A Class of Nonlocal Impulsive Problems for Integrodifferential Equations in Banach Spaces

Abstract

In this paper, we study the existence and uniqueness of the PC-mild solution for a class of nonlinear integrodifferential impulsive differential equations with nonlocal conditions $$\left\{\begin{array}{l} x'(t)=Ax(t)+f\left(t,x(t), \int_{0}^{t}k(t,s,x(s))ds\right), \quad t\in J=[0,b], \,\, t\neq t_{i},\\ x(0)=g(x)+x_{0},\\ \Delta x(t_{i})=I_{i}(x(t_{i})), \quad i=1,2,\ldots,p, \,\, 0=t_{0} < t_{1} < \cdots < t_{p} < t_{p+1}=b.\end{array} \right.$$ Using the generalized Ascoli-Arzela theorem given by us, some fixed point technique including Schaefer fixed point theorem and Krasnoselskii fixed point theorem, and theory of operators semigroup, some new results are obtained. At last, some examples are given to illustrate the theory.