Existence of the Mild Solution for Impulsive Semilinear Differential Equation

We study the existence of solutions of impulsive semilinear differential equation in a Banach space in which impulsive condition is not instantaneous. We establish the existence of a mild solution by using the Hausdorff measure of noncompactness and a fixed point theorem for the convex power condensing operator. 1. Introduction In a few decades, impulsive differential equations have received much attention of researchers mainly due to its demonstrated applications in widespread fields of science and engineering such as biology, physics, control theory, population dynamics, medicine and so on. The real world processes and phenomena which are subjected during their development to short-term external inuences can be modeled as impulsive differential equation. Their duration is negligible compared to the total duration of the entire process or phenomena. Impulsive differential equations are an appropriate model to hereditary phenomena for which a delay argument arises in the modelling equations. To further study on impulsive differential equations, we refer to books [1, 2] and papers [3–11]. In this paper, our purpose is to establish the existence of a solution to the following differential equations with non instantaneous impulses where is a closed and bounded linear operator with dense domain . We assume that is the infinitesimal generator of a strongly continuous semigroup in a Banach space . Here, , and , for all are suitable functions to be specified later. In [4], authors have introduced a new class of abstract impulsive differential equations in which impulses are not instantaneous and established the existence of solutions to the problem (1)–(3) with the assumption that operator generates a -semigroup of bounded linear operators. In this system of (1)–(3), the impulses begin all of a sudden at the points and their proceeding continues on a finite time interval [5]. To concern the hemodynamical harmony of an individual we think about the following simplified situation. One can recommend a few intravenous sedates (insulin) on account of a decompensation (e.g., high or low level of glucose). Since the presentation of the medications in the bloodstream and the ensuing retention for the form are progressive and continuous processes, we can depict this circumstance as an impulsive activity which begins abruptly and stays animated on a finite time interval. In [12], the generalization of the condensing operator as convex-power condensing operator has been introduced by Sun and Zhang and a new fixed point theorem for convex-power condensing operator has