A strong relaxation theorem for maximal monotone differential inclusions with memory. (English) Zbl 0817.34010

Papageorgiou, Nikolaos S.

Arch. Math., Brno 30, No. 4, 227-235 (1994).

A relaxation theorem for a maximal monotone differential inclusion with memory is proved. Namely, the author considers the following two problems: (1) \(-x'(t)\in Ax(t)+ F(t, x_ t)\), \(x(v)= \phi(v)\), \(v\in [- r, 0]\); (2) \(-x'(t)\in Ax(t)+ \text{ext }F(t, x_ t)\), \(x(v)= \phi(v)\), \(v\in [-r, 0]\). Here \(A\) is a maximal monotone operator on \(\mathbb{R}^ n\), \(F\) is a set-valued map, \(\text{ext }F\) denotes the set of extreme points of \(F\) and \(x_ t(v)= x(t+ v)\). It is proved that solutions of (2) do exist provided \(F\) is measurable in \(t\) and continuous in \(x\) and, if \(F\) is Lipschitz continuous they are dense in the solution set of (1).

Reviewer: V.Křivan (České Budějovice)

MSC:

34A60

Ordinary differential inclusions

Keywords:

relaxation theorem; maximal monotone differential inclusion with memory