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Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions
- 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