it follows
that T1(F n B) C F n B. By the minimality of F, we conclude that
F c B. This gives o(T1x. F) = r(F), whence, Tix E Fe . Therfore,
Ti(Fc) C Fe, and by hypothesis (ii), T 2(Fc) C Fe . We claim that Fe is
a proper subset of F.Suppose not, that is, F =Fe . Since 8(x, F)=r(F)
for each x E P, we obtain 8(F) = r(F) = o(x, F). From (i), we get
d(Tix, T2y) :;;;;;; max {[d(x, Tix) + d(y, T2y)]i2, {d(x, T2y)+d(y, T 1x)]/3,
[d(x, y) + d(x, Tix)+ d(y, T2Y)]/3} < max {3 o(F)/4, [o(F) + o(F)]/3,
[o(F) + o(F) + o(F)/2J/3}
= s o(F)/6. Sim1/arly, o(T1x, F) .;;;; 5 o(F)/6 < o(F),
',.;:;_.
[ 29
which is a contradiction. Consequently, if F contains. more than one
elen:ent, then Fr is a proper subset o'f F, a contradiction to the minimality of F. Hence, F contains exactly one element, &ay, x0 , whence
T 1x0 = x0 = T2x 0• Unicity of the fixed point is clear.
Theorem 2.3 Let K be a nonempty, closed, bounded, convex subset
of X. Suppose K bas normal structure. Let Tv T2 : K __,,. K be two map
pings satisfying :
(i) for each closed convex subset F of K invaria{lt under T1 and T2,
with S(F) > 0, there exists some ix 1(F), 0 < ix 1tF) < 1, such that d(T1x, T2y) ~max {[d(x, Tix) t d(y, T2Y)]/2, [d(x, Tzy) + d(y, T1x)]/3,
[d(x, Y) + d(x, Tix)+ d(y, T2y)J/3, a(x, F), ix1(F) a(F)} for x, y E F.
(ii) T1A C A if and only if T2A C A for each closed convex subset A
of K with a(A) > O,
(iii) for each closed convex subset B of K invariant under Th T2 with
8(8) > 0, there exists some a2(B), ~ ~ ix2(B) < 1, such that either
sup d(z, T1z) ~ max { r(B), a2(B) a(B) }. Then, T1 and T2 have a zeB col,llmon fixed point
Pro<lf As in the proof of Theorem 2.2, there exists a minimal element
F. Suppose F contains more than one element. By the definition of
normal structure, there exists xo e F such that
sup {d(xo, y) : y " F} = a(xo, F) = r xo (F) ~ a3 (F) a(F)
for some a3, 0 < a.3 < l. Without loss of generality, we assume that
sup d(z, T2z) ~ max { r(F), ix2 8(F)} for some a2, i ~ cx 2 < 1. If ZEF
d(T1x, T2y) ~ max { [ d(x, T1x) + d(y, T2y) ]/2,
[ d(x, T2y) + d(y, T1x)]/3, [ d(x, y) + d(x, Tix) + d(y, T2J)]/3, rx(F)}
for all x, y" F, let~ = max {ix2, as} and Fo = {x e F: rx(F) ~ ~a(F)}.
Otherwise, by hypothesis (i), there exists a 1(F),
30]
0 < rx 1(F) < 1, such that d(T1x, T2y) ~ cx1(F) 3(F) for some x, y E F.
Let l' = max { ai, a 2, a 3} and F0 = {x E F: rx(F) ~ l'3(F)}. Fo is non
empty, since x0 E F0 • Clearly, F 0 is convex and closed. Let x E Fo.
Then, d(Tix, T2y) ~ max { ! d(x, Tix) + d(y. T2y)J/2,
[ d(x, T2Y) + d(y, Tix)]/3, [ d(x, y) + d(x. Tix) + d(y, T2y)]/3, r.,(F),
cx1(F) o (F )} ~ !)3(F) for y E F. The same argument as in Theorem 2. 2
yields Fo E G. But.
o(Fo) ~ !'o(F) < o(F ), which contradicts the minimality of F. Hence,
F contains exactly one point, which is a common fixed point of T1
and T2•
Remark 2. 1 The following example shows that normal stru~ture
alone is not sufficient to establish the existenee of a com non fixed
point for T1 and T2• In other words, the condition ( iii ) cannot be
dispensed from Theorem 2 3.
Example 2. 2 Let X = R with the usual metric. and Let K = [O, 4].
Define T: K --? K as follows : To = 4 and Tx = 0 for . x =I= 0.
Clearly, K, being a compact convex subset, has normal structure.
Moreover, Kis \\Caklycompact and, hence, has property (C). Taking
T1 = T2 = T, it follows easily that T satisfies all conditions, except
(iii) of Theorem 2. 3, and is without fixed point .
REFERENCES
[IJ F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. A cad. Sci . . 54 (1965), 1041-1044.
[2] N. Dunford and J. T. Schwartz, Linear Operators, (Fart I: General Theory), Wiley Interscience, Ne\v York, 1958.
[3] M. D. Guay, K. L. Singh, and J. H. M. whitfield, Fixed point theorems for nonexpansive mappings in convex metric
~
~.
( 31
spaces, Nonlinear Analysis and Application, Marcel Dekker, Inc. , New York (1982), 179~189.
[4] s. Itoh, Some fixed point theorems in metric spaces, Fund Math. 102 (1979), 109-117.
[5] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 ( 1965 ) , 1004-1006.
[6] H. V. Machado, Fixed point theorems for nonexpansive mappings in metric spaces with normal structure, Ph. D. Dissertation University of Chicago (1971).
[7] S. A. Naim;>al!y, K. L. Singh, and J. H M. Whitfield, Fixed and common fixed points for nonexpansive mappings in convex metric spaces, Math. Seminar Notes (to appear).
(8] W. Takahashi. A convexity in metric spaces and nonexpansive mappings I, Kodai Math. Sem. Rep. 22 (1970), 142-149.
[9] L. A. Tallman, Fixed pomts for condeming multifunctions in metric spaces with convex structures, Kodai Math. Sem. Rep 29 (1977), 62-70.
Jfianabha, Vol. 16, 1986
A THEOREM ON THE EXISTENCE
OF POSITIVE EIGENVALUES OF k-SET CONTRACTIONS (*)
By
A. Carbone and G. Trombetta
Department of M"thematics University of Calabria
8/036 ARCAVACATA DI RENDE (Cosenza)
ITALY
( Received : August 8, 1985
In this note we prove a res.ult on the existence of non-zero
eigenvalues fork-set contractions which satisfy a Birkhoff-K~llog type
condition on a bounded open subset of a special wedge (see definition
below).
If we restrict our maps acting in c.ones, then analogous results
obtained previuosly by several authors (cfr. [3, 5, 6]).
To be more specific, . let X be a real Banach . space, P a special
wedge iw X, Q a bounded open subset of X containing 0 € X, . .. . ''···:
Q = 0 n P and/: .a - P a k-set contraction with k < I. Using some
elementary properties of the topological degree and a result of Fournier
Peitgen [2], \\e succeeded in showing the fol'lowing :result:
Let f: .Q - P beak-set contraction satisfying the condition
inf { 11 /(x) Ii : x e aO} > k sup { 11 x 11 : x € aO},
then there exist an eigenvalue A > 0 and an eigenvector x e oO with
j(x) ='= AX.
-·---··--------* Work supported by M. P. I., G. N. A. F. A. - C. N. R. <>f Italy.
34 ]
"1'
In order to prove the above result we recall, for sake of comple
teness, some basic definitions.
Let (M, d) be a metric space. By the Kuratowski measure of
non compactness of s~t A C M is meant the <XM(A)-number
n aM(A) = inf { e > 0: A = U Si, n < oo and diameter (S;) ~ e
i=l
for l ~ i ~ n}.
Let f: M ~ N be a continuous map where (M, d) and (N, d')
are metric spaces: By the measure of, non compactness. of the map
f(cfr. [IJ) is meant the ix(f)-number:
a(/)= inf {k: ctM(f(A)) ~ k IXM(A) for all A C M}. )
Let n be a subset of M. A continuous map/: n ~ M, is called
a strict-set contraction if a(f) ,< I. We will say that the map f is a
k-set contraction if k ~ ix(/).
Let X be a real linear normed space. Following Schaefer [7], a
nonempty closed, convex subset P of X is called a wedge if x € P imp
lies tx E P for t ·~ 0.
For r > 0, we set
B, = { x € P : 11 x II ~ r}
arid "J
oBr = { x € P: 11x11 = r}.
We recall that a linear ordering of X associataji with a wedge P
is the relation ~ defined by
x~y implies Y - x € P.
r
l 3~
By an ordered linear space X we shall mean a real linear space
with a linear ordering.
Let X be an infinite dimensional real Banach space with norm
II . II . We call a special wedge (see [21), if P is a wedge for which there
exists y0 € P, yo ::j:. 0, such that
11 x +Yo II> II x 11
for all x € P. This is a crucial assumption on deriving our main result.
In fact, in a special wedge P, the following lemma, proved by
fournier-Peitgen in {2, Lemma I. 5. I, p. 173], holds:
Lemnla 0- Let P be a special wedge in a" linear normed space X and
let r > 0. For all e > 0, there exists a retraction
F~ : . B, __,. aB, such that
p~/ oBr = Id a Br
and ~(pe) ~ l + e.
Let Q be a bounded, relatively open subset of a special wedge P
·( i. e., n = 0 .,() P for some open subset 0 of X ). By n and an we
denote the clousure and the boundary of Qin P, respectively. "
If f: Q __,.Pis a k-set contraction, we denote with i (f, Q) the
fixed point index off in n (see {41).
The'fir_st result that we shall prove is the following :
Theorem 1. Let (X, P} be an infinite dimensional ordered real Banach
space, P a special wedge and let n be a bounded, relatively open subset
of P containing 0. Let /:75. ~ P be a strict set-contraction.
36 ]
Assume that
i) a = inf { 11 f(x) II : x € an} > sup { 11 x 11 : x € an} = d:
Then
i( f, 0) = 0.
Poof. Let us note first of all that if condition i) of Theorem 1 is rep
laced by the stronger one:
i') a' = inf ii { (x) 11 : x € O} > sup { 11 x 11 : x € an} = d,
then 1he conclusion of the theorem. is straightforward, i. e.,
i(f, 0) ~ 0.
Indeed, if we assume the contrary, by the solution property of the
. index, there exist"s Xo € n such thatf(xo) -~ Xo • But then
8' ~ II f(xo) II ~ II xo II ~ d,
contradicting i').
Clearly, i) can be replaced by the following inequality :
i") 1l > (1 +-e:o)d, for some e:0 > O.
~ Being a(f) < I, there exists some positive e: < .e:0 such that
(! + e:) a(f) < I.
Now, by Lemma 0, we can choose a retract,ion_ pe of Bs onto its
boundary with a{pe) <;; 1 + e:. Define a map p: P - P by
{
PE(x) if x E Bs p(x) =
x if x E P/Bs.
C1early p is a (I + e: )-set contraction ..
i ~ -
I 37
··):'
Moreover, the m,.ap.g,: .!.1.+Pcefined by ·~ 1 :-, - '
. . -G.!; .·,_::.
·· .... \ . \ ~ .. .- .. ,·.
,.. . g(K) = p(f(x)) for x E· Q
has th~ following propertfos:
ii g(x) ii > 3 for any XE Q
and a(g) ~ a(;i) a(/) ~ (l + i;;) a(j ).
Hence, by our choice of i;;, we obtaih that a(g) < I, i.e., g is a
strict-set contraction satisfying i').
Thus i
i(g, 0) = o.
But the restriction of g to oO coincides with the map f. therefore
by the invariance property of the fixed point index (upon the boundary)
we obtain that
i(f, 0) = 0.
Remark. The same result holds in an infinite dim~nsional real
Banach space X for which for al! e > 0 there exists a retraction Pe of
the ball ~. ;: . '
B = {x E X: Ii x II < l} ·
onto its boundary
0B = {x E X: II x II = I} such that
ix(pe) ~ I + E,
The characterization of spaces with this properties will be the .· . ' . ;
object of a forthcoming paper.
38]
For a k-set contraction, k < I, we derive the foJlowing result:
CorolJary. Let (X, P) be an infinite dimensiona/ordered real Banach
space, Pa special wedge and let .Q a hounied, relatively open subset of
P containing 0. Let/: .Q-+ P beak-set contraction, 0 < k < I, such
that
io) 8 = inf { II f(x) II : x e ().Q} > k sup { II x II : x e c!l} = hi;
iio) i\x-::/= f(x)for k < ,\ < I and x e ().Q
then
i ( f, .Q) = 0.
Proof.
Let e: > 0 such that 8 > (k + e) d and k+e<I.
1 -Consider the map k + e f : .Q -+ P •
. Clearly, this map is a strict-set contraction which satisfies assum
ption i) of Theorem 1 and so we have that
i(·I f) k+e: '.Q = o.
Consider the homotopy H : .Q x [1. _I-]-+ P defined by k+e:
H(x, t) = x ~ tf(x).
By iio) His an admissible homotopy in the sense of [4J hence we have
that
i(J,.n) =; (-1- J, u) = o.
k+e:
This completes the proof.
r ) \
-At
,.·:.:o,
., q'-i"
[ 39
We show now how, using the previous results, we can obtain the
existence of positive eigenvalues for k-set contraction in special wedgs
with satisfying the Birkhoff-Kellog type condition (cfr. [3, 5, 6]).
Theorem 2. Let (X, P) be an infinite dimensional ordered real Ban
ach space, Pa special wedge, .Q a bounded relatively open subset of P
containing 0. If f: Q _,,. P is a k-set contraction for which
I) inf{ ilf(x) Ii: x € o.O} > ksuP { 11 x Ii: x € o.0},
then there exist.\ > 0 and x € c.O such that
,\x = f(x).
Proof. Assume the contrary, i. e., for all .\ > 0 and for all x E ()0,
.\x-::/= f(x).
Now, let e > 0 such that a > (I + e)d. Consider the map
1 k + e f: IT _,,. P.
Clearly, this map is a strict-set contraction satisfying assumption .
i) o.f Theorem 1. Hence
i (-1 f, .Q )· = 0.
k+e
On the other hand, for all
1 O~t~k+e'
40 1
the homotopy H: 0 x [o,-1-J _.,. p , ·· k + E '/·
defined ~Y H (x; t) = H1(i) · = >c ..::__ tf(x)
.·r for X € .Q a:rid'for'for .f .E f, --, . , .:'·' ! ·:· [--·, : .. ::·1 J
' 'k+e:' ' ' >: '·
is admissibl~ since .\x' # f(x) for all .\ ;;;.: k + e:, •. '
Hence i (·-1_·· f,·a ) ~ i(H0,.Q)=:' I, k + E . . . ,.. •
which is a contradiction.
REFERENCES
[1] G. Darco, Punti uniti in trasformazioni a codominio non compatto, Rand. Sem. Mat. Un. Padova, 24 (1955), 353-367.
[2] G. Fo~rnier-H. 0. Peitgen,: Leray End.omorphjs and Cone Maps, Ann. Sc. Norm. Sup. Pisa, Clas.s.e Scienze: Serie 5 (1978), 149-179.
[3] I. Massab9 and C. Stuart, Positive eigenvectors of k-set contractfons, Nonlinear Analysis, T. M. A. 3 (1979) 35-44.
[4] R. ti. Nussbaum, The fixed point index for local condensing maps, Ann. ii.fat. Pura Appl. 89 (1971), 217-258.
l5J R: D. NU._ssbaum, Eigenvectors of Nonlinear PQsitiVe Operators and the Linear Krein.-Kutman Theor~m Fixed Point Theory, Lecture Notes in Mathematics, 886. E. Fadell and G. Fournier, eds (1981), 309-331.
[6] W. v. Petryshin, On the s9lvability of x ET(.\'.) + .\F(x) in quasinormal cones with T and F k-set contractive. Nonlinear Analysis, 5 (1981), 585-591.
[7] H. H. Schaefer, Topological Vector Space, Springer Verlag, New York, 1971.
t !
Jiiiiniibha, Vol. 16, 1986
A DEGREE THEORY FOR WEAKLY CONTINUOUS
MULTIVALUED MAPS IN REFLEXIVE BANACH SPACES*
By
Giulio Trombetta
Department of Mathematics, University of Calabria
87036-Arcavacata di Rende, Italy
Received : September 9, 1985
ABSTTRACT
In this paper we propose a degree theory for multivah1.::d weakly
upper semicontinuous maps with convex valu~s in a reflexiv~ Banach
space with Schauder basis.
INTRODUCTION
A degree theory for singlevalued weakly continuous maps was
developed in [41 The aim of this note is to introfoce a degree theory
for multivalued weakly upper semicontinuous maps (see Definition
l. 5). Our results extend the degree theory of singlevalued weakly
('Ontinuous maps of (4J.
The paper is divided into three parts.
In the first part we give the notations and the basic definitions to
be used in the sequel and we introduce the so-called condition (A)
"'hi ch plays the same role in our theory as the usual condition "the
*Work supported by M. P. I., G. N. A. F. A. and C. N. R. of Italy.
42 ]
map has no fixed points on the boundary" i11 the case of maps upper
semicontinuous with convex compact values.
The second part is devoted to the degree theory for the multivalued
weakly upper semicontinuous maps with convex values, satisfying the
condition (A) with respect to some Schauder basis in reflexive Banach
spaces.
Finally, in the third part we introduce a class of multivalued maps
which satisfies the condition (A) with respect to every orthonormal
basis of a Hilbert space.
1. NOTATIONS, DEFINITIONS AND PRELIMINARY RESULTS
We recall that a multivalued map of a set X into a set Y is a tri
ple (G, X, Y), where G, the graph of T, is a subset of X >< Y such that
Tx = {ye Y; (x, y) e G} is nonempty for each x e X. TX= {Tx:x e X}
is the range of T, while Xis its domain. We shall use the symbol
T: X ~ Y to indicate a multivalued map. If D C X then
TD = {Tx: x .e D}.
Let X and Y be topological spaces and T : X ~ Y.
Definition 1. 1.
The map Tis called upper semicontinuous (u. s c.) at x0 "X if
for any open set. V containing Txo there exists a neighbourhood U of x0
such thatx.e U impl~es Tx CV.
Definition 1. 2.
The map T is called u. s. c. on X if it is u. s. c, at each point x € X
and Tx is compact for every x E X.
Definition 1. 3.
The map Tis called closed if its graph is closed in XxY.
t
l 43
Definition 1. 4.
The map T is called closed if its graph is closed in Xx Y.
A fixed point of a multivalued map T : X --? Y is a point x EX such that x E Tx.
Let X be a real refiexive Banach space with norm II · II , which
admits a Schauder basis. By Br we denote the ball {x E X : ii x II .,;;;;; r},
where r is a real positive number, with oBr the s11here {x E X:
II x II = r}. We shall also use the symbols "-o" and,,__,.,, to denote the
strong and the weak convergence in X respectively. Moreover, we shall
denote by -r the weak topology on X.
Definition l. 5.
The map T : B, --? Xis called weakly upper semicontinuons (w. u.
• 7" s. c.) at x0 ES,, if it is u. s. c. at xo, with respect to the weak topo~
logy'·
Definition 1. 6.
The map T: S, -+ Xis called w. u. s. c. on B,, if it is w. u. s. c.
at each po int x E B, and Tx is weakly compact (i. e. compact with res -
pect to -r) for every x E S,.
Defin.ition 1. 7.
The map T : B, --? X is called weakly closed, (fits graph is closed
in B, x X with respect to the weak topology on the product.
Remark 1. 1.
If T: S, -+Xis weakly closed then (see [I]) :
{xm} C. S,, Xn -"" X} )In --» y ==> y E Tx.
¥ n E N, y,. E Tx,. ·
44 I
Definition 1. 8.
Let D C X. The map T: D -J> Xis called bounded on D, if it maps
bounded sets of D into bounded sets of X.
Remark 1. 2.
Let D c X. If D is weakly compact then D is bounded and wea
kly closed (i. e. closed with respect to 1'), by the reflexivity of X.
Remark 1. 3.
If T: S, _,,.Xis w. u. c. on S, then it is weakly closed (see [I)).
Remark I. 4.
If T: X _,,. X is w. u. s. c. then it send.; weakly cJm;:iact sets of X
into weakly compact sets of X (see [I]).
Remark 1. 5.
If T: B, _,,. X is w. u. s. c. on S, then Tx is closed for any x E B,.
Proposition I. 1.
Let D C X. If T : D _,,. X is w. u. s. c. and D is weakly c!osfd
then Tis bounded on D.
Proof: Given a bounded subset K C D, its weak closure K ( i. e.
the c.losure of K with respect to -r) is weakly compact, by the reflexi-
vity of X. Moreover, K is contained in D, since D is weakly closed.
Consequently, TK is weakly compact (see Remar'.< l. 4). Form the
Remark I. 2 it follows that TK is bounded and we.ikly closed. Hence =-
TK C TK is bounded "
Let X be a real reflexive Banach space, { ei} a Schaujer basis of X and
{<f.;} the corresponding biorthogonal system in the dual space X*, i. e. ¢1 (e1) = a11. Given n E N, by 3,,;S, we denote the
'r-
.. [ 45
n n-dimensional sphere { x € as. : ~ I tfo; (x) 12 = ,2 }.
i=l
We say that the map T: B --+ X satisfies the condition (A), with res
pect to the Schauder basis {es}, if there exists no E N such that, for all
n > no and for all x E a..S, and y E Tx, there exists j ~ n such that
cfo; ( y) -::/=- cfo; ( x).
In particular, we say that the map T: B--+ X satisfies the cond
ition (B), if it satisfies the condition (A) with no = I.
Finally, the map T : B --+ X satisfies the condition (C), if
x ~ Tx for all x E a s ..
The following examples prove the mutual independence of the
condition (A) and ( C).
Example 1. 1. ((A) does not imply (C) ).
Let H be a real and separable Hilbert space with inner product (,)
and orthonormal ba~is {ei}. Fix a E H such that II a II = 2r - e, where
e: < r is a real and positive number. We define Tx = Sa + a - x',
where sf = { z E H: II z Ii ~ e:}, for all x E Sr. The map T has a fixed
point on as,. Let x = [r/(2r - e:)]a and z = [e:/(2r - e)]a, then we
have ,; = z + a - x with II x II = r and II z ii =e:. Consequently,
; is a fixed point of T on as, and thus (C) does not hold. On the
other hand the map T verifies the condition (A) with respect to every
orthonormal basis {ei} with the property that (a, e;) -::j=.0 for infinitely
many indices j. Suppose now that T does not satisfy the condition (A).
Then one can fmd x and z such that (z + a - x) E Tx, XE onSr, and
(z + a - x, e;) = (x, e;), for j = l, ... ,n. By the second and, third n
prop~rt y, lJ [ (z, e;) + (a, e;) ]2 = 4,.2; on the other hand j=l
46 ]
n ~ [(z, ei) + (a, e;)]2 < 4,.2, since (a, e:) =I= o for infinitely many j.
j=l .
This contradiction completes the proof.·
Example 1. 2. ( (C) does not imply (A))
Let {ei} be an orthonormal basis of H and e < r a real positive
number. We define, for all x E Sr
00
Lx = {h E H: h = x + ~ (y, e,) e;+l for some y ES, + x}. i=l
lt is easy to verify that the map Lis without fixed points on as,. The
map L, however, does not satisfy the condition (A) with respect to the
orthonormal basis {e.}. In fact, for all n E N, the element ren € onSr is
such that
00
(ren, e1) = (re,, + L: (re,., e,) e;+ 1, e;) for j = I, ,n. i=I
Remark I. 6.
Obviously, in the definition of condition (A), the choice of the incex
no depends, in general, on the basis. Moreover the example I. 1 shows
that no may tend to + oo, if the orthonormal basis {e;} changes.
Proposition l 2.
If T: B, --+Xis w. u. s. c. and does not sati~fy the condition (A) with
respect to a Schauder basis {e;}, then T has fixed point in S,.
Proof: Ii T does not satisfy the condition (A) there exists an infinite
sequence {111,} of positive intergers with m -o +oo and a sequence
{xk} with x" € o B,, llk
( 47
for all k e: N, such that </li(Y) = <fai(xk) for j= 1, .. ,n1, for some ye: Tx1 ..
. h . n,, T nk By contruct1on, t ere exists ze .: x ,
nk such that znk = xnk + / 1k with xn" = }.;
i=I
00
<fo; (xn1')ei
and y~~ = }.; <fo; (zm) e;, since {xnk} is bounded there i=nk+l
existg a subsequence {xnkcsi } of {:/11'} such that xnk<•> --+ x .: ST.
Moreover. the corresponding subsequence {znk<s1 } of { znk}
converges also weakly to x. In fact. this follows from the bounded
ness of the sequence {znk<•> }and the map T, (see Proposition l.l), since
( zn1'<•1 • e;) -o (x, t-;) for a!l j E N. Finally. since the graph of T is
weakly closed, one has x E Tx (see Remark I 1)
2. Definition of a Topological Degree for Multivalued
W. U. S. C. Maps
Let T : B~ __,. X be w. u. s. c., and {ei} a Schauder basis in X with
· biorthogonal system {,Pi} (see before). Let Xn = [e1 , , en] the linear
hull of the first n basis vectors, s: the sphere of radius r in Xn, and
08~ its boundary For fixed n.: N, we define Tn : B~ --o- Xn by
,Tn (x1 , .. ., Xn) = { (y1, · , y,.): y E Tx},
where Xi = .p;(x) and Y• = rfo;(y) for i = l , ... , n.
If T ; S ,--o- Xis w. u. s. c. and satisfies the c0nd1tion (A) with respect
to the Schauder basis {e;}, then Tn : B~ ~ Xn is.f~~ed point free on
48 J
cB;, for all n > n0• In fact, if we had x E Tnx for some x E oB~
there would exist y E Tx with x n ~ <Pi (x) ei and hence
i=I
q,j (y) = </J;(x) for j = 1 , .. , n, contradicting the condition ( 4). More-
over, Tn is u. s. c. for all n-~ N, as composition of map T with
imbeddings and projections.
Definition 2. I.
Let T: Br --? X be w. u s. c, with convex values. If T satisfies th~
condition (A), with respect to the Schauder basis {e;}, we d~fine
Deg(l-T, S,, 0), the degree of Ton S, with respect to 0, a~ follows: Let fl
L., be the set of all integers togethers with { + oo } and { -- oo }. Then h
Deg (I-T, S,, 0) is defined to be the subset of Z given by:
'· Deg (l-T, S,, 0) = {y E Z/there exists an infinte sequence {Ilk}
of positive integers with n,, -o = s'uch that Deg ( /-T11
k ,B~", O) - o y}.
Remark 2. 1.
h
Deg U-:-T, S,, 0) # q,, since Z is co.npact. In particular, if y is a
(finite) integer, then Deg (l-T , h~k. 0) -o y iff Deg (I-T , Bn" O) ni, nk r '
= y for all but a finite number of n""
Remark 2. 2.
The degree Deg (I-T , B~k, 0) used in Definition 2. I is the Cellina n"
-Lasota degree [3 ]. It is. well .defined for all nk > n0 •
In fact, the map T is w. u. s. c., bounded, with convex and closed
~·
[ 49
values (see Proposition 1. 1 and Remark 1. 5). Hence
1-T : B~" - X is u. s. c., with convex and compact values. nk n"
The following theorems gather the main properties of the degree of
Definition 2. 1.
Theorem 2. 1. (Solution property).
Let T : B, - X be w. u. s. c., with convex values, and such that condi
tion (A) holds with respect to the Schauder basis {ei}. If Deg (I-T, B,, 0)
:f=. {0}, then there exists an element x E B, such that x E Tx.
Proof: If Deg (I-T, B,, 0) ::/= {O} there exists an infinite sequence
{nk} of positive integers with mo -o + oo such that deg (I-T ,B~\O):f=.O. m
By the solution property of the Cellina-Lasota degree it follows that
for every nk there exists x n,, E B~7' such that xm E T xn,'. The remank
ining part of the proof is analogues to that of the Proposition I. 2.
Theorem 2. 2. (Homotopy invariance).
Let H: Br x [O 1] - X be aw. u. s. c. map such that H(x, t) ts convex
for every (x, t) E B, X [O, I]. If the family of the maps Ht = H(· , t):
Br -;. X verifies the condition (A) for all t E [O, I], with respect to the
Schauder basis {e1}, and if there exists n E N such that no (Ht) ~ n, for
all t E [O, l], then Deg (I-Ht, B,, 0) is independent oft E [O, l].
Proof: For all n ;;;;. n, H,.: B~ x [O, I]-;. Xn is a homotopy such that
0 ~ (·, t)) (a B~) for 0 ~ t ~ 1. By the homotopy property of the
Cellina-Lasota degree, we have that Deg (I-Hn (· , t), B~ , 0) is inde
pendent oft E [O, l]. Hence also Deg (I-Ht , B, , 0) is independent of
fE(O,l].
Theorem 2. 3 (Excision property) ,,,g;
50 1
Let T: B, - X be w. u. s. c., with c.onvex. values. If T sati:sfies, the
condition. (A), on aS).., with respect t.o the Schauder basis {e;J, for any A
with r ~A~ R, then Deg (l-T, B,, 0) = Deg (l-T, BR, 0).
Proof: We have that, for all n">no, deg(l-Tn, B~, O)=degEf-T, Bn, 0) R
0 0 sinee 0 ~ (l-Tn) (B; '-B; ), where B~ = B~'-aB~ .
Theorem 2. 4. (Borsuk type theorem).
Let T :.Br - X'be w. u. s. c., with convex values If T satisfies the
condition (A), with respect to the Schauder basis (e,}, and is odd on
3B, , then Deg(I-T, B, , 0) is odd (i. e., 2m ~ Deg(l-T, B, , 0) for any
integer m). In particular, {O} =f=. Deg(I-T, Br , 0) so that the equation
x i: Tx has a. sotutdon in Br •
Proof : For all n i: N, Tn : B~ - Xn is an u. s. c. map, with compact
and convex values, which is odd on 3B~. Hence, for all n >no,
deg(l-Tn,B;, O)is odd. In fact, the homotopy
H 11(x, t) = [l/{l + t)] (l-Tn) x + [t/(1 + t)] (-I+ Tn) x
is admissible in the sense of [3], hence
deg (Hn(", 0) B~, 0) =deg (H,.(",l), B~, 0)
and there exists ii i: N such that n ~A') ~· ii , for all Ai;fr , R],
r
[ 51
where H11(· , 1) is an odd map on aB;. The fact that deg U:fn(·, 1),B',1,0)
is an odd: integer can be shown by constructing a sequence
(see [5] , Lemma 2.3) of singlevalued continuous odd maps
f ~ : B~ __,,. co R(Hn (· , 1)) , where co R(Hn (" , J)) is the clo3ed
convex hull of the rangeR(l/,. (· , 1)) of the map H,.(· , l), converging
to Hri(· , I) in the sense of [3J. Consequently, Deg (l-T, Br,O) contains
only odd integers .
Theor.eJn 2.5. (Boundary value dependence).
Let T, L : B, __,,. X be two w. u. s. c. maps, with convex values, which
satisfy the condition (A), with respect to the Schauder basis { et}. If
Tx=Lxfor all x E aB,, then Deg (l-T, Br, O)=Deg (l-L, Br, 0).
Froof: For aB nEN and for all XEoB~,,Tn x=Lnx. For all n >ii
=max {no (T), no(L)}, deg (l-T,., B~,O) and deg (I-L,., B~, 0) are well
defir.ed. Hence the homotopy
H,. (x, t) = (l-t:) (l-T,,) x+t (I-L,,)x
is admissible in the sense of [3] for all n>n. Therefore
deg (I-Tn, B~, 0) =deg (I-Ln, B~, 0) for all n>n,
and hence Deg (I-T, B,, 0)= Deg (l-L, Br, 0) .
3. A SPECIAL CLASS OF MULTIVALUED MAPS
In this section, let H denote a real separab1e Hilbert space, We give a
sufficient condition for a map Tto satisfy condition (A) with respect
to every orthonormal basis { et } in H. To this and, we first prove
following
52 ]
Proposition 3. 1.
For a map T: B, --+ H the following assertions are equivalent
(i) The map T satisfies the condition (B), with respect to every
orthonormal basis {e;}.
(ii) For all x E 3B, and y E Tx we have (y , x) -::j=. II x II 2.
Proof: (i) =>(ii). Let x E 0B, and {ei} be an orthonormal basis of H,
where e1 =x/r . By assumption, the map T satisfies the condition (B),
with respect to the orthonormal basis {e;}, and x E 01 Br = {x E oBr :
(x, e1)2 = r2}. Hence (y, x) * II x It 2 for each y E Tx.
(ii) => (1). Let {e;} be an arbitrary orthonormal basis of H. lf T did
not satisfy condition (B) with respect to {e;}, then for all n E N we
could find x E on B. and ye Tx such that (y, e;) = (x, e;) for}= I, ... ,n,
hence
(y, x)= II x I! 2, contradicting (ii).
Theorem. 3. 1.
If the map T : B, _,,. H is bounded and satisfies the following
condition:
( I There exists a real number k > 0 and an element a E H such I I that, for all x E oBr and y E Tx, one of the following I
(b) ~ conditions hold : I j {I) (x, y) * 11 x 11 2 ;
I j (II) I (a, x-y) I > k ; l
then the map T satisfies the condition (A), with respect to every
~·
[ 53
Orthonormal basis {e,} in H.
Proof: Let {e;} be an orthonormal ba·sis in H. Since T is bounded,
there exists no e N such that
\ ~ (a, ej) (y, ej) / < k
j=l1o
for ally € TB,,
Now, if the map T did not satisfy the condition (A), with respect to
{e1}, then for all n E N we could find x € onBr ar:.d y € Tx such that
(x, e;)=(y, e;) for j=: I, .. •'!·
If x verifies condition I) of (b), this contradicts Proposition 3. 1. On '
the other hand, if x verifies condition II) of (b ), , this gives a contrad-
iction for n ~ no , since
k .,;;;, I < a , x - y ) I =J ~ ( a ,,ej ) ( y, e; ) · / <k . · j=m+l
Remark 3. I • ...... 11'
· If T : B, -+ II is a w. u. s. c. map which si,itisfies the condition (b) of
Theorem 3. 1, the~ de~ree of Defi~itiori'"3.1 is well defined with resp
ect to every orthonormal basis [ ei ] in H.
Remark 3. 2.
The class of singlevalued weakly continuous maps which satisfies the
.condition ( b) of Theorem 3. I strictly includes the class for which the
Canfora-Pacella degree theory applies ( see [ 2 ] and [ 6 J ) .
At last we give a fixed point theorem .
SA l
Theorem 3. 2.
te( T : S ~ H. bt;, w •. ii. ~. c .. • SupJJPSe that th.ere exis.ts q number fi suah
that , for all n > ii , the condition ( y , :>; ) .~ II :>; n 2 holds far any
x " on B, and y "Tx. Then T has afixedpoint in B,.
p..,oof. We consider the homotopy H (t, x) = t ( I~T) x. Then the
proof of the theorem follows from the proposition l '2.
REFERENCES
[ 1 ] C. Berge , Espaces topo/ogiques, fonctions multivoques, Paris,
Dunod, 1966 . .
[ 2 ] A. Canfora, la teoria de! grado tOpologico per una classe di
\.~
operatori non compatti in spazi di Hilbert , Ric. di Mat . , '
[ 3 ]
38( 1979)' 109-142.
A . Cf;lllin~-A, . ~a~Q.ta., ~ ~~w apprQ.acli. t.Q the defi11it.iott of
topological degree for multivalued mappings, Atti Accad. Naz.
Lincei Rend. Cl. Sci. Fis. Mat. Nat., (8) 47 (19()9) 434-440.
[ 4 ) E. De Pascale-G. Trombetta, Un approccio al grado topolog
ko per· le appHcazioni debolmente continue in spazi di Banach
riflessivi, Le Matemattche, (to appear).
[ 5 ) P. M. Fitzpatrick-W. V. Petryshyn, A degree theory, fixed
point theorems and mapping theorems for multivalued nonco
. mpact mappings Trans. Amer. Math. Soc., 94 (1974). 1-24.
l1 6 ], f .. :F.ac:~U~. U'gJJ~.QQ,t.QRolQg,(W;; P~.f Q,}1)~a:twi 11011: 1::0 .. mpatti 111 spa?i dj a~u_a~h q()n, iJ c;f;qq~ ~.tr~tt~A1~:11-t~ C-Qnves,~Q;, Rla• di. Mat., 29 (1980), 277-305,
)"
Jnanabha, Vol. 16, 1986
ON CERTAIN OPERATIONAL GENERATING; AND
RECURRENCE RELATIONS t
By
HUKUM CHAND AGRAWAL
Department of Mathematics, Bundelkhand' Post-Graduate Coll.ege,
Jbansi - 284002, U. P. , India
( Received : June 24, 1984 ; Revised: February 3, 1986)
ABSTRACT
In th is paper we have deduced certain operational generating
and operational recurrence relations for the polynomials sets, involv-(,0<) <n>
ing the operators R11
,,. and S°',,. defined below. We have also given
( as an illustration ) the, generating and recurrence relations for the
generalized Rice polynomials .
1. INTRODUCTION
In the recent literature several authors have used various combinations
of the differential operator to get generating and recurrence relations .
For example, Mittal* ( [3]. [4]) has given certain operational
t This work was carried out under a fellowship of the University Grants Commission ( Government of India ) .
* For a systematic presentation of val!ious:. classes of' gen:emtirtg: fun.e
tions that are rather easily derivable consequences of Lagrange's
expansion theorem, see Chapter 7 of the latest work on the subject
by Srivastava and Manocha ( [8] , p. 354· et seq~ ) •
56 1
generating and recurrence relations involving the differential operator
Tk = x ( k + xD. ), where k is a constant, for the polynomial sets; and,
as a sequel, he has· given some_ characterizations for special functions •
He has also given the generating and recurrence relations for the
generalized Laguerre, Hermite, Bessel, and _Jacobi polynomials (see
also Srivastava and Buschman [ 7 ] for a unified study of these
clas~es of generating functions) .
In this paper an attempt has been made to derive the operational '
generating and recurrence relations with the help of the operators
R~"'k and s~nl, which. are the com_binati.ons of the finite diffcrenc~ ' . ' - . "
operators !::,,"' ahd \}'"' , defined by (for n= 1,2, ... )
!::,,"' { f' (a) } == f( d + l )- f (a),
1::,,: { f ( a ) } = Lo. -{ 1::,,:-1 f ( o: ) } (I. 1)
and
\7"' { J ( a ) } = /( ~) - f _( o:-1 ) ,
~~ !
v: { f (a ) } v"' { v:-1 Jc o: ) } , (1. 2)
respectively.
In § 4, we obtain certain generating and recurrence relations for
. the generalized Rice polynomials [2].
2. Operational, Generating And Recurrence Relations
Consider the operator
)r
~
.l
I 57
n . R~~! = IT [ (a - j) k + ( a - j) ( 1>: -· j +. 1 ) /':;,"' ]
j= I
n :=(-)"(1-a)n TI [1>:6"'+k+j-l], (?..I)
j=l
where k is constant and n a positive integer. ·
It can be proved easily, that
R~~k { { a )r } = { k + r )n { a - n )n+r
and
\ [a, (c); x ll R~~k+mn l c+iFD (d); J~
=(a-n)n (k+mn) c F [ 1>:, (c), k+ (m+ 1) w x] n +2 D+l '
(d), k + mn" '
Consider the function G~"'> (x, y), defined by
00
G~"') ( x, y) = ~ (a)r Ar (x, s) _L , r=O r!
where, A, (x, s) are independent of a and n.
Using the Lagrange expansion formula [5]
00 tn IT 1>: (a + mn + l)n_J -- = (1 + v)"'
n=O n!
and the well-known result [5]
{l + v)"<Tl 1-m-v
oo (1>: + ( m + 1 )n) ~ tn
n=O n
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
58 l
where v = t( I + v )m+I, m being a constant. The following results can
be proved easily;
( 11. - n) R~~i'Hmn+I[ k G~rxi ( x, y) + 11.tc, .. G~rx> ( x, y,) ]
= R~~\+mn+I { G~ .. > (x, y)} - n (m+ 1) (11.-"--'n) R~~'r1<+mn+I{ G~ .. >(x,y) },
(2.7)
( 11. - n) R~°:.>1' 7c+mn+I [ k G~Ot.J ( X, y) + al:c,rx G~rxJ ( X, y )]
= R~~l k+mn { G~rx'(x,y) }- mn(11. - n) R~°:'l' k' mn+I { G~Ot.J (x, y) };
(2 8)
(11. - n) 6 .. [ R~~>k+mn ( G~ .. > ( x, y ) ]
I - -- R<rx> { G'rx> (' a-n-l n+I'k+mn s X,J)}-
- ( k + mn ) R~~\+,,.,. { G~ .. , ( x, y ) }, (2 9)
( IX - n ) tc, .. ( R~;> k+mn ( G~Ol.l( x, y) ]
I - . R<0t.> { a<rx> ( - 11.-n-r n+I' k+mn-1 s x, Y ) } -
- ( k + mn - n - I ) R~~>k+mn { G~ .. > ( x, y ) }, (2.10)
R~~>k+mn+l { G~Ot.) ( x, Y ) }
(Of.) (O<) (O<) (O<)
= n (a - n)R,.-p k+mn+I { G. ( x, y)} + K,.. Hmn { Gs ( x, y) },
(2.11)
oo tn . (O<)
E - r ( IX - n + I ) R,.-p k+mn+ 1 0
n! . n=
y-
cir
k Gs (x, Y) + [
(od 00
2: p=l
[ 59
' p (Ot) J (a)p /::,"' Gs (x, y)
00
(<Z)r A. (x, s) ( k + · 2: (-)P (-r)v] (y(I + v))" 00 p=l
= r (a) ( 1 + v )k 2: (k + r)r!
and
r=O
oo (o<)
2: ..!:..__ I'( a - n + I ) R,.-i• 1.:+"111+1 n=O n!
Cod (0t) ]
[kGs (x,y) + al:,oe Gs (x,y)
k (Ot)
= I'( a ) ( 1 + v) Gs ( x, y ( 1 + v ) ),
~ _ r( oc - n) R,.•1.:+mn+l (Gs (x, Y)) 00 t'TI [ (Ot) (0.)
n=O n! .
(Ot) (Ot) J - n(m + I) (a - n) R,,~r k+mn+l (Gs (x, Y)
(Ot)
= r(a) (I + v)k G. (x, y(l + v) ),
oo tn (o.) (Ot)
~ - I'(a - n) [R,., 1.:+mn ( Gs (x, y)) n=O n!
(Ot) (0.)
- mn (a - n) R,,.-1' k+mn+l ( Gs (x, y) ) ]
(Ot)
= r(a)(l + v)kG. (x,y(l + v)),
00
~ n=O
tn (Ot) (Ot)
-,- I'(a ,...-- n) Rn, k+mn+l{ G 8 (x, y)} = n.
(2.12)
(2.l3)
(214)
(2.15)
60 ]
=r(a) (l + v)k+l c:Ot.) (x, y (I + v) ), (2.16)
where, v = t (l + r)m+1.
3. THE OPERATOR S"' ,k
· In this section we deduce some operational generating and recur
<"'-> rence relations for the polynomial sets { Gs (x, y) }. But this time we
shall make the use of the operator S"', k, defined by
I S0t.,,, = I-ix [k + (1 - a - k) \7"' ], (3.1)
where the operator \7"' is given by ( I.2) and k is a constant. It is easily
seen, by induction hypothesis, that
s:.k { (ix)r} = (-)'! (k - r)n (oc)r_n (3.2)
where n is a positive integer and r is any integer.
Also, we have
s:,k l n
(T-) 11 [ (1 - IX - k) \7"' + k + j - I J. -a" j=l
(3.3)
One can easily, show that
s:+;•Hmn+l [ (a + n) !::,"' G~"'+ni (x, y) - k G~"'+n> ( x, y )
= ( a + n) s:+n+l' k+mn+I { G~"'+n+i ( x, y ) }
+ n ( m + 1 ) s:+1n' Hmn+I { G~"'+n> ( x, y) }, (34)
)--
,
1.
[ 61
s:+~· Hmn+l [(a + n) 60. G~°'+n> ( x, y ) - k G~°'+n> (x, y) ]
- ( + ) sn { c<0<+11~l> ( ) } - ot n 0<+n+l' k+mn s . x, y
+ Sn-1 { G<°'+n> ( ) } mn 0<+n' k+mn ~1 s x, y ' {3.5)
( a + n ) 6°' [ s:+n' ktmn ( G~°'+ni ( x, y ) ) ]
_ ( + )sn+l { G<°'+n+u ( ) } - ot n 0<+11+1' k+mn-1 s x, y
+ ( k + mn) s:+n' 1,+mn { G~°'+"> ( x, y ) } , (3.6)
[ n ( <o.+nl ) ] (IX+ n) 60< So.+n' k+mn Gs ( x, y )
n+l { (o.+n+l> } = {a +n) Sat+n+l' k+mn-1 Gs { X, Y)
n { <o.+nl } + ( k + mn - n - 1 ) So.+n' ktmn Gs ( x, y ) • (3.7)
n { <o.+nl } ( a + n - 1 ) So.+n' ic+mn+l Gs ( x, y )
n { <0t+nl } = ( a + n - 1 ) S or.+n' k+mn G • ( x, y )
n" 1 { <or.+11-1) } - n Sor.+n-1 k+mn+1 c. ( x, y) ·
' ' (3.8)
00 111 ( a )n }.; n' n=O .
n-1 [ (Ot-!-n) Sor.+n, kt-mn+l ( ot + n) 6°' G. .. ( x, y )
(or.+n> J k (0<+1l - k G 8 (x,y) =a(+ v) c. (x,y/ (l+v)), (3.9)
62 l
co tn ( a )n r(· ) S" . f 0'a.+n+1> ( . ) 1 ~ --i-- 0( + n 0<+n+l' k+mn+l s . x, y . n=O n. L
n-1 { (o.+n> } ] + n ( m + ! ) s .. + .. • k+mn+l a. (x, y )
(0<+1> = a ( 1 + v )k G 8 ( x, y / ( 1 + v ) ), (3.10)
~ --,- ( oc t n ) S"'+n+l' ktmn Gs ( x, y ) 00 tn (a )n r .. < f (O<+n+ll . }
n=O n.
n-1 f (0<+n> J + m n Sa.+n' Tc+mn+I G. ( x, y ) }
=oc(l+v)1' a. (x,y/(l+v)' (0<+1> )
(3.11)
and
00 (n ( oc )n " f (0<+n> }
~. n! S"'+n' k+mn+l c. ( x, y) n=O ·
( 1 + v )k+l G(O<) ( x, y I ( 1 + v)) = ( 1 - mv) s (3.12)
where,
v ;:::: - / c i + v r+1
.·. 4. APPLICATIONS
In this section, we give certain generating and recurrence relations
involving generalized Rice's polynomial [ 2 ]
(0<' 13> . ( 1 + !X )n H .. (p, q;x)= n!
;
,F, c- n, I+" + ~ + n,p; x] 1 + a, q;
)v·
~
. ~ .;
with the help of the operational relations obtained ~n § 2,
From the equation ( 2. I I ), we have
<"+P+l { <"+P+ll .. } Rn' k+mn+l Gs ( X, Y)
<0t+P+ll { <ai+P+ll } =n(a+(3-n+ I) Rn-1' k+mn+l Gs { X, Y)
. <0t.+P+ll { <"'+P+ll }. +Rn'k+mn. Gs (x,y)
where,
( 63
(4.1)
<0t+P+I> n ~ Rn' k = I1 · ( a + (3 + I - j) k + (a + (3 + 1-j)
j=l -'l
(oc+(3-j+2).6.(3 J Let,
(0<+!1+1> -s (0<-s'ill Gs ( x, y ) = ( - x) P s ( I -2 x) , (4.2)
- . (Ot'ill • • • where P s ( x ) 1s the Jacobi polynomial [ 6 ).
It is obvious that
{. (et+P+ll .. } . '!et+ P+2i
.6.p Gs (x,y) =Gs-1 (x,y) •
Using ( 2.2 ) , one can easily show that
<0t+ll+ll ·{ <0t-s'Pl } R,,,,. . . P • . . •. { 1 -:- 2 x ) . . . . .
(0<-s'l3l = ('.1 + (3- n + l)n (k)n H, (k + n, k; x). (4.3)
<et+P+ll Putting the value of Gs { x, y ) from ( 4.2 ) in ( 4.1 ) and
. making the use of ( 4 .3 ): we get
64]
(k + (m + 1) n) H!"'' P> (k + (m + 1) n + 1, k +' mn + 1; x)
= n H!°'' P1 (k + (m + 1) n, k + mn + 1; x)
+ (k + mn) Ht"' P1 (k + (m + 1) n, k + mn; x). (4.4)
Similarly, starting from (2.9), (2.10), (2.14), (2.15) and (2.16) and
using (4.2), (4.3) we can easily derive the following formulae :
1:.p [ (a + ~ - n + s + 1)11 H!°'' Pi (k + (m + 1) n, k + mn; x) J = ( a + ~ - n + s + 2 )n-1
( k + ( m + 1) n) H!°'' B> ( k + ( m + 1 ) n + 1, k + mn, x)
-(a+ ~-n + s + 2)11_1 (k + mn) Ht'B 1(k+(m+I)n, k+mn;x),
(4.5)
613 [ (a + ~- n + s + 1 )11 H~"''B (k + (m + 1) n, k + mn; x) J = (a + ~ - n + s + 2)n_1 (k + mn - 1)
H~"'·P (k + (m + I) n, k + mn-l; x)-(ix + ~-n + s + 2)n_t
(~ + mn - n - 1) H~"'·B> (k + (m + 1) n, k + mn; x), (4.6)
~ ~r(k + mn + n) [<k + (m + l)n)H!"'' 131 (k + m + l)n+I, n=O n! r(k + mn + 1)
k + mn + 1; x) - n (m + 1) H!"'•13 > (k+m + l)n, k+mn + l; x) J = (1 + v),. P~"'·B (1 - 2 x (l + v)), (4.7)
).--·
l
(65
00
tn r(k + mn + n) [ k + mn) H~°''P> (k + ( m + 1 ) ~. n:o n! r(k + mn + 1)
k + mn; x) - mn H!°'' 13> ( k + (m + 1) n, k + mn + 1; x) J
= (1 + v)7i P!°''P1 (1 - 2 x (1 + v ) ) (4 8)
and
oo (k + (m + I) n ) ~ t" . H!°'' 131 (k+(m+l).n+l,k+mn+_l;x)
n=O n ·
(1 + v)k+I p(°'•f3 (1 - 2 x (1 + v)), 8
1- v (4.9)
where, V = t (l + vr+I.
In a similar way more generating and recurrence relations can be
obtained for generalized Rice's polynomials from (3.4) to (3.12).
We conclude this paper ·with the remark that in (2.4), if y = 1
and Ar (x, s) = P~°.!.r ( x ) , then
G~°'> (x, 1) = ~ (a)r P!~r ( X), r=O r!
The polynomials G~°'> (x, 1), satisfy the functional relation
D,°' { G~°'> (x, 1) } = G~0:.!1 > (x, l),'
66 J
which has been studied by the author in bis recent paper. [ J ] . Th us
we can get the operational generating and recurrence relatrori~- for~ the
polynomials G~°'-> (x, 1), which satisfy the above functional relation,
directly from the results derived in § 2 and · § 3 .
Acknowledgments
My sincere thanks are due to Professor R. P. Agarwal for his'help during the preparation this paper. I am also thankful to Professor H. M. Srivastava for his valuable suggestions.
REFERENCES
[l] H. C. Agrawal, On a unified class of polynomials, Indian J pure Appl. Math. 12 (1981), 472-491.
[2] P. R. Khandekar, On a generalization of Rice's polyno01ial. I,proc. Nat. Acad. Sci. India Sect. A 34 \1904), 157-162.
[3] H. B. Mittal, Polynomials defined by generating relation,;, Trans. Amer. Math. Soc, 168 (1972), 73-84.
( 4] H. B. Mittal, Unusual· generating rtlations fur · polyno'm1al sets, J. Reine Angew. Math; 271 (1974), 122-IJ8.
[5] G. Polya and G. Szego: Aufgaben und Lehrs~ize aus der Analysis. J, Springer-Verlag, Berlin, 1964.
[6] E. D. Rainville, Special Functions, Macmillan, New York. 1960.
[7] H. M, Srivastava and R. G. Buschman, Some polynomials defined by generating relations, Trans. Amer.· Math. Soc. 205 (1975), 360-370. Addendum, ibid. 226 (1977), 393-394.
[8] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.
'!iv
~
Jfiiinabha, Vol. 16, 1986
SOME LINEAR AND BILINEAR GENERATING RELATIONS
INVOLVING HYPERGEOMETRIC FUNCTIONS OF
THREE VARIABLES
By
M. B. SINGH
Department of Mathematics, Janta College,
Rewa - 486001, M. P. , India
and
B. M. L. Shrivastava
Department of Mathematics, Government Science College,
Rewa - 486 001, M· P. , India
(Received: July 12, 1984 ; Revised: February 3, 1986)
ABSTRACT
The aim of this paper is to establish several linear and bilinear,
generating relations involving hypergeometric functions of three
variables · Some specializations, relevant to the present discussion,
are also discussed .
1. INTRODUCTION
If we use the notation
(a )n = a ( a + I ) ( a + 2 ) .. '( a + n - 1 ) ; ( a )0 = 1,
where a is arbitrary and n a positive integer, then the generalized
Horn functions of three variables, defined by Pandey [ 5, p. ll5],
Dhawan ( [ 2, pp. 241 - 242] , [ 3, p, 43 ( 1. 1)]), and Sriva5tava
[ 9, p. 105 ( 3. 5 ) ] , are as follows :
68 1
GA {a,~,.~'; y; x,y, z)
00
l: {a )n+p_m { ~ )m+P { ~' )n ( Y )n+P-m m! n! p!
xm yn zP, m,n,p=O
Gn ( IX, ~1> ~2' ~a; y; x, y. z )
00
~ m, n,p=O
( a )n+p_m ( ~1 )m ( ~2 )n ( ~3 )P ( y )n+p_m ml n!p!--
3GA<ll ( IX, ~l> y; x, y, z )
= ~ ( IX )n+p-m ( ~l )m+P .. xm yn zP, m,n,p=O ( Y )n+p_m m! n! p!
3GB<l> ( a, ~l• ~2; y; x, y, Z )
00
xm yn zP,
~ ( a )n+p_m ( ~1 )m ( ~2 )n
( Y )n+p_m m ! n ! p ! xm yn zP,
mn,p=O
Ge ( IX, 0:1, ~. ~1; y; x, y, z )
- ~ { a )p_m ( IX 1 )n ( ~ )m+P ( ~1 ),.
m,n,p=O { Y )n+p_m m ! n! p ! xm yn zP,
Ge* ( IX, ~. ~1; y; x, y, z )
~ ( IX )m+P ( ~ )m+n ( ~l )n p xm yn zP m,n,p=O (y)m+n-Pm!n!p! '
( I. I )
( 1.2)
(~:1.3)
{ 1.4 )
( 1,5)
( 1.6 )
For the precise regions of convergence of each of these triple
Gaussian hypergeometric series, see the latest work on the subject by
Srivastava and Karlsson ( [ 12 ] , p. 87, Section 3 .4 ) .
In the present investigation we also require the following relations
(See, e. g. Srivastava and Mariocha [ 13 ] ) :
jiY·
~
[ 69 ..
p(<1.-m-n, ~-m-n) (x = (-a-~)m+n(!_-:-.})m+n (x+ 1)"' m+n ) (m+n)! 2 x-1
2 2F1(m-<1.-{)+n, -a; 7a-{); x+l ), (m, n=O, 1, 2,. .. ), (1.7)
~ (u-l)n p(a.-n, {)+n) (~+u) 2F1(-n, 8-y; 8; t) n=O n 1-u
-ut ) _ r(8) r(a+a.-y) t"' (1-u+ut)-P 2F1({); y; 8+<1.; 1-u+ut ' - rca+°'> rca-r) (LS)
(1-t)-1-0.. exp (-t(x+y)) oF1 (-;1+°';~) 1-t (l-t)2
~ n' ~-·- L"''> ( ) L<"'> n=O (I +a)n n x 11 (y) 111
,
00
(1.9)
p~"',a>(x) = (I~~),, 2F1(-n, l+oc+{)+n; l+a; l 2
x), {1.10)
2F1(a, b; c; z) = (1-z)"-a-b 2F1(c-a, c-b; c; z), (1.11)
GA(a, {)1, {)2; a.; vt, v(l-2t), v(l-2t))
= ;;" n! (v-1) p(-n, {)i+{)2+n) (l+v ) n=O ({)1 +{)2)11 n 1-v
p(~1+{)2-l. -{)2-n) (1-2t), n
GB(a, {)i, {)2, {)3; a; vt, v(2t-l), v(2t-1))
(I.12)
_ ;;" n! (v-l)np(-n, l-{)2-{)a+n) (. l+v) n=O (l-{)2-{)a)11 n · 1-v .
70]
p(-!32-{33, !31 +!32+ {33..,.-l -n), {l-2t), n (1.13)
;; (,\)~ zn oF1 [-; -y] = (1-:z)->. oFJ-; .y(z-1)]. n=O n. 1-,\-n; Ll-,\;
i; (vb_ A+IFB(-n, (a); (b); x) ir~ n=O n!
= (f-t)-v A+lFB(-v, (a); (b); -~). t-I
where (I. 7), (1.8) are known results given by Srivastava
(I.14)
(1.15)
([IO, p. 92(6.4) J, [11, p. 28, (6.7) J ), l}.9), (1.10), (1.11) are given in
Rainville [6,p. 212,p. 254,p. 60], (1.12), (1.13), (1.14) are due to
'vV
Shrivastava [7,p. 154, (3.1), (3.2) J, [8,p. 131, (6. 3-37)), and (l.15) ~
is given by Chaundy [I, p. 62, (25) J. ,
2. BILINEAR GENERATING RELATIONS
We establish here the following bilinear generating relations :
~ (1-.:\)r GA ( >., a,.13; .:\ - r; xy (I -z) , r = 0 r!
x (1 - 2 y) ' x (I - 2 y) )~y 1-z 1- z
= (1 - z) .:\ - 1 ~ n! n = 0 (ix + !3)n (x - l)n Pn (-n, ix+{3+n)
( l + X ) Pn (a + 13 - I, - 13 - n) (1 - 2 y), (2.1) I+x
~ (I->.), Gl3 ( "'.a, ~' n " - r ; x y ( l - z ) , r=O r!
x ( 2 y - 1 ) x ( 2 _Y · 1 ) ) z• 1-z ' 1-z
,\ - l oo n1 = ( 1 - Z) 1: ·--n ( X -1 )n
n=O (l-~-Y)
p .. (- n, l - ~ - y + n) (~) 1 - x
[ 7l
p,.< - ~-y, IX + ~ + Y - J-n) ( 1- 2 y ), ( 2.2 )
00 1 ~ ----- 3GA<l> (ex fl. ( r = O ( I + ~ )r r! ' - I-' - Y ; a ; 1 - X) ( 1 - t ),
- t ( x + y ) 1) ( yt )' l - t ' ( 1 - t )3
oo I = xl3(1 - t )1+213 ~ n. Ln<l31 ( x )£11<!3> ( y) 111 , ( 2.3)
n=oO+~)n
~ ( ~1 ?' 3GBCI> ( a, ~1 + r, ~2; a; xy, x ( 1-2 y ), x ) ( x (1-y) )1' r = 0 r. ·
00 1 =exp(x) 1: n. (x-I)np11(-n,~1+~2+n)
n = 0 ( ~1 + ~2 )n
( ~ ~: ) p 11( ~1 + ~2 - 1, ~ ~2 - n) ( 1- 2 y ), ( 2.4)
To prove ( 2.1 ), consider
721
and
f:::. ~ ~ (1 - ,\), GA (,\· ct r.t· ,\ - r; x y (1 - z), ~ I ! ' I'• r=O r.
x (l - ? y) I - z
x (I - 2 Y) )z• 1 - z •
On expressing GA in series form, we have
00
6 = ~ (1- A)r r=O_r_!_
00
~ m, n,p = O
{ ,\ )n+P-"' ( ct )m+:P ( ~ )n ( .\ - r )n+:P-"' m ! n ! pi-
{ x y ( I - z ) }m ~ _x ( I - 2 y) }n+p l 1 - z -- zr.
Again using the results
(-_!_L, ( a )_k = ( 1 _ a )k
( " )r ( " + r )s = ( " )r+s = ( " )s ( " + s )r '
(1-x)-,\ = '; ~xt, i = 0 i!
(2.5)
(2.6)
(2.7)
we get
f:::. = ( ] - z ),\ - l ~ ( ,\ )n+i>_m {a )m+p { ~ )n (
0 ( 4 ) ml I I X yr m,n.p = n+P-m , n. p.
{ x ( 1 - 2 y) }":P Now applying the result ( I.I ), we obtain
,\- 1 ) 6 = ( 1-z) GA ( ,\, a, ~1 ,\, xy, x ( l - 2 y ), x ( 1 -- 2 y)
which, in light of ( l.12 ), provides ( 2.1 ).
~·
I 73
The proof of -the formula (2!f) would run parallel to what we have
obtained above in view of (1.2) and (1.13).
To prove (2.3), let
-r = l: I on> 00 ( r = O (l + ~)r r! 3 A a, - ~......, y; a; (1 - x) (1 - t) 1
_:t (x + y) t) (_J:!_- )' 1 - t • (1 - t)3
00
- ~ 1 r = 0 (1 + ~)r r!
00 ~ (-~-r)m+p
m, n,p=O m! n! p!
{ (l - x) (1 - t) } m f - t (x + y) }n t'P (_E__- .·)r 1 - t (1 - t)3 . ,
Now applying the formulae
00 ~ xi
i = o i! = exp (x) , (2.8)
and (2 5), we find that
-r =exp ( - t ( X + y) .. c; (- ~)-• (- ~ - r)m+'P .·
1-t m,p,r=O m!p!r!
m -.yt
1 ( )r
{ (1 --' x) (1 - t) tP (I _ t)3 •
Again, using the results (2.6), (2.7) and (2.5)
( .....; t (x + y)) 'i' · . ( - ~ )m
-r = {l - t)"' exp 1 _ t m;r..:.. 0 (1+ ~ -m), m!r!
74 ]
( ')'' ,, , (i - x)ii, 0, yt t)2, , ,
= (1 - t)P exp (- 1 (x + y)) ~ (- l3)m (I - x)m I - t m=O m!
0F1 [ ~; ~ _ m; - yt 1 (I-t)" J
Now employing the result (l.14), we get
't' = xP (I - t)P exp (, -=' t'(~ + Y)) 0 p1 ['- ; xyt J I - t , I + ~; (I - t)2
which, in view of (l.9), yields (2.3) •
To prove (2A), put
q, = ~ J ~~ )r 3G~(a, ~1 tr, ~2; a; xy; ¥{1..,...2y), x){:x(1-,,-2y)V r = 0 r.
00 k ( ~l )r
r= O rl
( ~1 + r )m ( !32 )n m,n,p=O m!n!p! (xyr
00
k
{ x ( 1 - 2 y ) }n x" { x ( I ..:.;:; 2 y )}r'.
Now applying the results ( 2.6 ) and ( 2.8 ),
00
cp = exp ( X ) ~ J ~I )m+r ( ~2 )n ( X m , ~m n, ,' = O m ! n ! r ! Y )
{ X ( 1 - 2 y ) }n+r
00
=exp ( x) ~ Iii, n = 0
; lit )m+r ( fa )n_r ( x y r r=O m!(n~r)!r!
,'-.
{ X ( 1 - 2 y ) }n •
"-:--
~
[ 75
Then;' on reversing the inner. summation, we get
oo n </> = exp ( x ) }:; }:;
m, n = 0 r = 0 ( ~1 )m+n_r (~2 )r ( X y )m m! r! ( n - r )!
{ x ( I - 2 y) }"
Now applying the relations ( 2.6 ), ( 2.5 ),
( n - k ) ! = ( - 1 )k n ! ( - n )k '
( 2A9 )
.and Exercise 4 of Rainville [ 6, p. 69 ] , we obtain
00
</; = exp ( x) ~ (~1)m+nJl - ~1 ·-~2 - m - n),, ( xy )m m, n = 0 ( l - ~1 - m - n ),, m ! n ! .
{ x (I - 2 y) }"
oo n = exp ( x ) ~ ~ ( ~1 )n ( I - ~1 - ~2 - n ) ,. m
n = 0 m = O ( I - ~1 - n )n_m m ! ( n - m ) !
( x y )"' { x ( I -.:. 2 y }"-m
which~ on reversing the inner summation and simplif yi:11g,
provides
oo n </> = exp ( x ) ~ ~ (}1 ),. ( - n )m ( 1 - ~l - ~2 -:-::- n )m ·
n = 0 m = 0 ( 1 - ~1 - n )m m ! n !--
( X y )"-m { X ( 2 y - 1 ) }m •
Expressing the inner series as a Jacobi polynomial, using the result
( 1.7) (with m = 0) and ( 1.11 ), one finds
</> = exp ( x) ~ <. ~1 +, ~2 )n 2F1(- n, ~2; ~1 + ~2; ( 1~ )) n = 0 n. x . .y
{ x ( 1 - y) }".
76 J
Finally, in the light of the results ( 1.15 ) ( with A = B = 1 ) a_nd
( 1.8 ), ( 2.4 ) is obtained •
3. LINEAR GENERATING RELATIONS
In this section we establish the following.generating relations :
00
I: r = 0
( 1 ~ ,\ )r GA ( ix, ~. Y,' ,\ - r; x, y, z ) t' r.
= ( 1 - t l- l GA ( a, ~. Y . ; A ; x .-,y( 1 - t ), z( 1-t), (3.1) -t
~ ( I - ,\ )r GB (a A, y, o; ,\ -r,· x, y, z ) tr I , t' -
r = 0 .r.
= ( 1 - t ),\_:_1 cB( a,~, y, O,' ,\; 1 x t' y (1-t), z ( 1-t) ), ( 3.2)
'i' (I ~ .\)r 3G'.J> (a, ~; ,\ - r; X, y, zW r=O r.
= (l - t) ,\ - 1 3G1> ( ix, ~; ,\;
12--, y (1 - t), z (l - t)_}
- t ' '
(3.3)
~ (1 -, .\)r aG'J-> (ix, ~l> ~2; ,\ - r; x, y, z)tr r=O r.
= (1 -t) ,\ - I aG'J-> ((ix, ~h ~2; ..\; l x t ,y tl-t), z (l-t)),
(3.4)
'i' (1-:-1 .\), Ge (a, ixi. ~u ~2; ,\ - r; x, y, zW
r=O r.
-"'-
[ 77'
= (1 - t) ,\ - 1 Ge (a, a1' ~h~2; l.;-x -.· , y ( 1-t), z (1-t) )• 1 - t .
~ (l-.~)r Ge* (a,~, y; ,\ - r; X, y, z. W r=O r!
. (3.5)
= (1 -- t) ,\ - I Ge *(a,~. y; >.; x (I -t), y (I-t), l \ )· (3.'5)
Derivation of (3.1) : consider
0= ~ ( l-,\), GA (a, ~. y; ,\ - r; x, y, z)t'. · r = 0 r!
On expressing G11 in series form, we get
0= 00
~ r = 0
(l-,\), -,.-!
~ (a)n+p_m (~)m+P (y)n xm Y" zP ir. m,n,p=O (A-y)n+P-mm!n!p!
Again using the results (2.5) and (2.6), we have
O = . ~ (c:)n+P-m (~)m+P (y)n (I-,\-n-p+m)r ;xfil yn zP tr. m,n,p, r=O (i\)n+P-m m! n! p! r!
Now applying (2.7), we find that
0 = (l-t)>..-1 ~ (a)n+P_m (~)m+P (y)n ( X )m m,n, p=O (i\)n+p_m ml n! pf -1-:f
{ y (1 - t) r { z (I - t) y which, in view of (I.I), provides (3.1).
Proceeding on similar lines as above results (3.2) to (3.6) can be
derived.
78 1
4. SPECIAL CASES
(i) On taking ~= 0 in (2.1) and simplifying, we obtain
G1 (a,,\, 1-,\; -xy, x (2y-1))
00
~ n! n=O 1--;y,;- (x-1)" p~-n, a+n) ( _!±_x ) 1-x
P!a-1, -n) (1-2y), (4.1)
where G1 is Horn's function [ 4, p. 224 (IO) J •
(ii) In (2.1) putting y= 0, letting a= 0 and solving,
we find that
2F1 [ ,\, ~; x J = 'i;' (x - 1)" P~-n, ~+n) ( 1 +x ·)· (4.2) ,\; n=O 1-x
(iii) In (2.2) setting y=O, putting a=O, replacing -x by x and
simplifying, we get
00
F1 (.\, ~; y; x, x) = ~ n=O
(1 - x)n f,(-n, l-~-y+n) ( 1-x ) ' I+x '
where F1 is Appell's double hypergeometric function [5, p. 265J. ·
(iv) Changingy to y/~1 and letting ~1 ~ oo, (3.4) yields
oo ( 1-A)r 012> ~ 3 A ,.,
r=O . ( a, ~2; ,\-r; x, y, z )t'
(4.3)
= ( l-t )'\- l aG~1 (a, ~2; ,\; _x_, y ( l-t), z (1-t)). 1-t
(4.4)
[ 79
where aGA!2> is given earlier ( 2, pp, 241-242] .
Acknowledgements
. Thanks are due to professor H. M. Srivastava ( University of
Victoria, Canada ) for suggesting a number of improvements in the
present paper .
REFERENCES
[ 1 ) T. W. Chaundy, An extension of hypergeometric functions ( I ), ··
Quart. J Math. (Oxford) 14 ( 1943 )s 55-78.
[ 2 ) G K. Dhawan, The confluent hypergeometric functions of three
variables, Proc. Nat. Acad. Sci. India Sect. A 39 ( 1969 ), 240-248.
( 3 ] G. K. Dhawan, Hypergeometric functions of three variables,
Proc. Nat. Acad. Sci. India Sect A 40 ( 1970 ), 43-48,
[ 4] A. Erdelyi, Higher Transcendental Functions, Vol. I, McGraw
H ill, New York, 1953.
[ 5 J R. C. Pandey, On certain hypergeometric transformations,
J. Math. and Mech. 12 ( 1963 ), 113-118.
[ 6 ] E. D. Rainville, Special Functions, Macmillan, New York, 1960.
[ 7 ) B. M. Shrivastava, Bilinear generating relations for Jacobi and
Laguerre polynomials, Vidya B 20 ( 1977 ), 150-156 .
[8 ) B. M. Shrivastava, A Study of Special Functions of One and Two
Variables with Some Applications, Ph.D. thesis, A. P. S.
University, Rewa, India, 1795 .
( 9 ) H. M. Srivastava, A note on certain hypergeometric differential
equations, Mat. Vesnik 9 ( 24) ( 1972 ), 101-107.
80]
[ 10 ] H. M. Srivastava, Certain formulas involving Appell functions,
Comment. Math. Univ. St. Paul. 21 ( 1972 ), 73-99 ,
[ II ] H. M. Srivastava and C. M. Joshi, Certain double Whittaker
transforms of generalized hypergeometric functions, Yokohama
Math. J. 15 ( 1967 ), 17-32 .
[ 12 J H. M. Srivastava and P. W. Karlsson, Multiple Gaussian
Hypergeometric Series, Halsted Press t John Wiley and Sons),
New York, 1985 .
[ 13 ] H. M. Srivastava and H. L. Manocha, A Treatise on Generating
Functions, Halsted Press (John Wiley and Sons), New York,
1984 .
~
~
Jfianabha, Vol. 16, 1986
ON CERTAIN POLYNOMIALS ASSOCIATED WITH
THE MULTIVARIABLE l-1-FUNCTION
By
P. C. MUNOT and RENU MATHUR*
Department of Mathematics, and Statistics, University of Jodhpur,
Jodhpur - 342()01, Rajasthan , India
( Received : September 4, 1984 ; Revised: November 22, 1985)
ABSTRACT
In an attempt to unify various bilateral generating functions
obtained earlier by Srivastava and Panda [ 6 ], Raina [ 2 ], and
Srivastava and Raina [ 10 ], we study here a few new sets of polyno
mials associated with the multivariable H-function of Srivastava and
Panda ( [ 6 ] ; see also [ 7 ] and [ 8 ] ), and give certain theorems
concerning the generating functions of these polynomials . In the
sequel we also show that these theorems can be applied to yield s~veral
bilateral generating functions for some polynomial sets .
1. INTRODUCTION AND PREREQUISITES
Foiconvenience, let ( ci, Yi )i. P and ( ai; ·ai',. , tX/•l )i.P abbrev
iate the p member arrays ( ci. Y1 ), ••• ,( cp, YP) and (a1; a1', .. :,or.1<s1), ... ,
( ap; or.p' ,.;.,0tv<si ), p > 0 respectively . The array being empty if p = O,
Then we recall the definition of multivariab]e H-function. introduced
by Srivastava and Panda [ 6, p. 27 l, eq, ( 4 1 ) ] iri ti1e follo .ving
form: ,_..,. __
+Present address : Department of Mathematics, Government
College; Ajmer-30500 I, Rajasthan, India .
82 ]
(1. 1) H<ll I : = H ! .
r. z1 J o. n: m'. n' ; ... ; m'·" n<·'[z1 I Lzs p, q: p', q'; ... ; p<•>, q<sl Zs
( . ' (•l) • (c ' y ') ' · · (c !sl y <•>) J aJ, a; , ••• ,a; l• 'P· ; , _; 1• P , •• ., ; , i 1,p<•l
(b . R' (.). (s)) • d , "' ') I • • (d (s)) "' (S)) j > t' i , ••. , t-J h Q• j ' Oj 1> Q , ••• ' j Oj 1, q!s)
'(21'~)" IL ··· J <I> (~1 ,. .. , ~.) . ~ ~ 0;(~;) z.~i d~, 1 Ls z=ll }·
where w = ¥-:=!, n s
(I. 2) <1>(~1 ,. •. , ~.) = TI r(I - a1 + ~ "/•> ~i) j= 1 i= 1
1 q. s p s J-1 ·j .TI r(l - b; + . ~ ~1 1 0 ~;) . TI P(a1 - . ~ a/t' ~1) L,.]=l l=l . ]=n+l l=l I
m<il n!'il (1. 3) 6;(~;) = TI r(d/i' - 8;<il ~1) TI r(l - c/il + y;<'> ~.)
j~l j=l
[
q!i)
• TI r(l - d/i> + 8/il ~t) . j=m1il+l
pW ~-1
• TI r(c/il - y/il ~;) J v i E {1 ,.. 's}, j=nw+ 1
an empty product is interpreted as unity, the coefficient
a;w, j= 1, 2, .. ., p; ~1w, j = 1, 2, ... , q; y/i>, j = I, 2, ... ,pw;
8J< 0 , j = 1, 2, .. , qw; i = 1, 2, ... , s are positive real numbers
and. n, p, q, mw, n<i>, p<i> and q<i> are integers such that
0 ~ n ~ p, 0 ~ m1il ~ q<il; q > 0, O ~ n<il ~ pli',
i = I, 2, .. , s. The multiple integral ( 1.1 ) converges absolutely if
I I 1 ( 1.4 ) U1 > 0 and are z; i < 27' u, ,
~
where
(1.5) U; = p q ~
j=n+I rt./i) - ~ ~/il +
j=l
p<i)
I 83
n<i> ~ y/il -
j=l
qW ~ y/i> +
mW ~
j=l 3/i> - ~ o/n,
j=nW+l j=m<i>+1
i E { I, 2. . .. ' s } .
Again, for convenience, let
[
Z- I ·< t: .• "-·' µ-<•) >1 J 1 '-:i1 ' r-J ' ••• ' ' .., u -H :
• ' (s) Zs \ ( '/Ji, Vj , ... , v; )1,v
denote the multivariable H-function
0 + U• m· n'. . m<•l (s) r z I { !:' • -• • •. (si -) • , n . , , .. ., . , n I · .1 c,;, Jl-1 , .. ., µ, 1, u (1.6) H :
+ + . I '· • (s) q!s) L z ( · .' (s) ·) ' p u, q v. p , q' .. ., p • s 'fj;, v,' ... , 'lj . 1' v
( • .' .(S),) • ( c·' "I I) I, • (c·(S) y·(s)) (S) J a4, a, , .. ., rt., i. :v, i , it 1' p , .. ., 1 -, 1 1' p •
( b;; ~;', ... , ~i(s) )i, a; ( d;') o;' )1. q'; ... ; ( d/sl, o;<s> h. q(s)
In all the theorems discussed in this ;paper we shall assume. the
following:
(i) The restrictions on the parameters of the H-function and ·the
convergence (and existence) conditions corresponding appropriately to
the ones detailed above are satisfied by each of the various
H-functions involved.
(ii) a/s, ~;'s, y/s, At's, i=l, 2, ••. ,rare arbitrary complex numbers
independent of ni, .. , nr. Also n; > 0~ q, E N, where N is the set of
natural numbers, i=l, 2, .. ., r and cr;W > O,j=l, 2, .. ., s;
i=l,2, .. .,r.
84 1
(iii) (nr) stands for the linear array n1, ... , n, with &imilar interpre
tations for (ar ), (~r) etc.
2. Generating functions for a general class of polynomials
Let f [(z,)] be a function of several complex variables zi, z2,. .. , z,
defined formally by the power series
00
(2.1) f [(zr )] = ~ ki,. .. ,k. = 0
r C [(k, )] II
i = 1
kt z:_ '
ki !
where the coefficients C [(k;)], k1 ;> O, i = 1,2, ... ,r are arbitrary
constants real or complex .
Also let a class of polynomiuls
[(a,); (~r)] (2.2) Q [(Ar); (xr ); (ys)] l:::e defined by
(n, ); (q,)
[(cxr); (~r)] [(.\,); (xr); (ys)] = Q [ ni I qi l ~ [ nr..J. q,] C [(k,)]
(n,); (qr) ki = 0 k, = 0
~ ) ( - n; ) q; k; i = 1 l
k; } Xi
kd
[ ~I \ ( - ex; - ( ~i + I ) n;. - .\;kt: c;,', ... ,c;;<•i )1,r. J-. • H • . .
J's , ( - a; - ~i n; - ( .\; + qi ) k; : <Ji' , ••• ,cr,<s> ) J,r
Then our first multidimensional generating fun..::tion is given by
Theorem 1. With the function f [(Zr )] defined by (2.1 ), let
(2.3) 0 (( n,.), (qr); (u,), (~r), (Yr); (Ar), (x,), (ys)]
oo r ~ C [(kr )] II } y;
ki,•••,kr = 0 i = 1 l k;
Xi
kd
J.,
·~
[ 85
.( n, + q; k, : Y• I ( ~1 + 1 ))-1 }
. H : ' [
J'l 1 (Yi -- 0t1 - A; k;; crt', ••• ,cr/•l )lo• J Yi J ( Yt - Uj + nt - Aik;'; cr/,. . .,crt(s) )1or .
and also let 00
(2.4) </> [(xr), (y.,); (v,)] = ~ 0 [(nr), (q,); (otr), (~r), (Yr), nh···onr = 0
( Ar); (Xr ), (ys )] 'i-J f _v~;J' i = I l nil
Then
oo r ( ) [(ar); (~r)] (2.5) ~ II YI Q [(Ar);(\'r),(Ys)]
n1,. • .,nr=0 i = 1 Yt + ( ~· + l ) n; (nr); (qr)
r n TI (~)
i = 1 nil
r a; r qi "-1 </> I X1 ( - V1) ( I + Vi) , ... , II ( 1 + V; )
i = l L
Xr ( - Vr )q' ( 1 + Vr ),\',
r cr;' r cr/•> Yi IT ( + Vi ) ,. ··, Js II ( 1 + V; )
i = I t = I
where Vt is a function oft; defined by
Vt = fi ( 1 + Vt ) ~i + l ,Vt( 0) = 0,
- V1
j + V1
- Vr J - ' ,. . ., } + Vr
8{) J
(( ar) ;(~r )J Proof: On replacing Q [(Ar) ; (x, ), (ys)] by its equivalent
(nr), (qr)
series from its definition (2.2) with the H-funtcion expressed in terms
of its Mellin-Barnes contour integral representation ( 1.1 ) in the left
member of (2.5), changing the order of summation and integration
which is assumed to be permissible and then applying a result due to
Srivastava and Panda [ 9; Th. 3, p. 34 ], and finally interpreting the
iesulting expression by means of (LI), we immediately arrive at the
right member of (2.5).
3. Two more classes of general polynomials
((e<r); (~r)J ( i) The polynomials T ((Ar); (xr), (y,,)J:
(nr ; (qr)
Associated with the power series (2.1), we may define a new set of general polynomials in the form
[(e<r); (pr)] (3.1) T {(Ar); (xr), (ys)] =
(nr ); (q,)
[ ni I qi J ~
ki = 0
[ nr / q, J ~ c [(k,)]
kr = 0 ~ \ ( - n; )qi kt
i = I l • Xi
H : , [
y 1 ,, ( l -e<; - ( p; + 1 ) n; - 1\i k;; cr;',. . .,crJCs) )I,,} ys ( - a; - pt n; - ( 1\1 + q;) f·u; cr,',. . .,cr/s> )1,r •
The following theorem gives a generating function of type
(2.5) with Yi= a;, i = 1,2,. . ., r for this set of polynomials.
Theorem 2. If we let
00
(3.2) ~ [(xr), (ys)] = ~ ki,. . .,kr = 0
C((k,)), h. 5 x;k;} 1 = I l
k;}
I~
[ 87
.H [ ~'1 I ( I - ai - { ~l + I ) q, k1 - .\, k;;cr/, ... ,cr/•l )i,r ]•
Ys ( - ix; - ( ~i + 1 )q; k; - .\; k,, cr/, ... ,cri<s> )i,r
ihen
00
~ [(otr); (~r)][(.\r); (xr), (Ys)] T r n ( t;n; )
(3.3) n1,. .. ,nr = 0 (nr); (qr) i = 1 -;;:;1
= . ~ ) ( 1 + V; )IX; l ~ ( X1 ( - V1 )q1 ( l + V1 ).\1,. .. ,
z = I l J
Xr ( - Vr )q' ( 1 + Vr ).\',
r , r cr;<•> Y1 Il ( I + v;)cr; ,. . .,y. Il ( 1 + v; ) ] ,
i=l i=l
The proof of Theorem 2 runs parallel to that of Theorem I.
[{ar); {~r)] {ii) The polynomials A [(.\r); (x,), (ys)] :
(n,); (µr)
Again with the meaning given earlier to C [(k, )], let us define
another class of more general polynomials
[(a,); (~r)] (3.4) A [\Ar);(xr),(ys)]=
(nr), (µ.r)
~ C [(kr)] Il Xi oo r { ~ 1 k1,. . .,k, = 0 i = I kd ( m + I ) w ki
[
Yl I ( - !Xi - ( ~; + I ) n; - .\; k;; cr;', .. .,cr;<s> )1,,J . • H : '
}'s ( - IX; - ~i n; - ( Ai - µ.i ) k;; <Ji 1 ,. • .,cr/sl)i,r
and
88 1
[(ar); (~r)] (3.5) i\. [(,\r); (xr), (Ys)] )+..,
(-n,); (µ.)
00 00
~
ki ~ [ ni I µi 1 ~ c [(kr)]
kr )> [ nr / µ.r ]
r l k; } Il Xi
i = I k;! ( - n; + I ) µ; kt
.Tl r yf I: { - ~i + (. ~i. + I ) .n; - ,\; k;'.. <1i~: .. ., cr;«:: )t,r J LY• ( - Qf.i + ~' n, ( ,\, - µ1 ) k,, cr, , .. .,cr1 )t,r
By using the known result [ 9, Th, p. 38 ] , we can establish the
following theorem giving a Laurent series expansion involving these
polynomials .
Theorem 3. in terms of the coefficients C [(kr)], k; )> 0,
i = 1,2,. . .,r, given by (2. I), let 6 and rp be defined by equal ions (2.3)
and (2.4), respectively, with q; = -µ;, i = 1,. .. ,r. Then the polynomials
defined by (3.4) and (3.5) are generated by
(3.6) ~ II y; tl; oo r l } n11 ... ,nr = - oo i = 1 Y; + (~; + 1 )n; n; f
[{atr ); (M] l(.\r); (Xr), (Ys)l i\. (nr); (µr)
- .~ f ( l + Vi) Qf.; 1 rp r Xt Vt-µ 1 ( 1 + Vi / 11 ,. . .,x,. v,-p.r z=l ~
,\,. r cr;' ( I + \ .. ) . YI II ( 1 + Vi ) ' '
i = 1
~
[ 89
Ys ~ ( 1 + '11 )cri<sl ' 1 :1 V1 •·:'.• I +V_r~,J i = I
4. SPECIAL CASES
(i) For r = l, all these theorems reduce to the results given earlier
by Srivastava and Raina [IO].
(ii) Theorem I, in its confluent case when Yi -+ oo, i = 1,2, ... ,r,
reduces to the result :
(4.1) 00 [( ar); (~r )] r ni
Q [(t\r); (xr), (ys)] . fl (_!L.) (nr); (qr) i = 1 nil
·~
ni.···· fir = 0
. ~ f ( 1 + V; )rJ.; + l (. f [ X1 ( - V1. )q1 ( 1 · + V1),\1
l = I I - ~i V; ~ .. •
.H<I>
r I Y1 I I I I Ys L
r II
i = 1
r II
i = 1
, .•. , Xr f - Vr )qr ( 1 + Vr )t\r J ·' l
( l + V; ) cr, I I. 1.
)cr/sl f (I + V; I
J
On making ,\;-+ 0, i = 1,2, . , r, the result (4)) gives another important result which, on takirig r = 1, a = p - i, ~ ,,;,, O,p = n
= q = O and cr1 = ... = crs = h, gives a known result of .Raina
[ 2; eq. (4.1), p. 303]. Moreover, each of Theorems I to 3 can be used
to derive bilateral· and multilateral generating functions . associated
with the multivariate H-functions by suitably specializing the coeffici _
ents C [(k,)] , k; ;> O,' i ~ I, 2; .. .,,.. Acknowledgments
The authors are thankful· to Professor H. M; Srivastava for his
suggestions leading to the present version of their work •
90 I
REFERENCES
[ 1 ] H. W. Gould, A series transformation for finding convol
ut1on identities, Duke Math. J. 28 (1961), 193-202.
[ 2 ] R. K. Raina, A formal extension of certain generating functios,
Proc. Nat. Acad. Sci. India Sect A 46 (1976), 300-304.
[ 3 ] R. K. Raina, Certain generating relations involving several
variables, J. Natur. Sci. and Math. 23 (1983), 33-42 .
[ 4 ) R. K. Raina, On some classes of generating relations involving
the H-functions, Acta Cien. Indica Math. 10 (1984), 32-36.
[ 5 J H. M. Srivastava and H. L. Manocha, A Treatise on Generating
Functions, Halsted Press (Ellis Horwood Limited, Chichester),
John Wiley and Sons, New York, 1984,
[ 6 ) H. M, Srivastava and R. Panda, Some bilateral generating
functions for a class of generalized hypergeometric polynomials,
J. Reine Angew. Math. 283/284 (1976), '265-274 .
[ 7 J H. M. Srivastava and R. Panda, E"<pansion theorems for the
fl-function of several complex variab!es. J. Rein~ Angew. Math.
288 {1976), 129-145 .
[ 8] H. M. Srivastava and R. Panda Some expansion theorems and
generating relations for the H function of several complex varia
bles. I and II, Comment. Math. Univ. St. Paul. 24 (1975),
fasc. 2, 119-137; ibid. 25 (1976), fasc. 2, 167:__197 .
[ 9 ] H. M. Srivastava and R. Panda, New generating functions
involving several variables, Rend. Circ. Mat. Palermo \2) 29
( 1980), 23-41 .
[ 10 ] H. M. Srivastava and R. K. Raina, New generating functions
for certain polynomial systems associated with the fl-functions,
Hokkaido Math. J. 10 (1981), 34-45.
Ii.._
_t \
Jfianabha, Vol. 16, 1986
ON SOME SELF - SUPERPOSABLE FLOWS IN CONICAL DUCTS
By
PRAMOD KUMAR MITTAL
Department of Mathematics, Government post - Graduate College,
Hamirpur - 210301, U. P., India
and
MOHD. IQBAL. KJI;\N
Department of Mathematics, Hindu College,
Moradabad - 244001, U. P., India
(Received: November 14, 1984; Revised :.November 25, 1985)
ABSTRACT
In finding out the exact analytic solutions of the basic equations
of fluid dynamics, the principle of self-super-posabilfty is of impor
tant use . In the present paper an attempt has been made to find some
se~f-superposable motions of incomprerssible fluid in the conical ducts.
No stress has been, given on boundary ·conditions and the solutions
thus determined containing a set of constants. Pressure distribution of
some of such flows has also been discussed. The curves along which the
vorticity of· the flow becomes constant have also been attempted •
The aim of the paper is to introduce a method for solving the basic
equations of fluid dynamics in conical system of co-ordinates by
using the property of self-superp0sability .
J. INTRODUCTION
It was shown by Ballabh ( [2], [3] ) that when a flow with
velocity q satisfies the condition
curl ( q x curl q ) = 0 ••. ( 1 )
92 J
it becomes self-superposability. In the present paper the authors have
attempted some fluid velocities for which ( q x curl q ) = p ( say)
can be represented as the gradient of a scalar quantity . As such the
flow will be self-superposable . Attention has been focussed on incom
pressible fluids only . For such fluids flowing in conical ducts, some
velocities have been determined which satisfy the continuity condition
and for which p can be represented as the gradient of a scalar quantity
0 (say) . The solutions thus found must be of self-superp ,sable type .
Each will have a set of constants which can be determined by the
boundary conditions .
It has further been sh()wn by Ballabh [ ~J that if 'ii and q2 are
self-superposabJe'flows and qi ± q2 are also self-super-posable then
q 1 and q2 are mutually superposable. By using this property some more
self-superposable flows have also been determined pressure distribut
ions for some of the flows have also been attempted . The curves along
which the vorticity of the· flow becomes constant have also been
found .
In this paper a method will be introduced to solve the basic equa
tions of fluid dynamics in conical co-ordinates by using the property
of self-superposability , Mittal ([6],[7]), and Mittal, Tbapaliyal and
Agarwal [8] have recently solve the equations for different co-ore! in ate
systems·.
2. FORMULATION OF THE PROBLEM
Let q x curlq = p ... ( 2 )
For incompressible fluids we have
div q ~ 0 ... ( 3 ) fl'. .
. Now, if a solution of equation (3) be found in such a way that p
.~
Jr
,'
[L~H::
can,, be ,i:epresented ~s Jbe , gradiet\-t, ~"°f a $calar , gradient of a <scalar . . "'" ,. .. - . .. . '- .
quantity ( say 0 }i kti•·: :; .. •• -~, l: Ci_] p:-~(~
P =grade - ... ( 4) , ' (; i
it wil I give a self- superposable flow It ·has been shown-by f\.garwal ( ""'·.: . . ; \ .i - - - ' ,i :.. ~
[I] that such solutions will also satisfy the equation of inotion for a
steady-flow . For determin.ing a flow of liqµid ~n conical ~luct let us
consider the flow ill conical CO-"Ordinates ( U~ V, W )'[4] •, -·------;-( ~~.: l · . ·. i. L .
If qu, qv, qw be the comp~nents of q at any point ( u, v, w ) in con-
ical co ordinates then equation (3), will become.· · , ~ i" . :J ·;:, ·..1 )" c -- Ji· ......
I o u2 2ii' ( u 2 qu ) + ~ - b2 ) ( c.2 ..,... y2 ) ] ! ,, , u ( v2 "'" u2 ) . . . ,
1 _i_ [( v2 - w2) 2 qv]
(JV
,,'. ·- - I! j :.
~- /
. . f . . . '
) t..
i !, "' ' + [( b2 - w~) ( c2 - w2)] _j_ [( v2 _· w2)i qw] = o ... (5) u ( v2 - 11- 2 ) ow .,
, ' ' -· i I, \ {n Order tO make eqU~tlOri (5) integrable We;, Jilay COnSid~f tthe
following cases : !'. .... ~ -.r-. . t.;. ;) ".i ,J.: 'J\.,,;
C:'1se L: ,let. qu,,= 0. In .tbis<;ase_equ~tioi:r(.5)wiU be satisfied by· "'''""·'•'· '" - - . ;; ... • <, ., ~ . • : •• '
a solution ... (: . -~.) :: ~ .~ . .J'
qu = 0 l
"" -~',;,: q v: =::
\4 U(i;) ·w(w) ·; '"' l "·:, >' --·-- ~-:L<( ~2 •-~ .\.j~-)t:''<''\t.'i.~.-
1 ~,~ , ...
'' -~.~~ f6 f' ·., ·. :Jr~~1
"·'"f.''· .... , .. ,""
B U1 (u) ''v( v) I · 1 I
(- v2 _ w2 )2 J qw =
whe,r~ U (u), U1 (u) are tpe integrable fhrfutioll's df ~;:' V ( v) and W (w)
the :o i~tegrable functiorls b.f 'V,and w, rrespecth>.e}yJ and , A, B the
constants .
94]
For this fluid ;yelocity; it caireasily be shown that p can be repre..:.
sented by the gradient of a scalar quantity 0 given by
fl·= [ A2W2U2 + B2V2Ui) + _A2
W2 f u2 du
2 ( v2 - w2 ) ( v2 - w2 ) J u
B2V2 + -
( v<1- w2 I [J2
-1
du + B2u2 J vv~. d u 1 ---( v2 - w2)
~AB [(b2-w2) (c2-w2)]! UU1W'
J Vdv .
( t2 - w2) [ ( v2 --b-2_)_( -c2 - v2)J!
+ A2U2 f WW' d~ - AB [ ( v2 - b2 ) ( c2 - v2 )}! UU1 V' j(v2-w2)
. Wdw
J
~
( v2 - w2 ) [ (-bi - w'! } ( c2 - w~ )]i J .. ( 7)
Here U ( u ), U1 ( u ), V ( v ) and W ( w) are represented by U,
U1, V and W, respectively, and U', U1 ', V', W' reprt>..sent the
differentials .
By cbosing different suitable sets of values of U, U1' V and W
we may get a number of self~superposable fluid velocities • One of
such velocities can be obtained by taking
U1 = U = u ) v = [ ( v2 - b2 ) ( c2 _ v2 )]! ~
W = [ ( b~ ·- w2 ) ( c2 - w2 )]!
A =B
I
j ... ( 8)
Jt
the fluid velocity will become
q,.. = 0
qv =Au [
qw =Au [
·For this flow
) I
( b2 - w.:_L~ c2 - w2) ]! j (v2-w2) }-
I ( v2 -:-:-_ b__: ) ( c2 - ~) _ ]! I
( v2 ..;_ w2 ) J .
[ 95
... ( 9)
0 = A'l. u2 [ ~ {( b2 - w2 ) (c2 -- }112 ) + ( v2 - b2) ( c2 _ vz )} 2 (v2-w2)
+ 2 {( b2 + c2 ) - ( v2 + w2 ) } log ( v - w ) - ( v2 + w2 ) J. ··'( 10)
When U, Ui. V and W are constant quantities, then
qu = 0 l I
q. = C1 I i I
[(v2-w2)]2 }-
qw=~ [( v2 - w2 )]!
For this fluid velocity
0 = Ki ( v2 - w2 ) log u •
1 I I J .
... ( 11 )
. •. ( 12 )
Cass II : When q~ = 0. In this case the self-superposable flows
may be
9()_ 1
(i) qu = AiV1 W1 lj2
c qv = 0
qw ~ B1U2V2 I . . ----· .l
' K v2 - w2 )]2 r ·.
1 --·' I I I ~
:I -- I
I -J \
. ; ~
... ( 13 )
and '-
(''
e ~ [ C ,, !__ w' l l Bi~ l u2u2 du + B1V! I : d~ ' ' v '
,, '•H";'\':-; .~;'\ '; :~J ."\.' . + A B r ( b - __ w H c :::::::-: w~>J v v "w I I U2 dul .. I 1 L ... ( 2 '> ) 1 2 1 3 .. - . v. - w~ u
: .t' ~·.... .. .• ). ·--· ( • .• ·_, _·--~ r tt .. :. ' . ~ .. ~ --- ( ~·. _.· ;., ~
.A1Wi
+ -ur f ViV1 '(/v . ·•··
j !C 112-:::- w2,)Jf , ----- •• ·-. •>' ,,. --
< - ~- - -- ',
2 - .l J . ViiV;' - dv
+ B~U2 [(v2- w2)]2 -f: AiV~ f ... W1W1' dw t ij4 j ,· [( v~ - w2)]l
'' 1: \ ~' /
(ii)
A1B1 ( U2 + U2' ) V1V2 u2 I
• .. _ W1 dw . J [( q~_-'w~')(c2-"-'W~ f ... (14)
>v:':!'- ... i .:: ~;_·_--·';,~::~tr ·c·:./'.J
qu = A1 V [(b2 _ w2) (c2 _ w2)]! 11 u2
''; 'J .. qi!= 0
, I J l •.. , -J : •• fts)·
~
lf.w: ... ,,r , .... 11, :u.~ ~l; : ·:'.: :: ~:_ J ~ : _·,.: I · [( v2 __:_ w2>J - J
,". ~·I_,'. i"I .'
·i .· ~- :_..;:·.~ - _}
"'.J...: ,fL~'J
"
-~
(iii)
97 ]
e = A2 [--u2v2 -~~ ) u + l _ u ( 2 w2 - b2 - c2) } 1 ( v2 - w2 ) u2 l ( v2 - w2 ) 3 / 2
+ ( v2 + w2) z ( b2 - w2 ) ( c2 - w2 ) + - v2 ( v2 - w2 ) 1 { 2 u4 3
- ( 2 v2 - b2 - c2 ) + u61 J
qu = _C2 u2-
qv =:= 0
qw = D2 ------( v2 - w2 )t
I I I I ?I 1 I
j
... { 16)
••• ( 17 )
Case HI : When qw = O, some self - superposable flows may be
( i)
e =[(
· A2VsW2 _ qu =
u2
B2U3W3 qv = 1
( 112 _ w2 )2
qw = 0
B~W§ .. f -. --2. ) )
U§
u
I I I I ?-I I I J
+ 1 B2W2
2 3
( v2 - w2)
.. { 18 )
J UsUs' du
[ ( v2 - b2 ) ( c2 - v2 )]t , ~ U3 du - AzB2 .-· - Vs W2 Ws --( 112 _ w2 ) 3/2 u3
98 ]
(ii)
A~W~I +tf!
VsVs' dv .l ( v2 _ w2 ) 2
AzB2Us'W2Ws -- u2 f
Vs dv
[ ( v2 -b2 ) ( c2 _ v2 )]fr
AzB2Us' W2 Ws J Vs dv - 1. u [ ( v2 - b2 ) ( c2 _ v2 ) p
A~ V§J + U4
W2W2' dw .l ( v2 _ w2) 2
+ B~ U~ f WsWs' dw ] ... (1 9) j ( v2 - w2)
qu = 42w [( v2 -b2) (c2 - v2 )]! \ u2 I
A2u ~ -.1
qv = (v2 - w2)2
qw = 0
I ~ I I I
J
0 = A2 r -z;2w - lu2 log ( v2 - w2 ) 2 L< ., ) 2 V2- w~
+ : 2v { __ b2 ± c2
32 ~-- 1 }
( v2 _ w2) / l
+
+
( v2 - w2 ) l { ( v2 _ b2 ) ( c2 - v2 ) u4
1 3
w4 ( b2 + c2 _ 2 v2 ) } J
... ( 20)
... ( 21 )
""
[ 9-9
( iii ) _ C3 l
qu- ---- I u2 I
I Ds ?- ... ( 22)
qv = ( Q _l
I v~ - w2 )2 I
I I
q,v = 0 J
Case IV : When qu = q" = 0 . In this case the possible solution
to the equation (5) may be
q,, = 0 I I
qv = 0 I ?-
... ( 23)
qw = B3U4V4 I ( v2 -- w2) i I
J
Case V : When qv = qw = 0, then a self-superposable flow may be
I AsV5W4 I qu = -u<i I ?-
qv = 0 I I
qw = 0 J Case VI: Wh~n qu = qw = 0. In this case a flow may be
q,, = 0
qv =
q,,, = 0
A4U5W5 1
( v2 - w2 )2
I I I ?I I J
.. ( 24)
... ( 25)
In all the above cases Un, V,,, W,, ( n = 1, 2, 3, ... ) are the
integrable functions of u, v and w respectively and A,,, B,., Cn, D,,
( n = 1, 2, 3 ... ) are the constants which may be determined by
boundary conditions .
100]
3. SUPERPOSABLE FLUID MOTIONS
It has already been shown that the hydrodynamic flows given by
equations (6) and (24) are self-superposable. By the method discussed
in preceding pages it can easily be shown that if q 1 aud q 2 be the
two flows (6) and (24) then p for q 1 + q 2 can also be represented by.
the gradient of a scalar quantity. Thus, according to Ba!labh [3 J, C(1
and q 2 will be mutually superposable and a flow
qu =
qv
qw =
AV(v)W(w) u2
BU ( u) W ( w) .l
{ v2 - w2 )2
CU(u)V(v) - .l ( v2 _ w2 )2
I I r I ! >I l I I J
... ( 26 )
is possible. The same flow can be determined by mutually superposing
the flows (13), (25) and (18), (23) .
4. PRESSURE DISTRIBUTION
It will be interesting to note that 0 is nothing but be rnoulli function [I] given by
e = __ <]_ + h + .l!._ 2g g
... ( 27)
where q, g, h and p denote velocity, acceleration due to gravity, height
above some horizontal plane of reference and the pressure head .
It is a well known fact for an incompressible fluid the pressure
head Pis given by [5]
p = P I po + constant, ... ( 28 )
where p is the pressure distribution .
,k_
~.
[ 101
Also, if the motion of the fluid .be stea:dy and ,slow then value of h can
be taken, without much loss of generality, as u for the flows (6),(9),
(1 ij,; as ~ f~; the fl~ws (13)' (l sj, (17) _and .w fo: (18), (20), c'22)' .'. . ' •. ' f :. . '- . . . . -·. . ( ·~.. ~ t
T~us for the flo .v (9) pressure ,distrihµtion can be determined as . - - -· .. - - ,_ - " '- ~ -
P = r K0u2 ( b2 + c2 - v2 - w2) + K1u2 ( v2 + 112) log ( v - w) :...
+ K<;.u2 + Kau + K4 ]
where Ko, K1, K~, Ka and K4 ae constants.
Similarly, for flow (11), taking C1 = D1 we have
p = l- _(Ko'_+ Ki' lo~ + K2'u +Ka'] ( v2 - w2)
·Here K0 ', Ki', K2 ', and Ka' are constants.
..• ( 29 }
Uo (30)
Similarly, the pressure distribution for the flows (6),, (13), {15),
(17), (18), (20) and (22) can easily ,be determined.
5. VORTICIT.Y OF THE FLOW .. It was shown by Ballabh [J] that foi: a. self .,..superposable flow,
vorticity is constant along its stream lines . If.T is a unit tangent,along
a stream line then Txq=O ••• ! ( 31 )
By eq~ations (17) atld (31) it can readily be'shown be that
' T == I ' '• ' 0 ' ' •. : ' ' 1 ... ( 32 ) r( v2 - w2 )i;2 u J
. L u2 + ~2 - w2 . [u2 + v2 _ w2]2 · '
Hence .the vorticity of the flow ( 17) is . c:onstant alon_g the curves
represented by equation (32) . Similarly the curves of constant vorticity
can also be found for other. flows .
102 1
Acluiowledgme~ts
The aathors take this o,pportunity to expi'ess·their cordial thanks
to Dr. R, N. Gupta for his inspiration and help in preparing this
paper . Thanks are due also to refereefor'his valuable sugg~Stibfis .
REFERENCES
[ I ] G. K. Agarwal, Some Studies in Non-Newtonian Fluids, Ph. D.
th'esis, Garhwal University, 1984 .
[ 2 1 R. Ballabh, Superposable fluid motiOns, Proc. Benares A-lath.
Soc. (N. S.) 2 (1940}, 69-79 .
[ 3 J R. Ballabh, On superposability J. Indian Math . Soc . 6 (1942),
33-40.
[ 4 J F. A. Hinchey, Vectors and Tensors for Engineers and Scientists,
Wiley Eastern, New Delhi 'f976 .
[ 5 ] R.. V. 'miSes, Maihemat°icai Theory of'Compressible ·Pfuid"Flow,
Academic Press, New York, }9 58 .
[ 6 ] P. K. Mittal, Lorentz force as the gradient of a scalar quantity
aricPt'h'e 'Nf-HD;ft6'w :un(ter ;forte .... ft~e1:in~fgnetic ~fieldin ;elHptic
'cylinder 'co..::ordinate.s &y.Sfein, '.Bull. Cl#eutra,Math . Soc . .r73 /
(1981), 213-2 7 .
[ 7] P. K. MHtal, On some magneto-hydrostatic configurations in
parabolic co-ordinates . Bull. Calcutta Math • Soc. (to appear).
[f8'] JP. K iMittal, P. S. Thapaliyal a11.d G. X. ·Agarwal, !A note on
some'self-superposable fluid motions in cylindrical du~ts having
'·e!Hptfc1cr0ss1'sect1on, 'J.'Math .iamf!Phys .-Sci .~,(to;appear).
[ 9] M. "R.. Spiegal, Mathematical Hand-Book of Formuals and
Tables, McGraw-Hill, New York, -1968 .
-~
-~
Jfiiiniibha, Vol. 16, 1986
THE GENERALIZED PROLATE SPHEROIDAL WA VE
FUNCTION AND ITS APPUCATJONS
/Jy
V.G. GUPTA
Department of Mathematics, University of Rajasthan,
Jaipur - 302004, 'Rajasthan, India
( Received·: January 17, 1985 ; Revised: September 13, 1985)
ABSTRACT
Here we employ the generalized p:i:pla,te s.ph~roidaJ \\\llYe,Jqpqt~on . . . . ' ' -
in obtaining the formal solution of the partial differential equation
related to a problem of heat conduction in ,an anisotropic material •
This type of problems occur mainly in wood teclinology, soil
mecban ics, and the mechanics of solids of fibrous stru<:ture. The
solution of the problem when the arbitrary function F(x) is giv.en in
tetms of Srivastava-Panda's multivariable H-function has also been
obtained . The results of this paper unify and extend a large number
of remits e:;tablished by various earlier .workers .
1. INTRODUCTION
The solution oflhefoHowing differential equation
( I - x2 ) -~~~, + [ \ ~ - ex) - ( ex + ~ + 2 ) x ) -~Y_ dx2 dx
+ [ i; ( s) + s2x2) y-·O ... ( l I )
is the generalized prolate spheroidal wave function ( .. see, e.g., Gupta
[ 3 ], p. 107 eqn. ( 2. l l ) ) which has been denoted as
a,~ oo ex,~ { ex,~ ) "' ( s,x) + ~
n p=O R (s)P (x) p,n p + n
... ( 1.2 )
104 ]
provided thats = O, ~ (0 ) = ( n t p )( 0t + {3 + n+ p + 1 ), p > 0.
Recently, Srivastava and Panda [7] have introduced and studied
the H-function of several complex variables by means of the multiple
Mellin-Barnes integral ( see also Srivastava, Gupta and Goyal [ 6 ],
p. 251 )
0, e : ( w' v' ) ; .. ; ( u<rl ' v<r>) ([ (a ) : 0' , ... , e<r> ] : H [ z1,. • .,zr ] = H
. .. A,C: ( B",D') ; ... ; ( B<r>' D<rl) [ ( c) .: ~' .... , ijJ<r>]:
[(b'):rp']; ... ;[(b<r>):rp<r>]; )
; [(d' j: 3' f;.:.; [ ( d<ri ): 3<r>]; z1, .. .,z,
= (2mo )7 r ... r U1(s1) ... Ur (Sr) v (s1, ... ,s. ) j Li . JL,
S1 Sr .y-Z1 ••• Zr ds1 ... dsr, where w = - 1,
u<o n r [ d/i> ,....... 3;<i> Si 1
j = I .
U; (SI)=----------DW n r[l-d;<i>+3/i>s;J
j = I + u(i)
vw n r [ 1 - b/i> + rfo,(i> s, 1
j = 1
B(i) n r [ b/il - rfo/i> s; 1
j = vW + 1
... ( 1.3 )
, i=:= I ··~·· r
)..__
[ iOS
V(si.····Sr) =
e r II r [ 1 - a; + ~ 0;<il s; J
j=l i=l
A r C r II. r [a;,- ~ ~/il Si] II r [I - c;+ ~- 'P'/il Si]
J=e+l i=l j=l i=l·
and, for convergence,
I arg z; I < ! Trrr., i, ... ,r ... (1.4)
where
A vii> B(i) T; = - ~ 0;(i) + ~ 01w - ~ 0;( 0
j = c: + I j = l j = v<i> + I
C u<il D(i) ~ 'f/O + ~ a;u1 -,... ~ a/ i)
j=l j=l j = u(il + l ·
> O,i=l, ... ,r ... ( 1.5)
We require the following result due to Chaurasia and G1._1pta
( [ 1 ], p. 82, eqn. ( 13 ) ) in our investigation :
rl a,~ h ( l - x )p ( l + x )a cf> ( s, x ) H [ z1 () - X ) 1 .
n
( I + x / 1 , ... , ~r ( 1, ---:- xl• ( x+l )"• J dx. ',;
oo a,~ p + a + ~ R (s)2
p = 0 p,n
n + P ( ~ ( - n-=p_)~+ ~ + n + p + l )m m ,,;,, 0 ( ot + 1 )m m !
106 J
H * [ ZJ 2 h1 + k1 ,. . ., Zr 2hr + kr J ... ( J.6 ) m
where
* 0, L\: + 2 : ( u', v,) , •••• ( u<r>, v<r> ) H [ Z1 , •• ., Zr J = H m A+ 2, C + 1; ( B', D' ), ... ,(BCrl, DCrl)
(
. [ - a : ki , .. ., kr J ,
[ { C ) ; 'Y' , ... , qr<r> J '
'[ - m - p: hi, ... , hr J' [(a): 0' , ... , e<•l J: [ ( b' .) : cf/ J j ... ;
[ - I -m - p - a : hi + ki ; .. ., h, + kr] : [ (d'): 8'] ; ... ;
[ ( bM ) : q,cr> ] ; h1 + kl r 2 hr + kr ) z12 , .. ' z
[ < d"> > : a<•l 1 :
The integral ( I. 6 ) is valid for
r Re ( p + ~ h; #'> / 8/i> ) > - I ,
i = I
r Re ( a + . ~ k, d/i! I aj(i) > - I '
i =I
... (1.7)
h, > 0 , ks > 0 and, of course, T1 > 0 , I arg z, l < ! 7tT; ,
Re (a) > - 1, Re ((3) > - 1, i = I , ... , r; J = I , .•. , u(i)
The orthogonality property of the generalized prolate spheroidal
wave function (Gupta [3], p, 107, eqn. (3.I) ) is recalled in the
following form :
11 cx,(3 a,(3
( I - x )"' ( 1 t x )a cf> ( s, x ) tfo ( s,x) dx -1 n t
a,(3 == N 8nt .... ( 1.8)
n,t
A
c·rn1
where
ct,~ 00 ct,~ .N = }.; ( R (s) )2 • n,t p=O p,n
2"'+P+l r ( n + a + p + 1 r ( n + ~+ p + 1 ) -------(2n +2p + « + ~ + 1) r (n+ p + 1) r ( n + p +a + ~ + 1)
and 8nt is the Kronecker delta .
2. FORMULATION AND SOLUTION OF THE PROBLEM
The partial differential equation related to' a problem of heat
conduction in an a:.nisotropic material has been obtained by Saxena
and Nageria [5] and is given below :
~- [ ( l -x2) ]!!__] - pc" ~ + qj_x)= pc Jl!__ ... (2.1) ox ox A ox A A ct
With the law of conductivity K = ..\ ( 1 - x2 ) , Q ( x ) is the intensity
of a continuous source of heat situated inside this solid • Let the
Initial temperature of the rod be given by
u ( x,O) = F ( x)
If we take
... (2.2)
a-~ ou pc " = - , Q ( x ) = - ( ct + ~ ) .\ x _ - s2xll >.u, q = _ q ox ,\
then the solution of (2.1) can be written in the form
oo -Bi.t «.~ u ( x,t ) = }.; Ai e
l = 0 cf> ( s,x)
I
Substituting this into equation ( 2.1 ) , we have
B1
= (! +P ) ( l + P + « + ~ + 1 ) q
... ( 2 3)
10s I
In order to find the value of Ai , we make use of the initial conditi.on
(2.2) . This gives
oo a,(3 F ( x ) = ~ Ai </> ( s,x )
I= 0 I
Multiplying both sides of (2.4) by
a;() ( 1 - x) °' ( 1 + x )a rp (s,x)
I
... ( 2.4)
a11d i~tegrating with respect to. x from. -I to 1, we obt~in .the
result Ai= Gi
a,(3 Ni,i
·1 I a,(3 where Gi = ( 1 - x r ( 1 + x )a rp ( s,x ) F ( x ) dx -1 I
... { 2.5)
... (2 6)
This is because the generalized prolate spheroidal wave function
possesses the orthogonality property (1.8) •
Hen'd: the solution of the problem may be expressed as
u ( x~·t) 00 cx,(3:- l
~ ~ Gi ( N exp [ _(./ +P) (I+ P +a + (3 +I) t] I= 0 /,I q
a (3 </> ( s,x) I
... ( 2.7)
If we takes= 0 in (2.7) , we get the. solution of the problem in
terms of the Jacobi polypo,mials given eadier by Saxena and. Nageria .. - r• - - • • •
( [5 ), pp. 1-6 ) . Again, setting a = (3 = 0 and s = 0 in (2. 7) , we get
the solution of the problem ip terms' of L_eg~ndre ·functions given by
Churchill ( [2], p. 224 ) .
.~
[ IQ9.
3 •. •EXAMPLES
Let
F( x) = ( 1 - x )P - ex ( 1 + x )cr - ~
H [ Z1 ( 1 - X )h1 ( l + X fi , ... , Zr ( 1 ~ X lr ( 1 + X lr]
then by ( 2.4) we have formally that
( l - x l - ex ( 1 + x )cr - ~
H [ Z1 { l - X )h1 { l + X ll , .. , Zr { 1 ""'"'.'" X ) hr ( I + X }k' ]
oo ex,~ ~ Ai cp ( s , x) ... ( 3.1 )
l = 0 l
Equation (3. l) is valid since F ( x ) is continuous and of bounded
variation in the open interval ( - 1,1 ) .
Now multiplying both sides of (3.1) by ( 1 - x )°' ( 1 + x )Ii
ex,~ cp ( s,x ), a > -'1, .. ~ > -1 and i11tegrating with respect to x from n
-1 tu l, change the order of integration and summation (which is
permissible) oh the right, we obtain
I 1 a,~
. ( 1 - x) P ( 1 + x )cr <P. ( s,x) -1 n
· H [ Z1 ( 1 - X )h1 ( l + X )k1 , ... , Zr ( 1 - X )hr ( 1 + x, )kr )dx
110].
oo I 1 "' a ot,~ a,~ = ~ A1 ( 1 - x) ( I + x) rjJ ( s, x ) rjJ ( s , x) dx
I = 0 -1 I n '·'
... ( 3.2)
Using the orthogonality property of the generalized prolate sphe
roid~l w,ave func~ion (1.8) on ~he right-h~nd side and the result ( • .6)
on the left hand side of (3 2), we obtain
00 (!'~ Ai = L R (s) 2P + cr +
p = 0 p,n
r ~ ( . - n - P Jm (·a + ~- + n + p + I )'.~'. n~p( { m = O ( a + 1 )m m !
i *. ii [ z1 2h1 + ki , ••. , Zr 2hr + k, ] m
. '*
()( ~ N · t ,I
rl .J
... ( 3. 3)
where H function is defined by ( 1.7) on supstituting the value m ,' .... : : ;-.~- · - .... ;
of Ai from (3.3)in (2.3) and using the lemma ( [4], p.57, eqn (2)),
we a~'rive at the solution of the problem (2.3) as
u ( x, t )= 00. p p 1; ~ L 2P + cr + I
p=O 1=0 m=O
ot,~ R (.~)
p - I, I { . ~ '} ·1 ( - P )m ( ot + ~ + P + l )m · Na• '. - .
( a + l )m m ! /, I
' / . - : . .
H* [ 21
2h1 + k1 , .. , Zr 2hr + krq'e-('pjq') ( jJ +·a -F ~ +. l/)t m , .
. . ,. a-,(3c; ·.. " ?> . ( s,x ) l
,, ., .. [. J \. I / .• :: ( 3.4)
.-~
~
[ Ill
valid under the condition mentioned with (I 6) .
We :notitb that th~ multivariable- Ji_:_function involved in our result·
(3 4)is quite general !n character; Jndeed, by: suitably~_sp.e~ializing the
parameters of this ft..nction we_,can t".asily .obtain solution of the '-.·.,· l :' ' . -
problem in terms of a large number of elementary functions of one
or more variables ( or produqt of several such functions ) : .
Acknowledgements
, ... i . .,
The author is highly thankful to Professor M. C. Gupta and
Dr. S. P. Goyal ( University of Raj::l:sthan; Jaipur:) for their keen • • . • . • .. :~ .•• ' • L· ··;· • . ' ·~ • • ' •
interest and useful suggestions in the prepal.'ation fo thi's 'paper ;;He is
also tl1ankful to Pr~fessor ~l. M.;· Srivastava fo.f"· t"h~ var;fous impr"ove
rnents in this paper .
REFERENCES
[ l ] V. B L. Chaurasia and V. G. Gupta . Some results on the
H functkn of several complex variables involving the
generalized hypergeometric function and the generalized prolate
spheroidal wave function, JD.anabha 13 (1983), 73-85.
[ 2 ] R. V. Churchill, Fourier Series and Boundary Values Problems,
McGraw-Hill, New York, 1942.
[ 3 ] R. K. Gupta, Generalized prolate spheroidal wave function,
Proc . Indian Acad. Sci. Sect . A 85 (19771, 104-114 .
[ 4] E. D. Rainville, Special Functions, Macmillan, New York,
1960.
112 1'
[ 5] V. P. Saxena and A. R. Nageria, Linear flow of heat in an
anisotropiC finite' solid moving in a conducting medium·,
Jfi~~·~bh~·s~ct . A.' 4 (1974); f:-6 .
[ 6 J H M. Srivastava, it. c. Gupfa a~d s. P. Goyal, The H-Funci
ions of One and Two Variables with Applications, South Asian
Publishers, New Delhi and Madras, 1982 .
[ 7] H. M. Srivastava and R. Panda, Some bilateral generating func
tions for a class of generalized hypergeometric polynomials
J. Reine Angew . Math . 283/284 {1976), 265-274 •
"~
J.iianabha, Vol. 16, 1986
ON A GENERALIZED MULTIPLE TRANSFORM. I
By
K. N. Bhatt and S. P. Goyal
Department of Mathematics, University of .Rajasthan,
Jaipur - 302004, Rajasthan, India
( Received: February 25, 1985; ReviseJ [Final]: February 3, 1986)
ABSTRACT
In the present paper we study a general multip1~;httegraltra:nsform
in which the kernel is the product of Fox's H- functions and the
JI- function of several variables which was introduced by H. M. Sriva
stava and R. Panda Our transform provides interesting extensions of
various classes of known integral transforms of many variables whose
kernels are expressible in terms of E, G and H- functions of one and
more variables. Several special cases of this transform have been discu
ssed briefly. we lastly give certain elementary and useful properties
for our transform.
1. DEFINifION
we study the multiple H- function tranform defined by
rfo{ S1 ,. •• ,Sr)= Hr { f: s1, .. .,s,} (1)
Joo I oo r l u; , O r I ~(s1 , •. .,s,) .... rr H i t;s;x;I Q Q l = l Vt , Wi ._.
(k/il, p/il)l ,Vi
(h,<" • .,w)l • "" JJ 0 '0 : m1 • ni; ... ; mr 'nr
H P • q ; Pl ' ql ; ... ;Pr' qr [~1 S1 X1
Ar Sr Xr
114]
( a1 ; a;' ,. .. , a/rl )v:i> :
( b1 ; ~/ , ••• , ~/rl )t>a :
·<rl (c ' y·') •• (c· y-<n>) 1 •
3 l, p1 ' ... ' ' ' ' 1, Pr I
(r) (rl
I I if ( X1 , ... , Xr) dx1 ... dxr. I
( d;', a;' )1, ql; ... ; ( d; ' a; )1, qr I
.J
The H-function of one variable appearing in (1.1) was first intro
duced by Fox [3] . The details of this function can be found in a
recent monograph by Srivastava, Gupta and Goyal [8] . On the other
hand, the multivariable H-function :
0, 0: m1' ni ; ... ; mr, nr (1.2) H
p, q; Pl' ql ; ... ;pr, qr [
z1 I ( llJ ; a;', .. , ot/r> )J,p :
~r ( b; ; ~( , ... , ~;<rJ )J>q :
( / ' ) . . (c (rl y (r) ) J C;,y; l.p1•···• i 'i l,pr = H1 [ z1 , ... , Zr ]
. ' ,1
• • .(r) Cr> ca,,a, )1 , ... , (d, , a1 ,)1 • ql , qr
. '
( 2rr;l w )-:;:- J L1 ... I Lr rp (s1 , ... , Sr ) 01 ( S1 ) ... 0, ( Sr) z/1 ... z/'
. ds1 ... ds,, w = v-1,
is a particular case of the H· function of several complex variables
( or the multi variable H. function ) which was introduced and studied
by Srivastava and Panda in a series of papers ( [5], [6] and [7]; See also
[8], p . 251, Eq. (C.l)).
The multiple integral on the right-hand side of ( l. I) exists and
defines the; multiple H-function transformation of a given function
f ( x1 , ... , xr ) under the following sufficient conditions :
.-~.
..1
Wi Vi
( i ) Ti = ~ cr;w - ~ p/i> < O ; j=l j=l
Ui WI Vi
(ii) Si = ~ j = 1
cr/il _ ~ cr;W _ ~ p/il > O, j = u;+ 1 j = 1
I arg t; St I < ( I /2 ) S1 l't;
( iii ) L\; > 0 , I arg A.;s; I < ( 1/2) L\t 'It , where
p L\; = _ ~ a/ii
j=l
q
ni
+ ~ j = 1
'
nu
Pt '(j(i) - ~
j~n1+l
q;
[ 115
yii)
~ ~;(O + ~ o/i> _ ~ Oj(i). > 0 , j = 1 j = 1 j =mi+ 1
\f i E { l , ... , Y } ;
( iv ) f ( x1 , ... , Xr) is a real-or complex-valued function of r varia
bles X1 ' •• , Xr defined on R : O< X1 < 00 , ••• , o~ Xr < 00 and
such that the product
r l ~. + n1 J . I1 Xi f ( X1 , .•. , x, ) is integrable ( in the sense of l = 1
Leb:!sgue ) over R ( Ri. R2 , ••• ,Rn ) : 0 < x; < R1; R; > 0,
where
~i = Re ( h;c n / cri< 0 ) and min l } 1 < j <ui
. .
min ·1 ' . } . . • - . (i) .. (i) . . • . .. . • -'i• - . Re ( d, t_a,, ) v 1 E {. 1, ... , 1 }.,
1 < j < mt •' . - .
116 I
and
( v) the limit of the finite form of the multiple integral in (1.1) with
f 00 Joo t Ri JR" j 0
,.; Q repJa-0ed by ) Q ••· Q exists at the point ( St , ... ,Sr )
when Rt , .. ., R, --+ oo . 2. SPECIAL CASES OF THE INTEGRAL TRANSFORM
DEFINED BY (1 l)
The integral transform (l.l) is quite general in character; indeed
its various specialized ·and confluent forms can be suitably related to
the several classes of known integral transforms of one, two and more
variables considered in the literature from time to time by various
research workers . Some interesting connections t of this transform
with the known integral transforms are mentioned below :
If we take u; = w, = I, w = 0. cr1<i) = I, h1<i> = 0 in (I.I) and,
using a known result [ 8, p. 18, Eq .·(2.6.2) ], we get
(2.1) </; (s1,. .. ,sr)=(s1···Sr) J00
••. f00
exp {- .~ (s;l;X;)} 0 0 l = 1
Hi [At s1 Xi, .•• ' Ar Sr Xr ] f ( X1 , •. 'Xr) dx1 dxr
provided that the multiple integral involved in (2.1) is absolutely
convergent •
t Similar multiple fl-function tr~nsformations have been studied .. by Y. N. Brasad and K. Nath [Istanbul Tek. Univ. Bu!. 36 (1983),
361-371 ; Vijnana Parishad Anusandhan Patrika 27 (1984), 23-43].
~
4
[ 117
The integral transform (2 1) is clearly related to the transform
studied earlier by Srivastava and Panda [7, p. 119, Eq. {I.I)] ..
Further, if we let t; - 0 ( i = 1 ,. . ., r ) , the integral transform (2.1)
will reduce to another integral transform studied by Srivastava and
Panda [ 7, p. 121, Eq. (l.15)].
(2.2) rP ( S1 ,. • ., Sr)= ( S1 ... Sr) r= .. (00
H1 [ ,.\1 S1 X1 ,. .. ,Ar SrXr] Jo Jo
. f ( X1 , ••• , Xr ) dx1 ... dxr ,
provided that the multiple integral in (2.2) converges absolutely.
Again, if we take r = 2 in (I.I), we shall obtain the following
double integral transform introduced and studied by Agarwal
[ I, p. 131] :
Ioo i= U1, 0 r I (k/, p/)1 v ] (2.3) di (sh s2 ) = s1 s2 H Lti s1 X1 ( h;', cr/ )
1
1
0 Q Vi, W1 · , W1
[
k II II ) J u2, 0 j ' p; I v . H t2 s2 x2 / h~ , ' ' a; ,, ) ' ,2
V2, W2 1, 142
H1 ( ..\1 s1 Xi. ..\2 s2 x2 J f ( x1. x 2 ) dx1 dx2 ,
provided that double integral (2.3) converges absolutely •
In (2.3), H 1 [ x,y ] stands for. the fl-function of two variables ( with n = 0) studied earlier by Mittal and Gupta ( [4] ; see also [8] ).
We should mention that the aforementioned transforms (2. l) and
(2.3) are quite general in nature and include a large number of other
well-known integral transforms of one, two and several variables .
For details, refer to the works by Srivastava and Panda [7], and also
by Agrawal [2] •
118 ]
3. ELEMENTARY PROP.ERTIES
Property I.
(3.1) J~··· J~ g ( S1 , ... ,Sr) H, { f: S1 , ... ,Sr 1 ~:1_ dsr S1·
=J~· .. J~f(s1, ... ,sr)Hr{g:s1 , ... ,s,} ~s1 ds., Sr
provided the multiple integrals converge absolutely .
Property II.
If the multidimensional H ,-integral transform of j ( x1 , ... , x, )
defined by (I. I) exists; then
(3.2) Hr{ J ( fl-1 -'t , ... , p.r Xr); S1 , ... ,Sr}
~ Hr {· f ( X1 , .. , Xr ) : ~ , .. , !~ } µl µr
provided that µ 1 , ... , µ.r > 0 .
Evidently, this property may be called similarity rule for the
multiple H,-transform defined ~y (I. I)
Property III.
If the multidimensional H,-transform of f ( x1 , ... , x, ) defined
by ( I. I) exi&ts, then
(3.3) . ~ /'::..; H, f f1: S1 , ... ,Sr }
J = 1 l . .~ H,. {61/i :s1 , .. ., Sr l, j=l j
where 6. 1 , ... , f'::..N are constants . This property may be looked upon as linearity rule for the multiple H,-integral transfurm .
-~-
[ 119
In our next communication we shall give an inversion formula
and study some other important properties of the transform ( 1.1 ).
Acknowledgements
Tbe second author is thankful to the University Grants
Commission for providing financial assistance for the present, work.
· The authors are also thankful to Prof. M. C. Gupta, Prof. K. C. Gupta
and Prof. H. M. Srivastava ( Canada ) for making some very useful
suggestions and encouragement in the preparation of this paper.
REFERENCES
[ 1 ] R. K. Agarwal, On a two-dimensional H-H integral trans-
form. II, Kyungpook Math J. 22 (1982), 131-140.
[ 2 ] R. K. Agrawal, A Study vf Two-Dimensional Integral
Transform and Special Functions Involving one or More
Variables, Ph. D. Thesis, University of Rajasthan,
India, 1980.
L 3 ] C. Fox, The G and H functions as symmetrical Fourier
kernels. Trans. Amer. Math. Soc. 98 (1961), 395-429.
[ 4] P. K. Mittal and K. C. Gupta, An integral involving genera-
lized function of two variables, Proc. Indian Acad. Sci.
Sect. A, 75, (1972), I 17-123.
[ 5] H. M. Srivastava and R. Panda, Some bilateral generating
functions for a class of generalized hypergeometric
polynomials, J. Reine Angew. Math., 283/284 (1976).
265-274.
120 1
[ 6 ] H M. Srivastava and R Panda, Expansion theorems for the
H-function of several complex variables, J. Reine !"', Angew.Math. 288 ( 1976) 129-145.
[ 7] H. M. Srivastava and R. Panda, Certain multidimensional integral transformations. I and II, Nederl. Akad.
Wetensch. Proc. Ser. A 81= lndag. Math, 40 (1978) ,
118-131 and 132-144.
[ 8 ) H. M. Srivastava, K. C. Gupta and S.P. Goyal, The fl-Funct-
ions of One and Two Variables with Applications,
South Asian Publishers, New Delhi and Madras, 1982.
~
,.
Jnanabha, Vol. 16, 1986
AN EXPANSION FORMULA FOR THE H-FUNCTION OF
SEVERAL VARIABLES
By
GEETA ANEJA
Department of Mathematics, University of Rajasthan,
Jaipur - 302004, Rajasthan, India
( Received : March 8, 1985; Revised : August 12, 1985)
ABSTRACT
The aim of the pre sent paper is to establish a general expansion
formula for H-function of several variables which was introduced and
studied in a series of papers by H. M. Srivastava and R. Panda (see,
for example, [4] and [5]). lt !S significant to observe that a large
number of finite and infinite series for this function can be easily
summed up by using the well-known summation theorems for ordinary
h) perf:rnmttric series in the main result . Also, by appropriately ·
si;::ecializing H.e parameters of the H-function of several variables, one
can easily obtain expansion formulas for simpler special functions of
one aLd more variables .
1. INTRODUCTION
We begin by recalling the definition of the H-function of several
variables introduced and studied by Srivastava and Panda ( (4] and
l5] ; see also Srivastava. Gupta and Goyal (3 .p. 251, Eq. (C.l)]):
H ( Z1 ,. •• , Zr ) 0, n: mi, n1 ; ... ; mr,nr l- z1
H ; p, q; Pl> ql ; ... ;Pr, q, Zr
122 l
( a . ,. 1 ,., (r) ) • ( C ' y.' ) . . ( C .(r) y r) ) ; ' ... ; , ••• , ""1 l,p " ; ' ' 1,pl , ... , 3 ' 1 l,pr J
( b . r:>. 1 r:>..lr)) • ( -"' "'-') •• I d-lr) "'.(r)) ; , l-'1 ,. •. , l-'' I q . u; , 01 I q , ···, \ ' , o, I q
• • 1 , r
= ( 2n:~ Y JL1··· f L' r/>1 ( ~1) ... rpr ( ~r) \}!' ( ~1 , ... , ~.)
~1 ~,. • Z1 ••• Zr d~l d~1
( 1. l)
where w = ..; -1 ,
rpi ( ~· )
and
m; n; 11 r ( J;<il - 8,W i )
j =I TI r ( I - c/-n + y/i1 ~; )
J=1
q; P• 11 I' ( 1-d/il+ 8/il ~; ) Il r ( c/i1 - y/i' ~1 )
j = m; + I j=n;+ I
i€ {I , ... ,r} ( I 2 )
n r 11 r ( I - a; + L ix;1
i) ~i )
j = I j=l
'Y ( ~1 , ... ' ~. )
p 11
j=n+l
r r (a; - L
i= I
q a/ 1> ~; ) II
j=l
r r ( 1 - b; + L ~;Ii> ~; )
i= I ( 1.3 )
The nature of the contours in ( I. I ) , the asymptotic expansions,
conditions of convergence of{ 1.l) and particular cases of the multi var
iable H-function can be found in the papers [4] and [5] . See also the
book by Srivastava, Gupta and Goyal [3, p 25 I] •
.~
2. MAIN RESULT
A II (a/ )k zk
co j =I ~ B
k = 0 II ( b; I )k k ! j = 1
m1 + F1, n1 + G1
0, n + C H
p + C + D, q + E:
, ... ;m, + F,, n, + Gr
I 123
Pl + G1 -t- H1, ql + Fi + /1 ; .. ;pr + Gr + Hr, q, + Fr + Ir
I :1 ,- 'J , ... i..., l,C' ,, ', ... ,~, l,p' r z I ( u· k• p·' . ,,.<r) ) (a·. a·' N (r))
'--Zr ( b;; ~/ , ... , ~/rl )1, q' ( W; - k; e:;' , ... , €/rl )l, E
( v; + k; a/ , ... , a/' 1 ) 1 D: ( g;· -k, YJ; )1 G • ' l•
: ( i;' + k, µ.;')I F, ( d1', 8;' )1 , ( !;' - k, 11;' ) 1 I ; ' l ,ql ' 1
( c;', y/ )1 p, ( h;' + k, !;;') 1 H ; ' 1 < ' 1
• ( i-c•> + k, µ/'> >1 F ' ••· ' ' ' r
• ( g;<r)_ k 'Yl.(r)) . ( c.(r) y·<rl ) ( h·<rl + k l:';(r)) J ·· ' '.,, l,Gr' ' ' ' l,p,' ' '"' l,H,
( dj{r) "'.{r) ) ( /.{r) k V .(r) ) '
03 1,qr • 3 - '
3 1,lr
= ! w )' 1 L1 ••• tr cfo1 ( ~l ) ... cpr ( ~r ) '¥ ( ~l ... ~r }
[
A1 ( ~1 , ••• , !;r ) l F f' A+ p
B1 ( ~t , ... , ~. ) j [
( a A ) , Ai ( i;1 , ••• , ~r ) J B+Q
( bn') , B1 ( ~l , •.• ; ~r )
~l .;. • Z1 ... Zr d .i;l ••• d ~r , (2.1)
v.bere
124 J
Ai ( ~1 , .. , ~' ) stands for
r ( I - u1 + ~ p/il ~;)IC' ( I - g/ + 'Y);' ~1 )1 G , ... ,
i = I ' ' 1
(I - g/r) + 'Y)/rl ~r )J. Gr ' ( jj - µ/ ~l ) I.Fi ' . ' ( i;\rl - µ;(r) ~r ) J.Fr
B1 ( ~1 ,. •• , ~' ) stands for
r r ( V; - ~ cr/il ~; ) I D ' ( I - w; + ~ e:,(i) ~; ) I E '
i=l ' i=l '
( h/ - ~/ ~1 )J,Jll , .. ,( h,frl - ~;(r) ~r )I.Hr'
I I - !·' + v·';: ) ( I - /.(r) + v1<r) i: ) ' ' ' -,1 l ,/1 , ... , ' ..,, I J,
Formula {2.1) bolds if any one of the following sets of conditions is
satisfied
(i) A+P<B+Q+ I
( ii ) A + p = B + Q + 1' I z I < I
( iii ) A + P = B + Q + I , Re ( 0 ) > 0 , z = I
( iv ) A + P = B + Q + I , Re ( 0 + l ) > 0 , z = - I
where
r P = C + ~ ( G; + F, )
i = 1
r Q = D + E + ~ ( H; t /;)
i = 1
.~
0 = B ~
j=I
/; + ~ /j(r)
j= l
A b/ - ~ a;'
j = 1
G, ·~ g/r>
j=I
{ i ~ F
( ~ 1 j ...:=
i/1>
H1 E C D J . ~ h/1
) ) + . ~ Wj - . ~ U; - . ~ V;
J=l 1=1 J=I J=l
r + E - C + ~ ( f, - Gi )
i =
[ 125
Derivation of (2.1). To evaluate (2.1), we first replace fl-function
of several variables, occurring on the left-hand side of (2.1) by the
contour integral in (I.I) , and change the order of integration and
summation [which is justified under the conditions mentioned with
(1.1)] ; then we get the required result (2.1) on using the definition of
the generalized hypergeometric function (see, e g., (2] , p. 40; [3J ,
p. 2 ) . The routine details may be omitted .
3. PARTICULAR CASES
( i ) lf in l2.l) A = 2, B = 0, P = 0, Q = l ( E = l ) z = 1, using
the Gauss theorem [ 2,p 243 ( III. 3) ] , we get the following
summation formula :
00
~ k=O
( a 1 ' )1. ( a21 h
- k !
f"z1\ (a1;a;', ... ,1X;<•>) 1 H, . ,p i ; . ' (r) :.... - r ( W1 - k , e1 , •• ., <:1 ), { b4 ' P. (rJ ) ~j , •• ., tJJ l,q
( 127
(r) ) -; ... ; c;<r>, Y1 I, Pr I ; ... ;(d/•>,8/<>)1,q, ..J (3.2)
It is thus seen that a large number of summation formulas can be
obtained from our main result (2.1) . Hence our main result may <ie
regarded as the key formula for the summation of series involving the
H-function of several variables . It is not out of place to mention that
the results obtained by Vashishta and Goyal [6] , Verma l7] , Parihar
[l] , and others, can be easily obtained as special cases of main result
(2.l) '
Acknowledgements
The autl-.or is thankful to Prof . M. C. Gupta and Dr. G. K.
Goyal for their valuable suggestions, and also to Dr. S. P. Goyal
for his help and guidance during the preparation of the paper .
REFERENCES
[ l ] C. L. Parihar, General expansion involving S-functions, Math.
Education 8 (1974), A 29 - 46.
l 2] L. J. Slater, Generalized Hypergeometric Functions , Cambridge
University Press, Cambridge and London, 1966 .
[ 3 ] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Funct
ions of One and Two Variables with Applications, South Asian
Publishers, New Delhi and Madras, 1982 .
128 J
[ 4] H. M. Srivastava and R. Panda Some bi!ateral·generating func
tiuns for a class of generalized hypergeometric polynomials
J. Reine Ar;gew . Math . 283/ 284 ( 1976 ), 265-274 .
[ 5 ] H M. Srivastava and R. Panda, Expansion theorems for the
H-function of several complex variables, J. Reine Angew.
Math. 28S ( 1976 ), 129-145.
[ 6 ) S. K. Vasbistha and S. P. Goyal, An·expansion formula for the
fl-function of two variables . Vijnana Pari:,/wd Anusa;1dhan
Patrika 21 (1978) , 237 - 246 .
[ 7] A. \'erma, General expanskns involving £-furctivns, Math
Ze itschr , 83 (1964 ), 29 - 36 .
~
·Iii<
Jfianabha, Vol. 16, 1986
SOME THEOREMS FOR A DISTRIBUTIONAL GENERALIZED
LAPLACE TRANSFORM
By
S. K. Akhaury
Department of Mathematics, Ranchi College,
Ranchi - 834008, Bihar, India
(Received: September 23, 1981; Revised [Final] March 2, 1984)
ABSTRACT
A generalization of the Laplace transform was given by professor
H. M. Srivastava in the year 1968. For this general integral transform
S. K. Sinha (1981) gave complex imersion and Tauberian theorems,
and A. K. Ti~ari and A. Ko 0982) gave several further properties in
the distributional sense including complex inversion and Abelian theo
rems . We extend this generalization to distributions and obtain similar
Abelian theorems of the initial and final value types [ see,also rhe
relent work of V. G. Joshi and R.K. Saxena (1931) involving an analo
gous study of a substantially more general fl-transform ] .
1. lntcoduction and Definitions
A gene1 alization of the Laplace transform
L(f(t) :p] = J~e-pt f(t)dt ,Rep>O. (I)
was given by Srivastava [5] in the form {see also [ 6] ) :
130 J
(p,cr) s [f(t):pJ
q, k, m
1: (pt )O" - 1/2 e -qpt/2 W1r,m ( ppt )f( t) dt (2)
where W1r,m (z) denotes the Whittaker function ( Whittaker and
Watson [8, p 339 ] ) . By analogy with Srivastava's definition (2), we
define the generalized Laplace transform of a distribution f (t), whose
support is bounded on the left , by (see also (3] and [7J )
( p,cr ) F(p)=S [f(t):p]
q,k,m
= <f(t), e-qpt/2 (pt)cr-l/2 W1;m (ppt) > (3)
In this paper we prove Abelian theorems of the initial and final
value types for the distributional (Srivastava 's) generalized Laplace
transform (3). For an analogous study of a substantially more general
H-tramform, see a recent work [2] .
Let us assume that th ere exists a real number c for which e -ct f(t)
is a distribution of slow growth . We may c )nvert (3) into
( p, O" ) F( p) = S [l (1): p ]
q,k,m
<e-ct j(t), .\(t)e-(qp- 2c)t/2(pt)cr-l/2 vv,,,m (ppl) >
(4)
where.\ ( t) is any infinitely smooth function (with support bounde.d
on the left ) which equals one ove1 a neighbourhood of support of
f ( t ) •
.~
~
,,,,
131 J
The definition ( 4) possesses a sense since
{ A (1) e - ( qp - 2c) t/2 (pt) cr - 1/2 ~·h,m ( ppt) }
is a testing function in St , which denotes the space of testing functions
of rapid descent .
Z. Abelian Theorems of the Initial Value Type
We relate here the asymptotic behaviour of f (t) as t-+ o+ to
the asymptotic behaviour of F(p) asp -+ oo • Let m assume that f (t)
is a locally integrable function and satisfies Condition A, that is,
( i ) f (t) = 0 for - oo < t < T, and
( ii ) there exists a real number c such that l is absolutely integrable over - oo < t < oo •
-ct e f (t)}
Theorem I t cf. [ 7, p. 3.'.-2, Theorem 4.1]) . If the locally inte
gruble function f ( t ) satisfies Condition A ·with T = 0 (that is,
the support of fl t) is in 0 < t < oo ) , and if there exist a complex
number a and a real number Y1 l .,, > - l ) such that
lim ) J~ l _ t _,. o+ l ,.. ~ - a,
(5)
then
{ pYJ + l F (p ) }
p -+ oo ~ ( "' m, k, p, cr, q)
lim a
(6)
where
132 I
and
~ ( °fJ· m, k, p; cr, q) =
r ( "fJ + m + (J + 1 ) r ( "fJ + (J - m + 1 ) pm+l/2 -------
r ( ri + cr - k + 3/2 ) [ ~ ( q + P ) 1 m + ri + cr + I
. 2F1 ( 'rJ + m + cr + 1 , m - k + 1/2, ''l + cr - k + 3/2/ - P ) q +P
Joo -qpt/2 cr - 1/2 (
Fl p ) = O e (pt ) W",m p pt) f ( t ) dt (7)
Proof. We first note that for 'rJ > -- l and p > O .
J~ r'fJ e -qpt/2 (pt )a- 1/2 Wk,m ( p pt ) dt
____ ~ ( 'r), m, k, p, cr, q) (8) p"'+l
provided that ''l + cr ± m + I > 0 and q -+ p > o (see Erddyi et al .
[ I, p 216 Eq. ( 16)] ) . By using this result and assuming that p > O
and y > 0 . we may write
'rJ + 1 Ip F(p)-tx~(·f),m,k,p,cr,q)j
= I p'IJ + 1 J~ e-qpt/l (pt)(] - 112 wk,.n ( ppt) {f(t) --a In }dt I
'1J + If Y - tj> - 1/2 < IP O e qp - (pt )cr W1c.m ( p pt) { f(t) - tx t"fJ } dt I
~
[ 133.
+ I p"fJ + I J~ e- qpt/2 (pt) cr - I/2 Wk,m (ppt) { f (t) - IX tn } dt I
=Ii+ h
Let us first consider Ii of (9)
11 = I p.,, + I f y t.,, e -qpt/2 (pt)a - 1/2 W1,,m ( FPt ) .o
{-·~~t) _ 1X 1 dt I
(9)
i f (t) ~ sup . --- a I . ~ { 11 , m, k, p, a, q) I l"' 0 ~l ~y
(IO)
Now for /2 , we note from the second part of Condittion A and
the assumption c > 0, that e-•t If (t) - at"' I is absolutely integrable
overO < t <=.So, for p > 2c q
1 _ I "') + I I 00
{ -qpt /2 ( t)cr -- I /2 W ( 1 ) ct } 2. - P e P 1'"" FP e
y •
- Cl r n r } e : l J (t) - IX f dt j
Now since the function
e- ( qp - 2c) t;2(pt )cr - 1/2 wk,m ( ppt)
is bounded in ( Y. oo ) , so it must have an upper bound . Let the /
upper bound of the function be attained at the point t = ~. Then
134 )
12 < p"' + 1 I e- ( qp- 2c ) ~/2 ( p~ ) (J -1 /2 wk,m ( PP~ ) I .
J~e-ct I f(t) ~at"' I dt
<KP "tJ+ l e- (qp - 2c) ~/2 (p~) cr - l/2 Wk,m (pp~) {l l) ;
where 100
-ct K = y e I f (t J - at"' I dt .
Now let e:> 0 . From relation (5) it is clear that y can be chosen
so small that the right-band side of ( 10), which is independent ofp,
becomes less than e: • By fixing y in this way , we can choose p very
large so that the right-hand side of ( l l) becomes less than e: Since e:
is arbitrary, combining the results of ti 0) and (l l ), we find that the
left hand side of (6) can be made arbitrarily small for large p . This
proves (6) . Thus
or
"'+ 1 IP F (p) - ex.~ ( "f),m,k,p,cr, q ) I < e: for p > ~c
q
p "'+1 F( p) ·---- --.;. ex. as p _,. oo •
~ ( "f),m,k, p, cr, q )
which proves the theorem .
Now weextend this theorem to a distribution w;1ich, in some neigh
bourhood of the origin, corresponds to an ordinary function . ln order
to make this extension , we shall need the following
~
~
~!
[ 135
Lemma. Let f (t) be a generalized Laplace transformable distrtbution
with support in y .;;;;; t < oo, where y > 0 . Let the real number c
be such that e-ot f (t) is in S', which denotes the space of all distrib
utions of slow growth . Then, for ( 1 +2c )/q < p < oo,
I F(p) I ,;;;;; M e-qp-r/2
where M is a constant and -r is any real number satsfying 0 < -r < y.
Proof. For p > 2c/q,
F(p) = < e-ct f(t)' .\(t) e-( qp- 2c )t/2 (pt)a-1/2 W,,,,}PPt)>
where .\(t) is an infinitely smooth function whose support in x.;;;;;t < oo •
Here x is a fixed number satisfying 0 < x < y • Again distribution
of slow growth satisfies a boundedness condition of the t) pe
I < f, <P > I ~ c Sup \ ( l + t2 y <P('> (t) I t
for given fin S and for q, in S, where the constant c and the nonne
gative intege1 r depend only on f.
Thus
I F(p) I= I < e-ct f(t), .\(t) le-{ qp-2c) t/2 (pt)cr-1/2
wk'm ( ppt )> I
.;;;;;c Sup I ( 1 +t2 )~ [ .\(t) e -(l,p--2c) t,2(pt) cr-1/2 X < t < oo dtn
• Wk,m ( ppl ) ] \
= c Sup I (I + t2 )" ~ 1.(t) = ~ ( - I)n. n (n) (n-r) r ( r )
x .;;;;; t < oo r = 0 r v= 0 v
136 J
~ I l' - v l-i--(q-p)p-c~ e-(qp-2c)t/2 P-m+l/2
~ (~_- m) ! v ( v ) u = o u ca - m -=--v+-uY!-- < -1 rcpt )a-v + <u-1)12
pv pm+(u-1)/2
Wk+u'2 , m_u ·2 ( ppt ) I , cr ;;;;. m + v ( see Slater [ 4, p . 25 J ) • ( 12 )
\\'here the constant c and the integer n < m depend only on f. Let
us consider
cf> ( ppt) =(pt )a - v + ( u-l )/2
wk+u/2, m-u/2 (pp!).
Here we see that the function <ji(ppt) is finite at zero and at
infinity. The function cf> ( ppt ) is continuous within the interval {O,oo ).
So the function </; ( ppt ) is bounded for all positive t.
Hence
I (pt )cr-v + (u-1 )/2 wk+u/2, m-u/2 ( ppt) I <,1H1
for all positive t •
Thus ([2) becorr.es
i F ( P ) I <cM1 Sup [ ( I + t2 )n 2 ( n ) x ~ t < oo r= 0 r
(n-r) ,\ ( t)
r 1;
v=O ( : ) ( -1 )'-v
~
~'
[ 137
- ( q-p )p-c ~ cr - m . (-l)u pv l l Jr-v 'V ( v ) ( } 1
2 u=O u ( cr-m- v + u )!
m + ( u-1 )/2 -( qp-2c) ( t-x )/2. p . e
e-( qp-2c) x/2 I • cr ;;;:;. m + v ;;;:;. O (13)
Now if -r is a fixed number such that 0 < -r < x, then
} I {r--v l 2 ( q-p) p-c ~ pv pm+ ( u-I )/2 e-( qp-2c) x/2
~A' e-< qp-2c )-r/2
for p > 2c/q • A' being a constant
Again, for p > ( 2c + I )/q ,
- ( pq-2c) ( t-x )/2 - ( t--x )/2 e ~ e fort ;;;:;. x .
So the right-hand side of (13) is dominated by
c A' M1 e-( pq- 2c) -r/2 Sup \ ( l+t2 )1' r~O ( : ) x~t<oo
(n-r) , ,\ (t) ~ . . ( r ) ( - I y-.,
=0 'V
~ ( cr - m) ! l u -( t-x )/2 'V I 'V )
u=O ( u ( cr - m - v + u) ! ( - ) e l ,cr;;;;.m+v.
But the function of t inside the Sup symbol is bounded for all
t ;;;:;. x , Consequently , the last expression is less than Me -qp-r/2 ,
where M is a sufficiently large positive constant .
138 I
Hence I F (p) I <Me -qpT/2
for ( 1 + 2c ) / q < p < oo •
Theorem 2 ( cf. [ 7, p. 354. Theorem 4.2 J ) . Letf (t) be a genera-
lized Laplace transformable distribution having its support · in
0 < t < oo and let us assume that, over some neighbourhood of the
origin , J(t) is a regular distrtbution corresponding to a (Lebesgue)
integrable function h (t) . Assume that there exist a complex number IX
and a real number ri > - l such that
fim { h (t) } = IX t~ o+ tn (14)
Then
lim ) p-+ool
p~.~1 F (p) } = .a
~ ( n, m, k, p, a, q) {b)
where ~ ( 'IJ, m, k, p, a, q) and F lp) are defined in Theorem I
Proof . Let the neighbourhood over which f(!) is a regular distri b
ution be - oo < t < T ( T > 0 ) and let 0 < y < T. Then f(t)
can be decomposed into /(t) = /1 (t) + f 2 (t) , where the support
of / 1 (t) is contained in the interval 0 < t < y and the support of
f2 (t) is contained in the interval y < t < oo •
Now it is clear that./1 (t) is a regular distribution that corresponds
to h(t) for 0 < t < y • Let us suppose that
( p, a ) ( p, a ) F1 (p) = S [Ji (t) : p] and F2 (p) S=
q, k, m q, k, m [ 12 (t) : p ]
_}..
\:
I 139
and
p ... +1 F(p) {3 ( 'IJ, m, k, p, a, q)
p ... +1 F 1 (p) ---
{3 ( 'tJ m, k, p, a, q )
P"'+l F2 (p) + - ---·-------{3 ( 'I), m. k, p, a, q )
( 16)
So, by the above lemma,
lim ~ p ... +i F2 (p) } - O P ~ 00 l {3 { "IJ, m, k, p, a, q )
(l 7)
Again we know that the ordinary generalized Laplace transform
is identical to the distributional generalized Laplace transform of the
corresponding regular distribu1ion . So. from Theorem I,
Ii m { P"' ' 1 F, (p) } - a p -;.oo {3 ( ·r,, m, k, p. cr, q )
(l 8)
l\ow frcm (16), (17). and (18) we have
lim \ yr.+I F(p) } = rx. P -+ 00 l ·~ ( ·1j. m, k, p, a, q )
3. A Final Value Theorem
l-!e1e v.e shall 1elate the behaviour off (t) as t ~ oo to the behaviour
of F (pl asp ~ 0-t- as in
Theorem 3 ( cf. [ 7. p 355, Theorem 5.1 ] ) . If a locally integrable
function f(r) satisfies Condition A, and if there exist a number rx. and
a real number ·r; ( 'tJ > - l ) such that
140 ]
lim ~ _j_Jf)_} t-+ 00 l t"' a ' (19)
then
I im f P"'+ 1 F_(..:.:_p_) __ x -+ o+ l ~ ( 'Y), m, k, p, a, q ) }=a (20)
where ~ ( 'YJ, m, k, p. a, q ) and F (p) are already defined above .
Proof. From the relation ( 19) it is clear that f(t) is a function
of slow growth and so its generalized Laplace transform has a region
of convergence that contains at least the half plane Rep> O • Now
let us assume that the support of f (t) is bounded on the left at t = r when T > 0.
Now, using the known result (g) • we have
I p .. +1 F (p) - ex ~ ( "IJ, m, k, p, cr, q) I
< P"'+I J~ e -qpt/2 (pt)a - 1/2
Wk,m ( pp! ) I f (t) - a I'" I dt
< p .. +1 J~ e-qptf2(pt)a -
112 w,,,m c ppt) if(t)- at"' I dt
+ P"'+l J7 e- qpt/2
(pt) a - l/2 w,,,,,, (pp!) I f(t)-a t" J dt
= Ii + /2 ' say . (21)
Now let us first consider /2 ,
~
\c;
141 ]
12 = p'"+l [00
e-qpt/2 (pt)(J - 1/2. w ]y ~m
( rPt) t.,., I Jtt)_(I. I dt
Sup ~y~t<w f(t) -ci I p'"+1 1--r;-
t'" e-qpt/. (pt) - Wk,m (ppt) dt J
oo '2 cr-1 /? .0
·~ Sup f(t) . y ~ _t < 00 I /-:0 - a; I p'"+l
!~ t'" e -qpt/ 2 (pt )(J - 112 wk,m ( ppt ) dt
= ~ l r,, m, k, p, cr, q ) Sup
Y~t<_oo I _((!l _ci I . f"
t'"
(22)
Let e: be a fixed arbitrary positive number. From relation (l9) it
is clear that we ean choose y so large that the right hand side of (22),
which is independent of P, can be made less than e: •
Now - p'"+l f Y
Ii- Jo e-qpt/2 (pl)cr - l/2 W1c,m (ppt) lf(t)-a t'" I dt
Having fixed y , we can choose p so small that integral Ii becomes
less than e: _ Hence
J p'"+l F (p) - a~ (ri, m, k, p, cr, q) I ~ e:.
This proves (20) for all cases when T > O .
If T < 0 , then the additional term''
142 ]
[3 = I p"l+l I~ e -qpt/2 (pt)(j - 112 wk,m ( ppt) /(t) dt I
occurs on the right-hand side of (21) . Putting
-qpt/2 (j - 1 /2 G (pt) = e (pt) Wk,m (ppt) •
We see that the function G (pt) has a fixed value at T and is also
finite at zero form > 0. The function G (pt) is continuous within
this interval . Hence the function G (pt) must be bounded in the inte
rval {T,O), T < 0. So let G (pt) attain its maximum in the interval
(TO) at the point t = ~ .
Thus
/ 3 = I p"l+l e -qp (,Ji (p~) - I /2 W11,m (pp~) I J; / f (t) / dt
Now /3 approaches zero asp ~ 0 + , so that
I p"+l F(p) - a~ ( 'YJ• m, k, p, cr, q) I ,,;;;; e:
for all values of T . Theorem 3 is thus proved .
Now we extend Theorem 3 to certain distributions . and we have
Theorem 4 ( cf. [7, p. 357, Theorem 5.2 J ) • Let f (t) be a gene-
' ralized Laplace transformable distrif;ution in DR and let us assu,ne
that, ove1 some semi-infinite interval -. < t < oo • j (t) is a regular
distribution corresponding to a locally integrable function h (t) that
satisfies the second part OJ Condition A. Also let us assume that there
exist a complex number a and a real number 'IJ ( 17> -1 ) such that
lim { h (t) 1 = a t ~ 00 /"I (23)
'k
~-
f 143
Then
lim \ -· p_"'+l F (p) {_ a p _,,.o+ l ~ ( '1), m, k, p, cr, q ) 5 - (24)
where F (p) and~ ( ·fJ, m, k, p, cr, q) are defined as before (p is real) .
Proof . Here f (t) can be decomposed into
f (t) = f1 (t) + h (t) '
where h (t) has its support contained in y ~ t < oo ( y > t" ) and ,.<>('
f 1 (t) has a bour.<led support that does not extend to the right of t= y.
Thus
( p, cr ) F1 (p) = s U1 (t) : p ]
q, k, m
in an entire function of p, so that
lim ) P "tJ + I F1
(p) l = 0 , p_,,. o+ l 5
Therefore, we have
lim { "tJ +I } p_,.0+ p F (p) lim f ri+ l } P_,,. 0 + l p F2(P)
1\ow from Theorem 3 the right-hand side of (25) equals to
a~ ( '1), m, k, p, cr, q ) . That is,
/im { pri+l F(p)} = ot~( 'I), m, k, p, cr, q} , P_,,. o+
(25)
144 ]'
so that
lim ~ p'fl + I F (p) t P ~o+ l ~ < ri.m,k,p:~ .~
which proves the theorem .
REFERENCES
Ot
[I] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,
Tables of Integral Transforms, Vol. I, McGraw'--Hill, New York
London and Toronto, 1954.
[2] V. G, Joshi and R. K Saxena. Abeliaa theorelll.S . f.or distributi
onal H-transform, Math Ann . 256 (1931). 311-321 .
[3] S. K. Sinha, A complex inversion formula and a Tauberian
theorem for the generalized Whittaker transform .Jiianabha 11
(1981) ' 31-39 .
[4] L. J, Slater, Confluent Hypergeometric Functions, Cambridge
Univ. Press, Cambridge, ~ondon and New York, 1960.
[5] H. M. Srivastava, Certain Properties of a generalized Whittaker
transform, Mathematica (Cluj) 10 (3~). (1968), 385-390 .
[6] H. M. Sriva~tava and 0. D. Vyas, A theorem relating generalized Hankel a11d Whittaker transform, Nederl. Akad. Wetensch. Proc. Ser. A 72=/ndag. Math. 31(1969),140-144.
[7] A. K. Tiwari and A. Ko Certain properties of a distrti butional
generalized Whitteker transform, Jndhn J. Pure Appl . Math . 12
(h 82), 348-361 .
f8] E.T. Whittaker and G. N. WatsJn, A Coune of Modern Analysis.
Fourth ed . , Cambridge Univ. Press, Cambridge, London and
New York, 1946 .
.~
Jnanabha, Vol, 16, 1986
ON ( -1,l ) RINGS WITH COMMUTATORS IN THE LEFT NUCLEUS
By
Y ASH PAUL and S. SHARADHA
Department of Mathematics, Indian Institute of Technology,
Hanz Khas, New Delhi - 110016, India
( Receil'ed : June 30, 1985; Revised : April 28, 1986)
ABSTRACT
In thio; paper we show that a left primitive ( -1,1 ) ring with
commutators in left nucleus is either associative or simple with right
identitiy element .
l. INTRODUCTION
A ( - I, l ) ring R is a non-associative ring in which the following
iclent ities hold :
( l ) ( x, v,z ) + ( x,z y ) = 0
( 2) t x.y,z) + ( y,:-,x ) + ( z,x,y) = 0
for all x,y,z belonging to R where ( x,y,z) = ( xy ) z - x ( yz ) .
The left nucleus LN is defined as the set :
LN = { x E R : ( x,y,z ) = 0 for all y,z E R} .
Throughout th is paper, we assume R to be a ( -1, I) ring of character
istic#- 2, #- 3 such that ( ( x,y), z,w) = 0 for all x,y,z,w E R where
the commutator ( x,y) = xy-yx.
That is, ( R,R) C LN •
146 J
Let U = { u E R : ( u,x) = o for all x E R} • Then by virtue
of ( ( R, R ) , x,y ) = 0 for all x, y E R we have
( 3) (R,R,R) C: U
where (R,R,R) is the set of all finite sums of elements of the form
(x,y,z). This is equation (II) in [4] .
The following identity given in [I] holds in a (-1,1) ring of
characteristic ::F 2 , ::F 3 •
( 4 ) (w,xy,z) + (w,zy,x) - (w,y,z) x - (w,y,x) z = 0
for all w,x,y,z E R .
2. CONSTRUCTION OF IDEALS
Lemma t. For any ring R, (R,R,R) + R (R,R,R) is a two - sided
ideal of R .
This is proved in l3]
Lemma 2. Let A be a left idqal of R. Then
(a) S = { s e A : sR C A } is a two - sided ideal of R,
( b ) (R,A,R) C: S .
Proof. (a) For any s E S and x,y e: R
(sx) y = (s,x,y) + s (xy)
= - (x,y,s) - ly,s,x) + s (xy) (using (2))
= - (,..,y,s) + (y,x,s) + s (xy) (using (I))
A is a left ideal and s " S implies
t
[ 147
- (x.y,s) + (y,x,s) + s (xy) <:A . Hence (sx) ye A . Also sR C A
implies sx e A . Hence sx e S. Now,
(xs) y = (x,s,y) + x (sy)
= - (x.y,s) + x(sy) (using (I))
Again, using the fact that A is a left ideal and s e S in the above
equation, we get (xs) ye A • Hence xs e S . So S is a two-sided ideal
of R.
( b ) Let x,w,z e Rand a e A . Then
(w,a,x) z = - (w,x,a) z (using (I) )
= -- z (w;x,a) (using (3))
Again, since A is a left ideal, - z (w, x, a) E A . Therefote,
(w, a, x) z E A, Hence (R,R,R) C S.
Theorem 1. If R has a maximal leji ideal A=F (0) which contains no
two-sided ideal of R other than (0) , then R is associative •
Proof. By .Lemma 2, S is a two-sided ideal of R contained in A.
Hence S = (0) . Since (R.A,R) C S , we have (R,A,R) = (0) . On the
other hand it is easy to verify that A +AR is a two-sided ideal of R .
Since A C A+AR we must have A +AR= R. Bence
(R,R,R) =(A +AR,R,R)
C (A,R,R) + (AR,R.R)
C (A ,R,R) + (RA,R,R) (since ( (R,R) , R,R) = (0) )
C (A,R,R) + tA,R,R) (since RA C A)
C (A,R,R)
148 ]
But (R,A,R) = (0) implies (R,R,A) = (0) (using ( 1) ) ,
and so (A,R,R) = (0) (using (2)).
Therefore, (R,R,R) C (0) . Hence R is associative ,
3. LEFT PRIMITIVE RINGS
Definition 1 . A left ideal A of R is called regular if there exists an
element geR such that x-xg e A for all xeR .
Definition 2. A ring R is called left primitive if it (contains a regular
maximal left ideal which contains no two-sided ideal of R other
than (0) .
Theorem 2. If R is a left primiiive ring, then either R is associa
tive or it is simple with a right identity element .
Proof , Let A be a regular maximal left ideal which contains no tv.o
sided ideal of Rother than (0) . If A =;t= (0) , then by Theorem !, R
is associative . Thus assume that (0) is a maximal regular left ideal
of R , which contains no two-sided ideal of .K other than (0) . (0)
is maximal implies , R has no proper ideal . Since (0) is regular,
there exists g e R such that x-xg e (il) for all x e R . That is, x=xg
for all x e R . This implies RR =I= lO) and g is a right identity
element . Hence R is simple with a right icentity element
Finally, at the referee's suggestion, we acid a theorem which was
proved in [4). This theorem states : If these (-1,1) rings are
semiprime, then they are associative . This result follows easily, since
in tnese rings (R,R ,R)2= (0): For the sake of ccmpltter.ess, the proof
of this theorem may be recalled here as outlined below (cf. [4] ) :
>-
( a) (R,R,R) is a11 ideal ;
( b ) If u E U, then (x,x,y) u = (x,x,yu) ;
( c ) If u E U, then {x,x,u) = 0 ;
( d ) Since (R,R,R) C U and (R,R,R) is an ideal, then
(x,x,R} (R,R,R) C (x,x,R (R,R,R) C (x,x,U) = O;
( e) {R,R,R) C { (x,x,R) : XE R} .
Thus, by (d) and {e), (R,R,R) ( R,R,R) = 0 .
These identities are all given in [2] •
Acknowledgements
r 149
The authors are thankful to the referee and professor H. M.
Srivastava for their valuable suggestions
REFERENCES
[ I] I. R. Hentze!. Nil semi-Simple (-1,1) rings, J. Algebra 22 (1972)
442-450 .
{ 2] I. R. Hentze!, The characterization of (-1,I) rings, J. Algebra
30 (1974)' 236-258 •
l 3 ] E. Kleinfeld, Asssosymmetric rings, Proc • Amer • Math • Soc.
88 ( 1957)' 933-986 .
{ 4] Yash Paul, A note on (-1,1) rings, Jninftbha 11 (1981),
107-109 .
Jiianabha, Vol. 16, 1986
CERTAIN MULTIDIMENSIONAL GENERATING
RELATIONS
By
B. D. AGRAWAL and SANCHITA MUKHERJEE
Department of Mathematics, Faculty of Science,
Banaras Hindu University, Varanasi - 221005, India
(Received: October 18, 1985; Revised: May 16, 1986 )·
ABSTRACT
In [ l] we defined a general polynomial system S~ [ ( Xm ). y ]
with the help of a generating relation involving the Lauricella function
FA . ln the present paper we derive a linear and a bilateral generating
relation for these polynomials . Finally, we apply our results to derive
several multilinear and multilateral generating relations
I. INTRODUCTION
The general polynomial system { s~ [ ( Xm) 'y 1 I n = 0, 1,2' }
is defined by means of the generating relation [ ] :
oo A;m,a, (bm), (µm) ~ S [ (xm) , y ] t'!i
n=O n;v, (rm;1), (cm)
evyt FA [a, (bm); (cm); µ1X1t, µ.2x2t'2 ,. •• ,µmXmlm ], (I.I)
where FA is one ofthe Lauricella functions [4,p.41] .
152 J
The conditions for validity of ( 1.1) are
( i ) (xm), y, t are all finite quantities
( ii ) m, (rm;1) are natural numbers ;
(iii) v, (µm) are finite numbers ;
( iv ) a, (bm) are non-zero numbers ;
( v) (c,,.) are neither zero nor negative integerals .
For this polynomial system, we have an explicit representation in
the form {I J :
m A n [ (n-m 1)/r2 J [ (n-mi-~ n 111; )/rm
·~ 2 S [(xm),yJ= J: ~ n 1111=0 1112=0 mm=O
(a) m n--m1- ~(r,-1 )mi
2
m·
(h1) m [(bm; 1)
n-m;- ~ r; 111i 2 .
] (111; I)
( n-m1 - ~ r;m; ) I (~ 1 ) .... m . [ (cm;1 )(m;I)] 2 n-m1 - ~ r;m,
2 m
m1 n-m1 - Lr; m; . (xy) (µ1X1) 2 [ (µ.m;1 Xm;i )mm,i J _____ _.:.__:_ ___________ (1.2)
[ ( mm ! ) J
which we shall use here •
Generating relations play an important role in the study of
polynomial sets . Recently, Srivastava and Manochn [9] (see also
McBride [5} have discussed a numper of useful methode of obtaining
generating functions . L. Carlitz and H. M. Srivastava ( [2] and [3] ) ,
R. Panda ( [6] and [7) ) , A, B. Rao and Anura Srivastava [8], and
[ 153
some others (s'ee [9] have studied polynomials of the form Fi,F2, Fv<r>
and F4 with the help Of generating relations .
In this paper we derive a Jinear and a bilateral generating relati
ons, which ultimate.Iy yields as its applications, various multidime
nsional generating relations .
We employ the following notations in our present investigations :
(i) ·. (m) = 1;2, ... , m.
(ii ) (a1,) = a1 • a2, ... ,ap .
(iii) [(ap) ]n = (a1) n (a2)n ··· lap),. .
(iv) [ (am:1)(p . ·)] =(a1)P .. (a2)p ... (a;_1)p· . m,1 . . . 1 2 '-1
(a•+l) ... (am) P1+1 Pm
( v ) !::, (r ;a) = _f!_ a+ 1 r ' -;.
a+r-l ' ... ' r
( . ) [ ( )Pm ] - ( ,pl - ( )P2 ( )Pm VI GmXm ·- a1X1; - G2X2 ••· amXm :
( ••) [( .. ··)prn;i] _I ) (. :i.P•-1(. )Pt+l Vll am,1 Xm,s - ,a1X1 ... ai_}Xi_n a1+tXi-1-l ·••
2. LINEAR GENERATNG RELATION
Using the series form, we finally have
OQ
L n=O
[ (ix'N) ]n [ (a"N') ]n --[ (~M) ]-,.-
A Sn [ (xn:), y ] t"
(amXm)Pm
154J'
. N+N' + 1 : 0; 1; ... ; 1 [ [ (a.'N): I, r2 ••• ,rm], = F . I
M: O; 1; ... ; I L [ (~M) ; 1, r2 , ••• ,rm ] ,
[ (oc'/N'): l, r2.-•:;., rm}, (a:J,0, 1, ... ,1):-; (b1: 1) ; ... ; (bm: 1);
--- ';__;; (C1: I);.;.; (cm: I);
t . t . r2 rm] . vy •'f.'1X1 •'f.'2X2t•: · , ... , µmXmt ·
It may also be written as
A ~ [ (a.'N) ]n ( (a"N') Jn S ( (Xm) , Y] l"
n = 0 l (~M)J,. n
00 00 00
( (ix'N)) ni-1 n+m1 + ~ Yi m-1
2 ~ ~ ~
( (~M) J n = 0 m1=0 mm_1=0 m-l n+m1 + ~
2
'I
[(ixN,)J m-1 n+m1 + ~ .. 2
Yi mi
(b1)n (a)
,., m~ n+m2 + ... +m,,._1
(c1)n n !
[ (bm-i·I) J (vy)m1 (µ1X1)11 [ (,n,n_i;1 Xm_1;1)
' (mm_1;1) · ·-
[(mm!)] [ (cm_1;i) · ] (mm_1;1)
(2.I)
~-
m-1 n+m1 + ~ ri m; · t 2
• I F r.,;, (N+N r+2 rmM+l
r m-1 I !:;, (rm; (ex: 1N) + n + m1 + ~ r; m;) , I 2 I I m-1 I !:;,(rm; l~M)+n+m1 + ~ Tim;), l 2
m~l
[ 155
( !:;, (rm;(cx:"N') + n+m1 + ·~ r;m;), a+n+m2+ .. +mm_1' bmL I 2 I l , , Cm ;
.. I rm I
gm Xm t j, J
where gm= µ,mrm(N+N,_M)rm . (2.2)
3. BILATERAL GENERATING RELATION
With the help of the generalized Lauricella function of Srivastava
and Daoust (cf [4J ( and using (2.2), we finally obtain the following
bilateral generating relation for our polynomial system
A ~ [ (or.'M) ]n [ {a"Ml_b Sn ( (xm), y·]
n=O [ (~N) ]n
A': Bl+ 2·; ... ; B<M> + 2 F
c : Dl ; ... ; D<Ml
;~ ,;
r [ (3): 1, ... ,1 ] : (a1' + n..: 1) .. (cx:1.'' +n: 1). [ (b'): 1) ; .. .
l [ (y) : 1,. .. , 1] : . . [ (d') : 1 ] ; .. .
156 )
( dM) ; J ) ; Z1 • .. , ZM }n ; (a'M +n: 1), (ocM" + n; 1), [ (bM): 1 ] ;
00 00 00
= 1: 1: 1: ' n=O m1=0 mm=O
[ (oc'M) J m [ (a"M)] m (a) n+m1+ ~nm; n+m1 + 1: r; m; n+m2+ •. +mm
2 2
[ (~N) ] . m .n ! (c1)n [ (mm !) ] n + m1 + 1: r; mi
2
) mm;t (bi)n [ (bm;1) (m"l;l) ]{ X t)n {vyt)m1 [ (µm;1xm;1 /m;1 J • ' -- /Al 1
[ (cm;1) (mm;t) l
A' : BI + 2 ; ... ; B<M> + 2 F , C: DI; ... ; D<m>
m m
[
[(8): 1, .. ., 1] : {a1 ; +n+m1+1: r1m;: 1), (oc1' 1 +n+m1 +E r;m;: 1), ' ' 2 2
[ ( Y) : 1, .. ., 1 ] :
, m ,, m [(b'): 1]; ... ; (a +n+m1+ ~ r;m; :1), a +n+m1+~ rtm~: l),[(b'Ml) :I];
M 2 M , 2
r Cd') : 1 1 ; ... ;
4. APPLICATIONS
, [ (dM) : I ] ;
., Zt ,. .. , ZM _] (3 I)
Formula (3.1) with v = 0, m = 2, = r2 = µi, (b2) = (c2), a= 1/2,
"'·
t I , 1 ,, ,. 1 l -z;
Xi = X, f!.2 = - , X2 = , a; = , IX; = t'j = , Z; = -' +z;
l 157
(j=l, ... M), B1 = ... = B1M> = 0, Dl = ... = D 1M> =I, A'= C, ll=y
yields :
}
-nM +I 2 2 1n ~ (n!)M p,. (x) Pn (z1) ... P" (zM)f tl ui) ... (I -t-zM)-
n=O
~ { (J)n+2m1 ~M ll/2)n+m1 (2xt)n (-t2)m1
n ! m 1 !
00
n,m 1=0
M jl +n+2m1 , l -f-n +2m1; _ 1 =!!_ -J. ' IT 2F1L . I +z;
J=I I , (4.1)
which is a multilinear generatir.g relation for Le~endre polynomials .
By specializing various parameters in (3. I) , we have.:
;j (1 +2ix),. }M+l { (1 +ix)n }M-1 ~ _L ... -_2 __ 1-( I +2ix+n)
n=O ' (n!)M lO+z1) (I+zm):j
(a,ix). (ix,ix) {a,a) . Pn (x)Pn (z1) ... Pn (zM) tn
00 }
M-1 l (I +2ot)n+m1. JM_·· I {I +o),; .· . (2xl)" (-1't1 !: ·m1,n=O n ! m1 ! (l)n+2m1
158]
M [ 1 +2a+n+2m1, I +cx+n+2m1 ; TI 2F1
j=2 l+a ;
With suitable conditions, we have :
J-z1
J+z; J . (4.2)
00
~ n=O
{ (2)n}M { (3/2)n }M
{~if {
2 2 }-(2+n) (1-) ... ( -) .
+z1 I +zM
. Ur. (x) U.,. (z1) ... Un (ZM) tn
00
~ n,m1=0
{ (Z)n+2m1 }M. { <3!2>n+2m1 !M n ! m1 ! { {l)n+2m1 lM
{l)n+m1
M .. [ 2+n+2mi. 3/2 + n+2m1 ; _ 1-z; J (2xt~n(-t~)ml (4.3) . _III 3/2 ; l+z; ]=
which is a rnultilinear generating relation for Tchebycheff polynomials.
Various other applications have similarly been deduced . Several
known bilinear, bilateral, trilinear, trilateral generating relations
happen to be special cases of our results (4.1), (4.2) and (4.3).
Acknowledgem.ents
The authors express their -sincere thanks to Professor H. M.
Srivastava for suggesting a numer of modifications in this paper •
;,,.
( 159
REFERENCES
[ I ) 8. D. Agrawal and S. Mukherjee, A multivariate polynomial
system and its characterizations, Banaras Hiudu Univ. J. Math .
l ( 1986), to appear .
[ 2 ) L Carlitz and H. M. Srivastva . Some hypergeometric
polynomials associated with the Lauricella function Fo of several
vanab!es. I, Mat. Vesnik l3 (28) (1976) , 41 -47 .
[ 3 ) L. Carlitz and H. M, Srivastava, Some hypergeometric poly
nomials associated with tne Lauricella function Fo of several
variables ll, Mat. Vesnik 13 (28) (1976), 143-152 .
[ 4) H. Exton, Multiple Hypergeometric Functions and Applications,
Halsted Press , John Wiley and Sons, New York; 1976.
[ 5) E. B. McBride, Obtaining Generating Functions, Springer Verlag,
Berlin, 197 l .
[ 6 ) R. Panda, Genenlized Polynomials A,, (x), Ph.D. thesis,
Banaras Hindu University , India , 1967 .
[ 7 ) R. panda, A theorem on bilateral generating functions, Glasnik
Mat . Ser. Ill 11 (31) (1976), 27-31 .
[ 8 ] A. Srivastava and A. B. Rao, On a polynomial of the form F4,
Indian J. Pure Appl .M.ith. 6 (1976), 1326-1334 •
[ 9] H.M. Srivastava and H.L.Manocha, A Treatise on Generating Fn
nctions , Halsted Press , John Wiley and Sons, New York , 1984 .
Jiiiiniibha, Vol. 16, 1986
DIFFERENTIAL EQUATIONS FOR A GENERAL
CLASS OF POLYNOMIALS
By
SHRI KANT* and R. N. PANDEY
Department of Applied Mathematics, Institute of Technology,
Banaras Hindu University, Varanasi - 221005, U. P., India
(Received : November 13, [985; Revised April 17, 1986)
ABSTRACT
The object of the present paper is to derive the differential equat
ions satisfied by a general class of polynomials generated by (I l)
below . We also present a bdef discussion about the complete solution
and verification of the solution for one of the various cases involved.
1. INTRODUCTION
In the present paper we derive the differential equations for a
general class of polynomials which provide a generalization of a
number of orthogonal and other polynomial systems . This general
polynomial set is defined by a generating relation (see (2], [3] and
[4] ) :
c c -.\ (l-l'1 X I y 2 tal)
;: : p; u r (ar) : (a,,*); {au*);
S : q; V _ (bs): (bq*);(~v*) ;
*Present Address : Scientific Analysis Group, Defence R and D
Organisation, Ministry of Defence, Metcalf
House, Delhi-110054, India.
162 J
C I µ, X 1 ta2 vyC2 ta3
J (v1 xc1 yc2 t"'I-I )1 (I - V1 xcl yc2 ta1)m2
oo ,\; µ,; V; C11
; C2' ; 1X2 ; aa; (a,); (ap*); !Xu*) ~ A (x,y)ta2n (I· I)
n=O ix2n; v1 ; C1 ; C2 ; m1; m2 ; a1 ; bs) ; bq*) ; ((3v*)
where F ~ :: ~ ~ ~ is a general double hypergeometric function (cf. , e.g ,
Srivastava and Manocha [ 6 , p. 63 , E-1 . ( 16) et seq . J ; ~ee also
Burcbnall and Chaundy {I, p. 112)).
The above general polynomial set contains a number of param
eters; therefore , for the sake of simplicity, we shall denote it by
A (x , y), unless there is some change in any one uf the parameters ix2n
(except ix2n) which is the order (degree) of the polynomial set . lf there
be-any· change m any of the parameters, only the changed parameter
will be indicated . For example
" ; µ, + I ; v ; C1' ... A (x,y)
a2n; v1 ; O ...
µ+I A
a2n; 111 ; 0 (x,y)
The explicit form of the polynomial set generated by ( l l) is
A (x,y) ix2n
00 00 [ (ar) ln-r1 k-d2 I { (ap*) Jn-d1 k-d2 I [(au*) )k
E ~
k=O l=O [ (bs)] k d l [ (b,*) J d k d I [ ((3v*) ]1, n-r1 - 2 n- i - 2
x
.)r.,_
( 163
n-di k-d2l c1' (n-d1k-d2l)+c1l y-c2'k+c2l vkv1i (.\) m1n+D1k.:...D2l µ x
(-lfm1 (n-d1k-d2l) k! l! (n-d1k-d2l) t O.)m1n+D
1k-m1d2l
(1.2)
where
d1 = aa/a2, d2 = CJ.i/a2, '1 = (di-1)
D1 = m2-m1d1 and D2 = m1d2-l,
d1 and d2 being non-negative integers (see also [2],(3] and(4] for various notations and conventions used here).
2. THE GENERAL DIFFERENTIAL EQ.UATIONS
The differential equations satisfied by the polynomials A (x,y) a 2n
are contained in the following theorems :
THEO.REM I, For D1 > 0 , and D2 = 0 = v , we have the following
differential equation with respect to x :
~-1-(0-ci'a) l (a,)+n+d2- _'!?_l0-c1 'n)-{d2) ] ( (ap*) +n+d2-l ei e1
e1 C2 d d2 1 V1X Y 2 , (
--(0-ci a) - (d2)] - d [ (bs)+n - - (0-c1 n) - d2)] ei . µ. 2 ei
[ (hq*) +n - .!!~ (0-c1'n) -ld2) l [l +n - d2 (0-c1'n)-(d2) ] ei ei
[ l ->.-m1n + _!_ (0-c1 'n)]} A* (x,y) = 0, e1 a2n (2.1)
where e1 = c1 -c1 'd2 , 0 = x ~x , and (for convenience )
164 ]
,\ ; µ ; v ; c1 ' ; ••• ; 02 ; ••• ; (a,) , (av*) ; A (x,y) = A* (x,y)
a2 n; V1 ; c1 ; C2; m1 ; ... ; a1 ; (bs) ; {bq*); ... a2n
(2.2)
Theorem 2. For D1 > 0 and D2 = 0 = v , we have the f ollowin7
differential equation with respect toy:
{ o/ [ (ar) +n+d2 - d2'f'' - (d2) J [ (av*) +n+d2-d2o/ - (d2) J
e1 C2
_l'.!::_~ [ (bs) + n-d2o/ - (d2)] [ (ba*) + n- d2o/ - (d2) J µ 2
( l +n -d2 o/--(d2) J [ l--.\-m1n+ o/ J} A\ n (x-,y) ·= 0. (2.3) 2
where o/ =L _d C2 dy
Proof. The proofs of Theorem I and Theorem 2 are fairly straight-
forward, and we omit the details .
The complete solution of (2.1) and (2.3) are given by (see
Rainville [5])
THEOREM:.\. For D1 > 0 and f>2 =0=i•, the complete solutions of
the differential equations (2.1) and ( 2.3) are
d2 r (I) (!) d2 p (1) (I) W= NoLo + 2: 2: N L + I: 2: N L
g=l h=l g,h g h g=l u=l g,u g,u
and d2 r (2) (2) d2 p
W * = No* Lo + 2: 2: N L + ~ 2:
{2) (2) N L
g,u g,u
g=I h=l g.lz g,h g=I u=l
(2.4)
(2.5)
).._ ..
(1) (I) r~speclively, where No, N , N and N0* ,
(2) (2) N ' .N
g,h g,u g,h g,u
are arbitrnry fimctions of y and x, respectively, and
Lo= A* a2n
(l) (x,y) , L
g,h = x1' mF,, [Ri)
(1) k (2) L = x 1 mF,. [R2] , L k2 F [R ] == y 1ll ·ri l
g.u g,h
[ Hi5
(2) k3 L = y .,,,F,.
g,u [R2], where k= !J
2 ( ta1i) + n + d2 - g) + c1 'n ,
('} d I k l ( • d k1 = -d ((a,.*)+ n+ 2 - g )+ c1 n, 2 = d-- \ai,) tn+ 2-g), 2 2
k 3 = -1-- ( (av*)+n+d2-g), and R1 and R2 are defined as folfows:
d2
R 1 ~.,,F
r I 1, 6. (d2 ; l-g (h,)+a,,+d2), 6. (dz; a"+d2-g), I
. I 6. (d2 ; I-g-(b,1*)+a1t +dz), (2-A-min Ho, +n-gl/d2); I
-d9 e c /k ~ 1'1 x 1 v 2
(-d2)d;--(p--q+-;:~~~
i .6 (d2; 1-g-(a,)+a,,+ d2),6. (d2; l-g-(a;."')+a"+d9.),
I ( (ah+n-g)/dd 2); L
i I I I I I \ I
.J
166 ]
and ~.
r I I I, !:::, (d2 ; 1-g-(bs)+au* + d2), !:::, (d2 ; au* + drg), I I I I !:::, (d2; 1-g-(bp*)+a,,*+d2), (2-A-m1n+(a,,* +n-g)/d2); I I I I -d2 e1 c2 I I fl- v1X y I
R2=F I d2 (p-q+r-s-1) I I (-dz) I I I I ,6(dz; l-g-(ap*) +a,.*+dz),,6(dz; l-g-(ar)+au*+d2). I I . l I ( (au*+n-g)/d2+2) ; I l J
Finally, we give
THEOREM 4. For D1 > 0. Dz > 0 and v > 0, we have the differ-
ential equations with respect to x and y :
{-1- ( 6-c1 'n) [ ( a,)+n td2-~(0-c/12)-(d2)] [ (ap*)+n+d2-
e1 e1
_d2_(0-c1'n)-(dz) ] X e1
D2 'J1Ycz xe1 [A+m1n+D2 - - (e-c1'n)-(Dz) J- [ (ar)+n-
e1 (-l)m1d2 µ.d2
dz (6-c1 'n)-(dz) ]x e1
[ (bq*) + n-3-2- (0-c1'n)-(d2)] [I +n- d2 (0-c1 'n)--(d2) J e1 ei
[.\+m1n-(m1d2/e1) (0-c1'n)-(m1d2)]} A* (x,y) = 0 (2.6) a2n
[ 167
and
{'¥ [ (ar) +n + dz-dz'¥ - (dz)] [ (ap*) + n + dz -- dz'¥ - dz)]
[ ,.\ + m1 n + Dz - Dz '¥ - ( Dz ) ]
c2 ei ~y x [ (b,)+n-d2 '¥ -(dz I] [ ( bq*) + n · d2 '11'-(d2)]
(-l)m1dz /2
[ l+n-d2'¥-(d2)] [ ,.\ +m1n-m1dzlfl'-(m,dz)]} A* (x.y)=O (2.7) azn
Acknowledgements
The authors express their gratitude towards Professor H M.
Srivastava (Department of Mathematics, University of Victoria) for
his constructive suggestions .
REFERENCES
[ I ] J. L. Burchnall and T. W. Chaundy, Expansions of Appcll's
double hypergeometric function (Il), Quart . J. Math. Oxford
Ser. 12 (:941), 112-128.
[ 2] S. Kant and R. N. pandey, A study of generalized polynomial
set { A (x.})} , Vijnana Parishad Anusandhan Patrika 24 otz'l
( l 9 8 I) , 14 7 -159 .
( 3 J S. Kant and R. N Pandey , Generating functions involving
generalized Appell-function of two variables , Istanbul Tek .
Univ. B~·l. 34 (1981), No. 2, 95-107.
168 l
[ 4 ] S. Kant and R. N. Pandey , Bilateral generating functions for
classical polynomials, Bull. Math . Soc. Sci Math . R. S.
Roumenie (N. S) 27 (75) (1983), 55-54 .
[ 5 ] E. D. Rainville, Special Function~ , Macmillan , New York,
1960.
[ 6 ] H. M. Srivastava and H. L. Manocha , A Treatise on -Genetatii1g
Functions , Halsted Press ( Ellis Horwood Ltd . , Chichester ) ,
John Wiley and Sons , New York , 1984 .
~'
.T'naniibha, Vol. 16, 1986
SLOW UNSTEADY FLOW 'v• ll:N A POROUS ANNULUS
WITH AXIAL WAVINESS
By
V. Bhatnagar and Y. N. Gaur
Department of Mathematics, M. R. Engineering College,
Jaipur - 302017, Rajasthan, India
( Received: May 27, 1985 ; Revised: October 25, 1985 )
ABSTRACT
Slow unsteady flow of a viscous incompressible fluid between
two porous coaxial circular cylinders with axial waviness, under the
influence of pulsating pressure gradient has been investigated . The
waviness has been taken to be small in comparision to the smooth
radii of the circular cylinders . The Fourier integral transform techn
ique bas been used for finding the expressions for axial velocity, radial
velocity and pressure in terms of modified Bessel functions . For a
particular case of sinusoidal waviness, the velocities have been ca!Cula
ted and shown graphically ,
1. INTRODUCTION
Viscous fluid fiow through porous wavy boundaries is of practical
importance , In aircraft propulsion systems, p0rous tubes are used as
injectors to increase the engine efficiency and as silencers in exhaust
systems . Several authors, lnamely, Citron [2], Khamrui [7], Verma and
Gaur [IO] and Gaur and Mehta [4]) have studied the slow viscous
fluid flow through rough circular tubes and rough coaxial infinite cyli
nders . Bauer [I] and Varshney (9] have investigated the puls atile flow
170 ]
in a circular pipe with reabsorption across the wall and fluctuating ~
flow through a porous medium'1~l:f6~1ri'ded by a porous plate, respecti-
vely . Hepworth and Rice ( [5], [6] ) have studied the flow between
parallel plates and circular /rectangular tubes with arbitrary time
varying pressure gradient . Recently, Gaur and Bhatnagar [3] have
discussed the slow unsteady flow in a porous tube with axial waviness.
The present investigation deals with slow unsteady flow of a
viscous incompressible fluid between two porous coaxial circular cyli
nders with axial waviness, under the influence of pulsating pressure
gradient . The waviness is taken to be small in comparison to the
smooth radii of the cylinders . Expressions for the velocity components
and pressure are obtained in terms af modified Bessel functions using
Fourier transform technique . A particular case of sinusoidal rough
ness is numerically investigated .
2. PROBLEM FORMULATION
In cylindrical polar coordinates, the Navier - Stokes equations in
the non - dimensional form, under the slow flow assumption, are
oU oT ~ + ~~ +
oil oil2 ii cU_ + 02u oA oZ'"
j_W = _ _}f!_ + 02W + _!_ _!~ + q2W oT oz oA2 ii oil oZ2 •
and
A o(Uil) + ow = 0 •
ol\ oZ
u i12 (I)
(2)
(3)
r 1n
where U and Ware the velocities in the radial (,\)and axial (Z) direc-
tions, P the pressure and T the time .
Using slow flow conditions (\72 P = 0), (2) becomes
r -~-+ L (J,\Z
c cT
,\
c (J,\
I 02 1 -+-",\2 ,\ L. (}
- 2
32 I + -2zz j w
c ax
2 I + _o_JI J!V.
oz~
The boundary conditions are
U=V1 CosnT, W=Oat ..\=l + E N1 (Z), T > 0
and U=V2 Cos nT, W=O at ..\=cr + E N2 (Z), T > 0 } '
(4)
(5)
where E < < I is the roughness parameter, N1 (Z) and N2 (Z) are arbi
trary functions of Z and cr is the ratio of the radius of the outer
cylinder to that of the inner cylinder . V1 and V2 are the known non
dimensional velocities of suction/ injection such that V1 =crV2 •
Let
P (,\,Z_T) =Po (Z,T) + P1 (,\,Z,T) I I
U (,\,Z,T) = Vo (,\,T) + Ur (,\,Z,'l) f-l
W (,\,Z,T) = Wo (,\,T) + W1 (..\,Z,T) j ,
(6)
where Pi. T11 and W1 are the variations caused by the roughness and
P 0, U 0 and W 0 are the known quantities for the porous coaxial cylinolers
without roughness
Assuming
cPo
oZ K Gos nT = Re [ K einT ] ,
172 ]
R [ ( inT d U R inT Wo= e !1 ,\) e ] an o= e [ 12 (,\) e ]
and solving the resulting equations under the boundary conditions
/ 1 (1) = 0 =f2(cr)andfi(l) = V1 andf2(cr) = V2,
we have
Wo (A,T) = Re [ ~ { (cr2-J) log_"-=_0_2-~log cr} einT l 4 log cr j
-'
(7)
and
Uo (,\,T) = Re { ------------} e1nT . [
V1 (cr2-,\2) + cr V2 (,\2-J) . J ,\ (cr2- J)
(8)
Using (6), (1) and (3) we have.
au1 0P1 a2 u1 a2 w1
-- = - -- + -- - --cT a A az2 aAaZ (9)
Equation ( 4) now becomes
a2 I a a2 a a2 i a a2 [ -- + - - + --- ]2 W1 =
aA2 " cA az2 ( - + - - + - - ) W1. (IO)
aT aA2 " aA az2
The boundary conditions now, are
Uo+U1=V1 Cos nT, W1 = - Woat A=l+e N1 (Z),T >0,Z>O
Uo+ U1 =V2 Cos nT, W1 = -Wo at A =cr te N2 (Z), T >0, Z >0
and
W1 =0 when { I+ e N1 (Z)} < ,\ < { cr + e N2 (Z) }, T <0, Z=O
3. METHOD OF SOLUTION
1 I I }-(1 J) I . I J
Using Fourier sine transform, the solution of the equation (IO),
gives
~
[ 17 J
Wi(A,Z,T) = Re [ if (2/rt) f ~ [ A (E,) lo (E,A) + B (E,) Ko (E,A)
+ C (q Io (mi\)+ D (i;) Ko (mi\)] sin (E,Z)einT dE, ]. (12)
Using (12), equation (9) gives
U1 {A,Z,T) =Re [ - .V (2/rt) J~ [A(.;) Ii (E,A) - B (E,} K1 (~A)
i; E, . T + - C {!;)Ii (mA) - - D (E,) Ki (mi\)] cos (i;Z) em dE, ] . ( 13)
m m
Using (12) and (13), pressure P 1 (A,Z,T) is given by
Joo I
Pi (t\,Z,T) = Re [ if(2/7t) in - [A (E,) lo (E,A) + B (~)Ko (i;A)] 0 E,
inT d x Cos (E,Z) e E, J + C , ( 14)
where A (E,), B (0, C (E,), D(~). and Care the constants of integration .
10 , Ko Ii and K1 are modified Bessel functions of first and second
kind of order zero and one .
We assume
A (E,) = Ao (E,) + E A1 (E,) + ··· , ( 15)
and similar expressions for B (E,) , C {i;) and D (E,) •
Using equations ( 11) to ( 15) and Fourier sine and cosine integral
theorems, the complete expressions for the axial velocity radial velocity
and pres sure are
W (A Z,T) = W o (A,T) + W1 (A,Z,T)
17.t ]
= ~ f ~~~~g ~ _ ,\2 + I] cos nT + € if (2/1t) 4 i__ log cr
Re J~ { [ .6An (1;) To(~,\)
+ .6 A31 (~)Io (;A) - .6 B12 (~) Ko (;A)
__: .6 B32 (0 Ko (;A) + .6 C13 (0 lo (mA) + .6 C33 (~)lo (m,\)
- .6 Du (0 Ko (m,\) - .6 D34 (~)Ko (m,\) J N1 (~)
- [ .6A21 (;)Io (~,\) + .6 A41 (~)lo(~,\) - .6 B22 {;)Ko(~,\)
- .6 B42 (~)Ko(~,\) + .6 C23 CO lo (mA) + .6 C43 (;)lo (m,\)
- .6 D24 (~)Ko (mA) - .6 D44 (~)Ko (m,\)] N2 (~) }
. ('Z) inT d' x szn t. e t, ,
U (A,Z,T) = U0 (it,T) + U1 (A,Z,T)
= l-~~cr2 _=:A2) _+_cr~~,\2-1~] cos PlT
,\ ( r;2 - l) ...
(16)
+ E Re [ - if (2/rt) J~ { [ .6 A11 (~) /1 (1;A) t- .6 Aa1 (~) 11 (~,\)
+ .6 B12 (0 K1 (~,\) t .6 B32 (;)Ki(~,\) +~/m.6 C13 (~)Ii (mA)
+~Im .6 Ca3 (~)Ii (mA) + ~Im D14 CO K1 (mA)
+;/m .6 D34 (;)Ki (mA)] N1 (;)- [ .6A21 (0 l1 (~A)
( 175
+ 6 A41 (C,) / 1 (C,A) t 6 B22 {(C,) K1 (~) + 6 842 (C,) Ki (~A)
+ ~/mt:,. C23 (C,) /1 (mA) +C,/m C43 (C,) /1 (m,\)
+ E,/m 6 D24 (f,) K1 (m..\) + E,/mt:,. D44 lf,) K1 (mA)] N2 (f,) }
C- ('Z) inT d" ] X OS <:; - e ; ( 17)
and
P (i\,Z,T) = Po (Z,T) + P 1 (,\,Z,T)
r 00
= c - KZ Cos ti'l + E v\2/rt) Re [Jo in 1/C, [ { 6 An(~) lo (f,tl)
+6 A31 (0 fo(~A) -6 812 (~)Ko (f,,\)--6 B32 (:;)Ko (f,A)} N1 (f,)
- { 6 A2l (f,) lo (f,tl) + 6 A41 m lo (f,A) -D. B22 (f,) Ko (~ti)
- /J.. 842 (2,) Ko (f,tl)} N2 m ) Cos (C,Z) einT df, ] , (18)
where C denotes a constant, and
6 A 11 (Z),, 6 B12 (Z), /J.. C13 (Z) and /J.. D14 (Z) are respectively
minors of Io (Z), K 0 (~), (m) and Ko (m') in /J.., each multiplied
by x I /J..,
/J.. A21 (Z), /J.. B22 (Z), /J.. C 23 (Z) and /J.. D24 (Z) are respectively
the minors of lo (crZ), Ko (crZ) / 0 (am') and Ko (crm') in /J.., each
multiplied by YI /J.. ,
/J.. Aa1 (Z), /J.. Ba2 (Z), /J.. C33 (Z) and /J.. D34 (Z) are respectively
176 ]
minors of !1 (Z). - K, (Z), (Z/m') Ii (m') and (-Z/m') K1 (m') in A,
each multiplied by R / A,
A A41 (Z), A 842 (Z), AC43 (Z) and A044 (Z) are respectively
minors of li(aZ), - K1(aZ), (Z/m') Ii (am') and - (Z/m') K1 (am')
in A, each multiplied by Q / A,
X= -12----- ,Y=- 2a-----, K r ( a2 - l) J k [ ( a2 - I) J 4 L log a 4 a log a
R = _ [~a2~1~ Vi _=--_2aV2 J ' Q = [ 2a V1 - (a2 + 1 ) V2 ] - -------------
(a2 - I ) a (a2 - I)
~=I lo (Z) Ko (Z) lo (m') Ko (m)
I lo (aZ) Ko (aZ) lo (am') K0 (am')
I ! 1 (Z) -K1(Z) Z/m' !1 (m') -Z/m' K1 (m')
I !1 (aZ) -K1 (crZ) Z/m' Ii (am') - Z/m' Kt (am') I
and m' = ( z2 + in )1/2 and N1 (~) and N2 are Fourier sine transfor.ns
of N1 (Z) and N2 (Z) respectively .
4. PARTICULAR CASE (SINUSOIDAL WAVINESS)
For this case we take the waviness function at the walls
N1 (Z) = N2 (Z) = sin (Z/!), ( 19)
where 27t/ is the wavelength of waviness at the walls .
We may formally write
N1 (~) = N2 (~)=if (7t/2) a(~ - 1/l), (20)
'•. '•.
r 111
where il is the Dirac delta function .
Substituting N1 (~)and N2 m from (2o) in (16) to (18) and using
the property of the Dirac delta function (Sneddon [8] we get
r ( a2 - l ) fag A ·1 W (A,Z,T) = K/4 -------- - A2 + l cos nT + ERe [ [f 6AJJ(i//) L log a. · j
and
+ 6 A31 (1//)- 6A2 1 (I//)- L,A41 (I//)} lo(A//)- { 6 B12(i/t)
+ 6 Bsz(l/l)- 6 B22<I/l)- 6 B42 (!//)}Ko(;\//)
+ { 6 C13 (l;/) + 6C23 (!//)- 6C23 (1//) - L C43 (!//)]lo (A//)
- ( 6D14 (l/l)+ 6Da4 (!//) - 6D24 {I//) - L,D44 (I//)} K<; (A//)]
>< Sin(Zil) einT ] (21)
. . - V1 l cr 2 A2) + cr V2 (A2- l) .,
U(AZ,T) = I-~ ·'-----------J cos nT ~ ,\(cr2-J)
-ERe [ [\ ,0, A11 (1//) +- 6 A31 (I/I)
- 6 A21 (1//) - A41 (I//)} / 1 (A/l) + { 6 B32 (1//) + 6BJ2 (I//)
- 6822(1/l)- L,B42(1//)} K1 (A//)+(!/!) { 6C13(l/l)+ 6C33(l//)
6 C2a (!/!) - ,0,C43 (1/l)} 11 (A/!)+ f'/l { 6 D14 (I//)
+ 6 D34 (1/l)
- 6 D24 (!//)- 6D44 (1//)} K1 (A//')] Cos (Z//) einT] (22)
P (A.Z,T) =C -KZ Cos nT + E Re [in l { [ 6. A 11 (1//) +6.Aa1 (l/l)
178 ]
- !':::, A21 (1 I!) - !':::,. A ill (I I!) ] I 0 (>../ /) - [ !':::, B12 (I I!)
+ !':::, Ba2 (!//)- !':::. B22 (I//) - !':::. B42 (!//)]Ko (A//)}
X cos (Z/l) einT ] ,
where
I//' = ( 1//2 + in )1/2.
5. NUMERICAL DISCUSSION
(23)
Axial and radial velocity profiles have been drawn for i;=O I,
a=2 at various sections of the annulus for· different values
of nT.
Fig. l shows axial velocity profiles at different sections of the
annulus for nT = 0 and nT = rt/3 for the injection / suction velocities
V1/k = I : V2/k = 1/2 and V1/k = 2: V2/k = I at the inner and outer
cylinders respectively. For nT = 0 they are shown in upper half (Which
will be the same in the lower half) and for nT = rc/ 3 in the lower half
(which will be !he same in the upper half) of the annulus . It is noted
that increase in the injection I suction velocities st the inner and outer
cylinders respectively results in the increase of the axial velocity at the
section Z = rt/2 while decr>!ase is observed at Z = 3rt/2. At the
scetion Z=O and rt, che axial velocity remains unaffected by injection/·
suction velocities • For the case nT = 0, the profiles are almost
parabolic.
Fig . 2 depicts axial velocity profiles in a similar way as shown in
Fig. I for suction/ injection velocities V1/k = - I : V2/k = -- (1/2)
and V1/k = - 2 : V2/k = - I at the inner and outer cylinders respec
tively . The axial velocity decreases with increase in the suction /
injection velocities at the section Z = rt/2, the decrease being much
A.
[ 179
sharper near the inner cylinder, while the axial velocity increases at
the section Z = 3n / 2. The profiles now are almost para'Joli;; at the
section Z = 3rc/2 .
Fig 3 (a), (Ii), (c) and (d) respectively represent the radial velo
cities· at sections Z = 0, :;/2, re and 3rc/2 of the annulus for Vifk = I :
V2/k = !/2, V1/k = 2: V2/k = I, V1/k = - l : V2/k = - (1/2) and
V1/k = - 2 : V2/k = - l and nT = 0 .
At all the four sections there is a uniform decrease in the radial
velocity from inner to tile outer cylinder .
Increase in we value of the waviness para.neter s , mcrea~es the
puiturbed pans of the vel,Kities and the pressure .
2 ' \ \ \
,, I
I ,
' , I
I
\ \
\
' I I I
I
i W/'f..-
1 i , ___ ......._ ______ .. _ _.L. ________ _.... ____ _.,. __
37\/2 27\"
,-:'.··~~,\ \ ; \ i I I I I
: I I
FIG AXIAL VELOCITY PROFILES AT DIFFERENT SECTIONS Of- T~ ANNULUS FOR rr: 2, £ :0·1 AND V1/K: J: .Vz/K:1/2 (----)ANO V1/K"2: V2/K:1 (-·) .
z
i ,<:
2
'--- w
W/K-
i I
I
-n'I" •'U
---
---a---------------------
........ ---,..,.._ .. -~
- '
L. ·-. ··-··---···----· ..... ·--·-0 iV2 II: 37\/2 27\ Z
\ I.
I ! t I
"hf./".'/~ .:.----------~ ·-\) --.
t t I
j/ 2~---.L.....,____------
FIG 2 AXIAL VELOCITV PROFILES AT DiFFt.RENT SECTIONS OF THE ANNULUS FOR a--• 2, ~=0·1 AND
V,IK=-1 Vz/K=-]12 (-----) AND v,11<:-2: V2/l<:-1 (-)
I· ;i
..... -r
....... - --,_
ar~
~ 11- '-..
• I I --------.:::. 0 tt- ------;------.. i ...... 0 -
..:."'. 0 1 7 ;\~~
_, r:=-11---. .,_
----I ./
:,,_~
I
I I
{'")
t~ ................... __ _
! of11 ----+--~ 2 -
-1
-2
(c)
:..:>
·1 ~
,/~
! -<I-
(b)
~.
l'·,~--.. ..il ·t . .._ -H ------- l
t:/ ~r
(d)
FIG. 3 RADIAL VELOCITY P~WF'ILES FOR u- :2, f =0·1, nT: 0 AND Viii<: l: Vz/I( :1/2 f-----), V1/K :2: Vz/K:l (--), V1/K:-1: Vzll<=-112 (-)
-
AND V1/K •-2: V2 /K: -t <-> AT DIFFERENT SECTIOt.IS Of THE ANNUL.US Z:Q, 7"i/2, 7'\ ANO l7V2 RESPEC'IVELV ( (a).{b),ic;) AND {d)l
I 183
REFERENCES
[ 1 ] H.F. Bauer, Pulsatile flow in a straight circular pipe with reab
sorption across the wall, Jngr. -Arch, 45 {1976), 1-15.
[ 2 ] J. S, Cit1on, Slow viscous flow between rotation concentric
cylinders with axial roughness, J. Appl. Meclz. 29 ( 1962), 188-192.
[ 3 ) Y N. Gaur and V. Bhatnagar, Slow unsteady flow through a
porous circular tube with axial wavine.>s, Indian J. Pure Appl. Ji.lath. 16 (!985). 90-101.
[ 4] Y. N. Gaur and D. M. Mehta Slow unsteady flow of a viscous
incompressible fluid between two coaxial cir~ular cylinders
with axial roughness, Indi:n1 J. Pure Appl. Math. 12 (1981),
l 160-1170 '
[ 5 J H. K. Hepv.orth and W. Rice, Laminar flow between parallel
plates with arbitrary time varying pressure gradient and 1rbit
rary initial velocity, Trans. ASMF J. Appl. Mech. 34 (1067),
215-2i6 .
l 6 J H. K. Hepworth and W. Rice, Laminar tNo-dimensional flow in conduits with arbitrary time varying pressure gradient, Trans. ASMF J. Af!pl. Mech. 37 (1970), 861-864 .
l 7 J S. 0
R. Khamru1, Slow steady flow of a viscous liquid through
a circular tuoe with axial roughness, hdiun J. Mech l (190.J),
l 8-21 •
[ 8 ] 1. N. Snedden, Fourier Transforms, McGraw-Hill, New York;
1951 .
[ 9] C. L. Varshney., Fluctuating flow of viscous fluid through a
porous medium bounded by a porous plate, lndian J Pure
Appl. Math . 10 (1979), 1558-1564 .
[ IO J P D. Verma and Y. N. Gaur, Slow unsteady flow or a viscous incompressible fluid through a circular tube with axial rough
ness, Indian J. Pure Appl . Math. l ([970), 492-501 .
Jiianabha, Vol. 16, 1986
RANDOM VIBRATION OF A BEAM TO STOCHASTIC LOADING
By
ASHOK GANGULY
Department of Applied Mathematics,
S. G. S. Institute of Technology and Science,
Indore - 452003, M. P. India
(Received : June JO, 1985 ; Revised : July 3 , 1986)
ABSTRACT
We study the random vibration of a beam of finite length under
stochastic loading . The technique of the Fourier transform is used
for the solution of the problem .
l. INTRODUCTION
In recent years, much attention has been devoted to the problems
involving stochastic vibrations of beams . The motivation for such
researches is due to various requirements such as wind effects and
earthquake effects on structures, etc . Stochastic vibration of different
types of beams has been studied by many authors (see [I], [3], [4],
[5] and [6] ) . In these problems, a mechanical system is excited by a
random type load and the response of the system is of random nature.
Its analysis is done with the help of stochastic p10cess , The analytical
tools for such study were borrowed srom the r.oise problems in comm
unication theory and adapted to problems of structural dynamics(l2],l3]).
In dealing with stochastic processes, the averages very often used are the
auto-correlation functions, and their Forier transforms are called the
power spectral densities
186 1
In the present paper, we illustrate the Subject-matter of stochastic
vibration by considering the vibrations of a beam of finite length,
which is freely hinged at its ends, subjected to stochastic loading .
The mean square values ·Of-; the output displacement and the velocity
are calculated with the help of auto-correlation function of appLed
load .
2, FORMULATION OF THE PROBLEM
Let a beam of finite length L, which is freely hinged at its ends.
be subjected to a stochastic load
P ( x,t) = qr (t, a ( x - x'), ( 0 < x' < L)
where qr ( t ) is stochastic in time t . Lf y ( x,t) is the transverse stoc
hastic displacement of any point x of the beam at time t, then differe
ntial equation governing the motion of the beam is ( 7, p. l l l, Eq.
(50)}
()4y + cx4 a2
22y -2[2-
P (x,t) -EI (1)
where a2 = EI/ (pS) , Eis Young's modulus of the material forming
the beam, I the moment of inertia of the cross-section of the beam with
respect to line normal to both the x and y axes aod passes thrcugh
the centre of mass of cross-sectional area S and p is the mass per
unit length of the beam •
The boundary conditions are [ 7, p. 116 Eq. ( o3) ]
a2y y = ox2
0 at x = 0 and x = L . (2)
~
l 187
3. SOLUTION OF THE PROBLEM
Applying the Fourier transforms to (I), we have, finally. the expression for the transverse displacement ( t7], p. 118 )
2La 00
• (mt"<). (n7tX') I y (x,t) = -,,- l: szn - szn - -2- X
1t" E I n = l L L n
~r'YC-r) sin [ n21t2as-'t") J d, l3)
giving the deterministic solution of displacement in terms of some integrals .
The auto-correlation function for the output process y( x,t)
is defined as
Rv (x, ~'ti. t2) == E [ y ( x, tr) y ( x,t2 )J (4)
where E is the expectation .
Using (3) and (4) and putting r = 11 - '1' s = t2-"2,
we get
Rv ( x, ~. ti. 12) = ( 2La) '; _l - sin (n1tX ') 'It~ E I m,n= l n2m2 L
sm - in - Ism -· (nn:x' ) s · (m1t~ \ . (nm~') L L ) L
! 11 J t 2
( 2 2 ) . n 7t a . x sm, -- rszn 0 0 >I L
( m2~a) s R'Y ( t1 -r, t2-:s ) dr ds l5)
188 I
where the definition for R'Y is used .
We prei;cribe the load '¥ (t), to be stationary stochastic process
of known spectral density S'Y (w) as (5J :
Slf (w) = 2 re SoS (w) (6)
where S 0 is a real constant and 3 (w) is Dirac s delta function .
Then, the auto-correlation function of '¥ (r) is
J +=
R'Y(-r) = -f- Sl}'. (w) eiw-r dw, re -oo
(6u)
frcm where, we get with the help of (6)
l< '¥ ( -r) = So • (7)
Since the proces5 'Y(t) is stati.rnarys we have
R'Y ( t1 -r, t2-s ) = R'Y (-r) = So
Using (7) in (5), v.e get
Ru ( x, ~' tl> t2 ) 2 00 y ~ 1 . nre >;; • 1 rex ' O --::fT .I/fl --Sin -
m,n= I rim L L
Sin -- Sin -- X --COS --- · -co, --. mre~ . mre~' l l n2re2at1 } { 1 . m'!n2at2 J
L L L · L (8)
giving auto-correlation function of the displacement ,
/~
~ I
! ,w
where y~ = 4L4 So / n2 £2 J2 . The series in (8) is convergent and
the sum is f;r.ite for all values of t1, t2 • Rv (x, ~' t 1 , t2) varies directly
as the constant S0 corresponding to the spectral density of the stoch
astic process tf(t) . It varies inversely as the square of I .
ln (ii), putting x = ~, x' = ~'and 11 = t2 = t, we have, the
n.ean square value of the displacement
Yi (X,t) - y~ 2; __ l __ 00 . nn;x . nn;x' . mnx . mrcx'
szn -- szn - -- szn -- szn -- X 111,n= I n4m4 L L L L
(
.x l 1-cos n2~2 at } {
m2rc2 at} 1--cos -L- ·
The velocity V (x,t) is given by
V (x,t) = ay ~,t) 2i -
The auto-correlation function of the velocity is
R, (x.<.t1t2) ~E [ V (x,11). V (x,t2) J
02 Rv (x,~,ti.t2)
at1 at2
Using (8), we have
R ( ~ t t ) _ v2 ~ 1 . nrcx . nn;x'
v x,c;, 1' 2 - 0 "'-' -- szn - szn -2 2 L m,n=l nm L
(9)
(10a)
(lOb)
190 1
. mrc~ . mn:~' . sin -- szn -- . sin L L
n2n:2 at1
L
2 . . where v
0 = 4 £2 So a2 / 'lt4 £2 12
sin m2n2a12 L
ti I)
Putting x = ~. x' = ~, and t1 = t2
velocity from (11) as
t, we get the mean square
2 2 00 1 . nnx . nnx' . nmx . mnx'
V (x, t) = V 0 ~ -.-- szn -- szn --- sm -- szn -- . I m.n = 1 n2m2 L L L L
. n2n2at . m2n2ut szn -- sm --- ,
L L
which obviously is zero at t=O .
4. NUMERICAL DISCUSSIONS AND GRAPHICAL
REPRI:.:SENT A TIONS
(12)
The calculations fot mean square values of displacement (9) atid
velocity (12) are done for different values of non-dimensional time
at/Lat the point x = x' = L/2 of the beam . The calculated results
are given in Table I . The tabulated results are plotted in Figures I
and 2.
It is seen from Figure l that mean square displacement is initially
zero . It increases with time and attains maximum value after some
time and then it gradually decreases to zero . It attains minima and
maxima at equal intervals of time,
'~ '
/
[ 191
Figure 2 represents mean square velocity . Starting from zero
value it increases to maximum value and then decreases to zero value •
This cyclee continues indefinitely .
TABLE I
For n = I to 20, m = I to 20, x = x' = L/2
at/L = 0 I 2 3 4 5 6 7 8 9 IO
2 ' 2 y I y = 0 3.6 0.15 l.5 l.4 0.2 3.6 0 3.6 0.14 1.5 1 0
2 2 0. 25 v /V = 0 0.63 0.9 1.0 0.6 0.3 0 0.2 0.6 09
1 0
at/L - 11 12 l3 14 15 16 17 18 19 20
2 2 1.4 1.2 2.5 0 3.6 0.12 y/y - 1.7 1.4 0.2 3.5
1 0
2 2 1.0 0.6 0.3 0 0.2 v /V - 0.6 0.9 LO 0·7 0.3
1 0
I
------~_J
....... ------~----~--. ~-,----------------------
194 J
Acknowledgements
The author is thankful to Dr. G. R. Tiwari for giving help . He
is also grateflll to Professor H. M. Srivastava and the referee for their
valuable suggestions in the improvement of this paper.
REFERENCES
[ l ] W. E. Boyce, Random vibration of elastic strin~s and bars,
Proc. Fourth U. S. Nat. Congr. , pp 77-8 5, Berkefoy, California,
(1962).
[ 2 J E. H. Crandall, Random Vibration, Vol. l, M. 1 T Press,
Cambridge, Massachusetts, 1958.
[ 3 J S. H. Crandall and W. M. Mark, Random Vibration in Mecha
nical Systems, Academic Press, New York, 1963 .
[ 4 ] l. B. Elishakav, Random vibration of two-span beam, Israel
J. Tech 11 (1973), 317-320.
[ 5 ) L. Fryba, Non-stationary response of a beam to a moving
random force, J. Sound Vibrations 46 (1976), 323-338 .
[ 6 J J. C. Samuels aud A. C. Eringen, Response of a simply suppo
rted Timoshenko beam to a purely random Gaussian process,
J Appl. Mech. 25 (1958), 496-500 .
[ 7 ] I. N. Sneddon, Fuurie1 Tran~Jorms, McGraw-Hill, New York,
1951 .
·-· ·- ~-
~-
Jiiiinabha, Vol. 16, 1986
MULTIPLE HYPERGEOMETRIC FUNCTIONS RELATEO TO
LAURICELLA'S FUNCTIONS
By
R. C. Singh Chandel and Anil Kumar Gupta
Department of Mathematics, D. V. Postgraduate College,
Orai-285001, U. P., India
( Received : July 19, 1985 ; Revised: July 14, 1986)
ABSTRACT
The papaer is based upon a commendable idea of interpolation
between Lauricella's functions. The authors introduce three multiple hypetgeometric functions related to Latrricella's functions a.lid derive various generating relations, integral .representations and recurrence realtions for them . Some special and confluent cases of these functions are also considered . Of course, as .remarked at the end of Section 1 below, each of these three multiple hypergeometric functions as well as a fourth one (defined in the following paper by P. W. Karlsson [ 12] ) is contained, as a special case, in the generalized Lauriclla hyp
ergeometric function introduced , almost two decades ago, by H. M.
Sriva~tava and M. C. Daoust [17].
1. INTRODUCTION
Lauricella ( 13] introduced the multiple hypergeometric functions
cni lnJ (n) (n)
FA , FB , Fe and Fn •
Exton {7) introduced the following two multilple hypergeometric (nl
functions related to Lauricella's FD
196 1
( 1.1)
(1.2)
<k>£< 11> ( a, b1 ,. •• , bn ; c, c' ; X1 , ... , Xn) (1) (D)
00 (a) + + (b1) ... (hn) 1111 m 1 ••• mn mi mn X1 ~ ~
/1111 Xn
(c) (c') m1 ! m 1 , ••• ,m,.=0 m1+ ... +mk m1<+1+···+m,.
• •• ;;;n !
<7'>£<"'> (a a', b1 , . , bn; C; XI , •.. , Xn) (2) (D)
00 (a) (a') + . (b1) ... (bn) m1+···+mk m1r+l ••. +m,. 1111 ll"!n
~ 1111 , ••• , 111n =0
(c) x m1+ ... +mn
m1 x ~-'···
ll"ln Xn
m1!: m,.!
Motivated by this work, Chandel [3] defined and studied the following <n>
function related to Lauricella's Fe :
(1.3) 1'£(") (a, a', b ; C1 , ••• , Cn ; XI, ... , Xn) (1) c
00 (a)m1 + (a') (b); • +mk tnk+ 1 + ... mn m 1 + ·· + m.n (c) ... (en)
1111 tnn ~
m1 , ... , mn=O
m1 mn Xi Xn
x -· ···--m1! mnl
x
In the same paper (3] , he also introduced certain other functions
related to E~">, and F1">, but they are trivially reducible to combina
tions of Lauricell's functions . Since then we have been interested in I .. , "
··rinding new m u1tip1e hypergeometric functions. i~1ate<l ;9 ·. f~,;) and - .. (I.~ ij" lj r • ' '• Iii t .
F1"> . Fortunately, in the present paper . we are able to introduce the
'
'
;
197
following multiple bypergeometric functions rel_a.ted to Lauricella's
functions [ 13 J :
(1.4) (kJF~"t ( n, b, b1ctl , ... , bn; C1 , •• ., Cn; X1 , ••. , Xn)
00 (a),,.11 + ... *mn (b) + + (b1,+i) ··· (bn) ; m 1 ••• m1c Fnk+ 1 111n X
2: m1 , ... , 111n=0
(c1) ... (en) m1 mn
m1 mn x X1 Xn. .. ,;;;:T ... mni··
For k=O, I it reduces to F~">, while for k=n it reduces to FJ:;">.
(I 5 (''>Fl"l ( b b. . . . ) • ) AD a, 1 , ••. , " , C ; Ck..;..J , ·., Cn , X1 , ... , Xn
00 (a) _J , (b 1 ) ·•· (bn) m1 T ··· 1mn 1111 . . Inn
~ ------·-···- x 0 (c) + (Ck+1) .•. (Cn)
111 1 ,. •• , IUn = tni +··· 1111: . tnk+I 111n
m1 111,, X X1 __ ••• Xn_
m1! 11111!
for k=O, I it reduces to F~">, while for k=n it reduces to F_b11l,
(J.6) (kl£1b (a, Gk+I , ••• ,an, b1 , ... , b,.; C; X1 ,. •• , Xn)
oo (a) + + (a1t+d ···(an) (b1) ••. (bn) ~ 1111 ... Ink m1o+J 111n 1111 111n
m1 .?"""• mn=O (c)1111+ ... +111.,.
m1 111n X X1, ••• ~.
m1 ! mn!
x
198 ]
Fork = 0, I, it reduces to F~">, and for I<, = 'n , it reduces to Fbn> .
Some confluent forms special cases, generating relations, integral
representations and recurrence relations have been discussed here .
It should be remarked here that, in the following paper, Karlsson
l12] has defined one more multiple hypergeometric series of this class
and has given the precise regions of convergence for all four of these
multiple hypergeometric series . He has also presented several other
properties of these functions, including, for example, Eulerian inetegral
representations, transformations, and reduction formulas .
Just as (I.I), (l.2) and ll.3), the multiple hypergeometric func
tions defined by ( 1.4), (I.5) and (1.6) as well as the fourth one defined
by Karlsson [12], are contained, as special cases, in the generalized
Lauricella hypergeometric function of Srivastava and Daoust
(cf. [17, p. 454]; see also [11, pp 106-IO?J and [18, p. 37]).
2. CONFLUENT FORMS
For confluent forms, we consider
(2 1 l im <l''F <"> (a b b b . c c . "' · ) b AC ' ' k+l , ••• , " > 1 ,. • ., " , ·"'l , .. ., Xk , bk+l ,. . ., ,,-?oo
00 (a) (b) m1 + ... + m,, m1 + ... + mk
(c1} ... (c,.) m1 m ..
L m1 ,. •. , mn=O
(k)<fa~'6 ( a, b ; Ct ,. • ., Cn ; X1 ,.. , Xn ) , k ;/:: n , (})
X!.i.1 ...&) -- ,. ... , bk+I b.,.
m1 X1
m1! Xn
m,.
m,,!
'
'
(2.2)
=
(2.3)
I 199
lim b -7 00
. x 1 X7, (k) FC:t (a, hk+l , ... , bn 'C1 , ... , Cn, -b , ... , T'
Xk+l , ••• , Xn )
OQ {a),,, + +m (b1,+i)m ... (b11)m·· !;. 1 · ··· n . k+l , n
·... -·(c1) ·· ... (c,,) -m1 , ... , m,,=0 ml mn
x m1 x l m1!
<kJcp!nJ (a, bk+1 , .•. , b,, , C1 , ••. , Cn ; X1 , ••. , X,.) • t2l AC
~.
mn Xn ... -,-mn.
lim (7'>F(") ( b . Ck+l , ••• , Cn-70 AD a' l , ... , bn, C 'C7,+l , ... , Cn, X1 , ••• , Xk,
Ck+l X<+l , ... , CnXn
00 (a) + + (b1) ... (bn) m 1 ... mn m1 mn ~
m1 ,., mn=O (c)m1+ +mk
(IC) cp ~~ ( a, bl , .. , bn ; C ; Xt , .• ·, Xn ), k -:j::. n • (])
x
m1 mn x ~ ••• ~
m1 ! Inn!
(2.4) lim (I') p<n) ( Gk+l , ••• , On-'>-=> BD a ' Gk+J , ••• , an , b1 , ... , b,, ;
Xk+l C ; X1 , ••. , Xk ' a1o+1
.' ' .. Xn··) , ... ,· .. ··~~ .
200 ]
and , ..
(2.5)
00 (a)m + + m (b1) •.. (bn )m 1 ... k m1 n
(c) m1 + + mn
x ~
m1 ,. •• , mn=o
x
(k).f.. (n) ( b b . . -.. .. ) I • ...:,_· '/' BD a, 1 ,. ·., n , C , X1 ,. ·., Xn , I~ -r- n
(1)
(I') pen) , a-+oo BD \. a • ak+1 , ... , an , b1 , .. ., bn ; c ; lim
Xk Xt
a , ... ,-a
m1 mn X1 Xn --···-, m 1 ! mn·
!:-- .•
, Xk+l , .. ., Xn )
~
(ak+l) .•. (an) (bi) ... (bn) mk+l mn m1 m,. 00
(c)m1 + ... + mn x
m1 , ••. , mn=O cii;
x m1 mn
X1 Xn ;n_;r ... m,.f
<llef> 1nh ( a1c+1 , ••. , On, b1 , ... , bn ; C ; A:J , ... , Xn ), k ::j:. 0 , (2)
3. SPECIAL CASES
For n = 4 and k = 3, (1.4) gives
·~ (4.1)
··! .-~~
l.3lF~~ (a, b, b4 ; c1 _, C2, C3, C4; X1, X2, X3, X4)
. .~?; 'f(
= K2 ( a, a, a, a ; b, b. b, b4 ; C1 ' C2 ' C3 • C4 ; X1 ' X2 ' X3' X4 )
·. "'
~
( 201
= ~~:Et41 Ut b4 ; c, C1 ' C2' C3' C4 ; Xi 'X2' X3. X4)
while; for n = 4 and k = 2, ( 1.4) reduces t.o
(3.2) c 21F~46 (a, b, b3 ' /;4; Ct' c~' C3 'C1 ; Xi 'X2 ' X3 'X4)
= Kio (a; a, a, a; b, b, b, b3' b4; C1: C2' C3' C4 ; X1, X2, X3. X4 ).
For n = 4 and k = 3, (l 5) gives
(3.3) c 3>f~4}; (a, bi, b2, b3 , b4 ; C, C4 ; X1 , X2 , X3, X4 )
Ku ( a, a, a, a, b1 ' b2 ' b3 ' b4 ; c, c, C, C4 ; X1 X2. X3, X4 )
c3l£14) ( a h1 • b2 h3 • b' ; c, C4 ; X1 ' X2' X3 'X4 ) lfr D ' · ' ,.
while, for n = 4 and k = 2, (1.5) gives
(3.4) l2IF~4~ (a. b1, b2, b3, b4; C, C3, C4; X1, X2, X3, X4
= Km (a, a. a, a ; bl 'b2' b3, b4; c, c, C3, C4; X1 'X2, X3, X4).
Further , for n = 4 and k = 3 , (I 6) yields
t3.5) l3JF~1j ( a, a4' b1 , b2' b3 'b4; c; X1 'X2' X3' X4)
- K15 (a, a, a, a4; b1 , b2 , b3, b4, c, c, c, c; x 1 , x2, x3, x 4 )
and, for n = 4 and k = 2, (1.6) gives
(3.6) 121 F1't (a, a3 , a4 . b1 , b 2 , b3, b4 ; c; x 1 , xz, x3, x4 )
202 ]
K21 ( a, a, a3, a4 ; bi. b2 , bs , b4 ; c, c, c, c ; xi , x2 , xs , X4) ,
where K2, K1o. K11, K13, K15, K21 are the multiple hypergeometric
functions of four variables studied by Exton ( [6] , [ 8 J, [ 1 I ] .) ,
For n = 3 and k = 2 , (l .4) reduces to
(3.7) <2>p~3t (a, b, bs; C1 'C2' C3; XI, X2' X3)
= FE ( a, a, a , bs b, b ; cs , c1 , c2 ; xs , x1 , x2 )
while, for n =3 and k = 2. ( 1.5) gives
(3.8) (2) (3)
F (a, b1 , b2 • bs ; c , c3 ; x 1 , x2 , xs ) AD
FG (a, a, a, bs , bi , b2 ; cs, c , c ; xs, x 1 , x2 )
:1l£~> (a, b3 , b1 , b2 , C3 , C ; X3 , X1 , X2 )
Further, for n = 3 and k = 2, ( 1.6) reduces to
(2) <31 (3.9) F 80 ( a, a3, b1 , b2 , bs ; c ; x1 , X2, xs )
Fs (a3 , a, a, b3, b1, b2 ; c. c, c; xs, x 1 , x2 )
~ g;E~) ( a3 ' a ' bs , b1 . b2 ; c ; X3 ' X1 ' X2 ) '
where FE, FG and Fs are Laur:celia's hypergeometric functions
of three variables studied, in these notations. by '-)a ran ( [14J ; see also
Srivastava and Karlsson [ 18, pp. 42-43 J).
Similarly, we can give special cases of confluent hypergeometric
functions also .
~
l '\
4. Generating Relations
From definitions, we obtain
(4.1) (J-t)-a (kJp1nJ (a, b, b1,_,_1 , ..• , bn ; CJ , ••• , Cn ; AC .
X1
1-=1 , ... ,
( 203
~) l-1
00 ~ (a)r
r= 0 rf r Ck>p<nJ (a+r, h, hk-'.-1 , .•• , bn ; C1 , ... , Cn, Xi, .. , X,.),
AC
(4.2) \l-t)-a ck'F~"Ji (a, b1 , ... , b,,.; C, Ck+l , ••• , Cn X1
l-t '~. ', ~)
l-t
00
_ ~ (a)r. i<ri <kJF~jy (a+r, b1 , •.. , b,. ; c, Ck+l , ... , c,. ; X1 , ... ,xn ),
r=O r !
(k) (r.)
(4.3) (I-1ra F8n (a,ak+i,····a,.,b1,---.b,,.;c;
00
~ r=O
and
(a). ir ,. !
X1
l-t ,. ... , 21'._ J Xk+l , ... , Xk )
l-t
''''F1"}y ( a+ r, a1:+1 ,. .. , a,. , b1 , ... b,. ; c ; X1 , ,Xn ),
(4.4) (l-1)- 0 ' 1"F~(! (a, b, b1<+ 1 ,. .. , bn ; c1 , ... , c,.;
00
~ r=O
--~.!.. , ... l-1 ~ ' X1r+1 , .. 'Xn )
1-t
.i!!]~ <k>F~~ (a, b+r , b11+1 , ... , bn; C1 , ... , Cn ; Xi, .•• , Xn) r !
Similarly, we can obtain generating relations for the confluent
hypergeorr.etric functions •
20~ I
5• l~tegral Representations
Making an appeal to [ 16, p 10 I ]
(5.1) (,\,m) = _1_ f oo e-t 1A+m-1 dt ' r(") lo
·and (5, p. 13 J
l (5.2) r (il+m) 270 J
IO+l
et i-,i.-m dt -oo
Re (,\) > 0, m = 0, 1,2 , ... , we derive the following integral
representations
{7q ('nJ
(5.3) F ( a,b, bk+1 •. > bn; C1 ' .. , Cn 'X1 , ... ,x,,) AC
I r= f oo r (a) r (b) Jo Jo e-ls+tl ,\a-1 tb-1 0F1 ( -·; C1; x st) ..
0F 1 (-; c,,, x.st) 1F1 (b"+l; c"H;Xk;-.1 s) ..
iF1 (b,, ; Cn ; Xn S ) d's dt ,
(k) <'')
(5.4) F (a, b1 , ... , bn ; c, Ck+1 , .•• , Cn; X] ,.. , Xn) AD
r (c) in:(r(a)- 1
00 110+> e-s+t sa-1 r" 1Fo ( b1
0 -oo
XkS F (b •• 1Fo ( b1c ;- ; --- ) I 1 l<+J ; Ctc+l ; Xkc-J S )
t
··· 1F1 ( bn ; C n ; Xn s ) ds dt ,
~!!..__
'
\
[ 205
where b1 ,. .• ,bk are negative integers, and
(k) ('l)
(5.5) F (a, Gk+t , •.• ,a,., bi, ... , bn; c ; X1 , .•• , Xn) BD
-r (a) r-(ak+i) ... r (a11) r (b1) ... r (bn) I~··· (2n-k+ I) ...
f 00 e -( S + Sk+l + ... + Sn + !1 + ... + tn) Sa-I
Jo a,.-1 b1-1 b,.-1 s ti ... t,. fl
oF1 (-; c; X1 t1 S + ··· + Xk fie S + Xk+i S'k+l tk1l
+ .. + Xn Sn tn ] ds dsk+I ... dsn dt1 .•• dt,. ,
Re (a), Re (ai,+ 1), , .•• , Re (an)'· Re (b1) , ... , Re (bn) > O.
dk+l-1 s k+l
Similarly, we can obtain integral representations for the confluent
forms .
6. Recurrence Relations
For further investigations, we shall use (Exton { [ 10, p. 115 (3.5) ] )
(6.l) oF1 (-; c-1 ;x)- oF1 (-; c; x) -
and (Slater [ 15, p. 19])
--~ oF1 ( - ; c + I ; x ) = O , c (c-1)
(6.2) c 1F1 (a; c ; x) - c 1F1 ( a,,---1 ; c; x) -x 1F1 (a; c+l, x)=O,
(6 3) (l f--a-c) 1F 1 (a; c; x) - a 1F 1 (a+ I ; c; x) + ( c - I ) 1F1 (a ; c- l ; x ) = 0 .
By an appe'.ll to (6.1 ), (6.2) a'1d l6.3), the relatio 1 (5.3) gives the
following recurrence re 1ations :
206 ]
(6 4) (k) (ri)
F ( a' b, bk+t ' .... b,. ; Ct , ...• Cn ; Xi , ••. ' Xn ) AC
= 1k>p~nt (a, b, bk+t , ••• , bn; Ct,. •• , CL1 , Cj-l, Ci+1 , ... , c,.; X1 , .. ., Xn)
(k) I") X; ab C; (c;- l)
F (a, b, bk+l , ... , b,.; c1 , .•. , c;_1 , c;+l, AC
C;+1 , ... , Cn , X1 , .• ., Xn) ,
where j = 1 , ... , k ;
(6.5) (k) F~"d ( a, b' bk+l , .• ., bn ; C1 , ••• , c,.; X1 , .•• , Xn)
Ck>p~~ (a, b, hk+t , ... , b,_1, b;-l , b;+l, .•. , bn ;
C1 , .•. , Cn ; X1 , , Xn )
a X;
C; Ck>p~nd (a, b, bk+t , .. ., bn ; C1 , .. ' CL1' c1+ I '
C;+1 , ••• , Cn ; X1 , ••• , Xn ) ,
where j = k + 1 , ... , n
ck> en) (6.6) (I +b;-C;) F (a, b, bk+t , ... , bn ; C1 ,. •. , Cn ; X1 , ••• , Xn)
AC
ck> en> ~ b; F ( a, b, bk+I , ... , b;_1 , b; + I , bH1 , .•. , bn ;
AC
C1 , · ··, Cn ; X1 , •• ., Xn )
(k) (ri)
- (c;-1) F (a, b, bk+1 , ... , b,. ; c1 , ... ,Ck, ... , CL1. AC
C;-1 , C;+l , ... , Cn ; Xi , .. ., Xn ) ,
"
( 207
where j = k + l , •.. , n •
Similarly, from (5.4) and (5.5) we derive
(6.7) <k>F',;h (a, b1 , •• ., b,.; C, Ck+I , •.• , c,.; X1 , .•• , Xn)
<k>p~"A (a, b1 , .•. ,bk , bk+I , ••• , b;_1 , h;-l , b;+1 , ...• b,.;
C , Ck+t ,. •• , Cn ; Xt , •• ., Xn )
X;a C;
<kJ F~i, ( a, b1 , ••• , b,. ; c , Ck+I , ••• , c;_1 , c; + I ,
C;+l , ••. , Cn ; XI, •••• Xn ) ,
where j = k + I , ... , n , and
(6.8) (k) F1nJ (a. llk+I ,. •• ,an' b1 , .•• , bn ; c; X1 ,. .. Xn)
<1' 1 F1n}i (a, llk+l , ••• ,a,,, b1 , .•• , bn; c-1 ; X1 , ••• , Xn)
a [ x1b1 <k 1F1] (a + I , ak+i , ... ,an, cl c-1 )
bi + l , ... , bn; C + 1 ; X1 , .•• , Xn )
+ ... + Xk bk <''>F';li (a + l , ak+l , ... ,a,. , bi, ... , bk-I,
bk + l • bk+l , ••• , bn ; C + 1 ; X1 ,. •. , Xn } ]
1 [ . b .. . (k)pC"l ( 1 --1-, XA+l ak+I k+l BD a, ak+I + ' c ( c-
ak+2 , .•• , a,. , bi , ... , bk , b1c+1 + l ,. . ., b., ; c + l ; X1 , .•• , X11 )
208 )
b <'<>p<n> ( + I + ... + Xn Gn" BD a, Gk+l,•••• Gn_1, Gn ,
b1 , ••• , bk , b1c+1 ,. •• , bn_l , bn + I ; C + 1 ; X1 ,. •. , Xn ) ]
Acknowledgements
Our sincere thanks are due to Professor H. M. Srivastava and Professor P. W. Karlsson for their suggestions.
REFERENCES
[ I ] R. C. Singh Chandel, The products of classical polynomials and the generalized Laplacian operator, Ganita 20 (1969) no. 2, 79-87 ; Corrigendum, ibid 23 (1972), no 1, 90.
[ 2] R. C. Singh Chandel, Fractional integration and integral representations of certain generalized hypergeometric functions of several variables, Jfianabha sect. A l ( 1971) , 45-56 .
[ 3 ] R. C. Singh Chandel, On some multiple hypergeometric functions related to Lauricella functions, Jnamibha Sect . A 3 (1973), 119-136 ; Errata and Addenda, ibid. 5 (1975), 177-180.
[ 4] R. V. Churchill, Fourier Series and Boundary Value Problems,
McGraw-Hill New York, 1961 .
[ 5 ] A, Erdelyi et al. , Higher Transcendental Fuuctions, Vol . I,
McGraw-Hill New York 1953 .
[ 6 ] H. Exton, Certain hypergeometric functions of four variables,
Bull. Soc. Grece (N. S. ) 13 (1972), 104-113 .
[ 7 ] H. Exton, On two multiple hypergeometric functions related to
Lauricella's Fv, Jfianabha Sect. A 2 (1972), 59-73.
~
I ._;
[ 209
[ 8 ] H. Exton, Some integral representations and transformations of
hypergeometric functions of four variables, Bull. Soc . Math . Grece (N. S) 14 (1973), 132-140.
( 9 ] H. Exton, A note on a recurrence formula for Appell's function
F 4 , JfJ.anabha Sect. A 5 (1975), 63-65,
[ IO] H. Exton, Recurrence relations for the Lauricella functions,
JfJ.anabha Sect A . 5 (1975), 111-123 •
[ 11 J H. Exton, Multiple Hyper geometric Functions and App/ ications,
John wiley and Sons, New York, 1976 .
[ 12 ] P. W. Karlsson, On intermediate Lauricella functions, JMinabha
16 {1986), 211-222
I
( 13 ] G. Lauriceila, Sulle Funzioni ipergeometriche a pi u variabili,
Rend. Circ. Mat. Pa!ermo1(1893),111-183.
[ 14 ] S. Saran, Hypergeometric functions of three variables, Ganita
5 (1954), 77-91 ; Corrigendum. ibid. 7 {1956), 65.
[ 15] L. J Slater, Corif!uent Hypergeometric Functions, Cambridge
Univ. Press , Cambridge, 1960 .
[ 16 ] H, M. Srivastava, Some integrals representing triple hyper -
geometric functions, Rend. Gire. Mat. Palermo Ser. 1116 (1967),
99-115.
f 17] H. M. Srivastava and M. C. Daoust, Certain generalized
Neumann expansions associated with the kampe de Feriet
function . Nederl. Akad. Wetensch. Proc • Ser. A 72 = Indag.
Math . 31 (1969), 449-457.
[ 18] H. M. Srivastava and P.W. Karlsson, Multiple Gaussian Hyper
geometric Serifs , John Wiley and Sons, New York, 1985 .
Jiianabha, Vol. 16, 1986
ON INTERMEDIATE LAURICELLA FUNCTIONS
By
Per W, Karlsson
Department of Physics, The Technical University of Denmark,
DK - 2800 Lyngby, Denmark
(Received: April 22, 1986 )
ABSTRACT
For the four intermediate Lauricella functions, of which three
were defined in the preceding paper [I] and a fourth one is di::fined
by the multiple series (I. I) below, we establish regions of convergence
together with some integral representations, transformations, and
reduction formulas. Of course, as remarked in [I] as well as at the end
of Section l below, each of these intermediate Lauricella functions is
contained in the generalized Lauricella function of several variables,
which was introduced eartier by H. M Srivastava and M C. Daoust
[6] .
l. INTRODUCTION
Cnl Cn) lnl In)
With the four Lauricella functions FA , Fs , Fe , FD in
mind, one might ask whether functions could be constructed that are
tin a suitable sense) intermediate between pairs of Lauricella functions.
Clearly, there are ( i) =6 cases to be considered ; and, in the preceding
paper, Chandel and Gupta [lJ in fact introduced three. intermediate
(lq (71)
functions F AC
(k) (k) (II) (R) ·
F , and E , see [I, Eqs. (1.4), (1.5), U.6)] . AD BD .
An attempt to construct AB-and BC-functions on similar lines leads
212 l
to products. In the sixth case, however, we obtain a new function
defined by the series
(k) (71)
(I.I) F [a, b, b1 •.•. ,bk; c, c,,+l , .. ., c., ; x 1 , ... , x .. J CD
00 (a) I (b) + (b1) ... (b1,) m1+ ... ,m,. mk+l ... +m,. 1111 mk ~
m1 ,. • ., m,.=0 (c + , (Ck+1) ... (en) m1 ... T mk m1,+1 tnn
x
m1 mn X ~ ... Xn_
m1 ! m,.
(71) (?'I)
It reduces to F fork = O, and to F for k = n . C D
In the present paper we shall supplement the invesiigation in [I)
by establishing regions of convergence for the four functions together
with some integral representations, transformations, and reduction
formulas . The four functions are, incidentally, contained in the wide
class of multiple hypergeometric functions introdnced by Srivastava
and Daoust ( [ 6, p. 454 et seq. ] ,: see also [7, p 37 J ) .
2. Regions of convergence
It is a well-kncwn consequence of Stirling's theorem thar the
region of convergence D in absolute space for a hypergeometric series
is independent of the parameters . (Exceptional values for which the
series becomes meaningless or terminating are tacitly excluded . ) lf
we choose suitable parameter values and take abs.1luk values of all
terms, a simpler series cr with region of convergence D emerges ;
and in many cases D is fairly easily obtained from cr . (Some exam
ples of the method may be found in [.!.], [3], and (4) . ) Frequently,
'
\
'
( 213
this approach is superior to the application of Horn's theorem, espe
cially because the latter grows complicated as the number of variables
increases, compare [7, Chapter 4 and Section 9.2].
In the present case, the following results were found
Series
(k) 1n)
FAc
( k) (tl)
FAD
(k) (fl)
FBD
(k) (tl)
FcD
Region of convergence
( .r i x1 i + .. +.r 1xk1 )2 + 1 xk+1 t + ... + 1x,.1 < 1
max \ I Xi 1 ,. •• , I Xk I ) + I Xk,1 I + ... + I x,. I < l
max ( I x 1 l , ... , I x,. I ) < l
max ( j X1 I, ... , I Xk I )+(-..,r \ Xk+l I+ ... +./ I Xn i )2 <1
Proofs are obtained by taking all parameters equal to unity. To facili
tate printing, we ~hall let
~ and ~ (k) (n-k)
denote summations over m1 , ... , rn1, and mk+J ,. .. , mn , respectively,
all indices run from 0 to oo . ( The trivial cases k = O and k = n are
excluded . ) For brevity we introduce
(2• l) {
K = m 1 + ... +-mk, N = nlk+1+ .•. +mn,
r =max ( I X1 I , ... , I X1c I ) ,z = I Xk+l I + ... + I Xn I .
214 J
(k) '"J In conjunction with FAc we now consider the series
O'J = ~ ~ ( K + J':!J ! K ~J_ Xt i mi ·•· I Xn I mn (k) (n-k) (mi ! ... mk !)2 mk+I ! ... m., !
r ~2
~ I K! J I X1 I mi ... I x1c I m" (k) L m1 ! ... mk !
~ (n-k)
( I+K)N Ix I m"H I Im,. _ ___ · k-; 1 __ ·.. Xn
m1c+1 ! ... mn ! ---- ---·
The inner series equals (l-Z)-1-K. Hence,
(k) ,...I I X1 I I Xk I J a1 = (t-zr1 Fe I, I; l. - 'I ; 1-Z ,. .. , 1--Z
L
The region of convergence is thus given by z < I and
.y ( I X} I I (1-Z) + ... + if ( I X1c I ) I (I- Z)) < I
(kJ (n)
A little rearrangement now yields the assertion for FAc. For the series
For tbe series (k) (n)
FAD we proceed as follows ,
0"2= ~ ~ (k) (n-k)
( K + N) ! I X1 I mi · .. i Xn I m,. --K! m1,+1! ... m,.! __ _
= ~ I X1 I m1 .. , I Xi I mk ~ ( 1 +K )N I Xk+i I m11+1 ... ) x,. I Inn
(k) tn-k) mk+l ! ... m,.! -- ·
Again, the inner series equals ( 1-zr1- K , and we find,
\
\
cr2 = l-Zp II ~ ~ l_ 1
( k oo [ Jm· j= I mj=O 1-Z .
Thus we must have Z < l and I x; I < 1-Z, j E {I , ... , k}, or
(!:) (1')
Y +z < 1, which is the assertion for FAD
In the third case we consider the series
cr3 =
Clearly,
'S ~ (k) (11-k)
K ! m,,_H ! ... m,. ! I . I m1 I I Inn --· X1 ... Xn • ( K+ N) !
cr3 < ~ ~ I X1 I m1 . . I Xn I m,.. (k) (11-k) '
[ 215 .
v.hich implies convergence of •k>F~}; if I x1 l , ... , I Xn I are all below
unity . Ou the other hand , upon discarding terms we obtain
a3 > 00
~ Ix; I m; m;=O
j E { I , .. , n } ,
which means that 11'> F';~ is divergent if the mod ulu:; of any variable
exceeds unity . This completes the third case .
Finally , for tk> F(;~ we consider
cr4 = ~ I (n-k) l
N 1 ]2 m1;+1 ! ... Inn ! I Xk+l I mk+l ... I Xn I m,.. cr5 •
216 1
where
0'5 = ~ (1 +N)K (k) K ! I X1 [ m1
I Xn I mn
00 ~ (1 +N)P
p=O p ! ~ I XJ. I mi . . . I Xk I m;.
k=p
Clearly , one of the term's in the inner series is YP . Thus ,
00
a5 > ~ (I + N)p Yv p=O p ! = (J-Y)-1-N '
which implies
a4 > (l-Y)-1 ptruq [I I· I c ' ' , ... , I Xk+l I 1-Y
I Xn J J ' 1-Y
On the other hand. the number of ternrs in the sum with K=p does
not exceed (p + 1 )7', and for any E > 0 we can find an A > O such
that (p + I )1' < A ( 1 + E )7' for all p > 0 . Consequently ,
0'5 < ~ (I+ N)p A (I +z)PYP = A ( I -(I +c:) y r1-N
p=O p !
and
1n_k) • j Xk+l I ! Xn / J a4<A (1-(1 +E)Y)-1 Fe (I' 1 ; l , ... , I ' l-(1 +·c:)Y , .. , 1-(l+e:)Y •
From the inequalities for a4 it follows, since E may be arbitrarily
small. that the region of convergence is given by
if ( I Xk+i / / ( 1-Y ) ) +,, · + if ( j Xn I / ( 1- Y ) ) < I .
The fourth assertion now follows .
\
\
{ 217
3. Integral representations of Euler's type
Since there is no simple Eulerian integral for F';!!> we can not expect
(k) {n) (I') 1n)
such representations to exist for F AC and F CD • In the other cases,
however, we have
(k) ('1)
(3.l) FBD [a, m+1 ,. ... lln 'b, ,. .. ,On; c; X1' .. , x,. J
r (c) ------ x r (J11) .•. r vJ,.) r (~-01 - ... - b,.)
i i bcl b,.-1 (l c-b1 - ... -b .. - I X U1 ••• Un -U1 - ··· - U,.) X
T,.
( 1 }-llk+l (l -an X -Uk+l Xk+l ··- -Un Xn) X
-a x ( l-u1 x1 - ... Uk x1, ) du1 ••• du,. ,
where
T,. = { (u1 , , Un) I 0 < U1 /\ ··· /\ 0 < Un /\ U1 + .. + Un< 1 },
Re c > Re ( b1 + ... + b,. ) , Re b; > 0 , j E {I , ... , n } ;
and
(k) (")
(3.2) FAD [a, b1 , ... , bn; c' c,,+1 , ... , Cn; X1 , ••• , Xn] =
_______ r___;__( C-")_r---'-( C_k+--=1"----) _· ·_· r ( Cn) _ X r (b1) ••• r (b,.) r (c-b1 - ••• - b1o) r (c,,+i - bk+1) ••• r(c,.-b,.)
218 1
X 1 U bi-I u b,.-1( l-u _ _ u .)c-b1 - ... - b1c-l l ... n 1 ... k
Sn,7c \
X ( l-u. )Ck+l-hk+rl (l Cn-b,.-1 1,+1 ••. -u,.) x
-a X ( l-U1X1 - •·• - UnXn ) du1 .•• dun,
11bere
Sn,k = { (u1 , ... ,Un) I 0 < U1 f>.. ·• /\ 0 < U7, /\ U1 -!-- ... +- U1.<I /\
/\ 0 < Ut.:+1 < 1 L\ ... /\ 0 ~ Un < ] } ,
Rec> Re ( b1 + ... +bk), Re bj > 0, j € {I, ... , k},
Re Ci > Re b, > 0 , i E { k + I , ... , n } •
The integral representations are proved in the usual way by expansion \
of factors containing x1 , ... , x,,. Incidentally, particular cases ( n =3,
k= 2 ) of both representations were given by Saran [ 5, Eqs (3.IO) and
(3.5) 1 .
4. Transformations
The domain of integration in \2.2) and the form of the integrand
are both preserved under the transformations
11 = l--Ui - ••• - U7,, t2 = U2 , ••• , !11 = Un ( k ;;;;: l) ,
and
11 = U1 , ... , ln_} , = Un_J , ln = I-Un ( k < n-1).
The resulting transformations of Eule1's type read ,
(k) (1i)
(4.l) FAD (a, bi •... , bn ; c, Ck+l , ... , c,.; X1 , .•• , Xn 1 ( l-x1 )a=
j
'
and
(71) 1n)
FAD [a, c-(b1 + ... +b1c)' b2 , ... , bn ; c, Ck+l .... , Cn ;
-X1
I-xi
(k) (fl)
' X2-X1 , ... , -:::k..:--"1, ~ , ... , ~ I I-xi I-xi l -x1 I -xi
-.J
[ 219
l4- '.) FAD [ a, bi, ... , bn ; c, C1c+1 , ••• , c,. ; X1 , ... , Xn J (1-xn)a =
(II) 1n1
FAD [a . b1 , ... , bn._] ' Cn --b,.; c 'Ck+l , •• ., Cn;
X1
1-xn , ... , Xn_1
1-Xn
1 __ ,,, I - .
J-Xn j
The~e are , however only two of several transformations which
can be obtained in a similar way . Particular cases ( n = 3 . k = 2 )
are due to Saran [ 5, Eqs. (5.3) and (5.2) ] .
5. Reducible cases
(Ill rn)
Except for the function FAc there is a reduction formula in case
the fir~t k variables are equal . The formulas read ,
(lq en)
(5 I) FAD [a~ b1 , ... , bn ; c' C1<+1 , ... , Cn; x , ... , x' Xk+I , ... , Xn J
Cn k+] F l [ a, b1 + ... + b,,, bk;·t , ... , b,. ; C , C1c+1 , ... , Cn ;
A
X, Xk+l , ... , Xn) ,
220 J
(k) (n)
(5 2) FBD [a' a1c+1 , ...• On' b1 , .•.• bn; c; x , ... , x' Xk+l , .. 'Xn] =
(5.3)
<n-k+V FB [a' a1:+1 , •. .,On' b1 + ... + b1c, b,,H , ... , bn; c;
X , X1:+1 , ••. , Xn ) ,
(kl (n)
F [a, b, b1 ,. .. , b,,; c. C1c+1 ,. •• , Cn 'x ,. .. , x, Xk+l , .• ., Xn ]= CD
(1) <n-k+ll , F [a, b, 01 + ... +bk; c, Ck+l , ... , Cn; x, Xk+1 , ... , Xn J.
CD
In all cases the function on the left-band side is written as a
series with summation indices m1c-1-1 , ••• , mn , such that the general term
'k> contains a FD with equal variables to which we apply Lauricella's
reduction formula
(//)
(5.4) FD (A b1 , .. ., bk; C; x ,. .. , x]= 2F1 (A, bi+ ... +b1c; C; .\]
Equality of parameters may give rise to reduction formulas , too . With
X = Xk+J + ... + Xn
for brevity , we have
(kl <n> (5 5) FAC (a, b, Cic+i , •• ., C11 ; C1 ,. •. , Cn ; X1 , ... , Xn J
C7dl r . ~ . , X7, J = (1-X)-a Fe a, b; Ci, .. ' C1c ' 1 X , .. ., l -X ' 1-
and
(k) (n)
(5.6) F [ o, bi , ... ' bn ; c ' b1c+1 , .. ., bn ; X1 , .• ' Xn ] AD
\
\
[ .221
(k) I X1 Xk ]' = (1-Xra FD I a, bi,. .. , bk; c; 1-X , ... , T-X . ,_
These are proved by the same series manipulations that we used
for cr 1 and cr2 •
Further reduction formulas could be obtained from the transform-
ations mentioned in Section 4 .
Acknowledgements
The author is indebted to Professor H. ,M. Srivastava for the
opportunity to see the manuscript of [I] before its publication and for !
/ the encouragement to undertake present inve<;tigation
REFERENCES
[ I ] R. C. S Chandel and A. K. Gupta, Multiple hypergeometric
functions related to Lauricella's functions, Jiiardibha 16 ( 1986 ), 195-210
[ 2 ] p W. Karlsson, On the conver5ence of certain hypergeometric
series in several variables, Jiianabha Sect A 5 ( 197:5), 125-127.
[ 3 J P. W. Karbson, On Certain Generalizations of Hypergeometric
Functions of Two Variables, Physics Laborator¥ II, The Technical
University of Denmark, Lyngby, 1976 •
[ 4 J P. W. Karlsson, Regions of convergence for some of Exton's
hypergeometric series, Nederl. Akad. Wetensch. Proc. Ser •
A 81 = Indag. Math, 40 (1978), 72-75.
222 1
[ 5 ) S. Saran, Hypergeometric functions of three variables, Gan ita
5 (1954), 77-91 ; Corrigendum ibid. 7 (1956), 65 .
( 6 ] H. M. Srivastava and Martha C. Daoust, Certain generalized
Neumann expansions associated with the kampe de Feriet
function , Nederl. Akad. Wetensch. Proc . Ser. A 72 = Indag.
Math. 31 (1969), 449-457.
[ 7] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian
Hypergeometric Series , Halsted Press ( Ellis Horwood Limited,
Chichester) John Wiley and Sons , New York,- Chichester,
Brisbane and Toronto, 1985 .
~
\
)
.lfianiibha, Vol. 16, 1986
A PROPERTY OF CONTRACTIVE TYPE MAPPINGS
IN 2-METRIC SPACES
Bv.
Cbih - Ru Hsiao
Department of Mathematics,
Statistics, and Computer Science,
University of Illinois at Chicago,
Chicago, Illinois 6'.>680, U. S. A.
(Received : October 9, 1985 ; Revised [Final] : September 23, 1986)
ABSTRACT
All the contractive type mappings in the theorems in [2] to [ 12]
have property (f-1) in a 2-metric space . If a mapping f possesses
property (H) in an Euclidean 2-metric space, then the points given by
the iterati,on of f at any point in the Euclidean 2-metric space are
collinear. Not all 2-metric spaces have property (H*) .
1. INTRODUCTION
The concept of 2- metric space has been investigated by S. Gabler
in [I]. Many contractive type principles on 2-metric spaces have been
proved by K lseki (2] , M. S Khan (3] , S. N. Lal and A. K. Singh
( [4], [SJ), S N. Lal and Mohan Das ( [6] ? [7] ) , B. E. Rhoades
[8] , A. K. Sharma (9], ( [I OJ, [ 11] ), and S. L. Singh [ 12] The purpose
of this article is to show that all the fixed point theorems in [2] to
[ 12] can apply only to a self-mapping/, which is defined in a
2-metric space (X, d) and has the following property denoted by
property (fl) :
"d l fix, fix , fkx ) = 0, for all i, j, k e /+
224 I
and all x E X , where /O x = x , v x E X ;
flx = fx; f2x = J0 fx; etc . "
It follows that all the theorems in [2] to [ 12} cannot apply to a
mapping which does not have the above rroperty .
2. MAIN RESULTS (PART I)
From [I] we have :
Definition 1. A 2-metric sp,ace is a space X with a non-negative
real-valued function d on X x X x X satisfying the following
conditions :
( i ) to each pair of distinct points x, y in X,
there exists a point z in X such tll~t
d ( x, y, z ) # 0 .
( ii ) d ( x, y, z ) = 0 , when at least two of x, y, z are equai
( iii ) d ( x, y, z ) = d ( ~· z, y ) = d ( y, z, x ) .
( iv ) d ( X, y, Z ) ~ d ( U, y, Z ) + d ( X, U. Z ) + d ( X, y, U )
( simplex inequality ) .
Definltion 2. Let ( X, d) be a 2-met.ric space, f: X -->- X.
If for any x E X, set : xo = x, Xi = fxo, X2 = fx1, · , Xn+i = fxn, .. ,
we have:
d ( X, , Xi , Xk ) = 0 , 'ef i, j, k E /+ ,
then we say that f has property (H) in ( X, d).
Definition 3. Let ( X, d) be a 2-m.etric space ,f, g: X ~ X.
\
\
j )
/
)
( 225
If for any x € X set : Xo = x, Xi. = fxo, x2 = gx1 , ... , X2n+1 = fx2n ,
x2n+2 = gx2,,+i , .. , we have d ( x; , x;, x" ) = 0 , Vi, j, k E /r , then
\'.e say that f and g have common property (H) in ( X, d).
Example. I I
Let J = ( 2
_ , l ] x [ T , l ] Define a 2-metric d
on J as:
. 1 I ( 1J1 a2 i I I d ( ( a1 , a 2 ), ( b1, b2 ) , ( c1 , c2 ) ) = - - I det I b 1 b2 I I I .
2 I l c 1 c2 I J l Define f: ( J, d )--'>- ( J, d ) as :
f ( ( a. b ) ) = ( al/2; bl/2 ) .
104 11 4 _ 10 2 II 2 Take xo = ( ( Ti) , ( i2 ) ),,, X1 = f'(o =· ( ( ii ) . ( i2 ) ) '
10 l I x2 = I'(1 = ( fr , i2 ) . Then d ( xo, x 1, x 2 ) =F 0 , l!-Dd f does not
have property (H) in ( J, d) .
Remark I. Let x : ( x1 , X2 ,. .. , Xn), Y : ( Yl> Y2 ., .. , Yn),
and z : ( z10 z2 ,. . ., z,. ) be three points in the Euclidean 2-metric space
(R11 , d),
d(x,y,z) I I ___,. ___,. 2 I xy I • I xz I . sin a j '
__..
where I xy I denotes the length of the vector from x to y ,. j xz I denotes the lenEth of the vector from x to z, aod ix is the angle between --'>- __,,_ xy and xz
The foJk;wing is a special case of Theorem 1 to be stated later .
226 1
_,.
If d ( x, y, z ) = 0 , then I xy I = 0 , or I xz I = 0 ,
or a = 0 or TI;, and x ,y , and z are collinear .
From the above argument we know : If a mapping f has property
(H) in a Euclidean 2-metric space , then the points given by the iter
ation of f at any point are collinear .
Remark 2 . Later we shall prove that , if a mapping f satisfie3 the
conditions of any fixed point theorem in [2] to ( 12], then f has property
(H) in a 2-metric space, and, if mappings/, g sausfy the CJnditions
of any common fixed point theorem in L2] to lJ2], then f and g have
common property (H) in a 2-metric space . So alt tne theorems in L2]
to p2] fail to discuss the mappings which do not have property (H)
in a 2-metric spac.:e ,
we are not interested in what the property ( H) leads to, since th~
mappings which have proverty (H) rn a 2-metnc are trivial We
prove :
Theorem l. Let (x, d) bea 2-mllric ~pa_ce. Mappings f,g, :X--i>X
satisfy the following conaition:
d ( fx, gy, a ) ~ <P ( d ( x, y, a ), d ( x, fx, a) d ( y, gy, a ) ,
d ( x, gy, a), d ( y, f'c, a)) .............. ( 1 *)
for all x,y.a " X, where the mapping <P : R~ ~ R+ is non - decre
asing in each variable, and
<P ( t, t, t, t, l ) < t, for each t > 0 ......... (2*)
Then f and g have common property (fl) in ( X, d) .
Proof . First Prove :
\
~\
1 I
)
d ( Xn+2, Xn+I , Xn ) = 0 for all n € [+
When n is even let n = 2i , from (I*) we have
d ( X2'+2 , X2i ,.} , X2i ) = d ( X2i+l , X2i+2 , X2i )
= d ( f'(2i , gX2i.T-l , X21 )
.;;;;; cp ( d ( X2i , X2i+J , X2i ) , d ( X2i , X2i +-1 ; X2J ) ,
[ 227
(A)
d ( A2i+J , X2i+2 , X2i ), d { X2;, X21+2 , X2i ), d ( X21+1 , X2i+l, Xzi ) )
= cp ( 0, 0, d ( X2i+2 , X21+1 , X2i ) , 0, Q ) ·
Suppose d ( x2;+2 , x2;+1 , x2; ) = t* > 0 . Since cf; is nondecreasing
by (2*) we have t* ,;;;;; cp ( t*, t*, t*, t*, t*) < t*, a contradiction
Hence d ( x21,.2 , x2H.1 , x2;) = 0 . Similaly we can prove
d ( X2i+3 . X2i+2 , X2i+l ) = 0 •
We now prove each of the following :
d ( Xn Xn H, Xo ) = 0, for all n e /+ . ········· (B)
d t Xn, Xn+I, X; ) = 0, for all n e [+ . ········· (C)
d ( Xo, X1, Xh ) = 0, for all k E /+ ........ (D)
d ( Xn, Xn-t-i , x1c ) = 0, for all n e / 1- , any k e /+ (E)
d ( X;, X;, Xk = 0, for all i, j, k E [+ (F)
Note that t B) is trivially true if n is 0 or l • Assume the induct
ion hypothesis~ suppose n is odd, and let n = 2m+ 1 • Using {l *)
we have:
d ( X2rn+1 , X2m+2 , Xo )
d ( fx2m , gx2mH , XJ )
228 J
~ <fa ( d ( X2m , X2m+l , Xo ) , d ( X2m , X2m+? , Xo ) , d ( X2m+1 ,
X2m+2 , XO ) t d ( X2m , X2m+2 , Xo ) , d ( X2m+1 , X2m+I , Xo ) ) •
= <fa ( 0, 0, d ( X2m+l , X2m+2 , Xo ) , d ( X2m , X2m+2 , XO ) , 0 )
Using the simplex inequlity and (A), we have :
d ( X2m , X2m+2 , Xo )
~ d l X2m+l• X2m+2• Xo )+d ( X2m, X2m+i Xo )+d ( X2m, X2m+2• X2m+1 )
= d ( X2m+l• X2m+2, XO ) •
If d l x2m, x2m+2, xo) :::j::. 0, then, from (2*) we obtain a contradiction.
The proof under ,the assumption that n is even is similar. as is the
proof of ( C) .
\
(D) is trivially true for k = 0 . Assume the induction hypl}th- ~ esi s . From the simplex inequality we have
d ( xo, xh Xk+l )
~ d ( Xk, Xi. Xk+1) + d( XQ, Xk, Xk+l) + d( XQ, X1, Xk).
The first two terms are zero by (B) and (C) and the third is zero by
the induction assumption . Hence d ( x0 , x1 , Xk+J ) = 0 and (D) is
proved .
(E) is true for n = 0 by (D) . Assume the induction hypothesis.
Suppose n is even and let n = 2m . From (I*) we have :
d ( X2m, X2m+I• Xk ) = d ( gX2m-1, fx2m, X1c )
~ cp ( d ( X2m, X2m-t· Xk ), d ( X2m• X2m+l• Xk) ,
d ( X2m-1, X2m, Xk ) , d ( X2m, X2m' Xk ) ,
d ( X2m-], X2m+1' Xk ) ) •
'·
f :!
I I !;
[ 229
Using the simplex inequality, the induction assumption, and (A),
d ( Xzn,,_ 1 , Xzm+1· X11) < d ( Xzm, X2m+l• X1c) • lf d ( Xzm, X2m+i. X11)
-::/:: 0 , then ( 2 *) leads to a contradiction . The proof under the assum
ption that n is odd is similar to the above argument .
To prove (F). we may assume, without loss of generality, that
i < j. Moreover. we may assume strict ineqnality, since, otherwise,
we obtain zero by the clefi nit ion of 2-metric spaces . Using the 8implex
inequality; \\e have
d ( X;, X; Xk )
< d ( X;_l, X;, X1,) + d ( X,, X;_ 1• Xie) + d ( X;, X;, X;_} )
= d (Xi, X; -1 Xk) ,
since the other t No tel m3 are zero by (E)
Continuing in this same manner, we obtain d ( X;, x;_i. X1c)
< d ( Xi. XJ 2• X1r ) ,,;;;; ... < d ( X;, Xi+I • X1; ) = 0 by (E) .
A different proof of Theorem I appears in the appendix . It is
included for its valu~ as a proof technique .
There is a special case of Theorem I; that is :
Theorem 2. Let f be a self-mapping on a 2-metric space ( X, d)
such that :
d ( fx, fy, a ) < <P (a ( x, y, a) , d ( x, f.,., a), d ( y.fy, a ) ,
d ( x,fy, a ) , d ( y, f", a)) (3*)
fo: all x. v, a EX. <P satisfies (2*) in Theorem I. Then f has property
(I-/) in ( X. d ) .
Remark 3. It is easy to check that all the contractive type mappings
in the theorems in (5], (8], (9], [ lO] and [ 12] have property (H) in (X, d)
230 ]
or common property (H) in ( X, d) , since they all are special cases of
(I*) in Theorem l or (3*) in Theorem 2 •
3. MAIN RESULTS (PART II)
Lemma l formalizes the fact that condition (£) implies (F) .
Lelllrna I. Let { Xn } be a sequence in 2-metric space ( X, d) and
d ( Xm, Xm+I> Xn) = 0, Vm, n E /+ . (1)
Then d( x;, X;, Xk) = 0, yi, }, k E /+ .
Theorelll 3. Let { Xn i be a sequence in 2-metric space ( X, d)
If there exists a real number k E f 0, I J such that, for all n " N
and all a EX,
d ( Xn, Xn+1, a ) .;;:; k • d ( Xn_}, Xn, a ) , (2)
then
d ( Xi , Xj , X1c ) 0, \f i, j, k E /+
Proof. From (2) we know :
d ( Xn, Xn+I> Xm ) .;;:; k • d l Xn_}, Xn, Xm ) ,
for all n E N and m " f+ •
JO When m = n or n + 1 , d ( Xn, Xn+I• Xm ) = 0
20 When m < n,
d ( Xn, Xn+1, Xm) .;;;; k. d ( Xn_} ~ Xn, Xm)
.;;:; k2 . d ( Xn_2 , Xn_1 , Xrn )
.;;;; kn-m , d ( Xm , Xm+i , Xm ) = 0
30 When m 'i>.n + 1,
\
[ 231
d ( Xn , Xn71 , Xrn )
< d ( Xm_}, Xn+; , Xm) + d ( Xn,. Xm_1, Xm ) + d ( Xn, Xn+h Xm_J ) •
By }O and 20 we have d ( Xm_ 1 , x,,,, Xn+i ) = 0 and
d ( Xm_J , Xm, Xn) = 0 . Then d ( Xn, Xn+J , Xm < d ( Xn, Xn+l , Xm_} ).
Continuing in this way we have :
d ( Xn , Xn+i , Am ) < d ( Xn , Xn+i , Xm_1 )
< d ( x,, , Xr.+l , Xm_z )
< d ( Xn , Xn+l , Xn+t ) = 0 ,
40 When n = 0 ,
( xo, X1, Xm ) < d ( Xz. x,,., Xo ) + d ( X1, X2, Xo) + d (Xi, X2, Xm ).
By JO, 20, 30 we have d ( x1, x2, x0 = 0 and
d ( X1, X2, Xrn ) = 0 Then d ( Xo, Xi, X n ) < d ( Xz. Xm, Xo )
Continuing in this way we have :
d ( Xo, Xi. Xrn ) < d ( Xz, Xm, Xo )
< d ( X3 , X.,. , Xo )
< d ( x,._t , X.n , Xo ) = 0 .
From ro '20' 30 and 40 we have: d(x,,, Xn+i 'Xai) = 0 ,
vm,n E /+. Then, by Lemma I, we have : d (Xi' Xj. X1c) = 0'
vi, j, k € [+ •
Remark 4 • By Lemma l and Theorem 3, it is easy to check that
all the contractive type mappings in the theorems in (4] and [ll] have
property ( H) in (X,d) or common property (H) in (X, d) .
232 1
4. MAIN RESULTS (PART III)
In this part we shall illustrate that all the proofs of the theorems
in [2], [3], [6], and [7] are :ncomplete or wrong .
Following liJ, we have Definitions 4 and 5.
Definition 4. A sequence {xn} in a 2-metric space (X, d) is convergent
and x EX is the limit of this sequence if
lim d ( x,. , x , a ) = 0 for all a E X . n~oo
Definition 5. A sequence {x11 } in a 2-metric space (X,d) is called a
Cauchy sequence if Jim d ( Xa,, x,,, a) = 0, for all a EX • m,n-+oo
Remark 5 .. From Definitions 4 and 5, we know : In the proof of
the Theorem of [2]. the author needs to prcve that the inequality
p ( x,. , x,.+t, a ) < q p ( x,._1 , x,. , a ) is true for all points a in X .
It is only shown to be true for some points a of X . Since the authors
of [3]. (6] and [7] use the same proof technique, those results are
also incorrect •
We are not interested in whether the statement
'' p ( x,, ; Xn+I , a ) < q p ( Xn_1 , Xn, a )' · is true for all a E X or not,
smce by Theorem 3 we know that :
''p ( Xn, Xn+i , a) < q p ( Xn_ 1 , Xn , a), for all a E X and all a EN''
implies that : p ( X; , X; , Xi. ) = 0 , fur all i, j, k E /+ .
Thus T has prop-::rty ( H) in (X, d)
5. CONCLUSIONS
In this article we have shown that all the theorems rn [2] to [ l 2]
only can discuss mappings that have property (H) in a 2-metric space.
\ '
\ ...
I 233
If f has property (H) in Euclidean 2-metric space ( R2, A ) , where
A ( x, y, z ) denotes the area of the triangle L',xyz formed by joining
the three points x, y, z in R2 . Then for each x in R2 , the iteration
off at x is in a straight Ii ne . So f is a trivial mapping. In fact, the
definition of a 2-metric space given by Gabler is not strong • The
Euclidean 2-metric space (R2, A) has the following property denoted
by property (fl*) :
''If A ( a, b, x ) = 0 = A ( a, b, y ) and A ( a, x, y ) =I= 0 ,
then a = b"
But, if we define a new 2-metric A* on R2 by A* (x, y, z ) = the area
of the intersection of L',xyz and the unit disc in R2, then we have:
A* ( ( 2, 0), ( - 2, 9 ) , ( 2, 9 ) ) = 0
A* ( ( -2 , 0 ) , ( -2, 9 ) , ( 2 , 9 ) ) = 0 and
A* ( ( -2, 0 ) , \ 2. 0) , ( 2, 9 ) ) =/= 0 , but
( 2, 9 ) =I= ( -2, 9 ) .
Hence ( R2, A* ) doesn't have property (H*) . Thus not all 2-metric
spaces have property (H*) as Euclidean 2-metric spaces have. Moreover,
in a 2-metric space ( X, d) if we want to prove " a = b '', we must
prove " d ( a, b, x) = 0 , for all x in ( X. d )'' .
We cannot establish a good fixed point theorem in a 2-metric space by considering iterations of a contra9tive type mapping . We
:,,uggest that we investigate 2-metric spaces that have property H* •
Then we have a better chance of establishing a useful fixed point
theorem in 2 metric spaces .
234 1
Acknowledge1nents
I am grateful to Professor H. M. Srivastava and the referee for
providing a short proof of Theorem I and for many valuabk
suggestions .
REFERENCES
[ I f' S. Gabler, 2-metrische Rilme und ihre Topologische struktur, Math. Il/achr. 26 (1963/<4), 115-148,
[ 2] K. Iseki, Fixed point theorem in 2-metric spaces, Math. Sem.
Notes KOhl:' U1,iv 3 (197.'), 133-136. [ 3 ] M. S. Khan, On fixed point theorems in 2-metric space, Pub/.
Inst. Math. tBeoKrad) ( N S) 27 (41) (1980), 107-112.
[ 4 ] S N. Lal and A. K. Singh, Invariant points of gen~rahzed nonexpensive mappings in 2 metrie spaces, Indian J. Pure Appl.
Math. 20 ( 19'8), 71-76 [ 5] S. N. Lal and A. K. Singh, An analogue of Banach's contraction
principle for 2-metric spaces, Bull. Austral. Math. Soc, 18
{1978)' 137-143 . [ 6 ] S. N. Lal and Mohan Das. Mappings with common invariant
l oints in :-metric spaces. Math . Sem . Notes Kobe Uni!'. 8 ( 1980), 8.l-89 .
[ 7 J S. N. Lal and Mohan Das. Mappings with common invariant points in 2-metric space. Ii Math. Sem. Notes Kobe Univ . HI (1982)' 691-595.
[ 8 ] B. E. Rhoade:>. Contraction type mappings on a 2-metric space. Math Atnhr 91 ( 1979), 151-15).
l 9 J A. K Sharma. On fixed points in 2-metric spaces, Math. Sem. Notes Kobe Univ. o ( 1971) ) , 467-473 .
[ 10 J A. K. Sharma, A generalization of Banach contraction prir.ciple to 2-metric space.;, Math. Sen. Notes Kooe Univ.
7 ( 1979)' 291-295. [ l l ) A. K. Sharma, On generalized contractions in 2-metric space.,,
Math. Sem Notes Ko!Je .Univ 10 (1982), 491-506. [ 12 ) S. L Singh, Some contractive type principles on 2-metric sp 1c-.:s
and applirntions, M.1th Sl:'ln Notes KObe Univ 7 (1979), l-11.
r 23s
APPENDIX
The following is a different way of proving Theorem I .
Proof. First prove :
d ( X11+2 . Xn+l , Xn ) = 0 for all n E [+ ••• (A)
When n is even let n = 2i , from (I*) we have
d ( X2i+2 , X2;+1 , X2; ) = d ( X2i-1 l , X2i+2, X2i )
= d ( fX21 , gX2i H , X21 )
~ rp ( d ( X2•+2 , X2i+l• X2i ), d ( X2; , X2i+i , X2; ) ,
d ( X2;. t, X2i+2, X21 ) , d ( X2i, X2i r2 , X2i ) , d ( X2i+l , X2;"·l X2i))
= rp ( 0, 0, d ( X2i i-2 , X2i+1 , X2; ) , 0 , 0) ·
Suppose d ( x2;+2 , x 2;+1 , x2i) = i* > 0 . Sirce rfo is nondecreasing
by ( 2*) we have t* ~ rfo ( t* , t*, t*, t*, t* ) < t*, a contradiction.
Hence d ( x2;+2 , x 2;+1 , x2, ) = 0 • Similarly , we can prove
d ( X2i+3 , x 2;+2 , X2i+t) = 0 . We now show that ;
d ( Xt1+3 , Xr. .. } , X11 ) = d ( Xn+3, Xn+2 , Xn ) = 0
for all n E 1+. ... (B)
By the simplex inequality
d ( Xn !-3 , Xn+i , Xn )
~ d l X'•+2 , Xn+J , Xn ) + d ( Xn+3 , Xn+2• Xn ) + d ( Xn+3 , Xn+b Xn+2 )
d ( Xn-i-3 , Xn., 2 , Xn )
~ d ( Xn+i , Xn+2 , Xn ) + d ( Xn+3 , Xn+1 , Xn ) + d ( Xn+3 , Xn+2 , Xn+i ) •.
From (A) we have
236 1
d ( Xn+3 ,Xn+1 , Xn ) .;;;;; d ( Xn+.3 , Xn+2 , Xn ) ,
and
d ( X11+3 , Xn+2 , Xn } .;;;;; d ( Xn J.3 • X11+1 , Xn ) '
then d ( Xn+3 , Xn+2 , Xn ) = d ( Xn+3 , Xn+t, Xn ) • When n is even,
let n = 2 i . Then
d ( X2i+3 , X2i+l , X2; ) = d ( Xv t-3 , X2; J.2 , X2i )
= d ( /X2i+2 , gX2i+J , X2; )
.;;;;; d> ( d ( x2;+2 , x 2;+1 , x2; ) , d ( X2i+2 , X21+a, x21 ) ,
d ( X2i+.J , X2i+2 , X2; ),( X2;+2 , X2i+2 , X21 ) t
d ( X2;+1 , X2i+3 , X2i ) )
= cP ( 0, d( X2i+3, X2i+2, X2i), 0, 0, d( X21+3 • X2i+l, X21)) •
Suppose d ( x21+a, .x2;+2 , x 2;) = d ( x21+a, x 2;+1 , x21) = t* > O,
then by (2*), 1* ~ <P (,t*, t*, t*, t*, t*) < t*, a contradiction.
This implies that (B) is true when n is.even.
Similarly, we can show that ( B 1 is true when n is odd . We now
show that d ( x;, x; , >.1, ) = 0, for all i, j, k e /+ •
[ st~p 1 1 • d < x; , x1 , x" ) = o , vi, j, k e { o, 1, 2, 3 } .
Since (A) is true, we only have to show
( JO ) { d ( X3 , XJ , Xo ) = 0
d ( .\3 , X2 , Xo) = 0 ,
Since (B) is true , ( Io ) is trivial .
( Step 2 J . d ( x; , x; , Xk ) = 0 , vi, j , k e { 0, 1, 2, 3, 4} •
By [ Step l ] and (A) we onJy have to show
3?8 ] ..
( d ( X5 ' X1 , Xo ) = 0 I I d ( X5 , x 2 , X.o ) = 0
( 30) ~ I d ( X5 , xa , xo ) = 0 I l d ( x5 , x4 , xo ) = 0
( d ( X5 , X2 , X1 ) = 0 I
( 21 ) ~ d ( X5 , X3 , X1 ) = 0 I L d ( X5 , X4 ' X1 ) = 0
{
d ( X5 , X3 , X2 ) = 0 ( )2)
d ( X5 , X4 , X2 ) =' 0 ·
The proof of ( 12) and ( 21) is similar to [Steil 2J; that is, "rewrite
{Step 2] by considering d (Xi, X1, x,,), Vi, j, k E { l, 2, 3, 4, 5 }
instead of considering d ( Xi , x; , Xk) , Vi, j, k E { 0, I, 2 3, 4 }",
since (B) is always true •
So we only have to show ( 30 )
d ( X5, X1 , Xo )
~ d ( Xl 'X1 ' Xo ) + d ( X5 t Xi ' Xo ) + d ( X5 • X1. X1 )
--+ d ( X", ' X1 , Xo ) ~ d ( X5 ' X1 , Xe) , for I = 2, 3, 4 .
d ( X5 , X1 , Xo )
~ d l X1 ' X1 ' Xo ) + d ( X5 ' X1 • Xo ) + d ( X5 ' X1 ' X1 )
=:> d ( X5 ' X1 ' Xo ) ~ d ( X5. X1 ' Xo ) ' for l = 2 ' 3, 4 .
Hence d ( x5 , x1 , Xo ) = d ( X5 , xi , xo ) , t = 2, 3, 4 • Since
for all i, j, E { 0, I, 2, 3. 4, 5} either d( x;, x;, x0 ) = 0
or d ( X; , x; , x0 ) appears in ( 30 ) , and
d ( x; , Xi , xo ) = d ( X5 , x1 , xo ) , then by (l *) and (2*) it's
easy to show that :
[ 239
d ( x5 , x1 , XJ ) = d ( x5, x1 , xu ) I ~ 2, 3, 4 , Su~pose cotltinue
in this way we can prove :
[ Step m -2 l , d ( Xi • x; • Xk ) = 0 , Vi, j. k, " { 0, l, 2, ... , m } .
To show :
[Step m-1 ] , d ( x,, \:;,Xi,) = 0, Vi, j. k E { 0, l, 2, .. , m,
m+ l } .
[ ( m·-1 )O l
{ ( m-2 )l ]
We only have to show
( <f { Xm+l , X1 , X3 ) = 0 I I I
~ I I l
a ( Xrn.+i , x2 • .X0 ) = O
d ( Xm+l, Xm , Xo) = 0
( d ( Xm+l , X2 , X1 ) = 0 I ~ I l_ d 1.. Xm +1 , Xm , X1 ) = 0
{
d ( Xm+] ' Xm_J , ·'""-2 ) = 0 ... ,[l".-2]
d ( X ,,_, 1 , Xm, Xm_1 ) = 0
Since the proof of [ ( m-2 )1] , ( ( m-3 )2] , •. ~ [ 2m~3] , [ }".-2]
is similar to [ Step m-2 ] , we only have to sh~w [ ( m-1 y:i ] . By simplex ineqtiaihy we can show that :
d ( Xm+i , X1 , Xo ) c-= d ( Xm-11 , Xi , Xo ) , I = 2 , 3 , ... , m ;
then by tl *) , (2*) we can prove [ ( m-l )1 ] • By ~ath'ematical induction for m the proof is completed .
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Indian
D. V. Postgraduate College
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·:1
CONTENTS
An Unified Approach for the Contractive mappings -D. Delbosco
Fixed Point of Set-Valued Mappings, of Convex Metric Spaces -M. D. Guay & K. L. Singh
Common Fi;.ed Point in Metric Spaces -Duane E. Anderson, K. L. Singh & J. H. M .. Whitfield
A Theorem on the Existence of Positive Eigenvalues of k-Set Contractions ·-A. Carbone and G. Trombetta
A Degree Th~ory for Weakly Continuous Multivalued Maps in Reflexive Banach Spaces -Giulio Trombetta On Certain Operational Generating and Recurrence Relations
-Hukum Chand Agrawal 55
Some Linear and Bilinear Generating
Functions of Three Variables On Certain Polynomials Associated with
Relations Involving Hypergeometric
-M.B. Singh the Multivariable H-Function
•. -P. C. it1unot and Renu Mathur
67
81
On Some Self - Superposable Flows in Conical Ducts
-Pramod Kumar Mittal 91
The Generalized Prolate Spheroidal Wave Function and its Applications
-V. G. Gupta 103 On a Generalized Multiple Transform • I
-K, N. Bhatt and S. P. Goyal 113 An Expansion Formula for the H-Function of Several Variables
· ~· -Geeta Aneja 121 Some Theorems for a Distributiooal Generalized Laplace Transform
-S. K. Akhaury 129 On (-1,1) Rings with Commutators in the Left Nutleus
-Yash Paul and S. Sharadha 145 Certain Multidimensional Generating Relations
-B. D. Agrawal and Sanchita Mukherjee 151
Differential Equations for a General Class of Polynomials
-Shri Kant & R. N. Pande_y 161
Slow Unsteady Flow in a Porous Annulus with Axial Waviness
-V. Bhatnagar and Y. N. Gaztr 169
Random Vibration of a Beam to Stochastic Loading
-Ashok Ganguly 185 Multiple Hypergeometric Functions Related to Lauricella's Functions
-R. C. Singh Chandel and Anil Kumar Gupta 195
Qn ~ntert~ediate Lauricella Functions
-Per W. Karfrson 211
A Property of Contractive Type Mappings in 2-Metric Spaces
-Chih-RuHisiao 223