Existence of solutions for equations involving iterated functional series
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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. 1. 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 ∞
λi Hi f i (x) = F(x),
(1.1)
i=1
where λi ’s are nonnegative numbers and f 0 (x) = x, f k (x) = f ( f k−1 (x)), k ∈ N. In (1.1) the functions F, Hi and constants λi (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 1 (I,I), the set of all continuously differentiable functions from I into I, is a closed subset of the Banach Space C 1 (I, R) of all continuously differentiable functions from I into R with the norm · c1 defined by φc1 = φc0 + φ c0 , φ ∈ C 1 (I, R) where φc0 = maxx∈I |φ(x)| and φ is the derivative of φ. Following Zhang [5], for given constants M ≥ 0, M ∗ ≥ 0, and δ > 0, we define the families of functions
1 I,M,M ∗ = φ ∈ C 1 (I,I) : φ(a) = a, φ(b) = b, 0 ≤ φ (x) ≤ M ∀x ∈ I, φ x1 − φ x2 ≤ M ∗ x1 − x2 ∀x1 ,x2 ∈ I
and Ᏺδ1 (I,M,M ∗ ) = {φ ∈ 1 (I,M,M ∗ ) : δ ≤ φ (x) ≤ M for all x ∈ I }. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) 219–232 DOI: 10.1155/FPTA.2005.219
(2.1)
220
Equations involving series of iterates
In this context it is useful to note the following proposition. Proposition 2.1. Let δ > 0, M ≥ 0, and M ∗ ≥ 0. Then (i) for M < 1, 1 (I,M,M ∗ ) is empty and for M = 1, 1 (I,M,M ∗ ) contains only the identity function; (ii) for δ > 1, Ᏺδ1 (I,M,M ∗ ) is empty and for δ = 1, Ᏺδ1 (I,M,M ∗ ) contains only the identity function. Proof. (i) Let φ ∈ 1 (I,M,M ∗ ), where 0 ≤ M < 1. Clearly φ is a strict contraction with Lipschitz constant M on I. So φ has a unique fixed point contrary to the assumption that φ has at least two fixed points a and b. If φ ∈ 1 (I,1,M ∗ ), then by the mean-value theorem and the hypothesis that φ (x) ≤ 1 for all x ∈ I, φ(b) − φ(x) ≤ b − x and φ(x) − φ(a) ≤ x − a for all x ∈ I. Since φ(a) = a and φ(b) = b, φ must necessarily be the identity function. (ii) Let φ ∈ Ᏺδ1 (I,M,M ∗ ), where δ > 1. Then by the mean-value theorem, φ(b) − φ(a) > b − a. This contradicts that a and b are fixed points of φ. The argument for the case when δ = 1 is similar to the case when M = 1. In view of the above proposition, one cannot seek solutions of equations such as (1.1) in 1 (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 ∈ 1 (I,M,M ∗ ). Then, for i = 1,2,..., (1) |(φi ) (x)| ≤ M i for
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