Iterative Method for a New Class of Evolution Equations with Non-instantaneous Impulses

Abstract

In this paper, we are concerned with the existence of mild solutions for the initial value problem to a new class of abstract evolution equations with non-instantaneous impulses on ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear evolution equation with non-instantaneous impulses is established. With the aid of this theorem, the existence of mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained by using perturbation technique and iterative method under the situation that the corresponding solution semigroup $T(\cdot)$ and non-instantaneous impulsive function $g_k$ are compact, $T(\cdot)$ is not compact and $g_k$ is compact, $T(\cdot)$ and $g_k$ are not compact, respectively. The results obtained in this paper essentially improve and extend some related conclusions on this topic. Two concrete examples to parabolic partial differential equations with non-instantaneous impulses are given to illustrate that our results are valuable.