A (CF)-mapping of integral functional of locally lipschitz functions

Abstract

We find a (CF)-mapping of the integral functional of locally Lipschitz functions f _t parametrized by $t \in T$. In the process of obtaining a (CF)-mapping, the hypothesis of upper semicontinuity of the set-valued map $t \mapsto C_{f_t} (x)$ is needed, where $C_{f_{t}} (x)$ denotes a convexificator of f _t at x. As a corollary of our result, we get (CF)-mappings which are obtained by Clarke subdifferentials and Michel–Penot subdifferentials, respectively. Finally, the examples specifically deriving a convexificator of the integral functional are provided.