Existence and Multiplicity of Positive Solutions for a System of Fourth-Order Boundary Value Problems

We study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problems , and where . We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some integral identities and inequalities and -monotone matrices. 1. Introduction In this paper we study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problems: where . It is well known that the deflection of elastic beams can be described by some fourth-order boundary value problems; for example, see [1, 2]. Consequently, fourth-order boundary value problems play a very important role in both theory and applications. Boundary value problems for systems of nonlinear ordinary differential equations have been also studied by several authors; for example, see [3–11] and the references therein. In [3], Li et al. use fixed point theorems on cones to establish the existence of positive solutions for a system of third-order boundary value problems: In [4], Lü et al. study the existence of positive solutions for a system of boundary value problems: where . In [11], the authors investigate the existence and multiplicity of positive solutions for the system The hypotheses imposed on the nonlinearities and are formulated in terms of two linear functions and . The main results in [11] are established by using fixed point index theory based on a priori estimates of positive solutions achieved by utilizing new integral inequalities and nonnegative matrices. This paper is organized as follows. In Section 2, we use the method of order reduction to transform (1) into a system of boundary value problems for second-order integrodifferential equations. Also, in this section, we develop some basic integral identities and inequalities that are useful in deriving a priori estimates in Section 3. Our main results, namely, Theorems 11–14, are stated and proved in Section 3. Finally, three examples that illustrate our main results are presented in Section 4. 2. Preliminaries Let , , . Then is a real Banach space and is a solid cone in . Let Define the linear integral operator by It is well known that is a completely continuous, positive, linear operator and where . Substituting , into (1), we transform (1) into the following system of seconed-order boundary value problems for the integrodifferential equations: which is equivalent to the system of nonlinear integral equations: Define the operators and by If , then and are completely continuous operators. Clearly, the existence of positive