Equivalents of Some Ordered Fixed Point Theorems

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.

Equivalents of Various Principles of Zermelo, Zorn, Ekeland, Caristi and Others

“…Note that this theorem implies several of the Caristi type and the Zermelo type theorems in this article; see also [11][12][13][14][15][16][17][18][19][20][21][22].…”

“…In Section 6, we derive Maximal (resp. Minimal) Element Principle in [18,19] from our new 2023 Metatheorem and a similar theorem from Metatheorem * in [21]. We also improve Jachymski's 2003 Theorem [25] on converses to theorems of Zsrmelo and Caristi.…”

“…All of the key results in the above can be unified by applying our Metatheorem and its variants. In fact, from our new 2023 Metatheorem in [21], we deduced the following prototype of Extreme Element Principles for non-empty valued multimaps: Theorem 13. Let (X, ) be a preordered set and A be a nonempty subset of X.…”

“…In the present article, we introduce our new 2023 Metatheorem in [21,22], its short history, its applications to various theorems of Zermelo, Zorn, Ekeland, Caristi, Takahashi, and related results. In fact, this is a historical supplement of [15] and organized as follows.…”

Variants of the New Caristi Theorem

“…The most recent version of them is a modification of the 2023 Metatheorem in [28]. This article is based on the modification given in [32]. It is applied to the traditional order theoretic results and, consequently, to the so-called Ordered Fixed Point Theory in [28].…”