Existence Of Solutions For Impulsive Equations In Banach Spaces

The theory of nonlinear impulsive differential equations is a new and important branch of differential equation, which originates from some mathematical model of biology, medicine. Because all the structure of its emergence has deep physical background, the research on nonlinear impulsive differential equations is very meaningful, which makes us research this subject seriously.In this paper, firstly, we consider the impulsive Volterra integral equation in a real Banach space E. Under a weaker compactness-type condition, using a new comparison result and recurrence method, through introducing the equivalent norm, Tonelii sequence and the locally convex topology, we obtain the existence of solutions for first order nonlinear impulsive Volterra integral equation on an infinite interval with an infinite number of impulsive times in a real Banach space E, which generalize and improve the results of paper [1]. And we obtain one sufficient condition of existence of maximal and minimal solutions. As applications of our results, the solutions of the infinite system of nonlinear impulsive integral equation of Volterra type be obtained.Secondly, we consider the IVP for first order impulsive integro-differential equation in Banach space E. By establishing comparison results and applying the Monch fixed point theorem, through introducing the equivalent norm, the recurrence method and Quasi-nilpotent operator, we obtain the existence of solutions of initial value problems of impulsive integro-differential equation in Banach spaces, which generalize and improve some related results. And we give some applications to the extremal solutions of impulsive integro-differential equation.