- Research Article
- Open Access

# Positive Solutions for Integral Boundary Value Problem with *ϕ*-Laplacian Operator

- Yonghong Ding
^{1}Email author

**Received:**20 September 2010**Accepted:**19 January 2011**Published:**23 February 2011

## Abstract

We consider the existence, multiplicity of positive solutions for the integral boundary value problem with -Laplacian , , , , where is an odd, increasing homeomorphism from onto . We show that it has at least one, two, or three positive solutions under some assumptions by applying fixed point theorems. The interesting point is that the nonlinear term is involved with the first-order derivative explicitly.

## Keywords

- Boundary Condition
- Differential Equation
- Real Number
- Partial Differential Equation
- Ordinary Differential Equation

## 1. Introduction

where , and satisfy the following conditions.

Moreover, , where denotes the inverse of .

(H2) is continuous. are nonnegative, and , .

where the nonlinear term does not depend on the first-order derivative and , . They obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of by applying Krasnoselskii fixed point theorem.

The main tools are the priori estimate method and the Leray-Schauder fixed point theorem. However, there are few papers dealing with the existence of positive solutions when satisfies (H1) and depends on both and . This paper fills this gap in the literature. The aim of this paper is to establish some simple criteria for the existence of positive solutions of BVP(1.1). To get rid of the difficulty of depending on , we will define a special norm in Banach space (in Section 2).

This paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence of one or two positive solutions for BVP(1.1) is established by applying the Krasnoselskii fixed point theorem. In Section 4, we give the existence of three positive solutions for BVP(1.1) by using a new fixed point theorem introduced by Avery and Peterson. In Section 5, we give some examples to illustrate our main results.

## 2. Preliminaries

It is obvious that is a cone in .

Lemma 2.1 (see [7]).

Let , then , .

Lemma 2.2.

Let , then there exists a constant such that .

Proof.

Denote ; then the proof is complete.

Lemma 2.3.

Assume that (H1), (H2) hold. If is a solution of BVP(1.1), there exists a unique , such that and , .

Proof.

From the fact that , we know that is strictly decreasing. It follows that is also strictly decreasing. Thus, is strictly concave on [0, 1]. Without loss of generality, we assume that . By the concavity of , we know that , . So we get . By , it is obvious that . Hence, , .

On the other hand, from the concavity of , we know that there exists a unique where the maximum is attained. By the boundary conditions and , we know that or 1, that is, such that and then .

Lemma 2.4.

Proof.

By a similar argument in [5], ; then the proof is completed.

Lemma 2.5.

is completely continuous.

Proof.

So is monotone decreasing continuous and . Hence, is nonnegative and concave on [0, 1]. By computation, we can get . This shows that . The continuity of is obvious since is continuous. Next, we prove that is compact on .

Hence, is uniformly bounded and equicontinuous. So we have that is compact on . From (2.13), we know for , , such that when , we have . So is compact on ; it follows that is compact on . Therefore, is compact on .

Thus, is completely continuous.

It is easy to prove that each fixed point of is a solution for BVP(1.1).

Lemma 2.6 (see [1]).

To obtain positive solution for BVP(1.1), the following definitions and fixed point theorems in a cone are very useful.

Definition 2.7.

for all and .

Theorem 2.8 (see [11]).

Let be a real Banach space and a cone. Assume that and are two bounded open sets in with , . Let be completely continuous. Suppose that one of following two conditions is satisfied:

(1) , , and , ;

(2) , , and , .

Then has at least one fixed point in .

Theorem 2.9 (see [12]).

for all . Suppose is completely continuous and there exist positive numbers , and with such that

and for ;

() for with ;

() and for with .

Then has at least three fixed points , such that

for ,

,

with ,

.

## 3. The Existence of One or Two Positive Solutions

where denotes 0 or .

Theorem 3.1.

- (i)
There exist two constants with such that

(a) for and

(b) for ;

(ii) ;

(iii) .

Then BVP(1.1) has at least one positive solution.

- (i)
Let , .

- (ii)

then for all , let . For every , we have . In the following, we consider two cases.

Case 1 ( ).

Case 2 ( ).

Then it is similar to the proof of (3.6); we have for .

- (iii)
The proof is similar to the (i) and (ii); here we omit it.

In the following, we present a result for the existence of at least two positive solutions of BVP(1.1).

Theorem 3.2.

Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.

Then BVP(1.1) has at least two positive solutions.

## 4. The Existence of Three Positive Solutions

In this section, we impose growth conditions on which allow us to apply Theorem 2.9 of BVP(1.1).

for all . Therefore, the condition (2.19) of Theorem 2.9 is satisfied.

Theorem 4.1.

Assume that (H1) and (H2) hold. Let and suppose that satisfies the following conditions:

for ;

for .

for ;

where defined as (3.1), .

Proof.

We will show that all the conditions of Theorem 2.9 are satisfied.

This proves that .

This shows that condition ( ) of Theorem 2.9 is satisfied.

Thus condition ( ) of Theorem 2.9 holds.

Hence condition ( ) of Theorem 2.9 is also satisfied.

## 5. Examples

In this section, we give three examples as applications.

Example 5.1.

where for .

Hence, by Theorem 3.1, BVP(5.1) has at least one positive solution.

Example 5.2.

where for .

Hence, by Theorem 3.2, BVP(5.5) has at least two positive solutions.

Example 5.3.

## Declarations

### Acknowledgments

The research was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.

## Authors’ Affiliations

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