Multivalued semilinear neutral functional differential equations with nonconvex‐valued right‐hand side

M Benchohra,E Gatsori,S K Ntouyas

doi:10.1155/s1085337504306202

Abstract

We investigate the existence of mild solutions on acompact interval to some classes of semilinear neutral functional differential inclusions. We will rely on a fixed‐point theorem for contraction multivalued maps due to Covitz and Nadler and on Schaefer′s fixed‐point theorem combined with lower semicontinuous multivalued operators with decomposable values.

Highlights

This paper is concerned with the existence of mild solutions defined on a compact real interval for first- and second-order semilinear neutral functional differential inclusions (NFDIs)
By means of a fixed-point argument and the semigroup theory, existence theorems of mild solutions on compact and noncompact intervals for first- and second-order semilinear NFDIs with a convex-valued right-hand side were obtained by Benchohra and Ntouyas in [1, 4]
We say that F is of l.s.c. type if its associated Niemytzki operator Ᏺ is l.s.c. and has nonempty closed and decomposable values

Summary

Introduction

This paper is concerned with the existence of mild solutions defined on a compact real interval for first- and second-order semilinear neutral functional differential inclusions (NFDIs). By means of a fixed-point argument and the semigroup theory, existence theorems of mild solutions on compact and noncompact intervals for first- and second-order semilinear NFDIs with a convex-valued right-hand side were obtained by Benchohra and Ntouyas in [1, 4]. We use a fixed-point theorem for contraction multivalued maps due to Covitz and Nadler [7] (see Deimling [8]) This method was applied recently by Benchohra and Ntouyas in [3], in the case when A = 0 and f ≡ 0. We use Schaefer’s theorem combined with a selection theorem of Bressan and Colombo [5] for lower semicontinuous (l.s.c) multivalued operators with decomposable values

Preliminaries

First-order semilinear NFDIs