# Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

## TL;DR

This paper establishes the equivalence of Ekeland, Takahashi, and Caristi principles in sequentially right K-complete quasi-pseudometric spaces, linking these variational principles to the space's completeness using Picard sequences.

## Contribution

It introduces a unified approach to these principles in asymmetric pseudometric spaces and proves their equivalence to the space's completeness.

## Key findings

- Proves the equivalence of the principles in quasi-pseudometric spaces.
- Shows the principles are equivalent to the space's completeness.
- Uses Picard sequences for set-valued mappings to unify the principles.

## Abstract

We prove versions of Ekeland, Takahashi and Caristi principles in sequentially right $K$-complete quasi-pseudometric spaces (meaning asymmetric pseudometric spaces), the equivalence between these principles, as well as their equivalence to the completeness of the underlying quasi-pseudometric space. The key tools are Picard sequences for some special set-valued mappings corresponding to a function $\varphi$ on a quasi-pseudometric space, allowing a unitary treatment of all these principles.