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Báo cáo hóa học: " Research Article Existence of Solutions to Anti-Periodic Boundary Value Problem for Nonlinear Fractional "Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2011, Article ID 915689, 17 pagesdoi:10.1155/2011/915689Research ArticleExistence of Solutions to Anti-Periodic BoundaryValue Problem for Nonlinear Fractional DifferentialEquations with Impulses Anping Chen1, 2 and Yi Chen2 1 Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, China 2 School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411005, China Correspondence should be addressed to Anping Chen, chenap@263.net Received 20 October 2010; Revised 25 December 2010; Accepted 20 January 2011 Academic Editor: Dumitru Baleanu Copyright q 2011 A. Chen and Y. Chen. 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. This paper discusses the existence of solutions to antiperiodic boundary value problem for nonlinear impulsive fractional differential equations. By using Banach fixed point theorem, Schaefer fixed point theorem, and nonlinear alternative of Leray-Schauder type theorem, some existence results of solutions are obtained. An example is given to illustrate the main result.1. IntroductionIn this paper, we consider an antiperiodic boundary value problem for nonlinear fractionaldifferential equations with impulses t ∈ 0, T , t / tk , k C Dα u t f t, u t , 1, 2, . . . , p, Δu|t Δu |t 1.1 I k u tk , J k u tk , k 1, 2, . . . , p, tk tk u0 uT 0, u0 uT 0,where T is a positive constant, 1 < α ≤ 2, C Dα denotes the Caputo fractional derivative oforder α, f ∈ C 0, T × R, R , Ik , Jk : R → R and {tk } satisfy that 0 t0 < t1 < t2 < · · · < tp
