%0 Journal Article
%T Positive Solutions for a System of Fourth-Order -Laplacian Boundary Value Problems
%A Lianlong Sun
%A Zhilin Yang
%J Chinese Journal of Mathematics
%D 2013
%R 10.1155/2013/915209
%X 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 JensenĄ¯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 [1¨C4] 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 [9¨C18] 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 [19¨C27] 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
%U http://www.hindawi.com/journals/cjm/2013/915209/