×

Functional-differential inclusions in Banach spaces with nonconvex right- hand side. (English) Zbl 0698.34067

The Cauchy problem for the functional-differential inclusion (*) \(x(t)\in F(t,x_ t)\), x(.)\(\in M\), \(F: [0,b]\times C([-r,0],X)\to C([-r,b],X),\) \(M\subseteq C([-r,0],X)\) compact, X separable Banach space, is investigated. It is shown, that under several assumptions on the orientor field F (concerning upper bounds, but without convexity assumptions) a weak solution of (*) exists.
Further, a relaxation theorem is proved, where it is shown that the closure of the solution set of (*) \((M=\{h\},X\) no longer separable) coincides with the solution set of the convexified problem. The stated existence theorem generalizes results of Fryszkowski for \(X={\mathbb{R}}^ n\).
Reviewer: L.Brüll

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34A60 Ordinary differential inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
PDF BibTeX XML Cite