Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 831960, 12 pages

doi:10.1155/2012/831960

Research Article

On the Nonhomogeneous Fourth-Order -Laplacian Generalized Sturm-Liouville Nonlocal Boundary Value Problems

Jian Liu¹ and Zengqin Zhao²

¹School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Shandong, Jinan 250014, China

²School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China

Received 3 August 2012; Accepted 12 September 2012

Academic Editor: Yanbin Sang

Copyright © 2012 Jian Liu and Zengqin Zhao. 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 the nonlinear nonhomogeneous -point generalized Sturm-Liouville fourth-order -Laplacian boundary value problem by using Leray-Schauder nonlinear alternative and Leggett-Williams fixed-point theorem.

1. Introduction

In this paper, we prove the existence of one and multiple positive solutions of the following differential equations:

where is -Laplacian operator, that is, , , , , , , , , , , .

Recently, much attention has been paid to the existence of positive solutions for nonlocal nonlinear boundary value problems (BVPs for short), see [1–4] and references therein. Such problems have potential applications in physics, biology, chemistry, and so forth. For example, a second-order three-point is used as a model for the membrane response of a spherical cap in nonlinear diffusion generated by nonlinear sources and in chemical reactor theory.

At the same time, the boundary value problems with -Laplacian operator have been discussed extensively, for example, see [1–3, 5–7].

In [1], Feng et al. researched the boundary value problem

they obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of by applying Krasnoselskii fixed-point theorem.

Zhou and Ma studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with -Laplacian operator in [3]:

they established a corresponding iterative scheme for (1.4) by employing the monotone iterative technique.

We would also like to mention the work of Zhang and Liu in [7], in which they considered the existence of positive solutions for

by virtue of monotone iterative techniques, and they established a necessary and sufficient condition of positive solutions for their problem.

However, to the best of our knowledge, there are not many results concerning about the existence and multiple solutions of fourth-order -Laplacian generalized Sturm-Liouville -point boundary value problems. In this paper, motivated by the study of [4, 8], we committed to consider the fourth-order -Laplacian generalized Sturm-Liouville nonlocal boundary value problem without assuming any monotonicity condition on the nonlinearity .

The rest of the paper is arranged as follows. We state some definitions and several preliminary results in Section 2 that we will use in the sequel. Then in Section 3 we present the existence of one positive solution of BVP (1.1) by Leray-Schauder nonlinear alternative. In Section 4 we get three solutions by Leggett-Williams fixed-point theorem.

2. Preliminaries and Some Lemmas

The basic space used in this paper is . It is well known that is a real Banach space with the norm .

Denote

Definition 2.1. A function is said to be a solution of the boundary value problem (1.1) if satisfies (1.1) and . In addition, is said to be a positive solution if for , and is a solution of BVP (1.1).

Throughout the paper, we assume the following condition is satisfied:

.

Let , then BVP (1.1) is divided into the following two parts:

It is not difficult that we can transform (2.2) into the following differential equations:

By routine calculations we can get the following three Lemmas.

Lemma 2.2. The BVP (2.4) has a unique solution

where

Lemma 2.3. The BVP (2.3) has a unique solution

where

The proof of Lemma 2.3 is similar to that of Lemma in [8], so we omit it here. From Lemmas 2.2 and 2.3 we can get that is a solution of BVP (1.1) if and only if

where

Lemma 2.4. Consider,

where

Denote

Lemma 2.5. Let

then

Proof. Firstly, we prove that . For , , then we can get , for all . Furthermore, condition leads to , and , thus, we get .

Secondly, for , we can get

Thus we can get that , which means .

We present here several definitions.

Given a cone in a real Banach space , a map is said to be a nonnegative continuous concave (resp., convex) functional on provided that is continuous and

for all, and .

Let be given, and let be a nonnegative continuous concave functional on . Define the convex sets and by

For the convenience of the reader, we present here the Leggett-Williams fixed-point theorem and the Leray-Schauder nonlinear alternative theorem.

Lemma 2.6 (see [9], Leggett-Williams fixed-point theorem). Let be a completely continuous operator, and let be a nonnegative continuous concave functional on such that  for all . Suppose there exist such that

, and for ; for ; for with .

Then has at least three fixed points , , and and such that , and , with .

Now we cite the Leray-Schauder nonlinear alternative.

Lemma 2.7 (see [10]). Let be a Banach space and a bounded open subset of ,. be a completely continuous operator. Then, either there exists , such that , or there exists a fixed point .

3. Results of One Nontrivial Solution

In this section, we study the existence of one nontrivial solution of BVP (1.1) by Leray-Schauder nonlinear alternative.

Denote

Theorem 3.1. Assume ,, and there exist nonnegative functions such that , then BVP (1.1) has a nontrivial solution .

Proof. If there exist two nonnegative functions such that , we can get that

thus, we get

In the same way, we obtain

Thus we have

Suppose that there exists such that

Therefore,

which leads to , and this contradicts , then by Lemma 2.7, has a fixed point ; since , the BVP (1.1) has a nontrivial solution . This completes the proof of Theorem 3.1.

4. Results of Multiple Positive Solutions

In the following parts, we will study the existence of multiple positive solutions of BVP (1.1) by using Leggett-Williams fixed-point theorem.

Denote

Define the nonnegative continuous concave functional on by

It is obvious that for each .

Let , , , and be defined in Sections 2 and 3.

We list the following three hypotheses:

, for all , ;, for all , ;, for all , .

Theorem 4.1. Assume hold, then BVP (1.1) has at least three positive solutions , , and , such that ,, and , with .

Proof. Firstly, we prove that . The operator is completely continuous.

From condition , we can get

Hence,

Thus we get ; therefore, . The operator is completely continuous by an application of Ascoli-Arzela theorem.

In the same way, condition implies that condition of Lemma 2.6 is satisfied. In the following, we show that condition of Lemma 2.6 is satisfied.

Let

then

thus, .

If , then , .

By condition , we obtain

Thus we get

Therefore, condition of Lemma 2.6 is satisfied.

Finally, we show that condition of Lemma 2.6 is satisfied.

If , and , then .

Therefore, condition of Lemma 2.6 is also satisfied. By Lemma 2.6, there exist three positive solutions , , and such that , and , with . Thus we completed the proof.

Acknowledgments

The first author is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AQ024) and the University Science and Technology Foundation of Shandong Provincial Education Department (J10LA62). The second author is supported by the National Natural Science Foundation of China (10871116), the Natural Science Foundation of Shandong Province of China (ZR2010AM005), and the Doctoral Program Foundation of Education Ministry of China (200804460001).

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