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).