Existence of solution of impulsive second order neutral integro- differential equations with state delay

Abstract

This paper consists of two parts. The first part deals with the existence of a mild solution of a class of instantaneous impulsive second order partial neutral differential equations with state dependent delay. The second part studies the non-instantaneous impulsive conditions on the same problem. The Kuratowski measure of noncompactness and M\'onch fixed point theorem are used to prove the existence of the mild solution. We remove the restrictive conditions on the priori estimation available in literature. The compactness assumption on the associated cosine or sine family of operators, nonlinear terms and associated impulsive term are also not required in this paper. The noncompactness measure estimation, the Lipschitz conditions and compactness on the nonlinear functions are replaced by simple and natural assumptions. We introduce new non-instantaneous impulses with fixed delays. In the last section, we study examples to illustrate the result presented.