Nonlocal Cauchy Problem for Impulsive Differential Equations in Banach Spaces

where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T (t) on a Banach space X; f : [0, b]×X → X; 0 < t1 < t2 < · · · < tp < tp+1 = b; Ii : X → X, i = 1, 2, · · · , p are impulsive functions and g : PC([0, b];X) → X . During recent years, the impulsive differential equations have been an object of intensive investigation because of the wide possibilities for their application in various fields of science and technology as theoretical physics, population dynamics, economics,etc. See [1, 4, 16] and the references therein for more comments. The existence and uniqueness of mild, strong and classical solution of nonlocal abstract Cauchy problem has been established by Byszewski [6, 7]. Subsequently, many authors are devoted to studying of nonlocal problems. Some papers have been written on various classes of differential equations [2, 8, 9, 18–21]. In this paper we will derive some sufficient conditions for the solution of differential equation (1.1)-(1.3),combining impulsive conditions and nonlocal conditions. Our results are achieved by applying the Hausdorff measure of noncompactness and fixed point theorem. Neither the semigroup T (t) nor the function f is needed to be compact in our results. So our work extends and improves many main results such as those in [1, 10, 13, 15].