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Journal of Applied Mathematics
Volume 2012, Article ID 382392, 7 pages
http://dx.doi.org/10.1155/2012/382392
Research Article

Positive Solution for a Class of Boundary Value Problems with Finite Delay

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received 25 May 2012; Accepted 13 September 2012

Academic Editor: Yansheng Liu

Copyright © 2012 Hongzhou Wang. 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 study a class of boundary value problems with equation of the form . Some sufficient conditions for existence of positive solution are obtained by using the Krasnoselskii fixed point theorem in cones.

1. Introduction

Because of applications in physics, bioscience, epidemiology and so on, most literature about differential equations with delay has focused on existence of periodic solution, oscillation, and so on. In recent years, there has been increasing interest in boundary value problems for differential equations with delay, see for example [18]. By using fixed-point theorems, the authors discussed the existence of one solution or twin, triple solutions for differential equations of the forms with boundary value conditions Here may be identically 0 in or .

Especially, Shu et al. investigated some sufficient conditions for the existence of triple solutions for the following boundary value problem in [1] by using Avery-Peterson fixed point theorem: For such a boundary value problem with delay in the first-order derivative, the differentiability of at is very important. In fact, a series of problems arise when we discuss boundary value problems like (1.3) without differentiability of at .

In this paper, we study the following boundary value problem with finite delay: where and are both continuous; and is not identically 0 on any subinterval of ; is continuous and .

With some semilinear and superlinear hypotheses about , we provide some sufficient conditions for existence of positive solution. The following lemma is fundamental in the proofs of our results.

Lemma 1.1 (Krasnoselskii fixed point theorem [9]). Let be a Banach space and a cone. Assume and are bounded open subsets of with , , and let be a completely continuous operator such that one of the following holds:(i), , and , ;(ii), , and , .Then has a fixed point in .

2. Main Results

In this section, we consider the existence of at least one positive solution to boundary value problem (1.4). Firstly, we present some necessary definitions and notes.

Definition 2.1. A function is said to be a positive solution of the boundary value problem (1.4) if it satisfies the following conditions:(i);(ii), ; ;(iii) holds for and holds for .
If is a positive solution of (1.4), we can easily check that it can be expressed by where .
Let with norm given by Obviously, is a Banach space.

Now we define a cone as ; is concave on . Then for all , , , and we have the following lemma.

Lemma 2.2. If , then , .

This lemma follows from the concavity of in a straightforward way, and we omit the details of the proof.

Now we define a map on by where . It shows straightforward show the following lemma.

Lemma 2.3. is completely continuous.

Our aim now is to show that has a fixed point. We introduce the following notations:

Theorem 2.4. Suppose the following conditions are satisfied:; , .Then boundary value problem (1.4) has at least one positive solution.

Proof. Let , then for all , , we will show .
With (H1), we have Equation (2.5) provides that for all , .
On the other hand, let , then for all , .
If , then . The concavity of provides . Therefore, for all , , .
If , then , and Lemma 2.2 provides . Therefore, for all , , .
In summary, for all , and both hold for . Then conditions (H2) guarantee That is, for all , . Then, Lemmas 2.3 and 1.1 guarantee the existence of a positive solution to boundary value problem (1.4).

Corollary 2.5. Suppose the following conditions are satisfied: for and ; for .Then boundary value problem (1.4) has at least one positive solution.

Proof. By (H3), there should exist a constant , for , and . Let , then (H1) is satisfied.
By (H4), there should exist a constant , so that for , and . Let , then (H2) is satisfied.

Theorem 2.6. Suppose the following conditions are satisfied:, , ; for , where ; for and any fixed .Then boundary value problem (1.4) has at least one positive solution.

Proof. Let , then for all , and for all , , . With (H5), we have that is, , for all .
On the other hand, we need two steps to show that and such that for all , .
Step 1. (H6) guarantees , for all , for and .
Let , . For all , we have , , . Then by (H6), we have
Step 2. For all , the concavity of implies , that is, . Suppose , then .
At the same time, (H7) guarantees there exists , for all , .
For all , define as follows:(I)if , then ;(II)if , then there should exist , . , let ; , let .
So for all , interval can be divided into three parts: , ; , , ; , , .
Obviously, let , then , the second part of (2.9) will tend to 0. So we can find , and , for all , . Together with (2.8), we have for all , . Then by Lemma 1.1, we can complete the proof.

Corollary 2.7. Suppose (H6) and (H7) are all satisfied, moreover, for and . Then boundary value problem (1.4) has at least one positive solution.

Proof. By (H8), there should exist such that for , and . Let , then (H5) is satisfied.

3. Examples

Example 3.1. Consider the following boundary value problem: Here , , . We can check that (H3) and (H4) are all satisfied. Then Corollary 2.5 implies that (3.1) has at least one positive solution.

Example 3.2. Consider boundary value problem: Here , , . We can check that (H6), (H7), and (H8) are all satisfied and so Corollary 2.7 implies that (3.2) has at least one positive solution.

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