An induction theorem and Ekeland's variational principle in partial metric spaces with applications

Abstract

We prove an induction theorem and versions of the Ekeland variational principle for functions on partial metric spaces. As the distance is a partial metric, these versions have additional terms in comparison with the original principle. We also relax the usual lower semincontinuity assumption. Selected applications are provided: in conditions for the existence of solutions of noncompact nonconvex minimization problems, equilibrium problems, minimax equalities, in a particular model of minimization with sharp solutions together with an explicit expression of the sharpness, and in fixed-point studies.

Keywords:

• Partial metric spaces
• induction theorem
• the Ekeland variational principle
• minimizers and sharp minimizers
• fixed-point theorems

AMS Classifications:

• 90K26
• 49J53
• 90B48

Acknowledgments

The paper is dedicated to Professor Boris S. Mordukhovich in honour of his 70th birthday. The authors are grateful to the anonymous referees for their valuable remarks and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This notion was first applied to investigate EVP in [32], aimed to relax the usual lower semicontinuity assumption.

Additional information

Funding

This work was supported by Vietnam National University-Hochiminh City [grant number B2018-28-02].