Double Perturbations for Impulsive Differential Equations in Banach Spaces

Abstract

In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.