Existence results for nondensely defined semilinear functional differential inclusions in Fréchet spaces

Johnny Henderson,Abdelghani Ouahab

doi:10.14232/ejqtde.2005.1.17

Abstract

Existence results for nondensely defined semilinear functional differential inclusions in Fréchet spaces

Highlights

This paper is concerned with an application of a recent Frigon nonlinear alternative for contractive multivalued maps in Frechet spaces [20] to obtain the existence of integral solutions of some classes of initial value problems for first order semilinear functional differential inclusions
For any continuous function y defined on [−r, ∞) and any t ∈ [0, ∞), we denote by yt the element of C([−r, 0], D(A)) defined by yt(θ) = y(t + θ), θ ∈ [−r, 0]
In the case where F is either a single or a multivalued map, and A is a densely defined linear operator generating a C0-semigroup of bounded linear operators, the problems (1)–(2) and (3)–(4) have been investigated on compact intervals in, for instance, the monographs by Ahmed [1], Hale and Lunel [21], Wu [31], Hu and Papageorgiou [22], Kamenskii, Obukhovskii and Zecca [24], and in the papers of Benchohra, Ntouyas [9, 10, 11] for the controllability of differential inclusions with different conditions; see the monograph of Benchohra, Ntouyas, Gorniewicz [13] and the papers of Balachandran and Manimegolai [6], Benchohra et al [7], Benchohra and Ntouyas [8], and Li and Xue [23]and the references cited therein

Summary

Introduction

This paper is concerned with an application of a recent Frigon nonlinear alternative for contractive multivalued maps in Frechet spaces [20] to obtain the existence of integral solutions of some classes of initial value problems for first order semilinear functional differential inclusions. In the case where F is either a single or a multivalued map, and A is a densely defined linear operator generating a C0-semigroup of bounded linear operators, the problems (1)–(2) and (3)–(4) have been investigated on compact intervals in, for instance, the monographs by Ahmed [1], Hale and Lunel [21], Wu [31], Hu and Papageorgiou [22], Kamenskii, Obukhovskii and Zecca [24], and in the papers of Benchohra, Ntouyas [9, 10, 11] for the controllability of differential inclusions with different conditions; see the monograph of Benchohra, Ntouyas, Gorniewicz [13] and the papers of Balachandran and Manimegolai [6], Benchohra et al [7], Benchohra and Ntouyas [8], and Li and Xue [23]and the references cited therein. These results extend some ones existing in the previous literature in the case of densely defined linear operators

Preliminaries

Functional Differential Inclusions

Controllability Functional Differential Inclusions

An Example