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¸s, Topology Appl. 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 φ on X.