Ekeland variational principle for quasimetric spaces

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Abstract

 In this paper, we study real-valued functions defined on quasimetric spaces. A generalization of Ekeland’s variational principle and a similar statement from the article [S. Cobzas, “Completeness in quasi-metric spaces and Ekeland Variational Principle”, Topology and its Applications, vol. 158, no. 8, pp. 1073–1084, 2011] is obtained for them. The modification of the variational principle given here is applicable, in particular, to a wide class of functions unbounded from below. The result obtained is applied to the study the minima of functions defined on quasimetric spaces. A Caristi-type condition is formulated for conjugate-complete quasimetric spaces. It is shown that the proposed Caristi-type condition is a sufficient condition for the existence of a minimum for lower semicontinuous functions acting in conjugate-complete quasimetric spaces.

About the authors

Richik Sengupta

Skolkovo Institute of Science and Technology; Derzhavin Tambov State University

Author for correspondence.
Email: r.sengupta@skoltech.ru
ORCID iD: 0000-0001-9916-8177

Candidate of Physics and Mathematics, Researcher

Russian Federation, 30 Bolshoy Boulevard, Territory of the Skolkovo Innovation Center, Moscow 121205, Russian Federation; 33 International St., Tambov 392036, Russian Federation

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