Local and global existence of mild solutions for impulsive fractional semilinear integro-differential equation M.H.M. Rashid ⇑ , A. Al-Omari Department of Mathematics and Statistics, Faculty of Science, P.O. Box (7), Mutah University, Alkarak, Jordan article info Article history: Received 11 November 2010 Received in revised form 19 December 2010 Accepted 31 December 2010 Available online 18 January 2011 Keywords: Mild solution Compact semigroup Fractional integro-differential equations Fractional calculus Impulsive condition abstract In this paper, we study the local and global existence of mild solutions for impulsive frac- tional semilinear integro-differential equations in an arbitrary Banach space associated with operators generating compact semigroup on the Banach space. Also, we review some applications of fractional differential equations. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of fractional differential equations is gaining much importance and attention. For details, see [1–8] and the references therein. Impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments. Our aim in this paper is to study local and global existence of mild solutions to a class of fractional semilinear integro- differential equation: u ðaÞ þ AuðtÞ¼ f ðt; uðtÞÞ þ Z t 0 qðt  sÞgðs; uðsÞÞds; t > 0; 0 < a 6 1; uðt 0 Þ¼ u 0 2 X; Duj t¼t k ¼ I k ðuðt  k ÞÞ k ¼ 1; ... ; m; ð1Þ where A is assumed to be an infinitesimal generator of a compact semigroup T(t), t P 0, the nonlinear maps f, g : I  U ? X, I = [0, T), 0 < T 6 1, are continuous where U is an open subset of X, q : I ! R and u 0 is in U. I k : X ! X; 0 ¼ t 0 ¼ 0 < t 1 <  < t m < t mþ1 ¼ T ; Duj t¼t k ¼ uðt þ k Þ uðt  k Þ; uðt þ k Þ¼ lim h!0 þ uðt k þ hÞ and uðt  k Þ¼ lim h!0  uðt k þ hÞ represent respectively the right and left limits of u(t) at t = t k . 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.12.043 ⇑ Corresponding author. E-mail addresses: malik_okasha@yahoo.com (M.H.M. Rashid), omarimutah1@yahoo.com (A. Al-Omari). Commun Nonlinear Sci Numer Simulat 16 (2011) 3493–3503 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns