Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 1, Pages 51-72
doi:10.1155/S104895330000006X
On the structure of the solution set of evolution inclusions with Fréchet subdifferentials
Perugia University, Department of Mathematics, Via Vanvitelli 1, Perugia 06123, Italy
Received 1 May 1998; Revised 1 November 1999
Copyright © 2000 Tiziana Cardinali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper we consider a Cauchy problem in which is present an evolution inclusion driven by the Fréchet subdifferential o ∂−f of a function
f:Ω→R∪{+∞} (Ω is an open subset of a real separable Hilbert space)
having a φ-monotone . subdifferential of order two and a perturbation
F:I×Ω→Pfc(H) with nonempty, closed and convex values.
First we show that the Cauchy problem has a nonempty solution set
which is an Rδ-set in C(I,H), in particular, compact and acyclic. Moreover, we obtain a Kneser-type theorem. In addition, we establish a continuity result about the solution-multifunction x→S(x). We also produce a
continuous selector for the multifunction x→S(x). As an application of
this result, we obtain the existence of solutions for a periodic problem.