- Research Article
- Open Access

# Existence of Solutions and Nonnegative Solutions for Prescribed Variable Exponent Mean Curvature Impulsive System Initialized Boundary Value Problems

- Guizhen Zhi
^{1}, - Liang Zhao
^{2}, - Guangxia Chen
^{3}, - Xiaopin Liu
^{1}and - Qihu Zhang
^{1, 4}Email author

**2010**:915739

https://doi.org/10.1155/2010/915739

© Guizhen Zhi et al. 2010

**Received:**10 May 2009**Accepted:**2 January 2010**Published:**23 January 2010

## Abstract

This paper investigates the existence of solutions and nonnegative solutions for prescribed variable exponent mean curvature impulsive system initialized boundary value problems. The proof of our main result is based upon Leray-Schauder's degree. The sufficient conditions for the existence of solutions and nonnegative solutions have been given.

## Keywords

- Sudden Change
- Euclidean Norm
- Differential System
- Positive Parameter
- Invariant Property

## 1. Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. On the Laplacian impulsive differential equations boundary value problems, there are many results (see [1–5]). Because of the nonlinearity of -Laplacian, the results about -Laplacian impulsive differential equations boundary value problems are rare (see [6]). In [7, 8], the authors discussed the existence of solutions of -Laplacian system impulsive boundary value problems. Recently, the existence and asymptotic behavior of solutions of curvature equations have been studied extensively (see [9–15]). In [16], the authors generalized the usual mean curvature systems to variable exponent mean curvature systems. In this paper, we consider the existence of solutions and nonnegative solutions for the prescribed variable exponent mean curvature system

where , with the following impulsive initialized boundary value conditions

where

For any , will denote the th component of ; the inner product in will be denoted by will denote the absolute value and the Euclidean norm on . Denote that , , and , , , where , . Denote that is the interior of , . Let

For any , denote that . Obviously, is a Banach space with the norm , and is a Banach space with the norm . In the following, and will be simply denoted by and , respectively. Denote that , and the norm in is .

The study of differential equations and variational problems with variable exponent conditions is a new and interesting topic. For the applied background on this kind of problems we refer to [17–19]. Many results have been obtained on this kind of problems, for example, [20–35]. If (a constant) and (a constant), then (1.1) is the well-known mean curvature system. Since problems with variable exponent growth conditions are more complex than those with constant exponent growth conditions, many methods and results for the latter are invalid for the former; for example, if is a bounded domain, the Rayleigh quotient

is zero in general, and only under some special conditions (see [25]), but the fact that is very important in the study of -Laplacian problems.

In this paper, we investigate the existence of solutions for the prescribed variable exponent mean curvature impulsive differential system initialized boundary value problems; the proof of our main result is based upon Leray-Schauder's degree. This paper was motivated by [6, 13, 36].

Let , then the function is assumed to be Caratheodory; by this we mean that

(i)for almost every the function is continuous,

(ii)for each the function is measurable on ,

We say a function is a solution of (1.1) if with absolutely continuous on , , which satisfies (1.1) on .

This paper is divided into three sections; in the second section, we present some preliminary. Finally, in the third section, we give the existence of solutions and nonnegative solutions for system (1.1)–(1.4).

## 2. Preliminary

In this section, we will do some preparation.

Lemma 2.1 (see [16]).

where ; .

then is a Banach space.

Lemma 2.2.

The operator is continuous and sends closed equiintegrable subsets of into relatively compact sets in .

Proof.

it is easy to check that is a continuous operator from to .

We want to show that is a compact set.

according to the continuity of , we can see that is convergent in . Thus we conclude that is convergent in . This completes the proof.

We denote that is the Nemytski operator associated to defined by

Define as

where .

Lemma 2.3.

where , .

- (i)

- (ii)

Hence is a solution of (1.1)–(1.4). This completes the proof.

## 3. Main Results and Proofs

In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)–(1.4).

We assume that

(H) for all

Theorem 3.1.

If (H ) is satisfied, then is an open bounded set in such that the following conditions hold.

(1) For any the mapping belongs to .

(2) For each , the problem

has no solution on .

Then (1.1)–(1.4) has at least one solution on .

Proof.

Denote that

Since is Caratheodory, it is easy to see that is continuous and sends bounded sets into equiintegrable sets. According to Lemma 2.2, we can conclude that is compact continuous on for any . We assume that for (3.4) does not have a solution on ; otherwise, we complete the proof. Now from hypothesis (2 ), it follows that (3.4) has no solution for , .

where obviously is the unique solution to this problem.

This completes the proof.

In the following, we will give an application of Theorem 3.1.

Denote that and

Obviously, is an open subset of .

Assume that

(H) , where is a positive parameter, and is Caratheodory.

(H) , for all .

Theorem 3.2.

If are satisfied, then problem (1.1)–(1.4) has at least one solution on , when positive parameter is small enough.

Proof.

where is defined in (3.3).

Obviously, is a solution of (1.1)–(1.4) if and only if is a solution of the abstract equation (3.8) when . We only need to prove that the conditions of Theorem 3.1 are satisfied.

(1 ) When positive parameter is small enough, for any , we can see that the mapping belongs to .

has no solution on .

- (i)

- (ii)
Suppose that , . This implies that for some .

It is a contradiction.

Summarizing this argument, for each the problem (3.8) with (1.4) has no solution on .

This completes the proof.

In the following, we will discuss the existence of nonnegative solutions of (1.1)–(1.4). For any , the notation means that for every .

Assume the following

where , and for all .

(H) and for all , .

(H) .

(H) satisfies for all .

Theorem 3.3.

If H H are satisfied, then the problem (1.1)–(1.4) has a nonnegative solution, when positive parameter is small enough.

Proof.

Hence is nonnegative. This completes the proof.

## Declarations

### Acknowledgments

This paper is partly supported by the National Science Foundation of China (10701066, 10926075 and 10971087), China Postdoctoral Science Foundation funded project (20090460969), the Natural Science Foundation of Henan Education Committee (2008-755-65), and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007).

## Authors’ Affiliations

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