Existence results for impulsive neutral functional differential equations with state-dependent delay

Mallika Mallika Arjunan,Veeraruna Kavitha

doi:10.14232/ejqtde.2009.1.26

Abstract

In this article, we study the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay. The results are obtained by using Leray-Schauder Alternative fixed point theorem. Example is provided to illustrate the main result.

Highlights

The purpose of this article is to establish the existence of mild solutions for a class of impulsive abstract neutral functional differential equations with state-dependent delay described by the form d dt
Where A is the infinitesimal generator of a compact C0-semigroup of bounded linear operators (T (t))t≥0 on a Banach space X; the function xs : (−∞, 0] → X, xs(θ) = x(s + θ), belongs to some abstract phase space B described axiomatically; 0 < t1.... < tn < a are prefixed numbers; F, G : I × B → X, ρ : I × B → (−∞, a], Ii : B × X → X, i = 1, 2, ..., n, are appropriate functions and ∆ξ(t) represents the jump of the function ξ at t, which is defined by ∆ξ(t) = ξ(t+) − ξ(t−)
The study of impulsive partial neutral functional differential equations with statedependent delay described in the general abstract form (1.1)-(1.3) is an untreated topic in the literature, and this fact, is the main motivation of our paper

Summary

Introduction

The purpose of this article is to establish the existence of mild solutions for a class of impulsive abstract neutral functional differential equations with state-dependent delay described by the form d dt [x(t). Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. The study of impulsive partial neutral functional differential equations with statedependent delay described in the general abstract form (1.1)-(1.3) is an untreated topic in the literature, and this fact, is the main motivation of our paper

Preliminaries

Existence Results

Example