Variational principles using a non-symmetric non-triangular distance

TL;DR

This work extends the Ekeland and Borwein-Preiss variational principles to a broad class of distance functions $d$ that need not be symmetric or satisfy the triangle inequality. The authors base their results on sequential convergence properties of $d$ and a Cantor-type intersection framework, rather than classical topology, to establish two principal variational principles and their weak forms. They also demonstrate two applications: a Caristi fixed point theorem for set-valued maps and an existence theorem for equilibrium problems, both formulated within the same non-symmetric, non-triangular distance setting. By replacing metric assumptions with sequential continuity and completeness concepts tied to $d$, the paper broadens variational analysis to non-metric contexts and expands tools for fixed-point theory and equilibrium analysis.

Abstract

We consider Borwein-Preiss and Ekeland variational principles using distance functions that neither is symmetric nor enjoy the triangular inequality. All the given results rely exclusively on the convergence and continuity behaviors induced synthetically by the distance function itself without any topological implications. At the end of the paper, we also present two applications; the Caristi fixed point theorem and an existence theorem for equilibrium problems.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Lemma 2.7
• proof
• Theorem 3.1: Borwein-Preiss variational principle