EXISTENCE OF SOLUTIONS FOR EQUATIONS INVOLVING ITERATED FUNCTIONAL SERIES

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1 EXISTENCE OF SOLUTIONS FOR EQUATIONS INVOLVING ITERATED FUNCTIONAL SERIES V. MURUGAN AND P. V. SUBRAHMANYAM Received 26 August 24 and in revised form 8 October 24 Theorems on the existence and uniqueness of differentiable solutions for a class of iterated functional series equations are obtained. These extend earlier results due to Zhang.. Introduction The study of iterated functional equations dates back to the classical works of Abel, Babbage, and others. This paper offers new theorems on the existence and uniqueness of solutions to the iterated functional series equation ( λ i H i f i (x = F(x, (. where λ i s are nonnegative numbers and f (x = x, f k (x = f ( f k (x, k N. In(. the functions F, H i and constants λ i (i N are given and the unknown function f is to be found. The above equation is more general than those considered by Dhombres [2], Mukherjea and Ratti [3], Nabeya [4], and Zhang [5]. 2. Preliminaries This section collects the standard terminology and results used in the sequel (see [5]. Let I = [a,b] be an interval of real numbers. C (I,I, the set of all continuously differentiable functions from I into I, is a closed subset of the Banach Space C (I,R of all continuously differentiable functions from I into R with the norm c defined by φ c = φ c + φ c, φ C (I,Rwhere φ c = max x I φ(x and φ is the derivative of φ. Following Zhang [5], for given constants M, M, and δ>, we define the families of functions ( I,M,M = { φ C (I,I:φ(a = a, φ(b = b, φ (x M x I, φ ( x φ ( x 2 M x x 2 x,x 2 I } (2. and δ (I,M,M ={φ (I,M,M :δ φ (x M for all x I}. Copyright 25 Hindawi Publishing Corporation Fixed Point Theory and Applications 25:2 ( DOI:.55/FPTA.25.29

2 22 Equations involving series of iterates In this context it is useful to note the following proposition. Proposition 2.. Let δ>, M, and M. Then (i for M<, (I,M,M is empty and for M =, (I,M,M contains only the identity function; (ii for δ>, δ (I,M,M is empty and for δ =, δ (I,M,M contains only the identity function. Proof. (i Let φ (I,M,M, where M<. Clearly φ is a strict contraction with Lipschitz constant M on I.Soφ has a unique fixed point contrary to the assumption that φ has at least two fixed points a and b. If φ (I,,M, then by the mean-value theorem and the hypothesis that φ (x forallx I, φ(b φ(x b x and φ(x φ(a x a for all x I. Sinceφ(a = a and φ(b = b, φ must necessarily be the identity function. (ii Let φ δ (I,M,M, where δ>. Then by the mean-value theorem, φ(b φ(a >ba. This contradicts that a and b are fixed points of φ. The argument for the case when δ = is similar to the case when M =. In view of the above proposition, one cannot seek solutions of equations such as (. in (I,M,M without imposing conditions on M. The following lemmata of Zhang [5] will be used in the sequel. Lemma 2.2 (Zhang [5]. Let φ, ψ (I,M,M. Then, for i =,2,..., ( (φ i (x M i for all x I, (2 (φ i (x (φ i (x 2 M ( 2i2 j=i M j x x 2 for all x,x 2 I, (3 φ i ψ i c ( i j= M j φ ψ c, (4 (φ i (ψ i c im i φ ψ c +Q(iM ( i j=(i jm i+j2 φ ψ c,where Q( =, Q(s = if s = 2,3,... Lemma 2.3 (Zhang [5]. Let φ δ (I,M,M. Then ( φ ( x ( φ ( x2 M δ 3 x x 2 x,x 2 I. (2.2 Lemma 2.4 (Zhang [5]. Let φ, φ 2 be two homeomorphisms from I onto itself and φ i (x φ i (x 2 M x x 2 for all x,x 2 I, i =,2. Then The following results are well known. φ φ c 2 M φ φ2 c. (2.3 Lemma 2.5 (see []. For each n N, let f n be a real-valued function on I = [a,b] which has derivative f n on I. Suppose that the infinite series n= f n converges for at least one point of I and that the series of derivatives n= f n converges uniformly on I. Then there exists a real-valued function f on I such that n= f n converges uniformly on I to f. In addition, f has a derivative on I and f = n= f n.

3 V. Murugan and P. V. Subrahmanyam 22 Lemma 2.6. Let f : J J be a differentiable function on an interval J in R satisfying the inequality <a f (x b, x J,forsomea, b in R. Then the inverse function f exists andisdifferentiable on J.Further,forallx J, Lemma 2.7. For n N, x R, b ( f (x a. (2.4 n ix i = (n +x n (x xn+ (x 2, x, n(n +, x =, 2 (2.5 and further ( x n x n n (n ix n+i2 (x 2 n x =, x, n(n, x =. 2 ( Existence In this section, we prove in detail a theorem on the existence of solutions for the functional series equation (.. Theorem 3.. Suppose (λ n is a sequence of nonnegative numbers with λ > and λ i =. Let F δ (I,λ ηm,m, H η(i,l,l, and H i (I,L i,l i for i = 2,3,..., where δ, η> and M, M, L i, L i for all i N. Assume further that (i M>, (ii K = (/(M λ i+ L i+ M i (M i and γ = λ η K M 2 >, (iii λ i L im i (M i <. Then the functional series equation λ i H i ( f i (x = F(x has a solution f in (I,M,M where M = (M + K M 2 /γ and K = λ i L im 2(i. Proof. For each φ (I,M,M, define the function ( (Lφ(x = λ i H i φ i (x for x I. (3. Since λ i, λ i =, and H i (x max{ b, a },(Lφ(xiswelldefinedforallx I. Further (Lφ(a = a and (Lφ(b = b.sinceφ and H i are differentiable with o H i (x L i and (φ i (x M i, λ i H i ( φ i (x ( φ i (x λi L i M i x I, i N. (3.2

4 222 Equations involving series of iterates and λ i L i M i converges in view of (ii. By Weierstrass M-test, λ i H i ( φ i (x ( φ i (x (3.3 converges uniformly on I. From Lemma 2.5, Lφ is differentiable on I and (Lφ (x = λ i H i ( φ i (x ( φ i (x x I, φ (I,M,M. (3.4 Since <η H (x L, it is clear that <λ η (Lφ (x λ i L i M i.writingk = λ i L i M i, we note that <λ η (Lφ (x K. (3.5 From Lemma 2.6,forx in I, <K ( (Lφ (x ( λ η. (3.6 In short, Lφ : I I is a nondecreasing self-diffeomorphism. For x,x 2 I, (Lφ ( x (Lφ ( x 2 = λ i H i ( φ i ( ( x φ i ( ( x H i φ i ( ( x 2 φ i ( x2 { λ i H i ( φ i ( ( x φ i ( ( x φ i ( x2 { + H i ( φ i ( ( x H i φ i ( ( x 2 φ i ( } x2 ( 2(i2 λ i L i M j=i2 } M i + λ i L im 2(i x x 2 (by the definition of H i s and using Lemma 2.2 { M = λ i+ L i+ M i( M i } + λ i L M im 2(i x x 2 = ( K M + K x x 2. (3.7 Thus, (Lφ ( x (Lφ ( x 2 K2 x x 2 x,x 2 I, (3.8

5 V. Murugan and P. V. Subrahmanyam 223 where K 2 = K M + K, K = /(M λ i+ L i+ M i (M i, and K = λ i L im 2(i.FromLemma 2.3, it follows that ( (Lφ ( x ( (Lφ ( x2 K 2 λ 3 η 3 x x 2 x,x 2 I. (3.9 We define T : (I,M,M C (I,Iby (Tφ(x = ( (Lφ ( F(x φ (I,M,M, x I. (3. Clearly (Tφ(a = a,(tφ(b = b,andby(3.6wehave δk (Tφ (x = ( (Lφ ( F(x F (x M x I. (3. So T is a sense-preserving diffeomorphism of I onto I.Forx,x 2 I, (Tφ ( x (Tφ ( x 2 = ( (Lφ ( ( ( F x (Lφ ( ( F x2 ( (Lφ ( ( F x F ( x ( F x 2 + ( (Lφ ( ( ( F x (Lφ ( ( F x2 F ( x 2 λ η x x 2 + K 2 λ 3 F ( ( x η 3 F x2 λ ηm M M λ η x x 2 + K 2λ 2 η 2 M 2 ( λ 3 x η 3 x 2 as F (x λ ηm ( M + K 2 M 2 = x x 2 λ η ( M + K M M 2 + K = M 2 x x 2. λ η (3.2 Since M (λ η K M 2 = M + K M 2, (Tφ ( x (Tφ ( x 2 M x x 2 x,x 2 I. (3.3 It implies that Tφ (I,M,M. Next we show that T is continuous. For arbitrary functions φ i (I,M,M, we denote f i = Tφ i, i =,2. Then f i (x M, f i (x f i (x 2 M x x 2, and

6 224 Equations involving series of iterates ( fi (x K /δ for x,x,x 2 I and i =,2. Hence, f f 2 c = f ( ( ( f f f2 c { ( = max f (x f f2 (x ( f ( f2 (x f 2 (x } x I { = max f (x ( f ( 2 f2 (x ( f f2 (x ( f ( f2 (x f 2 (x } x I { M max f (x ( f ( 2 f2 (x ( f f2 (x ( f ( f2 (x } x I { M max f (x ( f ( 2 f2 (x ( f ( f2 (x x I + ( ( f (x f f f2 (x ( f ( f2 (x } { M 2 ( max f ( 2 f2 (x ( f ( f2 (x x I + MK M δ max x ( f x I ( f2 (x }. (3.4 Thus, f f 2 c M 2 ( f ( f c 2 + MK M δ f f2 c. (3.5 Besides, by Lemma 2.4,wehave From (3.5and(3.6, it follows that f f c 2 M f f2 c. (3.6 Tφ Tφ c 2 = f f c 2 = f f c 2 + f f 2 c M f f2 c + M 2 ( f ( f2 c + MK M δ f f2 c. (3.7 Thus, Tφ Tφ c 2 E f f2 c, (3.8 where E = max{m + K MM /δ,m 2 }. Furthermore, since F δ (I,λ ηm,m, an application of Lemma 2.3 gives ( F ( x ( F ( x2 M δ 3 x x 2 x,x 2 I. (3.9

7 V. Murugan and P. V. Subrahmanyam 225 Now f f2 c = F ( Lφ F ( Lφ c 2 = F ( Lφ F ( Lφ 2 c + (( F ( Lφ ( Lφ (( F ( Lφ2 ( Lφ2 c. (3.2 Using Lemma 2.6 and the fact that F δ (I,λ ηm,m, f f2 c = δ Lφ Lφ c 2 + (( F ( ( (( Lφ Lφ F ( ( Lφ2 Lφ c + (( F ( Lφ2 (( Lφ ( Lφ2 c δ Lφ Lφ c 2 + K M δ 3 Lφ Lφ c 2 + ( ( ( Lφ Lφ2 c by(3.5and(3.9 δ ( δ + K M ( ( Lφ Lφ2 c. δ 3 (3.2 Thus, f f2 ( c E 2 ( Lφ Lφ2 c, (3.22 where E 2 = /δ + K M /δ 3. By the definition of Lφ,wehave Lφ Lφ 2 c λ i H i ( φ i λ i L i φ i φ2 i ( Hi φ i c 2 c ( i λ i L i M j φ φ c 2 j= i i+ L i+( M λ j φ φ c 2. j= ( H i (x L i, x I, i =,2,... ( by Lemma 2.2 (3.23 Thus, Lφ Lφ 2 c ( M λ i+ L i i+ φ φ c 2. (3.24 M

8 226 Equations involving series of iterates Further, ( Lφ ( Lφ2 c λ i ( H i ( φ i λ i { [ H i ( φ i + H i ( φ i 2 ( φ i H i ( φ i 2 [( φ i ( H i (φ i 2 ]( φ i c ( φ i c ( φ i ] c } { λ i M i L i φ i φ i ( 2 c + L i φ i ( φ i c 2 } ( using the fact that Hi (I,L i,l i, i N,andbyLemma (3.25 λ i M i L i ( i M j φ φ c 2 + λ i L i (i M i φ φ 2 c j= ( i2 + λ i L i Q(i M (i j M i+j3 φ φ c 2. j= Upon relabelling the subscripts in the above, we get ( Lφ ( Lφ2 c ( i λ i+ M i L i+ M j φ φ c 2 + λ i+ L i+ im i φ φ 2 c j= (3.26 ( i + λ i+ L i+ Q(iM (i jm i+j2 φ φ c 2. j= From Lemma 2.7, ( Lφ ( Lφ2 c ( M λ i+ M i L i i+ φ φ c 2 + λ i+ L i+ im i φ M φ 2 c { M + λ i+ L i+ Q(iM M i i (M 2 i } φ φ c 2. M (3.27

9 V. Murugan and P. V. Subrahmanyam 227 From (3.22and(3.24, it follows that Lφ Lφ 2 c = Lφ Lφ c 2 + ( ( Lφ Lφ2 c ( M λ i+ L i i+ φ φ c 2 M ( M + λ i+ M i L i i+ φ φ c 2 + λ i+ L i+ im i φ M φ 2 c { M + λ i+ L i+ Q(iM M i i (M 2 i } φ φ c 2. M (3.28 We can more conveniently rewrite this as Lφ Lφ 2 c λ i+ A i+ φ φ 2 c, where A i+ = max{((m i /(M (L i+ + M i L i++l i+ Q(iM M i [(M i /(M 2 i/(m ]; L i+ im i }. By hypotheses (ii and (iii of the theorem and with the fact that i (M i /(M, it is easy to see that the series λ i+ (L i+ + M i L i+((m i /(M, λ i+ L i+ im i,and λ i+ L i+ Q(iM M i {(M i /(M 2 i/(m } converge. Since the convergence of n= a n, n= b n for a n, b n foralln N implies that of n= max{a n,b n },weconcludethat λ i+ A i+ converges. We denote it by E 3.Thuswehave Lφ Lφ 2 c E 3 φ φ 2 c. (3.29 From (3.8, (3.22, and (3.29, it follows that Tφ Tφ 2 c E E 2 E 3 φ φ 2 c. (3.3 Consequently, T : (I,M,M (I,M,M is a continuous operator. Next we show that (I,M,M isaconvexcompactsubsetofc (I,R. The routine proof that (I,M,M isaclosedconvexsubsetofc (I,Risomitted. For φ (I,M,M, φ c = φ c + φ c max{ a, b } + M and for x in I, φ (x M. So (I,M,M is an equicontinuous family of functions bounded in the norm c.since φ (x φ (x 2 M x x 2 for all x,x 2 I and φ (I,M,M, {φ : φ (I,M,M } is also an equicontinuous family. From Arzela-Ascoli theorem and Lemma 2.5,weconcludethat (I,M,M isacompactconvexsubsetofc (I,R. T is a continuous map on (I,M,M into itself and by Schauder s fixed point theorem T has a fixed point in (I,M,M. Thus there is a function φ (I,M,M such that (Tφ(x = φ(x. So (Lφ (F(x = φ(x and λ i H i (φ i (x = F(x. Thus φ is a solution of the functional series equation (.in (I,M,M. Additionally,wenotethatifE E 2 E 3 <, then T is a contraction mapping on the closed subset (I,M,M ofc (I,R. So by Banach s contraction principle, T has a unique fixed point, which gives a solution of (.. This is restated in the following theorem.

10 228 Equations involving series of iterates Theorem 3.2. In addition to the hypotheses of Theorem 3., suppose that the number E E 2 E 3 is less than, where E = max {M + K MM },M 2, E 2 = δ δ + K M δ 3, E 3 = λ i+ A i+, K = λ i L i M i, A i+ = max {( M i M (Li+ + M i L i+ + L i+ Q(iM M i [ M i (M 2 i M ],L i+ im i }. (3.3 Then (.hasauniquesolutionin (I,M,M. Remark 3.3. When we are seeking a solution of (. withλ > and λ i = for given functions F δ (I,λ η,m, H η(i,l,l, by Proposition 2., λ ηm = λ η. Since λ andη, λ η =. So λ = = η. Furtherλ i = fori = 2,3,... Thus F and H are identity functions, and our equation reduces to f (x = x. Example 3.4. Consider the functional series equation ( e/27 f (x+ Here we have i!27 i sini [ π 2 f i (x] = x 2(e /2 e t/2 dt, x [,]. (3.32 λ = e/27, H (x = x, λ i = i!27 i, H i(x = sin i ( π 2 x, fori = 2,3,..., (3.33 and F(x = /2(e /2 x e t/2 dt, x I = [,]. Choose M = 3, M e /2 = 2 ( e /2, δ = 2 (, e /2 η =, L =, L =, L i = π 2 i, L i = π2 i(i, i = 2,3,... 4 (3.34 Then F( =, F( =, δ = /2(e /2 F (x e /2 /2(e /2 (55/27 e /27 3 = λ ηm, and F (x F (x 2 e /2 /2(e /2 x x 2 for x,x,x 2 I. So F δ (I,λ ηm,m, H (x (I,L,L, and H i (x (I,L i,l ifori = 2,3,...We note

11 V. Murugan and P. V. Subrahmanyam 229 that F is not differentiable on [,]. Now λ i = /i!27 i = e /27 28/27 and so λ i =. Also K M 2 = λ i+ L i+ M i+( M i M = π 2 (i +!27 i+ 2 (i +3i+( 3 i = π ( 3 2i+ 3i+ 4 i!33(i+ i!3 3(i+ = π [ ] 36 3!3 i 3!3 2i = π ( e /3 e /9 + = π ( e /3 e / (3.35 Thus we have λ η>k M 2.SinceL =, λ i L im i (M i = λ i+ L i+m i( M i+ π 2 = (i +!27 i+ 4 i(i +3i( 3 i+ ( = π2 3 2i+ 3i 4 33(i+ 3 3(i+ (i! < π2 4 (i! = π2 4 e. (3.36 As the positive series λ i L im i (M i converges and M>, K = λ i L im 2(i is finite. Since all the hypotheses of Theorem 3. are satisfied we conclude that there is a solution for (3.32in (I,M,M form = (M + K M 2 /(λ η K M 2. Example 3.5. Consider the functional series equation Setting ( 8e 4 e +2 ( f f (x+ i (x i i! = 8e4( e x, x I = [,]. (3.37 e λ = 8e4 e +2 8e 4, F(x = ex e, λ i = i!8e 4, H i(x = x i, i = 2,3,..., x I, (3.38 equation (3.37 canberewrittenas λ i H i ( f i (x = F(x. Clearly F( =, F( =, /(e F (x e/(e, and F (x e/(e for x I. Upon choosing M = 2, M = e e, δ = e, η =, L i = i, L i = i(i, i N, (3.39

12 23 Equations involving series of iterates it is readily seen that λ ηm = ((8e 4 e +2/8e 4 2 >e/(e. So F δ (I,λ ηm,m, H (x η(i,l,l, and H i (x (I,L i,l ifori = 2,3,...Also, K M 2 = λ i+ L i+ M i+( M i M = M (i +!8e 4 (i +2i+( 2 i i!4e 4 4i = ( e 4 4e 4 4. (3.4 Thus λ η = λ > /2 >K M 2.Further, λ i L im i( M i = λ i L im i( M i = λ i+ L i+m i+( M i+ = (i +!8e 4 (i +i2i( 2 i+ = ( 2 2i 8e 4 2 i (i! = 4 i e 4 (i! 2 i 4e 4 (i! = 4e 2. (3.4 Since all the hypothesesoftheorem3. are satisfied, we conclude that there is a solution for the given equation (3.37 in (I,M,M, where M = (M + K M 2 /(λ η K M 2 and K = λ i L im 2(i. Corollary 3.6. Suppose (λ n is a sequence of nonnegative numbers with λ > and λ i =.LetF δ (I,λ M,M,whereδ>, M, M. Assume further that (i M>, (ii K = (/(M λ i+ M i (M i and γ = λ K M 2 >. Then the functional series equation λ i f i (x = F(x has a solution f in (I,M,M, where M = M /γ. Proof. The proof follows from Theorem 3. upon setting H i (x x for each i N. Example 3.7. Consider the following functional series equation i f i (x = 2π [ 3 3 sinx, x I =, π ]. ( Here we have λ i = 26 2π, F(x = 27i 3 3, H i(x = x x I, i N. (3.43

13 V. Murugan and P. V. Subrahmanyam 23 Upon choosing M = 3, M = π 3, η =, δ = π 3 3, (3.44 it is readily seen that δ F (x 2π/3 3and F (x π/3forallx I. Nowλ ηm = (26/27(3 > 2π/3 3andsoF δ (I,λ ηm,m. Clearly λ i =, K M 2 = 3/24 < λ,andm = M /(λ K M 2 = 72π/9. Thus by Corollary 3.6, there is a function f in (I,3,72π/9 satisfying the functional series equation (3.42. The main theorem of Zhang [5] can be deduced as a corollary to Theorem 3.. Corollary 3.8 (Zhang [5]. Given positive constants δ, M, M, n N, wesupposethat M> and λ >K M 2,whereK = /(M n λ i+ M i (M i. Then for each F δ (I,λ M,M, there is a solution for the equation n λ i f i (x = F(x, λ >, λ i, i = 2,3,...,n, n λ i = in (I,M,M,whereM = M (λ K M 2. Proof. Setting λ i = foralli>nin Corollary 3.6, the result follows. The following theorem proves the existence of a solution for an iterative functional series equation. Given δ>, M, M, we define δ (I,M,M = { φ C (I,R:δ φ (x M x I and φ ( x φ ( x 2 M x x 2 x,x 2 I }. (3.45 Clearly δ (I,M,M δ (I,M,M. Theorem 3.9. Suppose (λ n is a sequence of nonnegative numbers with λ > and λ i =. Let F δ (I,λ ηm,m, H η(i,l,l, and H i (I,L i,l i for i = 2,3,..., where δ, η>and M, M, L i, L i for all i N. Assume further that (i M = ((b a/(f(b F(aM>, (ii K = (/(M λ i+ L i+ M i (M i and γ = λ η K M 2 >, (iii λ i L im i (M i <. Then the functional series equation ( λ i H i f i (x = (b af(x bf(a+af(b, x I, (3.46 F(b F(a has a solution f in (I,M,M, wherem = (M + K M /γ, 2 M = ((b a/(f(b F(aM, and K = λ i L im 2(i. Proof. For a function F δ (I,λ ηm,m, the mapping F defined by F(x = (b af(x bf(a+af(b F(b F(a x I (3.47

14 232 Equations involving series of iterates is readily seen to belong to δ (I,λ ηm,m, where δ = ((b a/(f(b F(aδ.From Theorem 3., it now follows that λ i H i (φ i (x = F(x forsomeφ (I,M,M. Corollary 3.. Let δ, η>, M>, L, λ i, i N with λ > and λ i =.Suppose that γ = λ η K M 2 >, where K = (L/(M λ i+ M i (M i. If F δ (I,λ ηm,m, H η(i,l,l, andh i (I,L,L for i = 2,3,..., then there is a solution function φ for the equation λ i H i (φ i (x = F(x in (I,M,M,whereM = M( + K M/γ and K = L λ i M 2(i. Corollary 3.. Let δ, η>, M>, λ i, i N with λ > and λ i =. Suppose that γ = λ η K M 2 >,wherek = (/(M λ i+ M i (M i.iff δ (I,λ ηm, M, H η(i,m,m, and H i (I,M,M for i = 2,3,..., then there is a solution function φ for the equation λ i H i (φ i (x=f(x in (I,M,M,whereM =M( + K M/γ and K = λ i M 2i. Acknowledgment The first author acknowledges the Council of Scientific and Industrial Research (India for the financial support provided in the form of a Junior Research Fellowship to carry out this research work. References [] R.G.Bartle,The Elements of Real Analysis, John Wiley & Sons, New York, 964. [2] J. G. Dhombres, Itération linéaire d ordre deux, Publ. Math. Debrecen 24 (977, no. 3-4, (French. [3] A. Mukherjea and J. S. Ratti, On a functional equation involving iterates of a bijection on the unit interval, Nonlinear Anal. 7 (983, no. 8, [4] S. Nabeya, On the functional equation f (p + qx + rf(x = a + bx + cf(x, Aequationes Math. (974, [5] W. Zhang, Discussion on the differentiable solutions of the iterated equation n λ i f i (x = F(x, Nonlinear Anal. 5 (99, no. 4, V. Murugan: Department of Mathematics, Indian Institute of Technology Madras, Chennai 6 36, India address: murugan@iitm.ac.in P.V. Subrahmanyam: Department of Mathematics, Indian Institute of Technology Madras, Chennai 6 36, India address: pvs@iitm.ac.in