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

- 1.1k Downloads

## 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

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

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 .

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.

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

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.

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.

Proof.

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

Lemma 2.5.

Open image in new window is completely continuous.

Proof.

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]).

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 Open image in new window and Open image in new window .

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]).

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 ,

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

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

Theorem 3.1.

- (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.

- (i)

- (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 ).

Case 2 ( Open image in new window ).

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

- (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).

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 ;

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 .

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.

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

## 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.

## Notes

### Acknowledgments

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

### References

- 1.Wang H: On the number of positive solutions of nonlinear systems.
*Journal of Mathematical Analysis and Applications*2003, 281(1):287–306.CrossRefMathSciNetMATHGoogle Scholar - 2.Wang H: On the structure of positive radial solutions for quasilinear equations in annular domains.
*Advances in Differential Equations*2003, 8(1):111–128.MathSciNetMATHGoogle Scholar - 3.Wang J: The existence of positive solutions for the one-dimensional
*p*-Laplacian.*Proceedings of the American Mathematical Society*1997, 125(8):2275–2283. 10.1090/S0002-9939-97-04148-8CrossRefMathSciNetMATHGoogle Scholar - 4.Ji D, Ge W: Multiple positive solutions for some
*p*-Laplacian boundary value problems.*Applied Mathematics and Computation*2007, 187(2):1315–1325. 10.1016/j.amc.2006.09.041CrossRefMathSciNetMATHGoogle Scholar - 5.Feng H, Ge W, Jiang M: Multiple positive solutions for
*m*-point boundary-value problems with a one-dimensional*p*-Laplacian.*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(8):2269–2279. 10.1016/j.na.2007.01.052CrossRefMathSciNetMATHGoogle Scholar - 6.Liu B: Positive solutions of three-point boundary value problems for the one-dimensional
*p*-Laplacian with infinitely many singularities.*Applied Mathematics Letters*2004, 17(6):655–661. 10.1016/S0893-9659(04)90100-0CrossRefMathSciNetMATHGoogle Scholar - 7.Wang Z, Zhang J: Positive solutions for one-dimensional
*p*-Laplacian boundary value problems with dependence on the first order derivative.*Journal of Mathematical Analysis and Applications*2006, 314(2):618–630. 10.1016/j.jmaa.2005.04.012CrossRefMathSciNetMATHGoogle Scholar - 8.Yang Z: Existence and uniqueness of positive solutions for an integral boundary value problem.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(11):3910–3918. 10.1016/j.na.2007.10.026CrossRefMathSciNetMATHGoogle Scholar - 9.Kong L: Second order singular boundary value problems with integral boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2010, 72(5):2628–2638. 10.1016/j.na.2009.11.010CrossRefMathSciNetMATHGoogle Scholar - 10.Boucherif A: Second-order boundary value problems with integral boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(1):364–371. 10.1016/j.na.2007.12.007CrossRefMathSciNetMATHGoogle Scholar - 11.Guo DJ, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar - 12.Avery RI, Peterson AC: Three positive fixed points of nonlinear operators on ordered Banach spaces.
*Computers & Mathematics with Applications*2001, 42(3–5):313–322.CrossRefMathSciNetMATHGoogle Scholar

## Copyright information

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.