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


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