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Chinese Journal of Mathematics
Volume 2013 (2013), Article ID 915209, 8 pages
http://dx.doi.org/10.1155/2013/915209
Research Article

Positive Solutions for a System of Fourth-Order -Laplacian Boundary Value Problems

Department of Mathematics, Qingdao Technological University, No. 11 Fushun Road, Qingdao, Shandong 266033, China

Received 29 August 2013; Accepted 22 September 2013

Academic Editors: S. A. Marano and G. Toth

Copyright © 2013 Lianlong Sun and Zhilin Yang. 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 investigate the existence of positive solutions for the system of fourth-order -Laplacian boundary value problems where and . Based on a priori estimates achieved by utilizing Jensen’s integral inequalities and nonnegative matrices, we use fixed point index theory to establish our main results.

1. Introduction

This paper is concerned with the existence of positive solutions for the system of fourth-order -Laplacian boundary value problems where and .

In recent years, boundary value problems for fourth-order nonlinear ordinary differential equations have been extensively studied, with many papers on this direction published. See [14] and the references cited therein. The so-called -Laplacian boundary value problems arise in non-Newtonian mechanics, nonlinear elasticity, glaciology, population biology, combustion theory, and nonlinear flow laws; see [5, 6]. That explains why many authors have extensively studied the existence of positive solutions for -Laplacian boundary value problems, by using topological degree theory, monotone iterative techniques, coincidence degree theory [7], and the Leggett-Williams fixed point theorem [8] or its variants; see [918] and the references therein.

To the best of our knowledge, problem (1) is a new topic in the existing literature. Closely related to our work here is [19] that studies the existence and multiplicity of positive solutions for the system of -Laplacian boundary value problems where . Based on a priori estimates achieved by utilizing the Jensen integral inequalities and -monotone matrices, the author used fixed point index theory to establish the existence and multiplicity of positive solutions for the previous problem. For more details of the recent progress in the boundary value problems for systems of nonlinear ordinary differential equations, we refer the reader to [1927] and the references cited therein.

We observe that a close link exists between (1) and the problem below and this link can be established by Jensen's integral inequalities for concave functions and convex functions. In other words, (1) may be regarded as a perturbation of (3). With this perspective, a priori estimates of positive solutions for some problems associated with (1) can be derived by utilizing Jensen's integral inequalities. It is the a priori estimates that permit us to use fixed point index theory to establish our main results.

This paper is organized as follows. In Section 2, we provide some preliminary results. Our main results, namely, Theorems 6 and 7, are stated and proved in Section 3.

2. Preliminaries

We first make the following hypothesis throughout this paper:(H1).

Let and then is a real Banach space and is a cone in . It is easy to see that (1) is equivalent to the system of nonlinear integral equations where Define the operators and by Obviously, under condition (H1), the operators and are completely continuous operators. In our setting, the existence of positive solutions for (1) is equivalent to that of positive fixed points of .

Below are some elementary inequalities.

Lemma 1. Let , then and

Lemma 2. Suppose that is concave on . Then

Lemma 3. Let . If , then If , then

Lemma 4 (see [28]). Let be a real Banach space, and is a cone in . Suppose that is a bounded open set and is a completely continuous operator. If there exists such that then , where indicates the fixed point index on .

Lemma 5 (see [28]). Let be a real Banach space, and is a cone in . Suppose that is a bounded open set, , and is a completely continuous operator. If then .

3. Main Results

Let , , , . Then the previous constants satisfy the simple relations We now list our hypotheses on and .

(H2) There exist five constants , , such that for all , , and the matrix is invertible with nonnegative, where

(H3) There exist five constants , such that for all , , and the matrix is invertible with nonnegative, where

(H4) There exist five constants and, a real number such that for all , , and the matrix is invertible with nonnegative, where

(H5) There exist five constants , such that for all , , and the matrix is invertible with nonnegative, where In the sequel, we set for all .

Theorem 6. If (H1)–(H3) hold, then (1) has at least one positive solution.

Proof. Let where is defined by . We want to show that is bounded. Indeed, if , then there exists such that and thus . Note that implies that are concave. So are with . Now Lemma 2 implies that hold for every . Note that for all and . By (H2), Jensen's integral inequalities, and Lemma 3, we obtain for all . Note . The same argument for the previous inequality can be used to obtain for all . Multiply the last two inequalities by , integrate over , and use Lemma 1 and (H2) to obtain for every , where . The last two inequalities can be written as Now (H2) implies Consequently, there exists a constant such that Recalling (25), we obtain Therefore, is bounded, as required. Taking , we have Now Lemma 4 yields Let Now we want to show that . Indeed, if , then there exists such that , and thus for all . Note that for all and . By (H3), Jensen’s integral inequalities, and Lemma 3, we obtain for all . Note that . The same argument for the previous inequality can be used to obtain for all . Multiply the last two inequalities by , integrate over , and use Lemma 1 and (H3) to obtain for all . The last two inequalities can be written as Now (H3) implies so that whence , and , as required. Thus we have Now Lemma 5 yields Combining (34) and (43) gives Consequently, has at least one fixed point on . Thus (1) has at least one positive solution. This completes the proof.

Theorem 7. If (H1), (H4), and (H5) hold, then (1) has at least one positive solution.

Proof. Let We now assert that . Indeed, if , then there exists such that . Thus . Note that for all and . By (H4), Jensen's integral inequalities, and Lemma 3, we obtain for all . Note that . The same argument for the previous inequality can be used to obtain for all . Multiply the last two inequalities by , integrate over , and use Lemma 1 and (H4) to obtain for all . The last two inequalities can be written as Now (H4) implies so that whence , and , as asserted. As a result of this, we obtain By Lemma 4, we have Let Now we want to show that is bounded. Indeed, if , then there exists such that . This implies . Note that for all and . By (H5), Jensen's integral inequalities, and Lemma 3, we obtain for all . Note that . The same argument for the previous inequality can be used to obtain for all . Multiply the last two inequalities by , integrate over , and use Lemma 1 and (H5) to obtain for all , where . The last two inequalities can be written as Now (H3) implies and thus there exists a constant such that for every . The concavity of and Lemma 2 imply The fact that , together with Jensen's inequalities, implies and, in turn, Recalling (61), we obtain for all . This proves the boundedness of . Taking , we have Now Lemma 3 yields Combining (53) with (66) gives Hence, has at least one fixed point on . Thus (1) has at least one positive solution. This completes the proof.

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