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Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem

DOI: 10.1155/2014/513051

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This paper is concerned with the existence and nonexistence of positive solutions for a nonlinear higher-order three-point boundary value problem. The existence results are obtained by applying a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O’Regan. 1. Introduction We consider the existence and nonexistence of positive solution to the nonlinear higher-order three-point boundary value problem (BVP for short): with , , are constants with . is continuous. Here, by a positive solution of BVP (1) we mean a function satisfying (1) and for . If , BVP (1) reduces to so-called boundary value problem which has been considered by many authors. For example, Agarwal et al. [1] considered the existence of positive solutions for the singular boundary value problem. Baxley and Houmand [2] considered the existence of multiple positive solutions for the boundary value problem. Yang [3] obtained some new upper estimates to positive solutions for the problem. Existence and nonexistence results for positive solutions of the problem were obtained by using the Krasnosel'skii fixed point theorem. In recent years, the existence and multiplicity of positive solutions for nonlinear higher-order ordinary differential equations with three-point boundary conditions have been studied by several authors; we can refer to [4–19] and the references therein. For example, Eloe and Ahmad in [4] discussed the existence of positive solutions of a nonlinear th-order three-point boundary value problem where , . The existence of at least one positive solution if is either superlinear or sublinear was established by applying the fixed point theorem in cones due to Krasnosel'skii. Under conditions different from those imposed in [4], Graef and Moussaoui in [5] studied the existence of both sign changing solutions and positive solutions for BVP (2). Hao et al. [6] are devoted to the existence and multiplicity of positive solutions for BVP (2) under certain suitable weak conditions, where may be singular at and/or . The main tool used is also the Krasnosel'skii fixed point theorem. Graef et al. in [7, 8] considered the higher-order three-point boundary value problem where is an integer and is a constant. Sufficient conditions for the existence, nonexistence, and multiplicity of positive solutions of this problem are obtained by using Krasnosel'skii's fixed point theorem, Leggett-Williams' fixed point theorem, and the five-functional fixed point theorem. Let be a real Banach space with norm and be a cone of . Recently, Zhang et al. [9] studied

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