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Journal of Convex Analysis 15 (2008), No. 3, 473--484
Copyright Heldermann Verlag 2008

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

Mohamed Ait Mansour
Université Cadi Ayyad, Faculté Poly-Disciplinaire, Route Sidi Bouzid, 4600 Safi, Morocco

Marius Durea
Al. I. Cuza University, Faculty of Mathematics, Bd. Carol I, nr. 11, 700506 - Iasi, Romania

Michel Théra
LACO, Université de Limoges, 123 Avenue A. Thomas, 87060 Limoges, France


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.

Keywords: Set-valued mappings, lower semicontinuity, regularization, approximate selections, fixed points, differential inclusions, variational inequalities.

MSC: 47A15; 46A32, 47D20

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