Smooth Selections of Convex-Valued Multifunctions

We establish a class of multifunctions having smooth ($C^\infty$) selections and formulate assumptions on a multifunction $F$ under which for any continuous selection $f$ of $F$ there is a sequence of smooth selections of $F$ converging uniformly to $f$. Moreover, we obtain a Castaing type representation of multifunctions by a sequence of smooth selections, i.e. we construct a sequence $\{f_k\}$ of smooth selections of $F$ satisfying the condition $F(x)=\overline{\cup_{k\geq 1} \ f_k(x)}$ for all $x\in X$.