By means of the fixed point index theory of strict-set contraction operator, we study the existence of positive solutions for the multipoint singular boundary value problem , , , , , , , in a real Banach space , where is the zero element of As an application, we give two examples to demonstrate our results. 1. Introduction The theory of ordinary differential equations in Banach spaces has become a new important branch (see, e.g., [1–13] and the references cited therein). In 1988, Guo and Lakshmikantham [4] discussed multiple solutions for two-point boundary value problems of second-order ordinary differential equations in Banach spaces. In [7], Guo obtained the existence of positive solutions for a boundary value problem of nth-order nonlinear impulsive integrodifferential equations in a Banach space by means of fixed point index theory and fixed point theory of completely continuous operators, respectively. Liu et al. in [6] obtained the existence of unbounded nonnegative solutions of a boundary value problem for nth-order impulsive integrodifferential equations on an infinite interval in Banach spaces by means of the Mddotoch fixed point theory in a Banach space. Zhang et al. in [9] dealt with the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear three-point boundary value problems of nth-order differential equations in Banach spaces. Zhao and Chen in [8, 12] investigated the existence of at least triple positive solutions for nonlinear boundary value problem by upper and low solution methods. In this paper, the author considers the existence of positive solutions of the following higher-order conjugate multipoint boundary value problems (BVPs): in a real Banach space , where is the zero element of . ？ is continuous and allowed to be singular at and . In scalar space, because of the widely applied background in mechanics and engineering, the nonlinear higher-order boundary value problems have received much attention (see Chyan and Henderson [Appl. Math. Letters 15 (2002) 767–774]). In [14], Eloe and Ahmad had solved successfully the existence of positive solution to the following nth-order boundary value problems: Recently, the existence of solutions and positive solutions of nonlinear focal boundary value problem and its special cases has been studied by many authors (see, e.g., [15–23]). By using the Krasnoselskii fixed point theorem, Eloe and Henderson in [15], Agarwal and O'Regan in [16], and Kong and Wang in [20] have established the existence of solutions for following the conjugate boundary value problem:
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