Abstract

A basic fact in real analysis is that every real-valued function ff admits a lower semicontinuous regularization f‾\underline{f}, defined by means of the lower limit of ff: f‾(x):=  lim inf⁡y→xf(y).\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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Université de Limoges
and: XLIM, UMR 6172, 123 Avenue A. Thomas, 87060 Limoges, France

michel.thera@unilim.fr

Keywords

  • Set-valued mappings
  • lower semicontinuity
  • regularization
  • approximate selections
  • fixed points
  • differential inclusions
  • variational inequalities

Mathematics Subject Classification

47A15; 46A32, 47D20

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Suggested citation

M. Ait Mansour, M. Durea, M. Théra. “A Lower Semicontinuous Regularization for Set-Valued Mappings and its Applications.” Journal of Convex Analysis 15 (2008), No. 3, 473–484. https://doi.org/10.68381/jca15034

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.