Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space

M Benchohra,J Henderson,A Ouahab,Sk Ntouyas

doi:10.1155/jia.2005.189

Abstract

The well-known Krasnoselskii twin fixed point theorem is used to investigate the existence of mild solutions for first- and second-order impulsive semilinear functional and neutral functional differential equations in Hilbert spaces.

Highlights

This paper is concerned with the existence of mild solutions of some classes of initial value problem for first- and second-order impulsive semilinear functional and neutral functional differential equations
For any continuous function y defined on [−r, b] − {t1, . . . , tm} and any t ∈ [0, b], we denote by yt the element of D defined by yt(θ) = y(t + θ), θ ∈ [−r, 0]
We study the second-order impulsive semilinear functional differential equations of the form y (t) − Ay(t) = f t, yt, a.e. t ∈ J = [0, b], t = tk, k = 1, . . . , m, ∆y|t=tk = Ik y tk−, k = 1, . . . , m, ∆y |t=tk = Ik y tk−, k = 1, . . . , m, y(t) = φ(t), t ∈ [−r, 0], y (0) = η, (1.2)

Summary

Introduction

This paper is concerned with the existence of mild solutions of some classes of initial value problem for first- and second-order impulsive semilinear functional and neutral functional differential equations. Differential and partial differential equations with impulses are a basic tool to study evolution processes that are subjected to abrupt changes in their state Such equations arise naturally from a wide variety of applications, such as space-craft control, inspection processes in operations research, drug administration, and threshold theory in biology.

Preliminaries

First-order impulsive functional differential inclusions

Second-order impulsive functional differential inclusions

First-order impulsive neutral functional differential inclusions

Second-order impulsive neutral functional differential inclusions