Abstract

We investigate two types of semicontinuity for set-valued maps, Painlevé-Kuratowski semicontinuity and Cesari's property (Q). It is shown that, in the context of convex-valued maps, the concepts related to Cesari's property (Q) have better properties than the concepts in the sense of Painlevé-Kuratowski. In particular we give a characterization of Cesari's property (Q) in terms of upper semicontinuity of a family of scalar functions σf( ⋅ )(y∗) ⁣:X→R‾\sigma_{f(\,\cdot\,)}(y^*) \colon X \to \overline\R, where σf(x) ⁣:Y∗→R‾\sigma_{f(x)} \colon Y^*\to \overline\R is the support function of the set f(x)f(x). We compare both types of semicontinuity and show their coincidence in special cases.

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Keywords

  • Semi-continuity
  • set-valued maps
  • property (Q)

Mathematics Subject Classification

47H04,58C07

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Suggested citation

A. Löhne. “On Semicontinuity of Convex-Valued Multifunctions and Cesari's Property (Q).” Journal of Convex Analysis 15 (2008), No. 4, 803–818. https://doi.org/10.68381/jca15052

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