%0 Journal Article
%T Existence of Positive Solutions for Fourth-Order Boundary Value Problems with Sign-Changing Nonlinear Terms
%A Xingfang Feng
%A Hanying Feng
%J ISRN Mathematical Analysis
%D 2013
%R 10.1155/2013/349624
%X the existence of positive solutions for a fourth-order boundary value problem with a sign-changing nonlinear term is investigated. By using KrasnoselskiiĄ¯s fixed point theorem, sufficient conditions that guarantee the existence of at least one positive solution are obtained. An example is presented to illustrate the application of our main results. 1. Introduction In this paper, we consider the existence of positive solutions to the following fourth-order boundary value problem (BVP): where is a positive parameter, is continuous and may be singular at , and is Lebesgue integrable and has finitely many singularities in . Boundary value problems for ordinary differential equations play a very important role in both theory and applications. They are used to describe a large number of physical, biological, and chemical phenomena. The work of Timoshenko [1] on elasticity, the monograph by Soedel [2] on deformation of structures, and the work of Dulcska [3] on the effects of soil settlement are rich sources of such applications. There has been a great deal of research work on BVPs for second and higher order differential equations, and we cite as recent contributions the papers of Anderson and Davis [4], Baxley and Haywood [5], and Hao et al. [6]. For surveys of known results and additional references, we refer the readers to the monographs by Agarwal et al. [7, 8]. Many authors have studied the existence of positive solutions for fourth-order boundary value problems where the nonlinearity takes nonnegative values, see [9¨C13]. However, for problems with sign-changing nonlinearities, only a few studies have been reported. Owing to the importance of high order differential equations in physics, the existence and multiplicity of the solutions to such problems have been studied by many authors, see [9, 12¨C17]. They obtained the existence of positive solutions provided is superlinear or sublinear in by employing the cone expansion-compression fixed point theorem. In [18], by using the strongly monotone operator principle and the critical point theory to discuss BVP the authors established some sufficient conditions for to guarantee that the problem has a unique solution, at least one nonzero solution, or infinitely many solutions. In [10], Feng and Ge considered the fourth-order singular differential equation subject to one of the following boundary conditions: where . By using a fixed point index theorem in cones and the upper and lower solutions method, the authors discussed the existence of positive solutions for the above BVP. However, most papers only focus
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2013/349624/