Topological properties of the solution set of integrodifferential inclusions. (English) Zbl 0836.34019

Avgerinos, Evgenios P.; Papageorgiou, Nikolaos S.

Commentat. Math. Univ. Carol. 36, No. 3, 429-442 (1995).

The following integrodifferential inclusion is considered: \(x' \in F(t,x, (Vx) (t), \lambda)\), \(x(0) = x_0 (\lambda)\). Here \(Vx(t) = \int^t_0 K(t,s) g(s,x(s)) ds\) is the nonlinear Volterra integral operator, \(\lambda\) is a parameter from a complete metric space. It is proved that the solution set of the above differential inclusion with non-convex values of \(F\) is a retract of the Sobolev space \(W^{1,1} (T, \mathbb{R}^n)\). If \(F\) has convex values, the solution set is a retract of \(C(T, \mathbb{R}^n)\). Using these results the existence of a continuous selector of the solution map is proved.

Reviewer: V.Křivan (České Budějovice)

MSC:

34A60

Ordinary differential inclusions

Keywords:

integrodifferential inclusion; retract; continuous selector