Noncompact Perturbation of Sweeping Process with Delay in Banach Spaces

F Aladsani ,Ahmed Ibrahim

doi:10.1155/2013/567094

F Aladsani , Ahmed Ibrahim

Open Access

https://doi.org/10.1155/2013/567094

Copy DOI

Abstract

We have proven an existence theorem concerning the existence of solutions for a functional evolution inclusion governed by sweeping process with closed convex sets depending on time and state and with a noncompact nonconvex perturbation in Banach spaces. This work extends some recent existence theorems concerning sweeping processes from Hilbert spaces setting to Banach spaces setting. Moreover, it improves some recent existence results for sweeping processes in Banach spaces.

Highlights

We have proven an existence theorem concerning the existence of solutions for a functional evolution inclusion governed by sweeping process with closed convex sets depending on time and state and with a noncompact nonconvex perturbation in Banach spaces
Differential inclusions represent an important generalization of differential equations
Moreau [5] proposed and studied the following differential inclusion governed by sweeping process of first order:

Summary

Introduction

Differential inclusions represent an important generalization of differential equations. In. International Journal of Mathematics and Mathematical Sciences a very recent paper, Aitalioubrahim [6] considered (2) when X is a Hilbert spaces C is a multifunction from I to the family of nonempty closed nonconvex subsets of X, and F is a multifunction defined on [0, T] × CX([−r, 0]) and with nonconvex noncompact values in X. Ibrahim and Aladsani [15] considered a second order sweeping process without delay in a separable p-uniformly convex and q-uniformly smooth Banach space X, and the perturbation F is an upper semicontinuous defined from [0, T] × X × X to the family of nonempty convex weakly compact sets of the topological dual space X∗ of X such that. Our work improves many results in the literature concerning the existence of solutions for some evolution inclusions governed by sweeping process in Banach spaces, for example, [14, 15]. In addition our technique allows us to discuss some sweeping process problems with noncompact perturbation in Banach spaces

Preliminaries and Notations

Conclusion