
Journal of Convex Analysis 15 (2008), No. 3, 473484 Copyright Heldermann Verlag 2008 A Lower Semicontinuous Regularization for SetValued Mappings and its Applications Mohamed Ait Mansour Université Cadi Ayyad, Faculté PolyDisciplinaire, Route Sidi Bouzid, 4600 Safi, Morocco maitmansour@hotmail.com Marius Durea Al. I. Cuza University, Faculty of Mathematics, Bd. Carol I, nr. 11, 700506  Iasi, Romania durea@uaic.ro Michel Théra LACO, Université de Limoges, 123 Avenue A. Thomas, 87060 Limoges, France michel.thera@unilim.fr [Abstractpdf] A basic fact in real analysis is that every realvalued 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 setvalued mappings. In this note, we first provide some counterexamples. We try further to define a kind of lower semicontinuous regularization for a given setvalued mapping and we point out some general applications. Keywords: Setvalued mappings, lower semicontinuity, regularization, approximate selections, fixed points, differential inclusions, variational inequalities. MSC: 47A15; 46A32, 47D20 [ Fulltextpdf (148 KB)] for subscribers only. 