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