Existence Results for Nonlinear Boundary Value Problems of Impulsive Fractional Integrodifferential Equations

In this paper, we investigate the existence result for nonlinear impulsive fractional integro-differential equations with boundary conditions by using fixed point theorem and Green's function. I. INTRODUCTION The topic of fractional differential equations has received a great deal of attention from many scientists and researchers during the past decades; see [1-7]. This is mostly due to the fact that fractional calculus provides an efficient and excellent instrument to describe many practical dynamical phenomena which arise in engineering and science such as physics, chemistry, biology, economy, viscoelasticity, electrochemistry, electromagnetic, control, porous media; see [8-13]. Moreover, many researchers study the existence of solutions for fractional differential equations; see [14-16] and the references therein. In particular, several authors have considered a nonlocal Cauchy problem for abstract evolution differential equations having fractional order. Indeed, the nonlocal Cauchy problem for abstract evolution differential equations was studied by Byszewski [17, 18] initially. Afterwards, many authors [19-21] discussed the problem for different kinds of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces. Balachandran et al.[22, 23] established the existence of solutions of quasilinear integrodifferential equations with nonlocal conditions. N'Guerekata [24] and Balachandran and Park [25] researched the existence of solutions of fractional abstract differential equations with a nonlocal initial condition. Ahmad [26] obtained some existence results for boundary value problems of fractional semi linear evolution equations. Recently, Balachandran and Trujillo [27] have investigated the nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. On the other hand, the theory of impulsive differential equations for integer order has emerged in mathematical modeling of phenomena and practical situations in both physical and social sciences in recent years. One can see a significant development in impulsive theory. We refer the readers to [28-31] for the general theory and applications of impulsive differential equations. Besides, some researchers [32-36] have addressed the theory of boundary value problems for impulsive fractional differential equations. Motivated by the aforementioned works, in this paper, we deal with the existence of solutions of nonlinear boundary value problem of fractional impulsive integro-differential equations: