- Research Article
- Open Access

# The Solutions of the Series-Like Iterative Equation with Variable Coefficients

- Yuzhen Mi
^{1}, - Xiaopei Li
^{1}Email author and - Ling Ma
^{1}

**2009**:173028

https://doi.org/10.1155/2009/173028

© Yuzhen Mi et al. 2009

**Received:**23 March 2009**Accepted:**6 July 2009**Published:**4 August 2009

## 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.

## Keywords

- Continuous Function
- Unknown Function
- Differential Geometry
- Variable Coefficient
- Structure Operator

## 1. Introduction

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.

for any in , where denotes .

Lemma 2.2.

Lemma 2.3.

for where as and as .

## 3. Main Results

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

thus .

For any , we have

Thus .

Define as follows:

Thus is a self-diffeomorphism.

Now we prove the continuity of under the norm . For arbitrary ,

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 mapping such that

Let we have as a solution of (1.2) in . This completes the proof.

Theorem 3.2 (Uniqueness).

Suppose that (P_{1}) and (P_{2}) 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

## Declarations

### Acknowledgments

This work was supported by Guangdong Provincial Natural Science Foundation (07301595) and Zhan-jiang Normal University Science Research Project (L0804).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.