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Holomorphic solutions of some functional equations IIIMiami Suzuki aa Department of Mathematics, Faculty of Informatics , Teikyo Heisei Unisersity , Uruido,Ichihara-shi, 290-0193, JapanPublished online: 29 Mar 2007.

To cite this article: Miami Suzuki (2000) Holomorphic solutions of some functional equations III, Journal of DifferenceEquations and Applications, 6:4, 369-386, DOI: 10.1080/10236190008808236

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Journal of Difference Equnrions and App/icorions, Q 2000 OPA (Overseas Pubhshers Assoaanon) N.V.

2OOn. Vol 6, pp. 369-7x6 Published by hcense under Repnnts ava~lable dtrectly from the pubhsher the Gordon and Breach Snence Photocopying pernutted by License only Pubhshers imprint.

Pnnted ir. Malaysta.

Holornorphic Solutions of Some Functional Equations Ill

Department of Mathematics, Faculty of Informatics, Teikyo Heisei University, Otani2289-23, Uruido, Ichihara-shi, Chiba, 290-0193, Japan

(Received I February 1999; In final form 17 June 7999)

We will concern with existence of solution $(x) of a functional equation $(X(x, $ ( I ) ) ) = Y(x, qixj) . where X(x , y) = i x -+ 2 p m n ~ m y z , Y(x, yj = px i Xmtn 2 2 g m ~ x n ' y n ,

i- s - important fioril i h ~ viewpin: of simultaneous systems ~f difTerence ~yuatims. The cases / X / > 1 5: O < / X / / ! have becn treated in a former paper. Hence we will consider here the case j X / = i. Further we wiii consider the cast that X = 1, / p 1 = 1 ( p # 1). The case X = p = 1 will be dealt within another paper.

Keywords: Difference equation; Functional equation; Analytic solution

1980 Mathematzcr Subject Clarrlficatlonr 39A-10. 39B-12. 39B-22

1 INTRODUCTION

We will consider here a simultaneous system of difference equations

x(s + 1 ) = X ( X ( S ) , Y ( S ) ) , Y ( S + 1 ) = Y ( x ( s ) , Y ( s ) ) , ( 1 . 1 )

where

* Tel.: 81-436-74-6139. Fax: 81-436-74-7872. E-mail: rn.suzuki@thu.ac.jp.

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For single difference equation,

it is knmm that there exists en5re sduticn x(s) with

We wish to consider the corresponding problems for system ( 7 .I) . Thus, we will seek a pair of entire solutions !x(s), y(s)) such that

(xis + rj, y(s + r j j - - jO,O) as r ---i -m (P E Rj, (1.3)

uniformly for s E K, where IK is any compact subset of the s-plane. Single difference equations of the first order are studied in some

extent [7], So, it would be worthwhile considering to reduce the system (1.1) to some single first order equation.

S??pp~se there is. a fiincti~n $ (x ) , h~!om~rphic in 1x1 : 6, which satisfies

for a 6 > 0. Let x(s) be a solution of

with x(s i- r) -+ 0 as r -+ -m. Put y(s) = $(x(s)), then (x(s), y(s)), is a solution of (1.1) with (1.3). Hence, solving the system (1.1) is reduced to solving the single Eq. (1.5). Therefore the functional Eq. (1.4) is of significance from the viewpoint of systems of difference equations (1.1).

We proved in [5] that, when I X I # 1, (1.4) admits solution $(x) which is holomorphic at x = 0. In this note, we will consider the case where I X I = 1. D

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Note the following fact: Put

in (1. I) , then we obtain

u is + 1 ) = U(u(s ) , v ( s ) ) , v(s + 1 ) = V(u(s ) , v(s)), (1.6)

with

a"2O - .-,, I \ Q ~ O = - ~ ~ ~ ~ ( / L - A ~ ) , _h

A l l B11 = bpi1 + - X ( 1 a - p ) , Q l l = aq11 +-p(1 b - A), (1-7)

b2 '402 2 B02 Po2 = _Po2 a + c4 (A - p ), Q02 = bgo2 + - b p(1 - p) .

Now we suppose 1 X / = 1 in (1.2). As in [5] , we assume that

Xk # p for any k E N, (1.8)

and

920 # 0.

Note that, under the assumption (1.8), the condition (1.9) may be sup- posed to hold without losing generality, as seen from (1.7). By these assumptions, non-zero formal solution $(x ) to (1.4) is determined uniquely, in the form

whether

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372 M. SUZUKI

Although the formal solution $(x) the form (1.10) gives holo- morphic solution for (1.4) when I X f 1 we will show here that, when X=ei"" with irrational a, there is no holomorphic solution 4(x) of (1.4) for most cases, if (1.1) admits solutions (x(s'),y(s)) with the condition (1.3) ("most cases" means the cases represented by the Siegel condition (I .I 1) below).

THEOREM 1 Suppose, in (1.2), that X = eiN". Further suppose that ct is irrational and satisfies the Siegel condition (1.1 1)

Suppose there exist holomorphic solutions (x(s), y(s)) with the condition (1.3). Then. for (1.41, lhere exists izo solution $(x) which is holomorphic at x = O .

Hence, if (I .4) admits a holomorphic solutions $(XI, then there is no solution, wiih (1.3),jbr (1.1).

When / X 1 = : and (1.4) does not admit the solution, we may exchange the role of x, y if 1 p 1 # 1. In fact, we may consider the equation for unknown 4(x),

instead of (1.4). Note that (1.12) may possess a solution, even if (1.4) does not.

It remains to consider the case that / X / = / p 1 = I . In connection with Theorem 1, we suppose X = e'(~'~)", p, q E Z.

Writing

we get by (1.4)

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HOLOMOP2HIC SOLUTIONS 173

Hence we can reduce the problem to the case X = I by considering x ~ ~ ( x , y) and ~ ~ ~ / v (A, .s) <A* A~.b,.~!, V/W ,,) V(V A k---,.y,l. ,A

We will deal in this note with the case X = I # p. The case X = p = 1 is considered in [6].

Wher! X = !, (1.4) does m t admit holomoqhic solution $!x)

in general. We consider the following example with 1 p # 1, for simplicity.

we obtain a, > 0 (n 2 2), and

and

Hence the formal power series is not convergent.

Therefore we seek a solution of (1.4) which is holomorphic, not at x = 0 but in an angular domain (1.13) below, and is asymptotically expanded to the formai solution (1.10) there. Put, for some K > 0 and S > O ,

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Now we state our results. First note that, if

in (1 3, then by (1.7) we can take a: b such that

So we may suppose that

THEOREM 2 Suppose X = 1 and 1 p 1 = 1 , p f 1 in (1 2). Further suppose that (1.15). Then we can determine a formal solution to (1.4) in the form of Eq. (1.10):

fci~ f i ; ~ ; ~ L 0 < K 5 ~ 1 2 , EX are 6 > 0 aid 6 d i i t i c i ~ $(XI .4) such that @jx) is holornorphic in D(n, 6) and is expanded asymptotically in the same domain as

From the viewpoint of difference equations, by Theorem 2, it remains to seek solutions x(s) for (1.5) such that x(s) E D(n, 6) for s

large.

2 PROOF OF THEOREM 1

Suppose that there exists a function $(x ) satisfying (1.4). Put

g ( x ) = X(x , $ ( x ) ) = Xx + C pm.xM($(x))", 1x1 < 6. m+n>2

Since X satisfies the condition (1.1 I), by [3] the equation

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HCLOMOLDM!C SOLUTIONS

admits a solution

We can choose numbers p > 0 and T > 0 SO that 8 ( z ) : O(z) is univalent in is/ < p and { 1x1 < r j c O({ / z i < p j ) . Let <x(sj,y(sj) be a soimion of (I. i j which satisfies ( l .3 ) . Let qo be a iiumbei a such that

Then, for any s E L(qO; a), there is a z, Iz 1 < p, such that

x(s)=@(z) , Z = Q - ' [ ~ ( s ) ] .

Put

Thus

and

X(S) = Q(7r(s)XS).

By (2.1) and @2), T(S) is bounded from above and below. Hence if s = E + iqO, < -+ -00, then I z I = I ~ ( s ) 1 . 1 e-a"70 1 is also bounded from above and below. Hence Q(.rr(s)Xs) cannot tend to 0 as Re[s] -+ m, which contradicts (1.3).

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376 M. SUZUKI

3 PROOF OF THEOREM 2

We can obtain a formal solution to (1.4) in the form (].lo), since A"= 1 fp.

Put

Since f(0, 0,O) = 0, f,(O, 0,O) = -p # 0, we obtain by the implicit function theorem that there is a function H(x, u) such that

which is ~o!oI?IoT"~~' Y--" i~ a C!OSP~ domain ] Y I 5 c i , f 2 5 EZ fer seine EI, i2 > 0. The raiige of H(x, u) contains a disc ly 5 E,. Put E = min(q, E Z , E ~ ) . Let +(x) satisfy (1.4): +(X(x, $(x))) = Y(x, $(x)), on a domain contained in 1x1 < E . If I$(x)I < r there, we obtain by (3.1) that

Conversely, if $(x) satisfies (3.2), it also satisfies (1.4). Hence it suffices to seek solutions $(x) of (3.2). We may suppose E > 0 is so small that there is a constant Go > 0 such that, noting that, JpJ = 1,

for 1x1 I E, J u J I E .

Let n (0 < n < 7r12) and N ( N E N) such that

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be fixed. Let g N ( x ) = be the truncation of the formal ~01utio11 jl . lOj . There is a constant GN siich that, for / x I < E ,

For a 5 > 0, consider D(K, 5) in (1.13). Take K > 0 and put

3 = 3 ( N , K, 6)

= ( @ ( x ) ; $ ( x ) 1s hoiomorphic and satisfies

I+(.)/ i ~ 1 x 1 " in D ( K , &)I, (3.5)

in which Kis determined by (3.10) below, if N is chosen. Take 5 < 1 so small that

We will show that, if K is fixed by @.If ) and 6 is determined sufficiently small as shown below, T [$I E 3.

Put v = X(x, gN(x ) + +(x)) = ~ ( 1 +pZOx + . . .) for a Q(x) E F , where by (3.61,

Then for x E D (K, 4,

Futher, if 6 is sufficiently small, we have 1 x 112 < I vl < I x 1 by the assumption pzO < 0 (Fig. 1).

Thus, if x E D(n, S), then we have v E D(n, 6) and $(X(x, gN(x) + $(x))) is well defined. Hence we can define the following map T : $ € F + T[$]as

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FIGURE 1

with v! = X(x, gN(x) + d(x)), that is?

with v2 = x(x, gN(x)j . Since gw(n) is the truncated formal solution, we get

for a constant K1, which is determined if N is fixed. We fix K so large that

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HOLOMOPuDR!C SOLUTIONS

Since, if 6, > 0 is small, we have for ! u ( , 1x1, 1 ~ 3 1 5 S1,

We take F so that

then we get for x E D(n, 6):

Further (noting that I p I = 1)

and, for u=gN(v1) + t+(vl) by (3.3)

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380 M. SUZUKI

hence

Thus we have, for 1 u: I = IgN(vl) 1,

Let S be taken so that

(GN+ K)S < 1 . (3.12)

Then, by (3.4) when Go < 1, we have

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and when Go > I , we have

Tkzs we hsve, fcr either csse?

and we have that T[I$] E F. i is obviousiy continuous and F is convex. Further F is d nuniid

f'amdy By Schauder's fixed point tineorem in [2,4]. we know the existence of a fixed pomts $fx) E F for the map T. That IS, we have, if we write w(xj = gN(x ) + qj(xj,

Next we will show that the fixed point 4 is unique for each N. In fact, suppose there would be q5j(x) E 3, j= 1,2 such that, writing vj = vj(x) = X(x, gN(xj + (Pi(x)), j = 1,2 , then

vi - v2 = (41 ( x ) - ~ Z ( X ) ) ( P I I X + higher terms)

= FI ( x ) (dl ( x ) - d2 ( x ) ) ,

g d v ~ ) - gN(v2)

= (41 ( x ) - 42(x))(2pll . a2x2 + higher terms), (3.14)

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M. SUZUKI

we get

Put Dl = D(rc/2,6'), (6' = (1 + sin(n12))-'6) and C= {(I I € - X I =

1x1 sin(rc12)) for x E Dl . Then C c D(K, 6) and by Cauchy integral [I] we see that

So that

Thus if N > 3,

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we obtain by (3.14)-(3.16) for x E D1\O:

for sufficiently small x. Since v l (x ) E D ( K , 6) for x E D ( K , S), we have

where v Y 1 ( x ) = V I ( V ~ ( X ) ) , v ~ ( x ) = X . Hence

Since Ih(x)l = j p+q l lx+0(x2)1 L 1 - lqllI 1x1 + 10(x2)I, we have I h(x) I 1 1 - blx 1 , with 3 = ] qll I $ 1 . for sufficiently smali x, Thus by (3.17) we get

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384 M. SUZUKI

Hence from (3.19) and (3.20) we get

Fwther, we have

if we take N so iarge that - dl - b > I and jx j is sufficiently smali, we get

since Ivf+'(x)l 5 I v ; ( x ) I 5 1x1 by (3.7). k Put p ( t ) = t ( l -d2t) , r ~ = r = 1x1, and rk=p ( r )=p(rk- ] ) for k > 1 .

By induction we have Iv,k(x) 1 5 rk-1. Since rk < rk-], we have some constani c (2 0) such that rk-+ c (as k -+ CO). And we see that c = c(1 - dzc). By d2 # 0, we have c = 0, i.e.,

rm = p m ( r ) -+ 0 (as m -+ 00).

Thus by (3.19)

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Thus

and c;il jx) = d2(x) if J X 1 is small. Hence we have the ilniqueness of the fixed point far the map T : & i (x) = C $ ~ ( X ) f ~ r x E D ( I E , ~ ) f a a 5xed N. We denote it by 4'"?(xj.

Let N' > N. Consider the families 3(N1), F (N) with constants IC(N1), S(N'); IC(N), S(N), respectively. Put 6=min(6(N1), 6(N)). Put $(x) = c$[~'](x) + ~N'(X) - g ,~(x) , then

where w = g,+) + $lN'](x), hence by the uniqueness we have

Therefore $IN1(x) =g&) + q5[N1(~) does not depend on N, which we denote by $(x). Obviously +(x) is a solution of (11.4) and

for large N, which shows that $(x) is asymptotically expanded in Db, 4.

Hence the proof is completed.

References

[I] L.V. Ahlfors, Complex Analysis (Third edition), McGraw-Hill Kogakusha, Ltd., Tokyo, Japan, 1979.

121 T. Kimura, Ordinary Differential Equation, Kyouritu shuppan Press, 1974. [3] C.L. Siegel, Iteration of analytic functions, Ann. Math. 43 (19421, 607-612.

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386 M. SUZUKI

D.R. Smart, Fixed Point Theorems, Cambridge tTniv. Piess, 1971. M . Suzuki, Holomorphic solutions of some functional equations, Nihonkai Marhr- matical Journal 5 (1994); 109- 1 14.

161 M. Suzuki, Holomorphic solutions of some functional equations 11, SEAMS Bull. Math. (to appear).

[q N. Yanagihara, Meromorphic solutions of some difference equations, Funkcial. Eqv. 23 (1980), 309-326.

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