Global Structure of Positive Solutions for a Singular Fourth-Order Integral Boundary Value Problem

We consider fourth-order boundary value problems , where is a Stieltjes integral with being nondecreasing and being not a constant on may be singular at and , with on any subinterval of ; and for all , and We investigate the global structure of positive solutions by using global bifurcation techniques. 1. Introduction Recently, fourth-order boundary value problem has been investigated by the fixed point theory in cones, see [1–4] ( ). By applying bifurcation techniques, see Rynne [5] ( ), Korman [6] ( ), Xu and Han [7] ( ), Shen [8, 9] ( ), and references therein. However, these papers only studied the nonsingular boundary value problems. In 2008, Webb et al. [10] studied the existence of multiple positive solutions of nonlinear nonlocal boundary value problems (BVPs) for equations of the form where are continuous and nonnegative functions and is a function of bounded variation. They treat many boundary conditions appearing in the literature in a unified way. The main tool they used is the fixed point index theory in cones. In 2009, Ma and An [11] studied the global structure for second-order nonlocal boundary value problem involving Stieltjes integral conditions by applying bifurcation techniques. Motivated by above papers, in this paper, we will use global bifurcation techniques to study the global structure of positive solutions of the singular problem where may be singular at and , and is a parameter. In order to prove our main result, let us make the assumptions as follows:(A1) is nondecreasing and is not a constant on , for , and with ;(A2) with on any subinterval of , and ;(A3) satisfies for all ;(A4) ;(A5) . Remark 1. For other results on the existence and multiplicity of positive solutions and nodal solutions for the boundary value problems of fourth-order ordinary differential equations based on bifurcation techniques, see Ma et al. [12–15] and Bai and Wang [16] and their references. The rest of the paper is arranged as follows: In Section 2, we state some properties of superior limit of certain infinity collection of connected sets. In Section 3, we will give some preliminary results. In Section 4, we state and prove our main results. 2. Superior Limit and Component In order to treat the case , we will need the following definition and lemmas. Definition 2 (see [17]). Let be a Banach space and let be a family of subsets of . Then the superior limit of is defined by Lemma 3 (see [17]). Each connected subset of metric space is contained in a component, and each connected component of is closed. Lemma 4 (see [11]). Let be a Banach space and let