Positive Solutions to Periodic Boundary Value Problems for Four-Order Differential Equations
Huantao Zhu1and Zhiguo Luo2
Academic Editor: G. Gripenberg
Received26 Nov 2011
Accepted09 Jan 2012
Published04 Apr 2012
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 [1β7] 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 ββis 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 . βHowever, relatively few papers have been published on the same problem for four-order differential equations. Recently, by using a maximum principle for operator ββin 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 βis continuous, , and satisfy , , . However, since there appears ββin 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 ββand suppose that
is a completely continuous operator. If one of the following conditions is satisfied:(i)for ββfor ,(ii)ββfor ββfor . 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 ,ββwe defined the operator
where
By a direct calculation, we easily obtain
Set
then , . Now, we consider the problem
Lemma 2.1. If ββis a (positive) solution of problem (2.5), then ββis a (positive) solution of problem (1.1). Moreover, the problem (2.5) is equivalent to integral equation
Proof. If ,ββthen ββand
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 ββis a cone in . For any , let ,ββthen . βFor any , define mapping by
then the fixed point of in is a positive solution of (2.5).
Lemma 2.2. For any , ββis completely continuous.
Proof. For any , βββand for all . Thus, if ,
It is easy to see that is continuous and completely continuous since ββis continuous. Next, we show that . βSince ββfor , . On the other hand,
The proof is complete.
In this section, we make the following hypotheses. There exist nonnegative functions , ββand , βsuch that
for all ββand
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 ββare satisfied, then there exist two positive constants , βββsuch 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 , , βββsuch that (1.1) has at least two positive solutions , with
Proof. First, we assume that and are satisfied. From the condition ,ββone can obtain that there exist a such that
For any , ββfor all and
Thus, for , from ,ββwe have
which implies that
From ββis satisfied, there exists a positive constant ββsuch 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 inββ with . βHence, ββis 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 , ββfor all and
which implies that
From (3.11) and (3.17) and Theorem 1.1, one can obtain that has a fixed point inββ with . βThus, ββis a positive solution of (1.1) with . Assume that , , and are satisfied. Repeating the above argument, one can obtain that has a fixed point inββββand a fixed point inββ with
Hence, , ββare 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 ββand ,ββwhere
Then,
and one easily check that (4.1) is equivalent to the problem
If is a (positive) solution of problem (4.4), then ββis 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 ββis 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 , ββsuch 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 ββis 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 ββas 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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