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Equivalents of maximum principles for several spaces

Equivalents of maximum principles for several spaces Topol. Algebra Appl. 2022; 10:68–76 Research Article Open Access Sehie Park* Equivalents of maximum principles for several spaces https://doi.org/10.1515/taa-2022-0113 Received 20 May, 2022; accepted 10 June, 2022 Abstract: According to our long-standing Metatheorem, certain maximum theorems can be equivalently re- formulated to various types of xed point theorems, and conversely. As examples of such theorems, in this paper, we list Zermelo’s xed point theorem, Brøndsted’s principle, Fang’s F-type theorem, related theorems for locally convex spaces and quasi-uniform spaces. Further we review some few works concerned with our Metatheorem. Keywords: Ekeland variational principle, pre-order, Zermelo, quasi-metric space, xed point, maximal ele- ment, locally convex space, quasi-uniform space MSC: 06A75, 47H10, 54E35, 54E50, 54H25, 58E30, 65K10 1 Introduction Recently we have obtained several equivalent formulations of Zorn’s lemma, the Banach contraction prin- ciple, Caristi’s xed point theorem, Ekeland’s variational principle, Takahashi’s nonconvex minimization theorem and some others in [1]. These are forceful tools in nonlinear analysis, control theory, economic the- ory, and global analysis. These theorems were already extended by a large number of authors. However, we are able to obtain their equivalent formulations. In 1985-2000, we had published several articles mainly related to the Ekeland variational principle for approximate solutions of minimization problems and its equivalent formulations with some applications; see [2-6]. From the beginning of such study, we made a Metatheorem for some equivalent statements on max- imality, xed points, stationary points, common xed points, common stationary points, and their locations. We applied the Metatheorem for various occasions in [1-6]. In our previous work [1], we obtained an extended version of Metatheorem and applied it to Zorn’s lemma, Banach contraction principle, Nadler’s xed point theorem, Brézis-Browder principle, Caristi’s xed point theorem, Ekeland’s variational principle, Takahashi’s nonconvex minimization theorem, some others and their variants, generalizations or equivalent formulations. In the present article, as a continuation of [1], we apply Metatheorem to Zermelo’s xed point theorem [7], Brøndsted’s principle [8], Fang’s F-type theorem [9], related theorems for locally convex spaces [10] and quasi-uniform spaces [11]. Further we review some works concerned with our Metatheorem. This article is organized as follows: In Section 2, we introduce our Metatheorem with its proof for com- pleteness. It is applied to pre-ordered sets in Section 3 as usual, and subsequent Sections 4-8 are devoted to applications of Metatheorem to various theorems mentioned above. In Section 9, we give some remarks on recent works of other authors in [11-14] related to our Metatheorem. Finally, Section 10 deals with some comments and conclusion. *Corresponding Author: Sehie Park: The National Academy of Sciences, Republic of Korea; Seoul 06579 and Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea, E-mail: park35@snu.ac.kr; sehiepark@gmail.com; parksehie.com Open Access. © 2022 Sehie Park, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. Equivalents of maximum principles for several spaces Ë 69 2 Metatheorem related to the Ekeland principle The well-known central result of I. Ekeland [15,16] on the variational principle for approximate solutions of minimization problems runs as follows: Theorem. (Ekeland [15]) Let V be a complete metric space, and F : V ! R [ f+∞g a l.s.c. function, ̸ +∞, bounded from below. Let ε > 0 be given, and a point u 2 V such that F(u) 5 inf F + ε. Then for every λ > 0, −1 there exists a point v 2 B(u, λ) such that F(v) 5 F(u) and F(w) > F(v) − ελ d(v, w) for any w 2 V , w ≠ v. When λ = 1, this is called the ε-variational principle. In order to obtain some equivalents of this principle, we obtained a Metatheorem in [2-6]. Later we found some additional statements and, consequently, we obtain a new extended version of the Metatheorem [1] in 2022. Now we add its proof for the completeness. Metatheorem. Let X be a set, A its nonempty subset, and G(x, y) a sentence formula for x, y 2 X. Then the following propositions are equivalent: (i) There exists an element v 2 A such that G(v, w) for any w 2 Xnfvg. (ii) If T : A ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying ¬G(x, y), then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : A ! X is a map such that for any x 2 A with x ≠ f (x), there exists a y 2 Xnfxg satisfying¬G(x, y), then f has a xed element v 2 A, that is, v = f(v). (iv) If f : A ! X is a map such that ¬G(x, f(x)) for any x 2 A, then f has a xed element v 2 A, that is, v = f(v). (v) If T : A ( X is a multimap such that ¬G(x, y) holds for any x 2 A and any y 2 T(x)nfxg, then T has a stationary element v 2 A, that is,fvg = T(v). (vi) If F is a family of maps f : A ! X satisfying¬G(x, f(x)) for all x 2 A with x ≠ f(x), then F has a common xed element v 2 A, that is, v = f (v) for all f 2 F. (vii) If F is a family of multimaps T : A ( X for i 2 I with an index set I such that ¬G(x, y) holds for any x 2 A and any y 2 T (x)nfxg, then F has a common stationary element v 2 A, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg satisfying¬G(x, z), then there exists a v 2 A \ Y. Here, maps and multimaps have nonempty values and ¬ denotes the negation. Proof. (i)=)(ii): Suppose v 2 / T(v). Then there exists a y 2 Xnfvg satisfying ¬G(v, y). This is a contradiction. (ii)=)(iii): Clear. (iii)=)(iv): Clear. (iv)=)(v): Suppose T has no stationary element, that is, T(x)nfxg ≠ ; for any x 2 A. By Axiom of Choice, we have a choice function f on fT(x)nfxg : x 2 Ag. Then f has no xed element by its denition. However, for any x 2 A, we have ¬G(x, f(x)). Therefore, by (iv), f has a xed element, a contradiction. (v)=)(vi): Dene a multimap T : A ( X by T(x) := ff(x) : f 2 Fg ≠ ; for all x 2 A. Since ¬G(x, f (x)) for any x 2 A and any f 2 F, by (v), T has a stationary element v 2 A, which is a common xed element of F. (vi)=)(i): Suppose that for any x 2 A, there exists a y 2 Xnfxg satisfying ¬G(x, y). Choose f (x) to be one of such y. Then f : A ! X has no xed element by its denition. However, ¬G(x, f (x)) for all x 2 A. Let F = ffg. By (vi), f has a xed element, a contradiction. (i)+(v) =) (vii): By (i), there exists a v 2 A such that G(v, w) for all w 2 Xnfvg. For each i 2 I, by (v), we have a v 2 A such that fv g = T (v ). Suppose v ≠ v . Then G(v, v ) holds by (i) and ¬G(v, v ) holds by i i i i i i i assumption on (vii). This is a contradiction. Therefore v = v for all i 2 I. i 70 Ë Sehie Park (vii)=)(v): Clear. (i)=)(viii): By (i), there exists a v 2 A such that G(v, w) for all w ≠ v. Then by the hypothesis, we have v 2 Y. Therefore, v 2 A \ Y. (viii)=)(i): For all x 2 A, let A(x) := fy 2 X : x ≠ y, ¬G(x, y)g. Choose Y = fx 2 X : A(x) = ;g. If x 2 / Y, then there exists a z 2 A(x). Hence the hypothesis of (viii) is satised. Therefore, by (viii), there exists a v 2 A \ Y. Hence A(v) = ;; that is, G(v, w) for all w ≠ v. Hence (i) holds. This completes our proof. In this proof, we notice that the element v 2 A is the same throughout (i)–(viii) and that (viii) gives some information on the location of v 2 A. 3 For pre-ordered sets Let (X,) be a pre-ordered set; that is, X is a nonempty set, is reexive and transitive. For each x 2 X, we denote S(x) = fy 2 X : x  yg and G(x, y) means x  y. We give a rst application of Metatheorem to pre-ordered sets as follows: Theorem 3.1. Let (X,) be a pre-ordered set, x 2 X and A = S(x ). Then the following eight propositions are 0 0 equivalent: (i) There exists a maximal element v 2 A such that v  w for any w 2 Xnfvg. (ii) If T : A ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying x  y, then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : A ! X is a map such that for any x 2 A with x ≠ f(x), there exists a y 2 Xnfxg satisfying x  y, then f has a xed element v 2 A, that is, v = f (v). (iv) If f : A ! X is a map such that x  f(x) for any x 2 A, then f has a xed element v 2 A, that is, v = f(v). (v) If T : A ( X is a multimap such that x  y holds for any x 2 A and any y 2 T(x)nfxg, then T has a stationary element v 2 A, that is,fvg = T(v). (vi) If F is a family of maps f : A ! X satisfying x  f (x) for all x 2 A with x ≠ f (x), then F has a common xed element v 2 A, that is, v = f(v) for all f 2 F. (vii) If F is a family of multimaps T : A ( X for i 2 I with an index set I such that x  y holds for any x 2 A and any y 2 T (x)nfxg, then F has a common stationary element v 2 A, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 A \ Y. Proof. In Metatheorem, put A := S(x ) and let G(v, w) be the statement v  w. Then each of (i)–(viii) follows from the corresponding ones in Metatheorem. This completes our proof. Remark 3.2. Note that we claimed that (i)–(viii) are equivalent in Theorem 3.1 and did not say that they are true. For a counter-example, think about the real line with the natural order. From now on, we are going to give examples that they are true based on their original sources. Equivalents of maximum principles for several spaces Ë 71 4 Extending Zermelo’s theorem We apply Theorem 3.1 to Zermelo’s xed point theorem [7] in 1908: Theorem 4.1 ([7]) Let (X,) be a partially ordered set in which each nonempty well-ordered subset has a least upper bound. Then every function f : X ! X satisfying x  f(x) for all x 2 X has a xed point. The Zermelo xed point theorem is also known as the Bourbaki xed point theorem or the Bourbaki- Kneser xed point theorem. It implies the Caristi xed point theorem, the Bernstein-Cantor-Schröder theorem, the Ekeland variational principle, the Takahashi minimization theorem, and others. Moreover, under the Axiom of Choice, it implies Zorn’s Lemma. Jachymski [15] noted: “Under the Axiom of Choice, the assumption of Theorem 4.1 can be weakened to “each nonempty well-ordered subset has an upper bound.” This is Kneser’s xed point theorem [18], which turns out to be equivalent to the Axiom of Choice as shown by Abian [19].” However we have the following by Zorn’s Lemma and Theorem 3.1: Theorem 4.2. Let (X,) be a partially ordered set in which each nonempty well-ordered subset has an upper bound. Let x 2 X and A = S(x ) = fy 2 X : x  yg. 0 0 0 Then the following eight equivalent conditions hold: (i) There exists a maximal element v 2 A, that is, v  w for any w 2 Xnfvg. (ii) If T : A ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying x  y, then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : A ! X is a map such that for any x 2 A with x ≠ f(x), there exists a y 2 Xnfxg satisfying x  y, then f has a xed element v 2 A, that is, v = f (v). (iv) If f : A ! X is a map such that x  f(x) for any x 2 A, then f has a xed element v 2 A, that is, v = f (v). (v) If T : A ( X is a multimap such that x  y holds for any x 2 A and any y 2 T(x)nfxg, then T has a stationary element v 2 A, that is,fvg = T(v). (vi) If F is a family of maps f : A ! X satisfying x  fx for all x 2 A with x ≠ f (x), then F has a common xed element v 2 A, that is, v = f(v) for all f 2 F. (vii) If F is a family of multimaps T : A ( X for i 2 I with an index set I such that x  y holds for any x 2 A and any y 2 T (x)nfxg, then F has a common stationary element v 2 A, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 A \ Y. Proof. In Metatheorem, put A := S(x ) and let G(v, w) be the statement v  w. Then (i) is Zorn’s lemma and each of (i)–(viii) follows from the corresponding ones in Theorem 3.1. This completes our proof. Note that (iii) and (iv) extend Zermelo’s Theorem 4.1. 5 Extensions of Brøndsted’s principle In a short note of Brøndsted [8] in 1976 observed that certain xed point theorems may be derived from the- orems on the existence of maximal elements in partially ordered sets. His main point is to show how certain xed point theorems can be deduced from the following simple observation: 72 Ë Sehie Park (A) Let (E,4) be a partially ordered set which admits at least one maximal element. Let f : E ! E be a map such that x 4 f(x) for all x 2 E. Then f admits at least one xed point. Now this can be called the Brøndsted principle. This is motivated by a certain type of such spaces consid- ered by Bishop and Phelps [20]. Brøndsted applied (A) to the Caristi-Kirk xed point theorem and other type of xed point theorems in (B) and (C) in [8]. In this section, by applying our Metatheorem or Theorem 3.1, we show that various extended types of converse of the principle holds: Theorem 5.1. Let (E,4) be a pre-ordered set. Then the eight equivalent conditions in Theorem 3.1 hold. For example: (i) There exists a maximal element v 2 X; that is, v 4̸ w for any w 2 Xnfvg. (ii) If T : X ( X is a multimap such that for any x 2 XnT(x) there exists a y 2 Xnfxg satisfying x 4 y, then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : X ! X is a map such that for any x 2 X with x ≠ f (x), there exists a y 2 Xnfxg satisfying x 4 y, then f has a xed element v 2 X, that is, v = f(v). (iv) If f : X ! X is a map such that x 4 f(x) for any x 2 X, then f has a xed element v 2 X, that is, v = f(v). Note that (i)()(iv) is later called the Brøndsted Principle by ourselves. 6 F-type spaces of Fang In 1996 Fang [9] introduced the concept of F-type topological spaces and established a variational principle and a xed point theorem in the kind of spaces, which extend Ekeland’s variational principle and Caristi’s xed point theorem, respectively. In [9], a characterization of F-type of spaces is given. The usual metric spaces, Hausdor topological vector spaces, and Menger probabilistic metric spaces are all the special cases of F-type topological spaces. Fang established a variational principle and a xed point theorem, and applied them to Menger probabilistic metric spaces. Now we recall the denition of F-type using a characterization of them proved by Fang (see [9, Theorem 2.1]). Denition 6.1. A topological space E is said to be F-type if there exists a familyfd g of semimetrics on E, i i2I I being a directed set, generating the topology on E, and satisfying the following conditions: (F1) d (x, y) = 0 for all i 2 I i x = y for all x, y 2 E; (F2) d (x, y) = d (y, x) for all i 2 I, for all x, y 2 E; i i (F3) d (x, y) ≤ d (x, y) if i < j for all x, y 2 E; i j (F4) for every i 2 I there exists j 2 I, with i < j, such that d (x, y) ≤ d (x, z) + d (z, y) for all x, y, z 2 E. i j j The following is [9, Theorem 3.1]: Theorem 6.2. ([9]) Let (X, T) be a sequentially complete F-type topological space and fd : λ 2 Dg be a generating family of quasi-metrics for T. Let φ : X ! R be a lower semi-continuous function, bounded from below, and k : D ! (0,∞) be a nonincreasing function. We dene a relation “” on X as x  y i d (x, y) ≤ k(λ)(φ(x) − φ(y)) for all λ 2 D. Then (X,) is a partially ordered set and it has a maximal element. From Theorem 6.2 and Metatheorem, we have the following: Theorem 6.3. For (X,) in Theorem 6.2, we have the following equivalent statements hold: Equivalents of maximum principles for several spaces Ë 73 (i) There exists a maximal element v 2 X such that v  w for any w 2 Xnfvg. (ii) If T : X ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying x  y, then T has a xed element v 2 X, that is, v 2 T(v). (iii) If f : X ! X is a map such that for any x 2 X with x ≠ f(x), there exists a y 2 Xnfxg satisfying x  y, then f has a xed element v 2 X, that is, v = f (v). (iv) If f : X ! X is a map such that x  f(x) for any x 2 X, then f has a xed element v 2 X, that is, v = f (v). (v) If T : X ( X is a multimap such that x  y holds for any x 2 X and any y 2 T(x)nfxg, then T has a stationary element v 2 X, that is,fvg = T(v). (vi) If F is a family of maps f : X ! X satisfying x  fx for all x 2 X with x ≠ f(x), then F has a common xed element v 2 X, that is, v = f(v) for all f 2 F. (vii) If F is a family of multimaps T : X ( X for i 2 I with an index set I such that x  y holds for any x 2 X and any y 2 T (x)nfxg, then F has a common stationary element v 2 X, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 XnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 X \ Y = Y. Proof. Note that (i) holds by Theorem 6.2. Then Theorem 6.3 follows from Metatheorem with X = A. Our Theorem 6.3(iii) and (iv) extend [9, Theorem 3.1]. Its equivalent form [9, Theorem 3.1 ] also holds. Fang showed that Caristi’s xed point theorem follows from [9, Theorem 3.1]. 7 For locally convex spaces Cammaroto et al. [10] showed that a Hausdor locally convex topological vector space X is a F-type topological vector space. Therefore Theorem 6.3 holds for such space X. They obtained the following [10, Theorem 3.2]: Theorem 7.1. ([10]) Let E be a complete, Hausdor, locally convex topological vector space and fp g a ba- i i2I sis of continuous seminorms generating the topology on E. Moreover, let φ : E ! R be a lower semicontinu- ous function bounded from below and k : I ! ]0, +∞[ a nonincreasing function respect to the order in I with sup k(i) < +∞. i2I (iii) If f : E ! E is a function such that p (x − f (x)) ≤ k(i)[φ(x) − φ(f (x))], 8i 2 I, 8x 2 E, then f has a xed point in E. This can be slightly improved by Theorem 6.3(iv) since (iii) implies (iv). Hence Theorem 6.3(i)–(viii) also hold for E. 8 Quasi-uniform spaces Recall that Statement (viii) in Metatheorem rst appeared in [6] in 2000. In this section, we introduce an example of our Theorem 3.1(viii) given by Fierro [11]: Let (X, U) be a Hausdor quasi-uniform space, i.e. U is a lter on X × X satisfying the axioms of a unifor- mity, with the possible exception of the symmetry axiom. For each x 2 X, we denote U[x] = fy 2 X : (x, y) 2 Ug, U 2 U. 74 Ë Sehie Park These sets form a neighborhood basis of a topology on X. Then Fierro [11] gave several examples of pre-ordered sets satisfying Theorem 3.1. Let p : X × X ! [0,∞) be a w-distance on (X, U), i.e. p is a function satisfying (C1) p(x, y) ≤ p(x, z) + p(z, y) for any x, y, z 2 X, (C2) p(x,·) is l.s.c. for each x 2 X, and (C3) for any U 2 U, there exists δ > 0 such that p(z, y) < δ and p(z, y) < δ imply (x, y) 2 U. Let Φ[X, U] be the set of all functions ϕ : X × X ! (−∞,∞] such that (C4) ϕ(x, y) ≤ ϕ(x, z) + ϕ(z, y) for any x, y, z 2 X; and (C5) ϕ(x,·) : X ! (−∞,∞] is l.s.c. for each x 2 X. Given ϕ 2 Φ[X, U], we consider the pre-order dened by x  y i€ x = y or ϕ(x, y) + p(x, y) ≤ 0. The following is [11, Theorem 3.9] extending a theorem of Oettli-Théra: Theorem 8.1. ([11]) Let ϕ 2 Φ[X, U], x 2 X, Y  X and suppose the following two conditions hold: (1) S(x , ) is p-complete; and (2) for each x 2 S(x , )nY, there exists y 2 X such that x  y and x ≠ y. 0 ϕ ϕ Then, S(x , )\ Y ≠ ;. Here, S(x , ) = fy 2 X : x  yg, which will be denoted by A in the following: 0 0 ϕ ϕ Theorem 8.2. Let (X, U) be a Hausdor quasi-uniform space, ϕ 2 Φ[X, U], x 2 X, Y  X such that A = S(x , ) is p-complete. 0 ϕ Then the equivalent statements (i)–(viii) in Theorem 3.1 hold. Here (i) and (viii) are as follows: (i) There exists an element v 2 A such that v  w for any w 2 Xnfvg. (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 A \ Y. Proof. Note that (viii) holds by Theorem 8.1. Therefore, by Metatheorem or Theorem 3.1, the proof is complete. 9 Some other remarks (I) Our original Metatheorem rst appeared in 1985. The one in [6] is a particular form without (iv),(v) and indicated that it holds for quasi-ordered (pre-ordered or pseudo-ordered) sets. The present Metatheorem is an extended new form by adding (v)–(viii). Metatheorem and Theorem 3.1 without (iv) were appeared in our previous article [1] with several applications. In the present article, their proofs are given for completeness. (II) Fierro [11] in 2017 stated: “Park in [6] states ve equivalent conditions to maximality with respect to a specic preordering dened on a metric space. In this paper, we prove that these equivalences hold for arbitrary preorderings, without metric considerations, and two additional conditions are added to this set of equivalences.” Fierro did not have a chance to read [2-5] for not necessarily metric spaces. Fierro’s new equivalence Theorem 2.1 is our Theorem 3.1 with the following additional ones: (2.1.2) there exists x 2 S(x ,) such that for each chain C in S(x ,), S(x,) ≠ ;; 1 0 1 x2C (2.1.3) there exist x 2 S(x ,) and a maximal chain C in S(x ,) such that S(x,) ≠ ;; 1 0 1 * x2C Equivalents of maximum principles for several spaces Ë 75 (2.1.8) for any subset Y of X such that S(x ,)\ Y = ;, there exists x 2 S(x ,)nY satisfying S(x,) = fxg. 0 0 The statement (2.1.8) is incorrectly stated since S(x ,) has a maximal element by (2.1.1) or our Theorem 3.1(i); compare (2.1.8) with our Theorem 3.1(viii). Fierro [11] also stated that “Due to Corollary 3.4, when (X, d) is a quasi-metric space, Theorem 1 in [6] follows in the more general form. This result is stated as Corollary 3.8.” This statement is incorrect and he did not recognize the identity of our element “v”. Anyway, although [11] is very informative, it needs certain corrections. (III) Boros, Iqbal, and Száz [12]: “An example shows, in particular, that a maximality theorem published by Fierro in 2017 is not true without assuming the antisymmetry of the corresponding preorder. A true particular case of this theorem improves and supplements a former similar theorem of Sehie Park, and has to be proved just after Zorn’s lemma and a maximality principle of H. Brézis and Browder. This example will show, in particular, that the implication (2.1.3) =) (ii) in Theorem 3.1 is not true without assuming the antisymmetry of the relation . A relational improvement of a true particular case of Fierro’s Theorem can be found in a subsequent paper [13], where the curious assertion (2.1.8) will also be reformu- lated. This improvement generalizes and supplements a former similar theorem of Park [6]. Moreover, it has to be treated just after the famous Zorn lemma and a useful maximality principle of Brézis and Browder [21, Corollary 2].” (IV) Boros, Iqbal, and Száz [13]: “In this paper, by using relational notations, we improve and supplement a true particular case of an inaccurate maximality theorem of Rául Fierro from 2017, which has to be proved in addition to Zorn’s lemma and a famous maximality principle of H. Brézis and F. Browder.” They replaced the inadequate condition (2.1.8) by (9) for any Y  X, such that S(x)nfxg ≠ ; for all x 2 S(x )nY ≠ ;, we have S(x )\ Y ≠ ;. 0 0 Recall that (9) is exactly given as Theorem 3.1(viii), which was rst appeared in [6] in 2000. (V) Iqbal and Száz [14]: The maximality principles of Brézis and Browder [21] in 1976 slightly generalized and improved. There are several maximum theorems which can be applied our Metatheorem. 10 Conclusion In this article, we introduce our Metatheorem in [1] plus one more statement and show that it can be applied to equivalent formulations of a number of known theorem as we did in our previous work [1]. In such equivalent formulations, certain maximal points are actually same to xed points, stationary points, collectively xed points, collectively stationary points, and we have some information on the location of such points. No one recognized this fact yet. Therefore, if we have a theorem on any of such points, then we can deduce at least seven equivalent theorems on other types of points. In many elds of mathematical sciences, there are plentiful number of theorems concerning maximal points or xed points that can be applicable our Metatheorem or Theorem 3.1. Some of such theorems can be seen in our previous works in [1-6]. Therefore, our Metatheorem or Theorem 3.1 is a machine to nd new equivalent theorems with trivial proofs. This is like an industrial revolution of making new equivalent state- ments. References [1] S. Park, Equivalents of various maximum principles, Results in Nonlinear Analysis 5 (2022), 169–174. [2] S. Park, Some applications of Ekeland’s variational principle to xed point theory, Approximation Theory and Applications (S.P. Singh, ed.), Pitman Res. Notes Math. 133 (1985), 159–172. 76 Ë Sehie Park [3] S. Park, Countable compactness, l.s.c. functions, and xed points, J. Korean Math. Soc. 23 (1986), 61–66. [4] S. Park, Equivalent formulations of Ekeland’s variational principle for approximate solutions of minimization problems and their applications, Operator Equations and Fixed Point Theorems (S.P. Singh et al., eds.), MSRI-Korea Publ. 1 (1986), 55–68. [5] S. Park, Equivalent formulations of Zorn’s lemma and other maximmumm principles, J. Korean Soc. Math. Edu. 25 (1987), 19–24. [6] S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), 881–889. [7] E. Zermelo, Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908), 107–128. [8] A. Brøndsted, Fixed point and partial orders, Shorter Notes, Proc. Amer. Math. Soc. 60 (1976), 365–366. [9] J.-X. Fang, The variational principle and xed point theorems in certain topological spaces, J. Math. Anal. Appl. 202 (1996), 398–412. [10] F. Cammaroto, A. Chinnì, and G. Sturiale, A remark on Ekeland’s principle in locally convex topological vector spaces, Math. Comput. Modelling 30 (1999), 75–79. [11] R. Fierro, Maximality, xed points and variational principles for mappings on quasi-uniform spaces, Filomat (Niş) 31 (2017), 5345–5355. [12] Z. Boros, M. Iqbal and A. Száz, An instructive counterexample to a maximality theorem of Raúl Fierro, manuscript. [13] Z. Boros, M. Iqbal and A. Száz, A relational improvement of a true particular case of Fierro’s maximality theorem, manuscript. [14] M. Iqbal and A. Száz, An instructive treatment of the Brézis-Browder ordering and maximality principles, manuscript. [15] I. Ekeland, Sur les problèmes variationnels, C.R. Acad. Sci. Paris 275 (1972), 1057–1059; 276 (1973), 1347–1348. [16] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. [17] J. R. Jachymski, Caristi’s xed point theorem and selections of set-valued contractions, J. Math. Anal. Appl. 227 (1998) 55-67. [18] H. Kneser, Eine direkte Ableitung des Zornschen lemmas aus dem Auswahlaxiom, Math. Z. 53 (1950), 110–113. [19] A. Abian, A xed point theorem equivalent to the axiom of choice, Arch. Math. Logik 25 (1985), 173–174. [20] A. Brøndsted, On a lemma of Bishop and Phelps, Pacic J. Math. 55 (1974), 335–341. [21] H. Brézis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. Math. 21 (1976), 355– http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Topological Algebra and its Applications de Gruyter

Equivalents of maximum principles for several spaces

Park, Sehie
Topological Algebra and its Applications , Volume 10 (1): 9 – Jan 1, 2022
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Topol. Algebra Appl. 2022; 10:68–76 Research Article Open Access Sehie Park* Equivalents of maximum principles for several spaces https://doi.org/10.1515/taa-2022-0113 Received 20 May, 2022; accepted 10 June, 2022 Abstract: According to our long-standing Metatheorem, certain maximum theorems can be equivalently re- formulated to various types of xed point theorems, and conversely. As examples of such theorems, in this paper, we list Zermelo’s xed point theorem, Brøndsted’s principle, Fang’s F-type theorem, related theorems for locally convex spaces and quasi-uniform spaces. Further we review some few works concerned with our Metatheorem. Keywords: Ekeland variational principle, pre-order, Zermelo, quasi-metric space, xed point, maximal ele- ment, locally convex space, quasi-uniform space MSC: 06A75, 47H10, 54E35, 54E50, 54H25, 58E30, 65K10 1 Introduction Recently we have obtained several equivalent formulations of Zorn’s lemma, the Banach contraction prin- ciple, Caristi’s xed point theorem, Ekeland’s variational principle, Takahashi’s nonconvex minimization theorem and some others in [1]. These are forceful tools in nonlinear analysis, control theory, economic the- ory, and global analysis. These theorems were already extended by a large number of authors. However, we are able to obtain their equivalent formulations. In 1985-2000, we had published several articles mainly related to the Ekeland variational principle for approximate solutions of minimization problems and its equivalent formulations with some applications; see [2-6]. From the beginning of such study, we made a Metatheorem for some equivalent statements on max- imality, xed points, stationary points, common xed points, common stationary points, and their locations. We applied the Metatheorem for various occasions in [1-6]. In our previous work [1], we obtained an extended version of Metatheorem and applied it to Zorn’s lemma, Banach contraction principle, Nadler’s xed point theorem, Brézis-Browder principle, Caristi’s xed point theorem, Ekeland’s variational principle, Takahashi’s nonconvex minimization theorem, some others and their variants, generalizations or equivalent formulations. In the present article, as a continuation of [1], we apply Metatheorem to Zermelo’s xed point theorem [7], Brøndsted’s principle [8], Fang’s F-type theorem [9], related theorems for locally convex spaces [10] and quasi-uniform spaces [11]. Further we review some works concerned with our Metatheorem. This article is organized as follows: In Section 2, we introduce our Metatheorem with its proof for com- pleteness. It is applied to pre-ordered sets in Section 3 as usual, and subsequent Sections 4-8 are devoted to applications of Metatheorem to various theorems mentioned above. In Section 9, we give some remarks on recent works of other authors in [11-14] related to our Metatheorem. Finally, Section 10 deals with some comments and conclusion. *Corresponding Author: Sehie Park: The National Academy of Sciences, Republic of Korea; Seoul 06579 and Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea, E-mail: park35@snu.ac.kr; sehiepark@gmail.com; parksehie.com Open Access. © 2022 Sehie Park, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. Equivalents of maximum principles for several spaces Ë 69 2 Metatheorem related to the Ekeland principle The well-known central result of I. Ekeland [15,16] on the variational principle for approximate solutions of minimization problems runs as follows: Theorem. (Ekeland [15]) Let V be a complete metric space, and F : V ! R [ f+∞g a l.s.c. function, ̸ +∞, bounded from below. Let ε > 0 be given, and a point u 2 V such that F(u) 5 inf F + ε. Then for every λ > 0, −1 there exists a point v 2 B(u, λ) such that F(v) 5 F(u) and F(w) > F(v) − ελ d(v, w) for any w 2 V , w ≠ v. When λ = 1, this is called the ε-variational principle. In order to obtain some equivalents of this principle, we obtained a Metatheorem in [2-6]. Later we found some additional statements and, consequently, we obtain a new extended version of the Metatheorem [1] in 2022. Now we add its proof for the completeness. Metatheorem. Let X be a set, A its nonempty subset, and G(x, y) a sentence formula for x, y 2 X. Then the following propositions are equivalent: (i) There exists an element v 2 A such that G(v, w) for any w 2 Xnfvg. (ii) If T : A ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying ¬G(x, y), then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : A ! X is a map such that for any x 2 A with x ≠ f (x), there exists a y 2 Xnfxg satisfying¬G(x, y), then f has a xed element v 2 A, that is, v = f(v). (iv) If f : A ! X is a map such that ¬G(x, f(x)) for any x 2 A, then f has a xed element v 2 A, that is, v = f(v). (v) If T : A ( X is a multimap such that ¬G(x, y) holds for any x 2 A and any y 2 T(x)nfxg, then T has a stationary element v 2 A, that is,fvg = T(v). (vi) If F is a family of maps f : A ! X satisfying¬G(x, f(x)) for all x 2 A with x ≠ f(x), then F has a common xed element v 2 A, that is, v = f (v) for all f 2 F. (vii) If F is a family of multimaps T : A ( X for i 2 I with an index set I such that ¬G(x, y) holds for any x 2 A and any y 2 T (x)nfxg, then F has a common stationary element v 2 A, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg satisfying¬G(x, z), then there exists a v 2 A \ Y. Here, maps and multimaps have nonempty values and ¬ denotes the negation. Proof. (i)=)(ii): Suppose v 2 / T(v). Then there exists a y 2 Xnfvg satisfying ¬G(v, y). This is a contradiction. (ii)=)(iii): Clear. (iii)=)(iv): Clear. (iv)=)(v): Suppose T has no stationary element, that is, T(x)nfxg ≠ ; for any x 2 A. By Axiom of Choice, we have a choice function f on fT(x)nfxg : x 2 Ag. Then f has no xed element by its denition. However, for any x 2 A, we have ¬G(x, f(x)). Therefore, by (iv), f has a xed element, a contradiction. (v)=)(vi): Dene a multimap T : A ( X by T(x) := ff(x) : f 2 Fg ≠ ; for all x 2 A. Since ¬G(x, f (x)) for any x 2 A and any f 2 F, by (v), T has a stationary element v 2 A, which is a common xed element of F. (vi)=)(i): Suppose that for any x 2 A, there exists a y 2 Xnfxg satisfying ¬G(x, y). Choose f (x) to be one of such y. Then f : A ! X has no xed element by its denition. However, ¬G(x, f (x)) for all x 2 A. Let F = ffg. By (vi), f has a xed element, a contradiction. (i)+(v) =) (vii): By (i), there exists a v 2 A such that G(v, w) for all w 2 Xnfvg. For each i 2 I, by (v), we have a v 2 A such that fv g = T (v ). Suppose v ≠ v . Then G(v, v ) holds by (i) and ¬G(v, v ) holds by i i i i i i i assumption on (vii). This is a contradiction. Therefore v = v for all i 2 I. i 70 Ë Sehie Park (vii)=)(v): Clear. (i)=)(viii): By (i), there exists a v 2 A such that G(v, w) for all w ≠ v. Then by the hypothesis, we have v 2 Y. Therefore, v 2 A \ Y. (viii)=)(i): For all x 2 A, let A(x) := fy 2 X : x ≠ y, ¬G(x, y)g. Choose Y = fx 2 X : A(x) = ;g. If x 2 / Y, then there exists a z 2 A(x). Hence the hypothesis of (viii) is satised. Therefore, by (viii), there exists a v 2 A \ Y. Hence A(v) = ;; that is, G(v, w) for all w ≠ v. Hence (i) holds. This completes our proof. In this proof, we notice that the element v 2 A is the same throughout (i)–(viii) and that (viii) gives some information on the location of v 2 A. 3 For pre-ordered sets Let (X,) be a pre-ordered set; that is, X is a nonempty set, is reexive and transitive. For each x 2 X, we denote S(x) = fy 2 X : x  yg and G(x, y) means x  y. We give a rst application of Metatheorem to pre-ordered sets as follows: Theorem 3.1. Let (X,) be a pre-ordered set, x 2 X and A = S(x ). Then the following eight propositions are 0 0 equivalent: (i) There exists a maximal element v 2 A such that v  w for any w 2 Xnfvg. (ii) If T : A ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying x  y, then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : A ! X is a map such that for any x 2 A with x ≠ f(x), there exists a y 2 Xnfxg satisfying x  y, then f has a xed element v 2 A, that is, v = f (v). (iv) If f : A ! X is a map such that x  f(x) for any x 2 A, then f has a xed element v 2 A, that is, v = f(v). (v) If T : A ( X is a multimap such that x  y holds for any x 2 A and any y 2 T(x)nfxg, then T has a stationary element v 2 A, that is,fvg = T(v). (vi) If F is a family of maps f : A ! X satisfying x  f (x) for all x 2 A with x ≠ f (x), then F has a common xed element v 2 A, that is, v = f(v) for all f 2 F. (vii) If F is a family of multimaps T : A ( X for i 2 I with an index set I such that x  y holds for any x 2 A and any y 2 T (x)nfxg, then F has a common stationary element v 2 A, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 A \ Y. Proof. In Metatheorem, put A := S(x ) and let G(v, w) be the statement v  w. Then each of (i)–(viii) follows from the corresponding ones in Metatheorem. This completes our proof. Remark 3.2. Note that we claimed that (i)–(viii) are equivalent in Theorem 3.1 and did not say that they are true. For a counter-example, think about the real line with the natural order. From now on, we are going to give examples that they are true based on their original sources. Equivalents of maximum principles for several spaces Ë 71 4 Extending Zermelo’s theorem We apply Theorem 3.1 to Zermelo’s xed point theorem [7] in 1908: Theorem 4.1 ([7]) Let (X,) be a partially ordered set in which each nonempty well-ordered subset has a least upper bound. Then every function f : X ! X satisfying x  f(x) for all x 2 X has a xed point. The Zermelo xed point theorem is also known as the Bourbaki xed point theorem or the Bourbaki- Kneser xed point theorem. It implies the Caristi xed point theorem, the Bernstein-Cantor-Schröder theorem, the Ekeland variational principle, the Takahashi minimization theorem, and others. Moreover, under the Axiom of Choice, it implies Zorn’s Lemma. Jachymski [15] noted: “Under the Axiom of Choice, the assumption of Theorem 4.1 can be weakened to “each nonempty well-ordered subset has an upper bound.” This is Kneser’s xed point theorem [18], which turns out to be equivalent to the Axiom of Choice as shown by Abian [19].” However we have the following by Zorn’s Lemma and Theorem 3.1: Theorem 4.2. Let (X,) be a partially ordered set in which each nonempty well-ordered subset has an upper bound. Let x 2 X and A = S(x ) = fy 2 X : x  yg. 0 0 0 Then the following eight equivalent conditions hold: (i) There exists a maximal element v 2 A, that is, v  w for any w 2 Xnfvg. (ii) If T : A ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying x  y, then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : A ! X is a map such that for any x 2 A with x ≠ f(x), there exists a y 2 Xnfxg satisfying x  y, then f has a xed element v 2 A, that is, v = f (v). (iv) If f : A ! X is a map such that x  f(x) for any x 2 A, then f has a xed element v 2 A, that is, v = f (v). (v) If T : A ( X is a multimap such that x  y holds for any x 2 A and any y 2 T(x)nfxg, then T has a stationary element v 2 A, that is,fvg = T(v). (vi) If F is a family of maps f : A ! X satisfying x  fx for all x 2 A with x ≠ f (x), then F has a common xed element v 2 A, that is, v = f(v) for all f 2 F. (vii) If F is a family of multimaps T : A ( X for i 2 I with an index set I such that x  y holds for any x 2 A and any y 2 T (x)nfxg, then F has a common stationary element v 2 A, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 A \ Y. Proof. In Metatheorem, put A := S(x ) and let G(v, w) be the statement v  w. Then (i) is Zorn’s lemma and each of (i)–(viii) follows from the corresponding ones in Theorem 3.1. This completes our proof. Note that (iii) and (iv) extend Zermelo’s Theorem 4.1. 5 Extensions of Brøndsted’s principle In a short note of Brøndsted [8] in 1976 observed that certain xed point theorems may be derived from the- orems on the existence of maximal elements in partially ordered sets. His main point is to show how certain xed point theorems can be deduced from the following simple observation: 72 Ë Sehie Park (A) Let (E,4) be a partially ordered set which admits at least one maximal element. Let f : E ! E be a map such that x 4 f(x) for all x 2 E. Then f admits at least one xed point. Now this can be called the Brøndsted principle. This is motivated by a certain type of such spaces consid- ered by Bishop and Phelps [20]. Brøndsted applied (A) to the Caristi-Kirk xed point theorem and other type of xed point theorems in (B) and (C) in [8]. In this section, by applying our Metatheorem or Theorem 3.1, we show that various extended types of converse of the principle holds: Theorem 5.1. Let (E,4) be a pre-ordered set. Then the eight equivalent conditions in Theorem 3.1 hold. For example: (i) There exists a maximal element v 2 X; that is, v 4̸ w for any w 2 Xnfvg. (ii) If T : X ( X is a multimap such that for any x 2 XnT(x) there exists a y 2 Xnfxg satisfying x 4 y, then T has a xed element v 2 A, that is, v 2 T(v). (iii) If f : X ! X is a map such that for any x 2 X with x ≠ f (x), there exists a y 2 Xnfxg satisfying x 4 y, then f has a xed element v 2 X, that is, v = f(v). (iv) If f : X ! X is a map such that x 4 f(x) for any x 2 X, then f has a xed element v 2 X, that is, v = f(v). Note that (i)()(iv) is later called the Brøndsted Principle by ourselves. 6 F-type spaces of Fang In 1996 Fang [9] introduced the concept of F-type topological spaces and established a variational principle and a xed point theorem in the kind of spaces, which extend Ekeland’s variational principle and Caristi’s xed point theorem, respectively. In [9], a characterization of F-type of spaces is given. The usual metric spaces, Hausdor topological vector spaces, and Menger probabilistic metric spaces are all the special cases of F-type topological spaces. Fang established a variational principle and a xed point theorem, and applied them to Menger probabilistic metric spaces. Now we recall the denition of F-type using a characterization of them proved by Fang (see [9, Theorem 2.1]). Denition 6.1. A topological space E is said to be F-type if there exists a familyfd g of semimetrics on E, i i2I I being a directed set, generating the topology on E, and satisfying the following conditions: (F1) d (x, y) = 0 for all i 2 I i x = y for all x, y 2 E; (F2) d (x, y) = d (y, x) for all i 2 I, for all x, y 2 E; i i (F3) d (x, y) ≤ d (x, y) if i < j for all x, y 2 E; i j (F4) for every i 2 I there exists j 2 I, with i < j, such that d (x, y) ≤ d (x, z) + d (z, y) for all x, y, z 2 E. i j j The following is [9, Theorem 3.1]: Theorem 6.2. ([9]) Let (X, T) be a sequentially complete F-type topological space and fd : λ 2 Dg be a generating family of quasi-metrics for T. Let φ : X ! R be a lower semi-continuous function, bounded from below, and k : D ! (0,∞) be a nonincreasing function. We dene a relation “” on X as x  y i d (x, y) ≤ k(λ)(φ(x) − φ(y)) for all λ 2 D. Then (X,) is a partially ordered set and it has a maximal element. From Theorem 6.2 and Metatheorem, we have the following: Theorem 6.3. For (X,) in Theorem 6.2, we have the following equivalent statements hold: Equivalents of maximum principles for several spaces Ë 73 (i) There exists a maximal element v 2 X such that v  w for any w 2 Xnfvg. (ii) If T : X ( X is a multimap such that for any x 2 AnT(x) there exists a y 2 Xnfxg satisfying x  y, then T has a xed element v 2 X, that is, v 2 T(v). (iii) If f : X ! X is a map such that for any x 2 X with x ≠ f(x), there exists a y 2 Xnfxg satisfying x  y, then f has a xed element v 2 X, that is, v = f (v). (iv) If f : X ! X is a map such that x  f(x) for any x 2 X, then f has a xed element v 2 X, that is, v = f (v). (v) If T : X ( X is a multimap such that x  y holds for any x 2 X and any y 2 T(x)nfxg, then T has a stationary element v 2 X, that is,fvg = T(v). (vi) If F is a family of maps f : X ! X satisfying x  fx for all x 2 X with x ≠ f(x), then F has a common xed element v 2 X, that is, v = f(v) for all f 2 F. (vii) If F is a family of multimaps T : X ( X for i 2 I with an index set I such that x  y holds for any x 2 X and any y 2 T (x)nfxg, then F has a common stationary element v 2 X, that is,fvg = T (v) for all i 2 I. i i (viii) If Y is a subset of X such that for each x 2 XnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 X \ Y = Y. Proof. Note that (i) holds by Theorem 6.2. Then Theorem 6.3 follows from Metatheorem with X = A. Our Theorem 6.3(iii) and (iv) extend [9, Theorem 3.1]. Its equivalent form [9, Theorem 3.1 ] also holds. Fang showed that Caristi’s xed point theorem follows from [9, Theorem 3.1]. 7 For locally convex spaces Cammaroto et al. [10] showed that a Hausdor locally convex topological vector space X is a F-type topological vector space. Therefore Theorem 6.3 holds for such space X. They obtained the following [10, Theorem 3.2]: Theorem 7.1. ([10]) Let E be a complete, Hausdor, locally convex topological vector space and fp g a ba- i i2I sis of continuous seminorms generating the topology on E. Moreover, let φ : E ! R be a lower semicontinu- ous function bounded from below and k : I ! ]0, +∞[ a nonincreasing function respect to the order in I with sup k(i) < +∞. i2I (iii) If f : E ! E is a function such that p (x − f (x)) ≤ k(i)[φ(x) − φ(f (x))], 8i 2 I, 8x 2 E, then f has a xed point in E. This can be slightly improved by Theorem 6.3(iv) since (iii) implies (iv). Hence Theorem 6.3(i)–(viii) also hold for E. 8 Quasi-uniform spaces Recall that Statement (viii) in Metatheorem rst appeared in [6] in 2000. In this section, we introduce an example of our Theorem 3.1(viii) given by Fierro [11]: Let (X, U) be a Hausdor quasi-uniform space, i.e. U is a lter on X × X satisfying the axioms of a unifor- mity, with the possible exception of the symmetry axiom. For each x 2 X, we denote U[x] = fy 2 X : (x, y) 2 Ug, U 2 U. 74 Ë Sehie Park These sets form a neighborhood basis of a topology on X. Then Fierro [11] gave several examples of pre-ordered sets satisfying Theorem 3.1. Let p : X × X ! [0,∞) be a w-distance on (X, U), i.e. p is a function satisfying (C1) p(x, y) ≤ p(x, z) + p(z, y) for any x, y, z 2 X, (C2) p(x,·) is l.s.c. for each x 2 X, and (C3) for any U 2 U, there exists δ > 0 such that p(z, y) < δ and p(z, y) < δ imply (x, y) 2 U. Let Φ[X, U] be the set of all functions ϕ : X × X ! (−∞,∞] such that (C4) ϕ(x, y) ≤ ϕ(x, z) + ϕ(z, y) for any x, y, z 2 X; and (C5) ϕ(x,·) : X ! (−∞,∞] is l.s.c. for each x 2 X. Given ϕ 2 Φ[X, U], we consider the pre-order dened by x  y i€ x = y or ϕ(x, y) + p(x, y) ≤ 0. The following is [11, Theorem 3.9] extending a theorem of Oettli-Théra: Theorem 8.1. ([11]) Let ϕ 2 Φ[X, U], x 2 X, Y  X and suppose the following two conditions hold: (1) S(x , ) is p-complete; and (2) for each x 2 S(x , )nY, there exists y 2 X such that x  y and x ≠ y. 0 ϕ ϕ Then, S(x , )\ Y ≠ ;. Here, S(x , ) = fy 2 X : x  yg, which will be denoted by A in the following: 0 0 ϕ ϕ Theorem 8.2. Let (X, U) be a Hausdor quasi-uniform space, ϕ 2 Φ[X, U], x 2 X, Y  X such that A = S(x , ) is p-complete. 0 ϕ Then the equivalent statements (i)–(viii) in Theorem 3.1 hold. Here (i) and (viii) are as follows: (i) There exists an element v 2 A such that v  w for any w 2 Xnfvg. (viii) If Y is a subset of X such that for each x 2 AnY there exists a z 2 Xnfxg such that x  z, then there exists an element v 2 A \ Y. Proof. Note that (viii) holds by Theorem 8.1. Therefore, by Metatheorem or Theorem 3.1, the proof is complete. 9 Some other remarks (I) Our original Metatheorem rst appeared in 1985. The one in [6] is a particular form without (iv),(v) and indicated that it holds for quasi-ordered (pre-ordered or pseudo-ordered) sets. The present Metatheorem is an extended new form by adding (v)–(viii). Metatheorem and Theorem 3.1 without (iv) were appeared in our previous article [1] with several applications. In the present article, their proofs are given for completeness. (II) Fierro [11] in 2017 stated: “Park in [6] states ve equivalent conditions to maximality with respect to a specic preordering dened on a metric space. In this paper, we prove that these equivalences hold for arbitrary preorderings, without metric considerations, and two additional conditions are added to this set of equivalences.” Fierro did not have a chance to read [2-5] for not necessarily metric spaces. Fierro’s new equivalence Theorem 2.1 is our Theorem 3.1 with the following additional ones: (2.1.2) there exists x 2 S(x ,) such that for each chain C in S(x ,), S(x,) ≠ ;; 1 0 1 x2C (2.1.3) there exist x 2 S(x ,) and a maximal chain C in S(x ,) such that S(x,) ≠ ;; 1 0 1 * x2C Equivalents of maximum principles for several spaces Ë 75 (2.1.8) for any subset Y of X such that S(x ,)\ Y = ;, there exists x 2 S(x ,)nY satisfying S(x,) = fxg. 0 0 The statement (2.1.8) is incorrectly stated since S(x ,) has a maximal element by (2.1.1) or our Theorem 3.1(i); compare (2.1.8) with our Theorem 3.1(viii). Fierro [11] also stated that “Due to Corollary 3.4, when (X, d) is a quasi-metric space, Theorem 1 in [6] follows in the more general form. This result is stated as Corollary 3.8.” This statement is incorrect and he did not recognize the identity of our element “v”. Anyway, although [11] is very informative, it needs certain corrections. (III) Boros, Iqbal, and Száz [12]: “An example shows, in particular, that a maximality theorem published by Fierro in 2017 is not true without assuming the antisymmetry of the corresponding preorder. A true particular case of this theorem improves and supplements a former similar theorem of Sehie Park, and has to be proved just after Zorn’s lemma and a maximality principle of H. Brézis and Browder. This example will show, in particular, that the implication (2.1.3) =) (ii) in Theorem 3.1 is not true without assuming the antisymmetry of the relation . A relational improvement of a true particular case of Fierro’s Theorem can be found in a subsequent paper [13], where the curious assertion (2.1.8) will also be reformu- lated. This improvement generalizes and supplements a former similar theorem of Park [6]. Moreover, it has to be treated just after the famous Zorn lemma and a useful maximality principle of Brézis and Browder [21, Corollary 2].” (IV) Boros, Iqbal, and Száz [13]: “In this paper, by using relational notations, we improve and supplement a true particular case of an inaccurate maximality theorem of Rául Fierro from 2017, which has to be proved in addition to Zorn’s lemma and a famous maximality principle of H. Brézis and F. Browder.” They replaced the inadequate condition (2.1.8) by (9) for any Y  X, such that S(x)nfxg ≠ ; for all x 2 S(x )nY ≠ ;, we have S(x )\ Y ≠ ;. 0 0 Recall that (9) is exactly given as Theorem 3.1(viii), which was rst appeared in [6] in 2000. (V) Iqbal and Száz [14]: The maximality principles of Brézis and Browder [21] in 1976 slightly generalized and improved. There are several maximum theorems which can be applied our Metatheorem. 10 Conclusion In this article, we introduce our Metatheorem in [1] plus one more statement and show that it can be applied to equivalent formulations of a number of known theorem as we did in our previous work [1]. In such equivalent formulations, certain maximal points are actually same to xed points, stationary points, collectively xed points, collectively stationary points, and we have some information on the location of such points. No one recognized this fact yet. Therefore, if we have a theorem on any of such points, then we can deduce at least seven equivalent theorems on other types of points. In many elds of mathematical sciences, there are plentiful number of theorems concerning maximal points or xed points that can be applicable our Metatheorem or Theorem 3.1. Some of such theorems can be seen in our previous works in [1-6]. Therefore, our Metatheorem or Theorem 3.1 is a machine to nd new equivalent theorems with trivial proofs. This is like an industrial revolution of making new equivalent state- ments. References [1] S. Park, Equivalents of various maximum principles, Results in Nonlinear Analysis 5 (2022), 169–174. [2] S. Park, Some applications of Ekeland’s variational principle to xed point theory, Approximation Theory and Applications (S.P. Singh, ed.), Pitman Res. Notes Math. 133 (1985), 159–172. 76 Ë Sehie Park [3] S. Park, Countable compactness, l.s.c. functions, and xed points, J. Korean Math. Soc. 23 (1986), 61–66. [4] S. Park, Equivalent formulations of Ekeland’s variational principle for approximate solutions of minimization problems and their applications, Operator Equations and Fixed Point Theorems (S.P. Singh et al., eds.), MSRI-Korea Publ. 1 (1986), 55–68. [5] S. Park, Equivalent formulations of Zorn’s lemma and other maximmumm principles, J. Korean Soc. Math. Edu. 25 (1987), 19–24. [6] S. Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), 881–889. [7] E. Zermelo, Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908), 107–128. [8] A. Brøndsted, Fixed point and partial orders, Shorter Notes, Proc. Amer. Math. Soc. 60 (1976), 365–366. [9] J.-X. Fang, The variational principle and xed point theorems in certain topological spaces, J. Math. Anal. Appl. 202 (1996), 398–412. [10] F. Cammaroto, A. Chinnì, and G. Sturiale, A remark on Ekeland’s principle in locally convex topological vector spaces, Math. Comput. Modelling 30 (1999), 75–79. [11] R. Fierro, Maximality, xed points and variational principles for mappings on quasi-uniform spaces, Filomat (Niş) 31 (2017), 5345–5355. [12] Z. Boros, M. Iqbal and A. Száz, An instructive counterexample to a maximality theorem of Raúl Fierro, manuscript. [13] Z. Boros, M. Iqbal and A. Száz, A relational improvement of a true particular case of Fierro’s maximality theorem, manuscript. [14] M. Iqbal and A. Száz, An instructive treatment of the Brézis-Browder ordering and maximality principles, manuscript. [15] I. Ekeland, Sur les problèmes variationnels, C.R. Acad. Sci. Paris 275 (1972), 1057–1059; 276 (1973), 1347–1348. [16] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. [17] J. R. Jachymski, Caristi’s xed point theorem and selections of set-valued contractions, J. Math. Anal. Appl. 227 (1998) 55-67. [18] H. Kneser, Eine direkte Ableitung des Zornschen lemmas aus dem Auswahlaxiom, Math. Z. 53 (1950), 110–113. [19] A. Abian, A xed point theorem equivalent to the axiom of choice, Arch. Math. Logik 25 (1985), 173–174. [20] A. Brøndsted, On a lemma of Bishop and Phelps, Pacic J. Math. 55 (1974), 335–341. [21] H. Brézis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. Math. 21 (1976), 355–

Journal

Topological Algebra and its Applications – de Gruyter

Published: Jan 1, 2022

Keywords: Ekeland variational principle; pre-order; Zermelo; quasi-metric space; fixed point; maximal element; locally convex space; quasi-uniform space; 06A75; 47H10; 54E35; 54E50; 54H25; 58E30; 65K10

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