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Abstract and Applied Analysis
Volume 2010, Article ID 236826, 8 pages
http://dx.doi.org/10.1155/2010/236826
Research Article

Positive Solutions for Second-Order Three-Point Eigenvalue Problems

Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China

Received 7 June 2010; Accepted 5 August 2010

Academic Editor: Chaitan P. Gupta

Copyright © 2010 Chuanzhi Bai. 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

With the help of the fixed point index theorem in cones, we get an existence theorem concerning the existence of positive solution for a second-order three-point eigenvalue problem ,  , where is a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result.

1. Introduction

Motivated by the work of Bitsadze and Samarskii [1] and Ilyin and Moiseev [2], much attention has been paid to the study of certain nonlocal boundary value problems (BVPs) in recent years.

In the last twenty years, many mathematician, have considered the existence of positive solutions of nonlinear three-point boundary value problems; see, for example, Graef et al. [3] Webb [4], Gupta and Trofimchuk [5], Infante [6], Ehrke [7], Ma [8], Feng [9], He and Ge [10], Bai and Fang [11], and Guo [12]. Recently, by applying the Avery-Henderson [13] double fixed point theorem, Henderson [14] studied the existence of two positive solutions of the three-point boundary value problem for the second-order differential equation where and is continuous.

In this paper, motivated and inspired by the above work and Wong [15], we apply a fixed point index theorem in cones to investigate the existence of positive solutions for nonlinear three-point eigenvalue problems where and .

We need the following well-known lemma. See [16] for a proof and further discussion of the fixed point index .

Lemma 1.1. Assume that is a Banach space, and is a cone in . Let . Furthermore, assume that is a completely continuous map, and for . Then, one has the following conclusions: if for , then ; if for , then .

2. Main Results

In the following, we will denote by the space of all continuous functions . This is a Banach space when it is furnished with usual sup-norm .

By [14], the Green's function for the three-point boundary-value problem is given by

From the Green's function , we have that a function is a solution of the boundary value problem (1.2) if and only if it satisfies

Lemma 2.1. Suppose that is defined as above. Then we have the following results: .

Proof. It is easy to see that (1) holds. To show that (2) holds, we distinguish four cases.(i)If , then (ii)If and , then (iii)If and , then (iv)Finally, if , then

Remark 2.2. If and , then and , respectively.
Define Obviously, is a cone in the Banach space .
Define an operator as follows:
It is easy to know that fixed points of are solutions of the BVP (1.2).
Now, we can state and prove our main results.

Lemma 2.3. is completely continuous.

Proof. For any , by Lemma 2.1 (1), we have , for each . It follows from Lemma 2.1 that Hence, , which implies . Moreover, it is easy to check that is completely continuous.

By simple calculation, we obtain that

Lemma 2.4. Suppose that there exists a positive constant such that holds. If , then

Proof. For , it follows from the definition of the cone that which implies Thus, we have by (H1) and (2.11) that since . This shows that It is obvious that for . Therefore, by Lemma 1.1 (), we conclude that .

Lemma 2.5. Suppose that there exists a positive constant such that where and . If where and , then

Proof. First, we claim that Suppose to the contrary that there exist and such that It is clear that (2.22) is equivalent to
Since and , it follows that there exists a such that . From on , we see that , on , and on . By (2.18) and (2.23), we have Multiplying (2.24) by and then integrating from to (), we get from that that is which implies that Thus, Hence, we obtain from (2.19) and (2.28) that This contradiction implies that (2.21) holds. By (2.21), we have for , and for . Thus, by Lemma 1.1 (), we obtain

For convenience, let

Theorem 2.6. Assume that there exist two distinct positive constants such that (H1)–(H3) hold. If then BVP (1.2) has at least one positive solution.

Proof. From Lemmas 2.4 and 2.5, and the property of the fixed point index, we can easily get that the operator has a fixed point in () or in (). Therefore, BVP (1.2) has at least one positive solution.

3. An Example

To illustrate our results we present the following example.

Example 3.1. Consider the following boundary value problem Let and . Choosing , , we have where and . Thus, , , and Hence, (H2) and (H3) hold. Moreover, we get which implies that (H1) holds. Therefore, it follows from Theorem 2.6 that BVP (3.1) has at least one positive solution if

Acknowledgments

The author would like to thank the referees for their valuable suggestions and comments. This paper was supported by the National Natural Science Foundation of China (10771212) and the Natural Science Foundation of Jiangsu Education Office (06KJB110010).

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