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 ut  ft, ut 0, u0 gu,

0 < t < T, 1 < α ≤ 2, uT uT ,

where c Dα denotes the Caputo’s fractional derivative.



Advances in Difference Equations

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

Dα ut  ft, ut 0,

u0 u0  gu,

0 < t < 1, 1 < α ≤ 2,

u 1 u1 


bi u ξi .


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