Eckland and Bishop-Phelps variational principles in partially ordered spaces

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Abstract

In this paper, an assertion about the minimum of the graph of a mapping acting in partially ordered spaces is obtained. The proof of this statement uses the theorem on the minimum of a mapping in a partially ordered space from [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Caristi-like condition and the existence of minima of mappings in partially ordered spaces // Journal of Optimization Theory and Applications. 2018. V. 180. Iss. 1, 48-61]. It is also shown that this statement is an analogue of the Eckland and Bishop-Phelps variational principles which are effective tools for studying extremal problems for functionals defined on metric spaces. Namely, the statement obtained in this paper and applied to a partially ordered space created from a metric space by introducing analogs of the Bishop-Phelps order relation, is equivalent to the classical Eckland and Bishop-Phelps variational principles.

About the authors

Zukhra T. Zhukovskaya

V.A. Trapeznikov Institute of Control Sciences of RAS

Email: zyxra2@yandex.ru
Candidate of Physics and Mathematics, Leading Researcher 65 Profsoyuznaya St., Moscow 117997, Russian Federation

Tatiana V. Zhukovskaia

Tambov State Technical University

Email: t_zhukovskaia@mail.ru
Candidate of Physics and Mathematics, Associate Professor of the Higher Mathematics Department 106 Sovetskaya St., Tambov 392000, Russian Federation

Olga V. Filippova

Derzhavin Tambov State University

Email: philippova.olga@rambler.ru
Candidate of Physics and Mathematics, Associate Professor of the Functional Analysis Department 33 Internatsionalnaya St., Tambov 392000, Russian Federation

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