Báo cáo hóa học: Research Article Solutions to a Three-Point Boundary Value Problem

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Báo cáo hóa học: " Research Article Solutions to a Three-Point Boundary Value Problem"Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2011, Article ID 894135, 20 pagesdoi:10.1155/2011/894135Research ArticleSolutions to a Three-Point Boundary Value Problem Jin Liang1 and Zhi-Wei Lv2, 3 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China 2 Department of Mathematics and Physics, Anyang Institute of Technology, Anyang, Henan 455000, China 3 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China Correspondence should be addressed to Jin Liang, jinliang@sjtu.edu.cn Received 25 November 2010; Accepted 19 January 2011 Academic Editor: Toka Diagana Copyright q 2011 J. Liang and Z.-W. Lv. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using the fixed-point index theory and Leggett-Williams fixed-point theorem, we study the exis- tence of multiple solutions to the three-point boundary value problem u t a t f t, u t , u t 0; u 1 − αu η λ, where η ∈ 0, 1/2 , α ∈ 1/2η, 1/η are constants, 0, 0 < t < 1; u 0 u0 λ ∈ 0, ∞ is a parameter, and a, f are given functions. New existence theorems are obtained, which extend and complement some existing results. Examples are also given to illustrate our results.1. IntroductionIt is known that when differential equations are required to satisfy boundary conditions atmore than one value of the independent variable, the resulting problem is called a multipointboundary value problem, and a typical distinction between initial value problems andmultipoint boundary value problems is that in the former case one is able to obtain thesolutions depend only on the initial values, while in the latter case, the boundary conditionsat the starting point do not determine a unique solution to start with, and some randomchoices among the solutions that satisfy these starting boundary conditions are normallynot to satisfy the boundary conditions at the other specified point s . As it is noticedelsewhere see, e.g., Agarwal 1 , Bisplinghoff and Ashley 2 , and Henderson 3 , multipoint boundary value problem has deep physical and engineering background as well asrealistic mathematical model. For the development of the research of multi point boundaryvalue problems for differential equations in last decade, we refer the readers to, for example, 1, 4–9 and references therein. Advances in Difference Equations2 In this paper, we study the existence of multiple solutions to the following three-pointboundary value problem for a class of third-order differential equations with inhomogeneousthree-point boundary values, ut a t f t, u t , u t 0, 0 < t < 1, 1.1 u 1 − αu η u0 u0 0, λ,where η ∈ 0, 1/2 , α ∈ 1/2η, 1/η , λ ∈ 0, ∞ , and a, f are given functions. To the authors’knowledge, few results on third-order differential equations with inhomogeneous three-pointboundary values can be found in the literature. Our purpose is to establish new existencetheorems for 1.1 which extend and complement some existing results. Let X be an Banach space, and let Y be a cone in X . A mapping β is said to be anonnegative continuous concave functional on Y if β : Y → 0, ∞ is continuous and 1 − t y ≥ tβ x 1−t β y , x, y ∈ Y, t ∈ 0, 1 . β tx 1.2Assume that H 1 a ∈ C 0, 1 , 0, ∞ , 1 − s sa s ds < ∞, 0< 0 1.3 f ∈ C 0, 1 × 0, ∞ × 0, ∞ , 0, ∞ .Define f t, u, v max f0 lim max sup , v v → 0 t∈ 0,1 u∈ 0, ∞ f t, u, v min f0 lim min inf , v v → 0 t∈ 0,1 u∈ 0, ∞ 1.4 f t, u, v max f∞ lim max sup , v → ∞ t∈ 0,1 v u∈ 0, ∞ f t, u, v min f∞ lim min inf . v → ∞ t∈ 0,1 u∈ 0, ∞ v This paper is organized in the following way. In Section 2, we present some lemmas,which will be used in Section 3. The main results and proofs are given in Section 3. Finally, inSection 4, we give some examples to illustrate our results ...