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Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions

Authors :
P.S. Kenderov
M. Ivanov
Julian P. Revalski
Source :
Set-Valued and Variational Analysis. 30:559-571
Publication Year :
2021
Publisher :
Springer Science and Business Media LLC, 2021.

Abstract

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game); The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X). In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset; the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X). If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).

Details

ISSN :
18770541 and 18770533
Volume :
30
Database :
OpenAIRE
Journal :
Set-Valued and Variational Analysis
Accession number :
edsair.doi...........087d7cdc0c579dab24b7be2bc1c6c53e