zbMATH — the first resource for mathematics

Continuous selections for a class of non-convex multivalued maps. (English) Zbl 0534.28003
Let S and T be compact spaces and let $$\mu_ 0$$ be a nonnegative, regular, Borel measure on T. Denote by $$L^ 1(T,Z)$$ the Banach space of $$\mu_ 0$$-integrable functions from T into the separable Banach space Z.
A set $$K\subset L^ 1(T,Z)$$ is called decomposable, if for all u,$$v\in K$$ and A $$\mu_ 0$$-measurable, $$u\cdot \chi_ A+v\cdot \chi_{T\backslash A}\in K$$ holds. The main result states, that lower semicontinuity of the map K defined on S with closed and decomposable values in $$L^ 1(T,Z)$$ implies the existence of a continuous function $$k:S\to L^ 1(T,Z)$$ (called continuous selection), such that $$k(s)\in K(s)$$ for all s. Moreover, there exists a countable family of continuous functions $$k_ n:S\to L^ 1(T,Z)$$ such that for all $$s\in S K(s)=closure\{k_ n(s):n=1,2,...\}$$ holds.
This theorem generalized the theorem of Antosiewicz and Cellina, who proved the existence of a continuous selection for the map in the form $$K(s)=\{u\in L^ 1([a,b],R^ m):u(t)\in F(t,s(t))\quad almost\quad everywhere\quad in\quad [a,b]\}.$$ The main result can be generalized in the case, when the map K has decomposable values in Orlicz space.

MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 54C60 Set-valued maps in general topology 28B05 Vector-valued set functions, measures and integrals 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: