- Research Article
- Open Access

# Positive Solutions for Fourth-Order Singular -Laplacian Differential Equations with Integral Boundary Conditions

- Xingqiu Zhang
^{1, 2}Email author and - Yujun Cui
^{3}

**2010**:862079

https://doi.org/10.1155/2010/862079

© The Author(s) Xingqiu Zhang and Yujun Cui. 2010

**Received: **7 April 2010

**Accepted: **12 August 2010

**Published: **19 August 2010

## Abstract

By employing upper and lower solutions method together with maximal principle, we establish a necessary and sufficient condition for the existence of pseudo- as well as positive solutions for fourth-order singular -Laplacian differential equations with integral boundary conditions. Our nonlinearity may be singular at , , and . The dual results for the other integral boundary condition are also given.

## Keywords

## 1. Introduction

where , , , , , , , , and is nonnegative. Let , . Throughout this paper, we always assume that , and nonlinear term satisfies the following hypothesis:

Remark 1.1.

Condition is used to discuss the existence and uniqueness of smooth positive solutions in [1].

Conversely, (1.4) implies (1.2).

Conversely, (1.5) implies (1.3).

Remark 1.2.

Typical functions that satisfy condition are those taking the form = , where , , ; .

Remark 1.3.

Boundary value problems with integral boundary conditions arise in variety of different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. They include two point, three point, and nonlocal boundary value problems (see [2–5]) as special cases and have attracted much attention of many researchers, such as Gallardo, Karakostas, Tsamatos, Lomtatidze, Malaguti, Yang, Zhang, and Feng (see [6–13], e.g.). For more information about the general theory of integral equations and their relation to boundary value problems, the reader is referred to the book by Corduneanu [14] and Agarwal and O'Regan [15].

where , , , , is nonnegative, symmetric on the interval , is continuous, and are nonnegative, symmetric on .

To seek necessary and sufficient conditions for the existence of solutions to the ordinary differential equations is important and interesting, but difficult. Professors Wei [16, 17], Du and Zhao [18], Graef and Kong [19], Zhang and Liu [20], and others have done much excellent work under some suitable conditions in this direction. To the author's knowledge, there are no necessary and sufficient conditions available in the literature for the existence of solutions for integral boundary value problem (1.1). Motivated by above papers, the purpose of this paper is to fill this gap. It is worth pointing out that the nonlinearity permits singularity not only at but also at . By singularity, we mean that the function is allowed to be unbounded at the points and .

## 2. Preliminaries and Several Lemmas

Definition 2.1.

Definition 2.2.

To prove the main results, we need the following maximum principle.

Lemma 2.3 (Maximum principle).

Proof.

Lemma 2.4.

Proof.

then from (2.29) and (2.30), we have (2.25).

Lemma 2.5.

then the solution of BVP (1.1) is a pseudo- positive solution.

Proof.

By condition , we have that is continuous.

By (2.33), it is easy to verify that is continuous and is a bounded set. Moreover, by the continuity of , we can show that is a compact operator and is a relatively compact set. So, is a completely continuous operator. In addition, is a solution of (2.35) if and only if is a fixed point of operator . Using the Shauder's fixed point theorem, we assert that has at least one fixed point , by , we can get

which contradicts the assumption that Therefore, is impossible.

Similarly, we can show that So, we have shown that (2.38) holds.

Using the method of [21] and Theorem .2 in [22], we can obtain that there is a positive solution of (1.1) such that and a subsequence of converging to on any compact subintervals of .

In addition, if (2.32) holds, then . Hence, is absolutely integrable on . This implies that is a pseudo- positive solution of (1.1).

## 3. The Main Results

Theorem 3.1.

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

which is the desired inequality.

Sufficiency.

Thus, we have shown that and are lower and upper solutions of BVP (1.1), respectively.

From (3.1), we have So, it follows from Lemma 2.5 that BVP (1.1) admits a pseudo- positive solution such that

Remark 3.2.

under the following condition:

Lei et al. [25] and Liu and Yu [26] investigated the existence and uniqueness of positive solutions to singular boundary value problems under the following condition:

for all , where and is nondecreasing on and nonincreasing on .

Obviously, (3.21)-(3.22) imply condition and condition implies condition . So, condition is weaker than conditions and . Thus, functions considered in this paper are wider than those in [23–26].

If satisfies one of the following:

Theorem 3.3.

Proof.

The proof is similar to that of Theorem 3.1; we omit the details.

Theorem 3.4.

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

which is the desired inequality.

Sufficiency.

Thus, we have shown that and are lower and upper solutions of BVP (1.1), respectively.

From the first conclusion of Lemma 2.5, we conclude that problem (1.1) has at least one positive solution .

## 4. Dual Results

By analogous methods, we have the following results.

Theorem 4.1.

Theorem 4.2.

Theorem 4.3.

Theorem 4.4.

Theorem 4.5.

Theorem 4.6.

Theorem 4.7.

Theorem 4.8.

Theorem 4.9.

## Declarations

### Acknowledgments

The project is supported financially by a Project of Shandong Province Higher Educational Science and Technology Program (no. J10LA53) and the National Natural Science Foundation of China (no. 10971179).

## Authors’ Affiliations

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