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ISRN Mathematical Analysis

Volume 2012 (2012), Article ID 830983, 12 pages

http://dx.doi.org/10.5402/2012/830983

## Positive Solutions to Periodic Boundary Value Problems for Four-Order Differential Equations

^{1}Hunan College of Information, Changsha, Hunan 410200, China^{2}Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

Received 26 November 2011; Accepted 9 January 2012

Academic Editor: G. Gripenberg

Copyright © 2012 Huantao Zhu and Zhiguo Luo. 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.

#### Abstract

We apply fixed point theorem in a cone to obtain sufficient conditions for the existence of single and multiple positive solutions of periodic boundary value problems for a class of four-order differential equations.

#### 1. Introduction

In this paper, we investigate the existence of positive solutions of the following periodic boundary value problem: where , and are positive constants with , .

In recent years, the nonlinear periodic boundary value problems have been widely studied by many authors, for example, see [1–7] and the references therein. Many theorems and methods of nonlinear functional analysis, for instance, upper and lower solutions method, fixed point theorems, variational method, and critical point theory, and so on, have been applied to their problems. When positive solutions are discussed, it seems that fixed point theorem in cones is quite effective in dealing with the problems with singularity. In [8], Zhang and Wang proved periodic boundary value problems with singularity have multiple positive solutions under some conditions, where is singular at , that is, Relying on a nonlinear alternative of Leray-Schauder type and fixed point theorem, Chu and Zhou [9] discussed the existence of positive solutions for the third-order periodic boundary value problem where . However, relatively few papers have been published on the same problem for four-order differential equations. Recently, by using a maximum principle for operator in periodic boundary condition and fixed point index theory in cones, Li [10] considered the existence of positive solution for the fourth-order periodic boundary value problem where is continuous, , and satisfy , , . However, since there appears in nonlinear term , the method in [9] cannot be directly applied to (1.1). The main aim of this paper is to establish sufficient conditions for the existence of positive solutions to the problem (1.1).

To prove our main results, we present an existence theorem.

Theorem 1.1 (see [11]). *Let be a Banach space and a cone in . Suppose and are open subsets of such that and suppose that
**
is a completely continuous operator. If one of the following conditions is satisfied:*(i)*for for ,*(ii)* for for . Then has a fixed point in .*

#### 2. Preliminaries

In this section, we present some preliminary results which will be needed in Section 3.

Let , and for any function , we defined the operator where By a direct calculation, we easily obtain Set then , . Now, we consider the problem

Lemma 2.1. *If is a (positive) solution of problem (2.5), then is a (positive) solution of problem (1.1). Moreover, the problem (2.5) is equivalent to integral equation
*

*Proof. *If , then and
Thus,
Then,
On the other hand, ,
Hence, if is a solution of problem (2.5), then is a solution of problem (1.1). And, if is a positive function, noting that for any , , we have
Noting that, for any function , linear problem
has a unique solution
one can easily obtain that (2.6) holds. The proof is complete.

In the following application, we take with the supremum norm and define where .

One easily checks and verifies that is a cone in . For any , let , then . For any , define mapping by then the fixed point of in is a positive solution of (2.5).

Lemma 2.2. *For any , is completely continuous.*

*Proof. *For any , and for all . Thus, if ,
It is easy to see that is continuous and completely continuous since is continuous. Next, we show that . Since for , . On the other hand,
The proof is complete.

#### 3. Positive Solutions of (1.1)

In this section, we make the following hypotheses. There exist nonnegative functions , and , such that for all and where is nonincreasing and is nondecreasing on . One has One has

Under the above hypotheses, we can obtain the following result.

Theorem 3.1. *Assume that and are satisfied, then there exist two positive constants , such that (1.1) has at least positive solution with
**Assume and are satisfied, then there exist two positive constants , such that (1.1) has at least positive solution with
**Assume , , and are satisfied, then there exist positive constants , , such that (1.1) has at least two positive solutions , with
*

*Proof. *First, we assume that and are satisfied. From the condition , one can obtain that there exist a such that
For any , for all and
Thus, for , from , we have
which implies that

From is satisfied, there exists a positive constant such that
For any , for all and
which implies that
From (3.11) and (3.14) and Theorem 1.1, one can obtain that has a fixed point in with . Hence, is a positive solution of (1.1) with .

Next, we assume that and are satisfied. In this case, we have (3.11).

Suppose that is satisfied, there exists a positive constant such that
For any , for all and
which implies that
From (3.11) and (3.17) and Theorem 1.1, one can obtain that has a fixed point in with . Thus, is a positive solution of (1.1) with .

Assume that , , and are satisfied. Repeating the above argument, one can obtain that has a fixed point in and a fixed point in with
Hence, , are two positive solutions of (1.1) with
The proof is complete.

#### 4. A Similar Problem

In this section, we use the idea in Sections 2 and 3 to consider the following problem: where , .

Let and , where Then, and one easily check that (4.1) is equivalent to the problem If is a (positive) solution of problem (4.4), then is a (positive) solution of problem (4.1). Moreover, the problem (4.4) is equivalent to integral equation For any , define mapping by For any , one can obtain that is completely continuous.

Similar to the proof of Theorem 3.1, we can obtain the following result.

Theorem 4.1. *Assume that and are satisfied, then there exist two positive constants , such that (4.1) has at least positive solution with
**Assume and are satisfied, then there exist two positive constants , such that (4.1) has at least positive solution with
**Assume , , and are satisfied, then there exist positive constants , , such that (4.1) has at least two positive solutions , with
**
where is condition obtained by replacing and by in the condition defined in Section 3. *

*Example 4.2. *Consider the differential equation
where is a constant.

Let
Then, for all ,
Noting that
we obtain that holds when is sufficiently large. On the other hand, it is easy to check that is satisfied since as for any and . Hence, (4.10) has at least a positive solution when is sufficiently large.

#### Acknowledgment

A Project Supported by the NNSF of China (10871063) and ScientificResearch Fund of human Provincial Education Department (10C0258).

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