On the solution set of second-order delay differential inclusions in Banach spaces

A Sghir

doi:10.5802/ambp.124

Abstract

In this paper, we consider the second-order delay differential inclusion x (t) E Ax(t) + F(t, x t ) in a Banach space and we study some properties of its solution set. We prove a relaxation theorem which reveals the connection between the solution sets of a second-order delay differential inclusion and its convexified version, under some weak conditions.

Highlights

Where A is the infinitesimal generator of a Co-propagator of linear operators (C(t))tER on a Banach space (E, |.|E) and F is a nonlinear multimapping, satisfying assumptions to be specified in the third section
This paper is concerned with the second-order delay differential inclusion (1) and its mild trajectories
We show that many results which allow us to apply differential inclusions, see for example ~l, 3, 8,10,13~ and references therein, are valid as well for

Summary

Introduction

Many problems in applied mathematics, such as those in control theory, lead to the study of second-order delay differential inclusions x" (t) ~ Ax(t) + F(t, xt) ,. Where A is the infinitesimal generator of a Co-propagator of linear operators (C(t))tER on a Banach space (E, |.|E) and F is a nonlinear multimapping, satisfying assumptions to be specified in the third section. As particular cases of relations of the form (I) we have: i) The second-order delay differential equation x" (t) = Ax(t) + f (t, xt) where F(t, xt) = J(t, xt). Second-order delay differential equation x" (t) = + f(t, xt~ u(t)), u(t) E U(t). This paper is concerned with the second-order delay differential inclusion (1) and its mild trajectories. The assumption of integrale boundedness (condition (H4)) will be replaced by an integrability condition (condition (H3)). We give some properties of the solution set of the inclusion (1)

Preliminaries

We have:

Some properties of the solution set