- 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

- Linear Operator
- Maximum Principle
- Operator Equation
- Integral Condition
- Lower Solution

## 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).

If , such that , , then , ,

Proof.

By (2.23), we can get that ,

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.

where .

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

where is given by (2.27).

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

## References

- Du X, Zhao Z:
**Existence and uniqueness of smooth positive solutions to a class of singular****-point boundary value problems.***Boundary Value Problems*2009,**2009:**-13.Google Scholar - Graef JR, Webb JRL:
**Third order boundary value problems with nonlocal boundary conditions.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(5-6):1542-1551. 10.1016/j.na.2008.12.047MathSciNetView ArticleMATHGoogle Scholar - Graef JR, Yang B:
**Positive solutions of a third order nonlocal boundary value problem.***Discrete and Continuous Dynamical Systems. Series S*2008,**1**(1):89-97.MathSciNetMATHGoogle Scholar - Infante G, Webb JRL:
**Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations.***Proceedings of the Edinburgh Mathematical Society. Series II*2006,**49**(3):637-656. 10.1017/S0013091505000532MathSciNetView ArticleMATHGoogle Scholar - Webb JRL, Infante G:
**Positive solutions of nonlocal boundary value problems: a unified approach.***Journal of the London Mathematical Society. Second Series*2006,**74**(3):673-693. 10.1112/S0024610706023179MathSciNetView ArticleMATHGoogle Scholar - Gallardo JM:
**Second-order differential operators with integral boundary conditions and generation of analytic semigroups.***The Rocky Mountain Journal of Mathematics*2000,**30**(4):1265-1292. 10.1216/rmjm/1021477351MathSciNetView ArticleMATHGoogle Scholar - Karakostas GL, Tsamatos PCh: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electronic Journal of Differential Equations 2002, (30):1-17.Google Scholar
- Lomtatidze A, Malaguti L:
**On a nonlocal boundary value problem for second order nonlinear singular differential equations.***Georgian Mathematical Journal*2000,**7**(1):133-154.MathSciNetMATHGoogle Scholar - Yang Z:
**Positive solutions to a system of second-order nonlocal boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2005,**62**(7):1251-1265. 10.1016/j.na.2005.04.030MathSciNetView ArticleMATHGoogle Scholar - Yang Z:
**Existence and nonexistence results for positive solutions of an integral boundary value problem.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(8):1489-1511. 10.1016/j.na.2005.10.025MathSciNetView ArticleMATHGoogle Scholar - Yang Z:
**Positive solutions of a second-order integral boundary value problem.***Journal of Mathematical Analysis and Applications*2006,**321**(2):751-765. 10.1016/j.jmaa.2005.09.002MathSciNetView ArticleMATHGoogle Scholar - Feng M, Ji D, Ge W:
**Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces.***Journal of Computational and Applied Mathematics*2008,**222**(2):351-363. 10.1016/j.cam.2007.11.003MathSciNetView ArticleMATHGoogle Scholar - Zhang X, Feng M, Ge W:
**Symmetric positive solutions for****-Laplacian fourth-order differential equations with integral boundary conditions.***Journal of Computational and Applied Mathematics*2008,**222**(2):561-573. 10.1016/j.cam.2007.12.002MathSciNetView ArticleMATHGoogle Scholar - Corduneanu C:
*Integral Equations and Applications*. Cambridge University Press, Cambridge, UK; 1991:x+366.View ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D:
*Infinite Interval Problems for Differential, Difference and Integral Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001.View ArticleMATHGoogle Scholar - Wei Z:
**Positive solutions for****th-order singular sub-linear****-point boundary value problems.***Applied Mathematics and Computation*2006,**182**(2):1280-1295. 10.1016/j.amc.2006.05.014MathSciNetView ArticleMATHGoogle Scholar - Wei Z:
**A necessary and sufficient condition for****th-order singular super-linear****-point boundary value problems.***Journal of Mathematical Analysis and Applications*2007,**327**(2):930-947. 10.1016/j.jmaa.2006.04.056MathSciNetView ArticleMATHGoogle Scholar - Du X, Zhao Z:
**A necessary and sufficient condition of the existence of positive solutions to singular sublinear three-point boundary value problems.***Applied Mathematics and Computation*2007,**186**(1):404-413. 10.1016/j.amc.2006.07.120MathSciNetView ArticleMATHGoogle Scholar - Graef JR, Kong L:
**Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems.***Journal of Mathematical Analysis and Applications*2007,**331**(2):1467-1484. 10.1016/j.jmaa.2006.09.046MathSciNetView ArticleMATHGoogle Scholar - Zhang X, Liu L:
**A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with****-Laplacian.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(10):3127-3137. 10.1016/j.na.2007.03.006MathSciNetView ArticleMATHGoogle Scholar - Zhang Y:
**Positive solutions of singular sublinear Emden-Fowler boundary value problems.***Journal of Mathematical Analysis and Applications*1994,**185**(1):215-222. 10.1006/jmaa.1994.1243MathSciNetView ArticleMATHGoogle Scholar - Hartman P:
*Ordinary Differential Equations*. 2nd edition. Birkhäuser, Boston, Mass, USA; 1982:xv+612.MATHGoogle Scholar - Lin X, Jiang D, Li X:
**Existence and uniqueness of solutions for singular fourth-order boundary value problems.***Journal of Computational and Applied Mathematics*2006,**196**(1):155-161. 10.1016/j.cam.2005.08.016MathSciNetView ArticleMATHGoogle Scholar - Lin X, Jiang D, Li X:
**Existence and uniqueness of solutions for singular****conjugate boundary value problems.***Computers & Mathematics with Applications*2006,**52**(3-4):375-382. 10.1016/j.camwa.2006.03.019MathSciNetView ArticleMATHGoogle Scholar - Lei P, Lin X, Jiang D:
**Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(9):2773-2779. 10.1016/j.na.2007.08.049MathSciNetView ArticleMATHGoogle Scholar - Liu Y, Yu H:
**Existence and uniqueness of positive solution for singular boundary value problem.***Computers & Mathematics with Applications*2005,**50**(1-2):133-143. 10.1016/j.camwa.2005.01.022MathSciNetView ArticleMATHGoogle Scholar

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