Moduli of Continuity of Selections from Non-convex Maps

Summary: The paper offers a technique for the construction of selections in the following problems. 1. Given a Holder continuous set-valued map $F(t,x)$, with compact values in $Rsp n$, find a selection $f(t,x)$ such that the mapping $uto f(t,u(t))$ is continuous from the space of Lipschitz functions $u(·)$ on $[0,T]$ to $Lsb 1([0,t])$. 2. Given a differential inclusion $dot xin F(t,x)$, with Lipschitz, compact-valued right-hand side, find for every initial condition $x(0)$ a solution $x(t,x(0))$ which depends continuously on $x(0)$. In both cases the construction given in the paper yields a computation of estimates for the moduli of continuity of the resulting maps. The two problems arise in the study of differential inclusions, but have, in particular, in view of the estimates, far-reaching applications.