Smooth Selections of Convex-Valued Multifunctions

We establish a class of multifunctions having smooth (

) selections and formulate assumptions on a multifunction

F F

under which for any continuous selection

f f

of

F F

there is a sequence of smooth selections of

F F

converging uniformly to

f f

. Moreover, we obtain a Castaing type representation of multifunctions by a sequence of smooth selections, i.e. we construct a sequence

\{f_k\} { f k }

of smooth selections of

F F

satisfying the condition

F(x)=\overline{\cup_{k\geq 1} \ f_k(x)} F ( x ) = ∪ k ≥ 1   f k ( x ) ‾

for all

x\in X x ∈ X

.