Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 15 (2008), No. 3, 473--484Copyright 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 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 [Abstract-pdf] 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 [ Fulltext-pdf  (148  KB)] for subscribers only.