Equivalents of various maximum principles

Abstract

Certain maximum principles can be reformulated to various types of fixed point theorems and conversely, based on Metatheorem due to ourselves. Such principles are Zorn's lemma, Banach contraction principle, Nadler's fixed point theorem, Brézis-Browder principle, Caristi's fixed point theorem, Ekeland's variational principle, Takahashi's nonconvex minimization theorem, some others, and their variants, generalizations, or equivalent formulations. Consequently, we have many new theorems equivalent to known results on fixed point, common fixed point, stationary point, common stationary point, and others. We show that such points are all maximal elements of certain ordered sets. Further, we introduce our earlier related works as a history of our Metatheorem.