Abstract
By constructing a structure operator quite different from that ofZhang and Baker (2000) and using the Schauder fixed point theory, the existence and uniqueness of the solutions of the series-like iterative equations with variable coefficients are discussed.
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1. Introduction
An important form of iterative equations is the polynomial-like iterative equation
where is a given function, is an unknown function, and is the th iterate of that is, The case of all constant was considered in [1–10]. In 2000, W. N. Zhang and J. A. Baker first discussed the continuous solutions of such an iterative equation with variable coefficients which are all continuous in interval In 2001, J. G. Si and X. P. Wang furthermore gave the continuously differentiable solution of such equation in the same conditions as in [11]. In this paper, we continue the works of [11, 12], and consider the series-like iterative equation with variable coefficients
where are given continuous functions and We improve the methods given by the authors in [11, 12], and the conditions of [11, 12] are weakened by constructing a new structure operator.
2. Preliminaries
Let , clearly is a Banach space, where , for in .
Let , then is a Banach space with the norm , where, for in .
Being a closed subset, defined by
is a complete space.
The following lemmas are useful, and the methods of proof are similar to those of paper [4], but the conditions are weaker than those of [4].
Lemma 2.1.
Suppose that and
where and are positive constants.Then
for any in , where denotes .
Lemma 2.2.
Suppose that satisfy (2.2).Then
Lemma 2.3.
Suppose that satisfy (2.2) and (2.3).Then
for where as and as .
3. Main Results
For given constants and , let
Theorem 3.1 (existence).
Given positive constants and if there exists constants and , such that
,
,
then (1.2) has a solution in .
Proof.
For convenience, let
Define such that , where
Since , it is easy to see that for all , and for all It follows from that is uniformly convergent. Then is continuous for . Also we have
thus .
For any , we have
By condition , we see that is convergent, therefore is uniformly convergent for , this implies that is continuously differentiable for . Moreover
By Lemma 2.1,
Thus .
Define as follows:
where . Because , and are continuously differentiable for all , is continuously differentiable for all . By conditions and , for any in we have
We furthermore have
Thus is a self-diffeomorphism.
Now we prove the continuity of under the norm . For arbitrary ,
Let
Then we have
which gives continuity of .
It is easy to show that is a compact convex subset of . By Schauder's fixed point theorem, we assert that there is a map** such that
Let we have as a solution of (1.2) in . This completes the proof.
Theorem 3.2 (Uniqueness).
Suppose that (P1) and (P2) are satisfied, also one supposes that
then for arbitrary function in , (1.2) has a unique solution .
Proof.
The existence of (1.2) in is given by Theorem 3.1, from the proof of Theorem 3.1, we see that is a closed subset of , by (3.12) and , we see that is a contraction. Therefore has a unique fixed point in , that is, (1.2) has a unique solution in , this proves the theorem.
4. Example
Consider the equation
where It is easy to see that
For any in ,
thus By condition , we can choose and by condition , we can choose . Then by Theorem 3.1, there is a continuously differentiable solution of (4.1) in .
Remark 4.1.
Here is not monotone for , hence it cannot be concluded by [11, 12].
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Acknowledgments
This work was supported by Guangdong Provincial Natural Science Foundation (07301595) and Zhan-jiang Normal University Science Research Project (L0804).
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Mi, Y., Li, X. & Ma, L. The Solutions of the Series-Like Iterative Equation with Variable Coefficients. Fixed Point Theory Appl 2009, 173028 (2009). https://doi.org/10.1155/2009/173028
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DOI: https://doi.org/10.1155/2009/173028