Applications of Several Minimum Principles

Abstract

In our previous works, a Metatheorem in ordered fixed point theory showed that certain maximum principles
 can be reformulated to various types of fixed point theorems for progressive maps and conversely. Therefore,
 there should be the dual principles related to minimality, anti-progressive maps, and others. In the present
 article, we derive several minimum principles particular to Metatheorem and their applications. One of
 such applications is the Brøndsted-Jachymski Principle. We show that known examples due to Zorn (1935),
 Kasahara (1976), Brézis-Browder (1976), Taskovi¢ (1989), Zhong (1997), Khamsi (2009), Cobzas (2011) and
 others can be improved and strengthened by our new minimum principles.

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.…”

“…Later in 2022, we came back to our Metatheorem after 22 years have passed and obtained its extended versions in [11][12][13][14][15] with a large number of their consequences [11][12][13][14][15][16][17][18][19][20]. These are applied to the traditional order theoretic results and, consequently, there have appeared the so-called Ordered Fixed Point Theory [15].…”

Equivalents of Some Ordered Fixed Point Theorems

“…Our Metatheorem was originated from the Ekeland Principle which has equivalent forms like the Caristi fixed point theorem, Takahashi's minimization theorem, and many others. Our recent applications of Metatheorem to those theorems were given in [7,8,10,13,14,15,20,21,22].…”

Variants of the New Caristi Theorem

“…A maximal or minimal element will be called an extreme element. From our new 2023 Metatheorem in [32], we deduce the following prototype of Extreme Element Principles as in [29] for multimaps having nonempty values: Theorem A. Let (X, ⪯) be a preordered set and A be a nonempty subset of X.…”