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

**2009**:173028

**DOI: **10.1155/2009/173028

© Yuzhen Mi et al. 2009

**Received: **23 March 2009

**Accepted: **6 July 2009

**Published: **4 August 2009

## Abstract

## 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.

Lemma 2.2.

Lemma 2.3.

## 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.

Thus is a self-diffeomorphism.

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

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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