## Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order

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Research Article Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order Liu Yang1, 2 and Haibo Chen1 1 2

Department of Mathematics, Central South University, Changsha, Hunan 410075, China Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan 421008, China

Correspondence should be addressed to Liu Yang, [email protected] Received 18 September 2010; Accepted 4 January 2011 Academic Editor: Mouﬀak Benchohra Copyright q 2011 L. Yang and H. 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. We study a nonlocal boundary value problem of impulsive fractional diﬀerential equations. By means of a fixed point theorem due to O’Regan, we establish suﬃcient conditions for the existence of at least one solution of the problem. For the illustration of the main result, an example is given.

1. Introduction Fractional diﬀerential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, aerodynamics, chemistry, and engineering and biological sciences, involves derivatives of fractional order. Fractional diﬀerential equations also provide an excellent tool for the description of memory and hereditary properties of many materials and processes. In consequence, fractional diﬀerential equations have emerged as a significant development in recent years, see 1–3. As one of the important topics in the research diﬀerential equations, the boundary value problem has attained a great deal of attention from many researchers, see 4–11 and the references therein. As pointed out in 12, the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena. There are three noteworthy papers dealing with the nonlocal boundary value problem of fractional diﬀerential equations. Benchohra et al. 12 investigated the following nonlocal boundary value problem c

Dα ut  ft, ut 0, u0 gu,

0 < t < T, 1 < α ≤ 2, uT uT ,

where c Dα denotes the Caputo’s fractional derivative.

1.1

2

Zhong and Lin 13 studied the following nonlocal and multiple-point boundary value problem c

Dα ut  ft, ut 0,

u0 u0  gu,

0 < t < 1, 1 < α ≤ 2,

u 1 u1 

m−2 

bi u ξi .

1.2

i 1

Ahmad and Sivasundaram 14 studied a class of four-point nonlocal boundary value problem of nonlinear integrodiﬀerential equations of fractional order by applying some fixed point theorems. On the other hand, impulsive diﬀerential equations of fractional order play an important role in theory and applications, see the references 15–21 and references therein. However, as pointed out in 15, 16, the theory of boundary value problems for nonlinear impulsive fractional diﬀerential equations is still in the initial stages