A Lower Semicontinuous Regularization for Set-Valued Mappings and its Applications

A basic fact in real analysis is that every real-valued function $f$ admits a lower semicontinuous regularization $\underline{f}$, defined by means of the lower limit of $f$: $$\begin{aligned}\underline{f}\left( x\right):=\;\displaystyle\liminf_{y\rightarrow x}f\left( y\right).\end{aligned}$$ This fact breaks down for set-valued mappings. In this note, we first provide some counterexamples. We try further to define a kind of lower semicontinuous regularization for a given set-valued mapping and we point out some general applications.