Minimal element theorems revisited

TL;DR

This paper revisits and strengthens various variational principles and minimal element theorems, clarifying assumptions and introducing new results involving set-valued maps and set relations.

Contribution

It provides stronger versions of existing theorems, clarifies assumptions, and introduces new results with set-valued maps and set relations.

Findings

Stronger versions of variational principles established

Clarification of assumptions related to metric and boundedness

New results involving set-valued maps and set relations

Abstract

Starting with the Brezis-Browder principle, we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are discussed and clarified, i.e., assumptions to the metric structure of the underlying space and boundedness assumptions. New results involving set-valued maps and the increasingly popular set relations are obtained along the way.