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

Abstract

We prove versions of Ekeland, Takahashi and Caristi principles in pre-ordered 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. Cobzaş, Topology Appl. 265 (2019), 106831, 22, for quasi-metric spaces. The key tools are Picard sequences for some special set-valued mappings on a pre-ordered quasi-metric space X, defined in terms of the preorder and of a function φ on X.