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, and Yi Chen2 Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, China 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 Introduction In this paper, we consider an antiperiodic boundary value problem for nonlinear fractional differential equations with impulses C Dα u t Δu|t tk f t, u t , I k u tk , u u T t ∈ 0, T , t / tk , k Δu |t 0, tk J k u tk , u u T 1, 2, , p, k 1, 2, , p, 1.1 0, where T is a positive constant, < α ≤ 2, C Dα denotes the Caputo fractional derivative of order α, f ∈ C 0, T × R, R , Ik , Jk : R → R and {tk } satisfy that t0 < t1 < t2 < · · · < < T , Δu|t tk u tk − u t− , Δu |t tk u tk − u t− , u tk and u t− represent the right k k k and left limits of u t 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, 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, 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 We give three results, 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 Preliminaries In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper Definition 2.1 see The Caputo fractional derivative of order α of a function f : 0, ∞ → R is defined as C Dα f t Γ n−α t t−s n−α−1 f n s ds, where α denotes the integer part of the real number α n − < α < n, n α 1, 2.1 Advances in Difference Equations Definition 2.2 see The Riemann-Liouville fractional integral of order α > of a function f t , t > 0, is defined as t Γ α I αf t t−s α−1 f s ds, 2.2 provided that the right side is pointwise defined on 0, ∞ Definition 2.3 see The Riemann-Liouville fractional derivative of order α > of a continuous function f : 0, ∞ → R is given by Γ n−α Dα f t n d dt t t−s n−α−1 f s ds, 2.3 where n α and α 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 0, t1 , Ji ti , ti , i 1, 2, , p−1, Jp , T J J \{t1 , t2 , , } Let J 0, T , J0 u tk , ≤ We define PC J {u : 0, T → R | u ∈ C J , u tk and u t− exists, and u t− k k k ≤ p} Obviously, PC J is a Banach space with the norm u supt∈J |u t | Definition 2.4 A function u ∈ PC J is said to be a solution of 1.1 if u satisfies the equation c α D u t f t, u t for t ∈ J , the equations Δu|t tk Ik u tk , Δu |t tk J k u tk , k 1, 2, , p, and the condition u u T 0, u u T Lemma 2.5 see 20 Let α > 0; then I α C Dα u t for some ci ∈ R, i u t 0, 1, 2, , n − 1, n c0 α c1 t c t2 ··· cn−1 tn−1 , 2.4 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 ∈ U and A : U → C is continuous, compact map Then either A has a fixed point in U, or there exists u ∈ ∂U and λ ∈ 0, with u λAu Lemma 2.7 Schaefer fixed point theorem 22 Let S be a convex subset of a normed linear space Ω and ∈ S Let F : S → S be a completely continuous operator, and let ζ F {u ∈ S : u λFu, for some < λ < 1} Then either ζ F is unbounded or F has a fixed point 2.5 Advances in Difference Equations Lemma 2.8 Assume that y ∈ C 0, T , R , T > 0, < α ≤ A function u ∈ PC J is a solution of the antiperiodic boundary value problem C Δu|t Dα u t y t , t ∈ 0, T , t / tk , k Δu |t 1, 2, , p, 1, 2, , p, 2.6 ⎧ p ti t ⎪ 1 ⎪ ⎪ t − s α−1 y s ds − ti − s α−1 y s ds ⎪ ⎪Γ α 2Γ α i ti−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p ti ⎪ ⎪ ⎪ − ⎪ T − ti ti − s α−2 y s ds ⎪ ⎪ 2Γ α − i ⎪ ti−1 ⎪ ⎪ ⎪ ⎪ ⎪ p ti p ⎪ ⎪ T − 2t ⎪ ⎪ ti − s α−2 y s ds − T − ti J i u t i ⎪ ⎪ ⎪ 4Γ α − i ti−1 2i ⎪ ⎪ ⎪ ⎪ ⎪ p p ⎪ ⎪ T − 2t ⎪ ⎪ J i u ti − Ii u ti , t ∈ 0, t1 , ⎪ ⎪ ⎪ i 2i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t ⎪ k ti ⎪ ⎪ ⎪ t − s α−1 y s ds ti − s α−1 y s ds ⎪Γ α ⎪ Γ α i ti−1 ⎪ tk ⎪ ⎪ ⎪ ⎪ ⎨ p ti k 1 − ti − s α−1 y s ds t − ti ⎪ 2Γ α i ti−1 Γ α−1 i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ti ⎪ ⎪ ⎪ ⎪ × ti − s α−2 y s ds ⎪ ⎪ ⎪ ti−1 ⎪ ⎪ ⎪ ⎪ p ⎪ ti ⎪ ⎪ ⎪ − ⎪ T − ti ti − s α−2 y s ds ⎪ ⎪ 2Γ α − i ⎪ ti−1 ⎪ ⎪ ⎪ ⎪ ⎪ p ti k ⎪ ⎪ T − 2t ⎪ ⎪ ti − s α−2 y s ds t − ti J i u t i ⎪ ⎪ ⎪ 4Γ α − i ti−1 ⎪ i ⎪ ⎪ ⎪ ⎪ p p ⎪ ⎪ T − 2t ⎪ −1 ⎪ J i u ti T − ti J i u t i ⎪ ⎪ 2i i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p k ⎪ ⎪ ⎪ ⎪ I i u ti − Ii u ti , t ∈ tk , tk , ≤ k ≤ p ⎩ 2i i 2.7 tk I k u tk , u u T J k u tk , tk u 0, u T k 0, if and only if u is a solution of the integral equation ut Proof Assume that y satisfies 2.6 Using Lemma 2.5, for some constants c0 , c1 ∈ R, we have u t I α y t − c0 − c1 t Γα t t−s α−1 y s ds − c0 − c1 t, t ∈ 0, t1 2.8 Advances in Difference Equations Then, we obtain t Γ α−1 u t α−2 t−s y s ds − c1 , t ∈ 0, t1 2.9 If t ∈ t1 , t2 , then we have t Γα u t t−s y s ds − d0 − d1 t − t1 , 2.10 t Γ α−1 u t α−1 t1 t−s α−2 y s ds − d1 , t1 where d0 , d1 ∈ R are arbitrary constants Thus, we find that t1 Γα u t− α−1 t1 − s −d0 , u t1 t1 Γ α−1 u t− t1 − s u t1 − u t− t1 −d0 −d1 I u t1 t1 Γα t1 − s 2.11 α−2 y s ds − c1 , −d1 u t1 In view of Δu|t y s ds − c0 − c1 t1 , and Δu |t α−1 u t1 − u t− t1 y s ds − c0 − c1 t1 J1 u t1 , we have I u t1 , Γ α−1 t1 2.12 t1 − s α−2 y s ds − c1 J u t1 Hence, we obtain u t Γα t t−s α−1 y s ds t1 t − t1 Γ α−1 I u t1 t1 t1 − s α−2 Γα t1 α−1 y s ds y s ds − c0 − c1 t, t1 − s t ∈ t , t2 t − t1 J u t 2.13 Advances in Difference Equations Repeating the process in this way, the solution u t for t ∈ tk , tk t Γα u t t−s α−1 y s ds tk k Γ α−1 t − ti ti − s i can be written as ti−1 α−2 α−1 ti − s k y s ds t − ti J i u t i y s ds 2.14 i − c0 − c1 t, I i u ti ti ti−1 i k ti k Γ α t ∈ t k , tk , k 1, 2, , p ti − s α−1 i On the other hand, by 2.14 , we have T Γα u T T −s α−1 Γα y s ds p Γ α−1 ti T − ti p ti i ti−1 p ti − s α−2 T − ti J i u t i y s ds ti−1 i y s ds i p − c0 − c1 T, I i u ti 2.15 i T Γ α−1 u T α−2 T −s Γ α−1 y s ds p ti i ti−1 ti − s α−2 y s ds p − c1 J i u ti i By the boundary conditions u c0 p 1 2Γ α ti c1 α−1 y s ds ti−1 i T − 4Γ α − 1 2i ti − s u T p ti ti − s α−2 ti−1 i p 0, u u T p 2Γ α − T y s ds − T − ti i T − ti J i u t i 2Γ α − p i ti ti−1 ti ti − s α−2 y s ds ti−1 p J i u ti i p 0, we obtain 2.16 I i u ti , 2i p ti − s α−2 y s ds J i u ti 2i Substituting the values of c0 and c1 into 2.8 , 2.14 , respectively, we obtain 2.7 Conversely, we assume that u is a solution of the integral equation 2.7 By a direct computation, it follows that the solution given by 2.7 satisfies 2.6 The proof is completed Advances in Difference Equations Main Result In this section, our aim is to discuss the existence and uniqueness of solutions to the problem 1.1 Theorem 3.1 Assume that H1 there exists a constant L1 > such that |f t, u − f t, v | ≤ L1 |u − v|, for each t ∈ J and all u, v ∈ R; H2 there exist constants L2 , L3 > such that Ik u − Ik v ≤ L2 |u − v|, Jk u − Jk v ≤ L3 |u − v|, for each t ∈ J and all u, v ∈ R, k 1, 2, , p If 3p T α 2Γ α L1 p Tα 4Γ α p L2 7T L3 < 1, 3.1 then problem 1.1 has a unique solution on J Proof We transform the problem 1.1 into a fixed point problem Define an operator T : PC J → PC J by Tu t t Γ α − t−s Γα f s, u s ds tk 2Γ α p i 1 Γ α−1 − α−1 2Γ α − T − 2t 4Γ α − ti ti − s 0