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Research Article Existence of Solutions to Anti-Periodic Boundary Value Problem for Nonlinear Fractional Differential Equations with Impulses

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Research Article Existence of Solutions to Anti-Periodic Boundary Value Problem for Nonlinear Fractional Differential Equations with Impulses Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 915689, 17 pages doi:10.1155/2011/915689 Research Article Existence of Solutions to Anti-Periodic Boundary Value Problem for Nonlinear Fractional Differential Equations 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, [email protected] 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. Introduction In this paper, we consider an antiperiodic boundary value problem for nonlinear fractional differential equations with impulses C α D ut ft, ut,t∈ 0,T,t/ tk,k 1, 2,...,p, Δu| I ut , Δu| J ut ,k , ,...,p, ttk k k ttk k k 1 2 1.1 u0 uT 0,u0 u T 0, C α where T is a positive constant, 1 <α≤ 2, D denotes the Caputo fractional derivative of α f ∈ C ,T × R, R I J R → R {t } t <t <t < ··· <t < order , 0 , k, k : and k satisfy that 0 0 1 2 p t T Δu| ut − ut− Δu| ut − ut− ut ut− p 1 , ttk k k , ttk k k , k and k represent the right and left limits of ut at t tk. Fractional differential equations have proved to be an excellent tool in the mathematic modeling of many systems and processes in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, 2 Advances in Difference Equations electromagnetic, porous media, and so forth. In consequence, the subject of fractional differential equations is gaining much importance and attention see 1–6 and the references therein. The theory of impulsive differential equations has found its extensive applications in realistic mathematic modeling of a wide variety of practical situations and has emerged as an important area of investigation in recent years. For the general theory of impulsive differential equations, we refer the reader to 7, 8. Recently, many authors are devoted to the study of boundary value problems for impulsive differential equations of integer order, see 9–12. Very recently, there are only a few papers about the nonlinear impulsive differential equations and delayed differential equations of fractional order. Agarwal et al. in 13 have established some sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo farctional derivative. Ahmad et al. in 14 have discussed some existence results for the two-point boundary value problem involving nonlinear impulsive hybrid differential equation of fractional order by means of contraction mapping principle and Krasnoselskii’s fixed point theorem. By the similar way, they have also obtained the existence results for integral boundary value problem of nonlinear impulsive fractional differential equations see 15. Tian et al. in 16 have obtained some existence results for the three-point impulsive boundary value problem involving fractional differential equations by the means of fixed points method. Maraaba et al. in 17, 18 have established the existence and uniqueness theorem for the delay differential equations with Caputo fractional derivatives. Wang et al. in 19 have studied the existence and uniqueness of the mild solution for a class of impulsive fractional differential equations with time-varying generating operators and nonlocal conditions. To the best of our knowledge, few papers exist in the literature devoted to the antiperiodic boundary value problem for fractional differential equations with impulses. This paper studies the existence of solutions of antiperiodic boundary value problem for fractional differential equations with impulses. The organization of this paper is as follows. In Section 2, we recall some definitions of fractional integral and derivative and preliminary results which will be used in this paper. In Section 3, we will consider the existence results for problem 1.1.Wegivethreeresults,the first one is based on Banach fixed theorem, the second one is based on Schaefer fixed point theorem, and the third one is based on the nonlinear alternative of Leray-Schauder type. In Section 4, we will give an example to illustrate the main result. 2. Preliminaries In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper. Definition 2.1 see 4. The Caputo fractional derivative of order α of a function f : 0, ∞ → R is defined as t CDαft 1 t − sn−α−1f nsds, n − <α<n,n α , Γn − α 1 1 2.1 0 where α denotes the integer part of the real number α. Advances in Difference Equations 3 Definition 2.2 see 4. The Riemann-Liouville fractional integral of order α>0ofafunction ft, t>0, is defined as t Iαft 1 t − sα−1fsds, Γα 2.2 0 provided that the right side is pointwise defined on 0, ∞. Definition 2.3 see 4. The Riemann-Liouville fractional derivative of order α>0ofa continuous function f : 0, ∞ → R is given by d n t Dαft 1 t − sn−α−1fsds, Γn − α dt 2.3 0 where n α 1andα denotes the integer part of real number α, provided that the right side is pointwise defined on 0, ∞. For the sake of convenience, we introduce the following notation. J ,T,J ,t ,J t ,t ,i , ,...,p− ,J t ,T J J\{t ,t ,...,t } Let 0 0 0 1 i i i 1 1 2 1 p p . 1 2 p . J{u ,T → R | u ∈ CJ,ut ut− ut−ut , ≤ We define PC : 0 k and k exists, and k k 1 k ≤ p}. Obviously, PCJ is a Banach space with the norm u supt∈J |ut|. Definition 2.4. A function u ∈ PCJ is said to be a solution of 1.1 if u satisfies the equation cDαutft, ut t ∈ J Δu| I ut Δu| J ut ,k for , the equations ttk k k , ttk k k 1, 2,...,p, and the condition u0 uT0,u0 u T0. Lemma 2.5 see 20. Let α>0;then IαCDαu t u t c c t c t2 ··· c tn−1, 0 1 2 n−1 2.4 for some ci ∈ R, i 0, 1, 2,...,n− 1, n α 1. Lemma 2.6 nonlinear alternative of Leray-Schauder type 21. Let E be a Banach space with C ⊆ E closed and convex. Assume that U is a relatively open subset of C with 0 ∈ U and A : U → C is continuous, compact map. Then either 1 A has a fixed point in U,or 2 there exists u ∈ ∂U and λ ∈ 0, 1 with u λAu. Lemma 2.7 Schaefer fixed point theorem 22. Let S be a convex subset of a normed linear space Ω and 0 ∈ S.LetF : S → S be a completely continuous operator, and let ζF {u ∈ S : u λFu, for some 0 <λ<1}. 2.5 Then either ζF is unbounded or F has a fixed point. 4 Advances in Difference Equations Lemma 2.8. Assume that y ∈ C0,T,R,T>0, 1 <α≤ 2. A function u ∈ PCJ is a solution of the antiperiodic boundary value problem C α D ut yt,t∈ 0,T,t/ tk,k 1, 2,...,p, Δu| I ut , Δu| J ut ,k , ,...,p, ttk k k ttk k k 1 2 2.6 u0 uT 0,u0 u T 0, if and only if u is a solution of the integral equation ⎧ p ⎪ t 1 ti ⎪ 1 α−1 1 α−1 ⎪ t − s ysds − ti − s ysds ⎪Γα 2Γα t ⎪ 0 i1 i−1 ⎪ p ⎪ ti ⎪ 1 α−2 ⎪ − T − ti ti − s ysds ⎪ 2Γα − 1 t ⎪ i1 i−1 ⎪ ⎪ p t p ⎪ T − t 1 i ⎪ 2 t − sα−2ysds − 1 T − t J ut ⎪ Γα − i i i i ⎪ 4 1 i ti− 2 i ⎪ 1 1 1 ⎪ p p ⎪ T − t ⎪ 2 J ut − 1 I ut ,t∈ ,t , ⎪ i i i i 0 1 ⎪ 4 i 2 i ⎪ 1 1 ⎪ ⎪ t k ti ⎪ 1 t − sα−1ysds 1 t − sα−1ysds ⎪ i ⎪Γα t Γα t ⎪ k i1 i−1 ⎪ ⎨ p 1 ti k ut 1 α−1 1 ⎪ − ti − s ysds t − ti 2.7 ⎪ 2Γα t Γα − 1 ⎪ i1 i−1 i1 ⎪ ⎪ ti ⎪ α−2 ⎪ × ti − s ysds ⎪ ⎪ ti− ⎪ 1 ⎪ p ⎪ ti ⎪ − 1 T − t t − sα−2ysds ⎪ i i ⎪ 2Γα − 1 t ⎪ i1 i−1 ⎪ ⎪ p ⎪ T − t 1 ti k ⎪ 2 α−2 ⎪ ti − s ysds t − tiJiuti ⎪ Γα − t ⎪ 4 1 i i−1 i ⎪ 1 1 ⎪ p p ⎪ 1 T − 2t ⎪ − T − t J ut J ut ⎪ i i i i i ⎪ 2 i 4 i ⎪ 1 1 ⎪ k p ⎪ ⎪ I u t − 1 I u t ,t∈ t ,t , ≤ k ≤ p. ⎩ i i i i k k 1 1 i1 2 i1 y c ,c ∈ R Proof. Assume that satisfies 2.6 .UsingLemma 2.5, for some constants 0 1 , we have t ut Iαyt − c − c t 1 t − sα−1ysds − c − c t, t ∈ ,t .
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