Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces

TL;DR

This paper extends classical variational principles like Ekeland, Takahashi, and Caristi to preordered quasi-metric spaces, establishing their equivalence and linking them to completeness properties of these spaces.

Contribution

It generalizes existing results by proving these principles in preordered quasi-metric spaces and demonstrating their equivalence to each other and to completeness.

Findings

Proved versions of Ekeland, Takahashi, and Caristi principles in preordered quasi-metric spaces.

Established the equivalence among these principles.

Linked these principles to completeness results in the underlying spaces.

Abstract

We prove versions of Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces, the equivalence between these principles, as well as their equivalence to some completeness results for the underlying quasi-metric space. These extend the results proved in S.~Cobza\c{s}, Topology Appl. \textbf{265} (2019), 106831, 22, for quasi-metric spaces. The key tools are Picard sequences for some special set-valued mappings on a preordered quasi-metric space $X$, defined in terms of the preorder and of a function $\varphi$ on $X$. Key words: preordered quasi-metric space; completeness in quasi-metric spaces; variational principles; Ekeland variational principle; Takahashi minimization principle; fixed point; Caristi fixed point theorem.