Equivalents of Some Ordered Fixed Point Theorems
Sehie Park *
The National Academy of Sciences, Seoul 06579, Korea and Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea.
*Author to whom correspondence should be addressed.
Abstract
Some ordered fixed point theorems on metric spaces are equivalent to completeness and existences of maximal (or minimal) elements, common fixed points, common stationary points, etc. Some known or new theorems related to the Caristi fixed point theorem can be equivalently formulated. Consequently, dual versions of the Ekeland principle, the Caristi theorem, theorems of Bae-Park, Takahashi, Chen-Cho-Yang, Jachymski, Cobzas, and others are substantially improved and strengthened.
Keywords: The 2023 metatheorem, Brndsted-Jachymski principle, preorder, metric space, anti-progressive maps, fixed point, stationary point, maximal element