Existence of a mild solution for an impulsive nonlocal non-autonomous neutral functional differential equation

Alka Chadha,Dwijendra N Pandey

doi:10.1007/s11565-016-0242-2

Abstract

This paper considers an impulsive neutral differential equation with nonlocal initial conditions in an arbitrary Banach space E. The existence of the mild solution is obtained by using Krasnoselskii’s fixed point theorem and approximation techniques without imposing the strong restriction on nonlocal function and impulsive functions. An example is also provided at the end of the paper to illustrate the abstract theory.

Highlights

In recent years, impulsive differential equations have become an active area of research due to their demonstrated applications in widespread fields of science and engineering such as biology, physics, control theory, population dynamics, economics, chemical technology, medicine and many others
Many physical systems which are characterized by the occurrence of an abrupt change in the state of the system can be described by impulsive differential equations
The existence, uniqueness and stability of mild solutions to functional differential equations with impulsive conditions have been considered by many authors in literatures

Summary

Introduction

Impulsive differential equations have become an active area of research due to their demonstrated applications in widespread fields of science and engineering such as biology, physics, control theory, population dynamics, economics, chemical technology, medicine and many others. In [1], authors have considered a class of abstract impulsive mixedtype non-autonomous functional integro-differential equations with finite delay in a Banach space and obtained sufficient conditions for controllability of considered system by virtue of semigroup theory via Monch fixed point theorem technique and measures of noncompactness. An impulsive neutral integro-differential equation of Sobolev type with time varying delays has been considered by authors in [27] and the sufficient condition for existence of the mild solution has been provided by using the Monch’s fixed point theorem. In [16], authors have considered an impulsive stochastic functional integro-differential inclusions with nonlocal conditions in a Hilbert space and provided existence results for mild solution by using approximation technique and BohnenblustKarlins fixed point theorem.

Preliminaries

Main result

Example