Equivalents of Ordered Fixed Point Theorems of Kirk, Caristi, Nadler, Banach, and others

Abstract

Recently, we improved our long-standing Metatheorem in Fixed Point Theory.
 In this paper, as its applications, some well-known order theoretic fixed point theo-
 rems are equivalently formulated to existence theorems on maximal elements, com-
 mon fixed points, common stationary points, and others. Such theorems are the
 ones due to Banach, Nadler, Browder-Göhde-Kirk, Caristi-Kirk, Caristi, Brøndsted,
 and possibly many others.

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].…”

“…(h) For each α ∈ (0, 1), there exists a partition of X, X = γ∈Γ Xγ, and complete metrics dγ on Xγ such that all the sets Xγ are nonempty; T-invariant and pairwise disjoint; and dγ(T x, T y) ≤ αdγ(x, y) f or all x, y ∈ X. (b) Zermelo's theorem is improved by Theorem C, Theorem D(iii), Section 10(I) in [10] and its equivalents there. Note that its conclusion should be as above (a).…”

Applications of Several Minimum Principles

“…In order to get some equivalents of the well-known central result of I. Ekeland [5,6] on the variational principle for approximate solutions of minimization problems, we obtained a Metatheorem in [10][11][12] and its applications in 1983-2000. Later in 2022 we found an extended version of the Metatheorem [13][14][15]17]. Now the following is the 2023 version in [22].…”

“…In 2022, we found the extended version of Metatheorem in [12][13][14][15]17] and applied its several particular forms to various results in ordered xed point theory, nonlinear analysis, and other elds. In fact, we applied Metatheorem to Zorn's lemma, Banach contraction principle, Nadler's xed point theorem, Ekeland's variational principle, Brézis-Browder principle, Caristi's xed point theorem, Takahashi's nonconvex minimization theorem, some others and their variants, generalizations or equivalent formulations.…”

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

“…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].…”