Geodesic
A Lower Semicontinuous Regularization for Set-Valued Mappings and its Applications
Journal of convex analysis, Tome 15 (2008) no. 3, pp. 473-484.

Voir la notice de l'article provenant de la source Heldermann Verlag

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{align*} \underline{f}\left( x\right) :=\;\displaystyle\liminf_{y\rightarrow x}f\left( y\right). \end{align*} 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.
Classification : 47A15, 46A32, 47D20
Mots-clés : Set-valued mappings, lower semicontinuity, regularization, approximate selections, fixed points, differential inclusions, variational inequalities 
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     title = {A {Lower} {Semicontinuous} {Regularization} for {Set-Valued} {Mappings} and its {Applications}},
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M. Ait Mansour; M. Durea; M. Théra. A Lower Semicontinuous Regularization for Set-Valued Mappings and its Applications. Journal of convex analysis, Tome 15 (2008) no. 3, pp. 473-484. http://geodesic.mathdoc.fr/item/JCA_2008_15_3_JCA_2008_15_3_a2/