Boundary Value Problems

, 2011:827510 | Cite as

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

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Abstract

We consider the existence, multiplicity of positive solutions for the integral boundary value problem with Open image in new window -Laplacian Open image in new window , Open image in new window , Open image in new window , Open image in new window , where Open image in new window is an odd, increasing homeomorphism from Open image in new window onto Open image in new window . 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 Open image in new window is involved with the first-order derivative explicitly.

Keywords

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

1. Introduction

We are interested in the existence of positive solutions for the integral boundary value problem

where Open image in new window , and Open image in new window satisfy the following conditions.

(H1) Open image in new window is an odd, increasing homeomorphism from Open image in new window onto Open image in new window , and there exist two increasing homeomorphisms Open image in new window and Open image in new window of Open image in new window onto Open image in new window such that

Moreover, Open image in new window , where Open image in new window denotes the inverse of Open image in new window .

(H2) Open image in new window is continuous. Open image in new window are nonnegative, and Open image in new window , Open image in new window .

The assumption (H1) on the function Open image in new window was first introduced by Wang [1, 2], it covers two important cases: Open image in new window and Open image in new window . The existence of positive solutions for two above cases received wide attention (see [3, 4, 5, 6, 7, 8, 9, 10]). For example, Ji and Ge [4] studied the multiplicity of positive solutions for the multipoint boundary value problem
where Open image in new window , Open image in new window . They provided sufficient conditions for the existence of at least three positive solutions by using Avery-Peterson fixed point theorem. In [5], Feng et al. researched the boundary value problem

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

As for integral boundary value problem, when Open image in new window is linear, the existence of positive solutions has been obtained (see [8, 9, 10]). In [8], the author investigated the positive solutions for the integral boundary value problem

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 Open image in new window satisfies (H1) and Open image in new window depends on both Open image in new window and Open image in new window . 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 Open image in new window depending on Open image in new window , 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

The basic space used in this paper is a real Banach space Open image in new window with norm Open image in new window defined by Open image in new window , where Open image in new window . Let

It is obvious that Open image in new window is a cone in Open image in new window .

Lemma 2.1 (see [7]).

Let Open image in new window , then Open image in new window , Open image in new window .

Lemma 2.2.

Let Open image in new window , then there exists a constant Open image in new window such that Open image in new window .

Proof.

The mean value theorem guarantees that there exists Open image in new window , such that
Moreover, the mean value theorem of differential guarantees that there exists Open image in new window , such that
So we have

Denote Open image in new window ; then the proof is complete.

Lemma 2.3.

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

Proof.

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

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

Lemma 2.4.

Assume that (H1), (H2) hold. Suppose Open image in new window is a solution of BVP(1.1); then

Proof.

First, by integrating (1.1) on Open image in new window , we have
According to the boundary condition, we have

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

Now we define an operator Open image in new window by

Lemma 2.5.

Open image in new window is completely continuous.

Proof.

Let Open image in new window ; then from the definition of Open image in new window , we have

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

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

Thus, Open image in new window is completely continuous.

It is easy to prove that each fixed point of Open image in new window is a solution for BVP(1.1).

Lemma 2.6 (see [1]).

Assume that (H1) holds. Then for Open image in new window ,

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

Definition 2.7.

The map Open image in new window is said to be a nonnegative continuous concave functional on a cone of a real Banach space Open image in new window provided that Open image in new window is continuous and
for all Open image in new window and Open image in new window . Similarly, we say the map Open image in new window is a nonnegative continuous convex functional on a cone of a real Banach space Open image in new window provided that Open image in new window is continuous and

for all Open image in new window and Open image in new window .

Let Open image in new window and Open image in new window be a nonnegative continuous convex functionals on Open image in new window , Open image in new window a nonnegative continuous concave functional on Open image in new window , and Open image in new window a nonnegative continuous functional on Open image in new window . Then for positive real number Open image in new window , and Open image in new window , we define the following convex sets:

Theorem 2.8 (see [11]).

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

(1) Open image in new window , Open image in new window , and Open image in new window , Open image in new window ;

(2) Open image in new window , Open image in new window , and Open image in new window , Open image in new window .

Then Open image in new window has at least one fixed point in Open image in new window .

Theorem 2.9 (see [12]).

Let Open image in new window be a cone in a real Banach space Open image in new window . Let Open image in new window and Open image in new window be a nonnegative continuous convex functionals on Open image in new window , Open image in new window a nonnegative continuous concave functional on Open image in new window , and Open image in new window a nonnegative continuous functional on Open image in new window satisfying Open image in new window for Open image in new window , such that for positive number Open image in new window and Open image in new window ,

for all Open image in new window . Suppose Open image in new window is completely continuous and there exist positive numbers Open image in new window , and Open image in new window with Open image in new window such that

Open image in new window and Open image in new window for Open image in new window ;

() Open image in new window for Open image in new window with Open image in new window ;

() Open image in new window and Open image in new window for Open image in new window with Open image in new window .

Then Open image in new window has at least three fixed points Open image in new window , such that

Open image in new window for Open image in new window ,

Open image in new window ,

Open image in new window with Open image in new window ,

Open image in new window .

3. The Existence of One or Two Positive Solutions

For convenience, we denote

where Open image in new window denotes 0 or Open image in new window .

Theorem 3.1.

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

    There exist two constants Open image in new window with Open image in new window such that

     

(a) Open image in new window for Open image in new window and

(b) Open image in new window for Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window .

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

For Open image in new window , we obtain Open image in new window and Open image in new window , which implies Open image in new window . Hence, by (2.12) and Lemma 2.6,
This implies that
Next, for Open image in new window , we have Open image in new window . Thus, by (2.12) and Lemma 2.6,
From (2.13), we have
This implies that
Therefore, by Theorem 2.8, it follows that Open image in new window has a fixed point in Open image in new window . That is BVP(1.1) has at least one positive solution such that Open image in new window .
  1. (ii)
     

then for all Open image in new window , let Open image in new window . For every Open image in new window , we have Open image in new window . In the following, we consider two cases.

Case 1 ( Open image in new window ).

In this case,

Case 2 ( Open image in new window ).

In this case,

Then it is similar to the proof of (3.6); we have Open image in new window for Open image in new window .

Next, turning to Open image in new window , there exists Open image in new window such that
Then like in the proof of (3.3), we have Open image in new window for Open image in new window . Hence, BVP(1.1) has at least one positive solution such that Open image in new window .
  1. (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 Open image in new window which allow us to apply Theorem 2.9 of BVP(1.1).

Let the nonnegative continuous concave functional Open image in new window , the nonnegative continuous convex functionals Open image in new window , Open image in new window , and nonnegative continuous functional Open image in new window be defined on cone Open image in new window by
By Lemmas 2.1 and 2.2, the functionals defined above satisfy

for all Open image in new window . Therefore, the condition (2.19) of Theorem 2.9 is satisfied.

Theorem 4.1.

Assume that (H1) and (H2) hold. Let Open image in new window and suppose that Open image in new window satisfies the following conditions:

Open image in new window for Open image in new window ;

Open image in new window for Open image in new window .

Open image in new window for Open image in new window ;

Then BVP(1.1) has at least three positive solutions Open image in new window , and Open image in new window satisfying

where Open image in new window defined as (3.1), Open image in new window .

Proof.

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

This proves that Open image in new window .

To check condition ( Open image in new window ) of Theorem 2.9, we choose
It is similar to the proof of assumption (i) of Theorem 3.1; we can easily get that

This shows that condition ( Open image in new window ) of Theorem 2.9 is satisfied.

Thus condition ( Open image in new window ) of Theorem 2.9 holds.

So like in the proof of assumption (i) of Theorem 3.1, we can get

Hence condition ( Open image in new window ) of Theorem 2.9 is also satisfied.

Thus BVP(1.1) has at least three positive solutions Open image in new window , and Open image in new window satisfying

5. Examples

In this section, we give three examples as applications.

Example 5.1.

where Open image in new window for Open image in new window .

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

Example 5.2.

where Open image in new window for Open image in new window .

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

Example 5.3.

Let Open image in new window , Open image in new window ; consider the boundary value problem
It is easy to check that
Thus, according to Theorem 4.1, BVP(5.8) has at least three positive solutions Open image in new window , and Open image in new window satisfying

Notes

Acknowledgments

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

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Copyright information

© Yonghong Ding. 2011

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.

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouChina

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