Far Eastern Mathematical Journal

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A criterion for the approximation of a semicontinuous functional by Lipschitz functional

Prudnikov V. Ya., Podgaev A. G.

2022, issue 1, P. 84-90 DOI: https://doi.org/10.47910/FEMJ202208

Abstract

It is proved in [1, 2] that a functional semi-continuous from below and bounded from below in the metric space X is represented as the limit of a non-decreasing family of Lipschitz functionals. In the lemma from [3], a sufficient condition for such a representation is given for a function semi-continuous from below with respect to one of the variables in a finite-dimensional space. This paper contains a criterion for approximation of a semi-continuous functional from below in a metric space by Lipschitz functionals.

Keywords: lower semicontinuous functional, metric space

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References

[1] F. Khausdorf., Teoriia mnozhestv, Ob"edinen. nauchn.-tekhn. izd-vo NKTP SSSR, Gl. red. tekhn.-teoret. lit., M.-L., 1937. [2] V. V. Gorokhovik, “O predstavlenii polunepreryvnykh sverkhu funktsii, opredelennykh na beskonechnomernykh normirovannykh prostranstvakh, v vide nizhnikh ogibaiushchikh semeistv vypuklykh funktsii”, Tr. IMM UrO RAN., 23:1 (2017), 88–102. [3] V. Ia. Prudnikov, “Neravenstvo Iensena v ideal'nom prostranstve”, Sib. zhurn. industr. matem., 10:2 (2007), 119–127. [4] Zh. P. Oben., Nelineinyi analiz i ego ekonomicheskie prilozheniia, Mir, Moskva, 1988.

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