Monotone Principle of Forked Points and Its Consequences

This paper presents applications of the Axiom of Infinite Choice: Given any set P , there exist at least countable choice functions or there exist at least finite choice functions. The author continues herein with the further study of two papers of the Axiom of Choice in order by E. Ze rme l o [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107–128; translated in van Heijenoort 1967, 183–198], and by M. Taskov i ć [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897–904]. Monotone principle of forked points is a direct consequence of the Axiom of Infinite Choice, i.e., of the Lemma of Infinite Maximality! Brouwer and Schauder theorems are two direct censequences of the monotone principle od forked points.


Introduction and history
Let X := (X, M ) be a topological space and T : X → X, where M : X → R 0 + := [0, +∞). In connection with this, in 1985 we investigated the concept of TCS-convergence in a space X, i.e., a topological space X satisfies the condition of local TCS-convergence iff x ∈ X and if M (T n x) → 0 (n → ∞) implies that {T n (x)} n∈N has a convergent subsequence. Theorem 1.1. (Localization Monotone Principle, Tasković [1990, Th. 1]). Let T be a mapping of a topological space X := (X, M ) into itself, where X satisfies the condition of local TCS-convergence. Suppose that there exists a mapping ϕ : R 0 + → R 0 + such that ∀t ∈ R + := (0, +∞) ϕ(t) < t and lim sup z→t+0 ϕ(z) < t (ϕ) and the following inequality holds in the form as M (T (x)) ≤ ϕ(M (x)) for every x ∈ X, (1) where M : X → R 0 + is a T -orbitally lower semicontinuous function and M (u) = 0 implies T (u) = u. Then T has at least one fixed point in X.
For x ∈ X the set σ(x, ∞) := {x, T x, T 2 x, . . .} is called the orbit of x. A function f mapping X into the reals is f -orbitally lower semicontinuous at p if {x n } n∈N is a sequence in σ(x, ∞) and x n → p (n → ∞) implies that f (p) ≤ lim. inf f (x n ). A space X is said to be T -orbitally complete iff every Cauchy sequence which is contained in σ(x, ∞) for some x ∈ X converges in X.
Let X := (X, A) be a topological space and T : X → X, where A : In 1985 year we investigated the concept of TCS-convergence in a space X, i.e., a topological space X satisfies the condition of TCSconvergence iff x ∈ X and if A(T n x, T n+1 x) → 0 (n → ∞) implies that {T n (x)} n∈N has a convergent subsequence. As an immediate consequence of Theorem 1.1 we have the following statement on topological spaces. Tasković [1990, Th. 2]). Let T be a mapping of a topological space X:=(X, A) into itself, where X satisfies the condition of TCS-convergence. Suppose that there exists a mapping ϕ : R 0 + → R 0 + such that (ϕ) and where x → A(x, T (x)) is a T -orbitally lower semicontinuous function and A(u, v) = 0 implies u = v. Then T has a unique fixed point ζ ∈ X and T n (x) → ζ as n → ∞ for each x ∈ X.
Proof. Let M (x) := A(x, T (x)), then it is easy to see that A and ϕ satisfy all the required hypotheses in Theorem 1.1. Uniqueness follows immediately from condition (2). The proof is complete.
Survey of facts. For the preceding monotone principles, specially for Localization Monotone Principle of Fixed Point, J a m e s D u g u n d j i, in the letter for me of October 5 in 1984 year, briefly among the rest writes that he is convinced of the role of Localization Monotone Principle in the fixed point theory (and nonlinear functional analysis).
This opinion of J. Dugundji has been confirmed many a time, via various phenomena, as one can see from many results proven in nonlinear analysis and nature.
In this paper we considered and formulated some new monotone principles for fixed points and for fixed apices as a new way in the nonlinear functional analysis.
We notice that Djuro Kurepa in 1971, first version of my Monotone Principle of Fixed Point, has been sent to Professor J e a n L e r a y (Paris) for the opinion. Some of Leray's ideas I am to realize in several published papers. In general form for the first time, fundamental elements of Monotone Principle I give in: Proc.
History of TCS-convergence. For the first time in 1985 I introduced the  conditions of TCS-convergence and local TCS-convergence with the intention to  transmit it to the properties of Cauchy sequence from metric spaces on topological spaces, see: Ta s k o v i ć [1990].
This conceptions are very operational and useful for "calculation" on topological spaces. In this sense after this viewpoint appears in most of my papers and books from fixed point theory (see: Ta s k o v i ć [1986], [1990] and [2001]). We can briefly say, in connection with this, that the results of forked points are based on RBSconvergence and BCS-convergence. It is a new viewpoint which is an extension of the TCS-convergence.
At the interval of the next seven years more authors have considered appearance of TCS-convergence as a special case od the property TCS-convergence, precisely, in this way,

Monotony and Forked Points
Further, by the "Axiom of Infinte Choice" we mean a statement in the following form as: Given any set S, there exist at least countable choice functions or there exist at least finite choice functions.
In general, equivalents of the Axiom of Infinite Choice appear frequently in almost all branches of mathematics in a large variety of different forms.
In this part of the paper we present an equivalent form of the Axiom of Infinite Choice which is expressible in the following sense. ). Let P be an inductive partially ordered set with ordering , then P has at least countable maximal elements or P has at least finite maximal elements. Further, in this part we introduced a fundamental result of a new forks theory which unified and connected three known theories on fixed point, transversality and von Neumann's minimax theory. In classical von Neumann's theory fundamental notions is saddle point. In new general convex forks theory its role plays transversal and forked points. In this sense we formulate a new way in nonlinear functional analysis.
Let X be an arbitrary nonempty set, T be a mapping from X into X, and P := (P, ) a nonempty partially ordered set. A mapping f : X → P (or f : X → X) has a forked point (or furcate point) p ∈ X if the following equality holds in the form frequently, we say that in this case (Ra), the mapping f : X → P or f : X → X has a pair (p, T p) of bifurcation points, or that T : X → X has a forked (or forks) point p ∈ X.
We notice that many problems in nonlinear functional analysis (as and in the fixed point theory) are reducible to the existence of forked points of certain mappings.
Further, let P := (P, ) be a partially ordered set with a minimum (or with the property that every nonempty subset in P has an infimum) such that every decreasing sequence {x n } n∈N in P has a limit in P , denoted by lim n→∞ x n .
In connection with this, we shall introduce the concept of lower ordered RBS-convergence in a topological space X for B : X → P , i.e., a topological space X satisfies the condition of lower ordered RBS-convergence iff {a n (x)} n∈N is an arbitrary sequence in X with arbitrary x ∈ X and if B(a n (x) In this part of the paper, we apply the technick of maximal elements to the equations of the forks theory. As an immediate consequence of Lemma 2.1 we obtain the following ordered principle.
In this sense, let X be a topological space, Define a relation B,con on X by the following conditions: where B : X → P is a function with the given conditions.
It is verify that B,con is a partial ordering (asymmetric and transitive relation) in X. The poset X together with this partial ordering, is denoted by X B,con . Theorem 2.1. (Ordering Principle). Let X be a topological space with the poset X B,con . If X satisfies the condition of lower ordered RBS-convergence, then X B,con has at least countable or finite minimal elements z k ∈ X B,con with z k B,con x for given x ∈ X B,con .
Proof. Let C be a chain in X B,con and now let t ∈ C be given. Denote by α := inf{B(x) : x ∈ C}. If B(m) = α for some m ∈ C, then m is a lower bound in C. For, if x B,con m for some x ∈ C\{m}, then B(x) B(m), which yields B(x) ≺ α, which is a contradiction. Therefore, one san assume B(x) = α for all x ∈ C. Then the set M (x, n) of all y ∈ C with y B,con x and α ≺ B(y) ≺ α n (α n → α) is nonempty for each n ∈ N and x ∈ C. In fact, there is a y ∈ C satisfying α ≺ β(y) ≺ α n , and so y belongs to M (x, n) if y B,con x; if x B,con y then since B(x) B(y) we have α ≺ B(x) B(y) ≺ α n , which shows that x belongs to M (x, n). Let I be a choice function for the family of all nonempty subsets of C. Then, by the recursion theorem, there is a sequence {x n } n∈N in C such that x 0 = t and x n+1 = I(M (x n , n)) for n ∈ N. Since x n+1 B,con x n for all n ∈ N, we have This implies (from lower ordered RBS-convergence) that there exists ξ ∈ X B,con such that B(ξ) · · · B(x n ) for n ∈ N. Now let x ∈ C. Then we can find an i ∈ N such that B(ξ) B(x i ) ≺ α i B(x). Since x and x i are in the chain C, we obtain ξ B,con x. This shows that ξ is a minorant of C. By the nature of C (by Lemma of Infinite Maximality) it follows that there is at least countable or finite z k ∈ X B,con which are minimal in X B,con .
We notice that the proof of this statement is totally an analogy with the former proofs of ordered principles.
As an immediate consequence of Theorem 2.1 (Ordering Principle) we obtain the following result in the forks theory.
Theorem 2.2. (Monotone principle of forked points). Let T be a mapping of a topological space X into itself, where X satisfies the condition of lower ordered RBS-convergence. If then for T there exist at least countable or finite forked points ξ k ∈ X, i.e., then the following equalities hold in the form for some sequences {b k n (x)} n∈N in X which converges to the forked points ξ k ∈ X.
Proof. Consider the partially ordered set X B,con and let ξ k be minimal elements. Using (B), it also following from (fk) that T x B,con x for all x ∈ X and T ξ k B,con ξ k in X B,con and, because ξ k are minimal it follows that T ξ k = B,con ξ k . The proof is complete. Proof of a special case of Theorem 2.2. Let x be an arbitrary point in X. Then from the inequality (B) we obtain the following inequalities in the form for every n ∈ N ∪ {0} and for every x ∈ X. Thus, for the sequence {B(T n x)} n∈N∪{0} from (3), we obtain that B(T n x) → b ∈ P (n → ∞) with arbitrary x ∈ X. This implies (from the lower ordered RBS-convergence) that its sequence {T n x} n∈N∪{0} contains a convergent subsequence {T n(k) (x)} k∈N with a limit point ξ ∈ X. Since X satisfies the condition of lower ordered RBS-convergence, from (3), we have i.e., B(T ξ) = B(ξ) = α. This means that (Ri) holds, i.e., that the mapping T : X → X has a forked point ξ ∈ X, where the existing sequence {b n (x)} n∈N , de facto, is the preceding subsequence of the sequence of iterates {T n(k) x} k∈N . The proof is complete. Let X be an arbitrary nonempty set, T : X → X, and P := (P, ) be a nonempty poset. A mapping f : X → P (or T : X → X) has a k-forked point (or k-furcate point) p ∈ X if for arbitrary fixed integer k 1 the following equalities hold in the form f (T k p) = · · · = f (T p) = f (p) for some p ∈ X; (Rk) Frequently, we say that in this case (Rk), the mapping f : X → P or f : X → X has cycle or k-pair (p, T p, . . . , T k p) of bifurcation points, or that T : X → X has a k-forked or a k-forks point p ∈ X.
In connection with this, from the proof of Theorem 2.2, we obtain, as a direct extension of the preceding result, the following general statement. then for T there exist at least countable or finite k-forked points ξ t ∈ X, i.e., then the following equalities hold in the form for an arbitrary fixed integer k 1 and for some sequence {b t n (x)} n∈N in X which converges to ξ t ∈ X.
Interpretation and facts. We notice, first, that Theorem 2.1 hold even we are to make weaker the condition of lower ordered RBS-convergence, in the sense that this condition holds only for iteration sequences.
In this sense, let X be an arbitrary nonempty set, T : X → X, P := (P, ) be a nonempty poset, and B : X → P . A topological space X satisfies the condition of orbital lower ordered RBS-convergence iff {T n (x)} n∈N is an arbitrary iteration sequence in X with arbitrary x ∈ X and if B( Also, we shall introduce the concept of RBS-completeness in a space X for a function B : X → P , i.e., a topological space X is called lower ordered RBS-complete (orbital lower ordered RBS-complete) iff {an(x)} n∈N is an arbitrary sequence (an arbitrary iteration sequence) in X with arbitrary x ∈ X and if B(an(x)) → b = b(x) ∈ P as n → ∞ implies that {an(x)} n∈N has a convergent subsequence in X.
On the other hand, a function B : X → P is lower ordered RBS-continuous (orbital lower ordered RBS-continuous) at p ∈ X iff {an(x)} n∈N is an arbitrary sequence (an arbitrary iteration sequence) in X with arbitrary x ∈ X and if an(x) → p (n → ∞) implies that Second, we are now in a position to formulate the following explanations of the preceding theorems as corresponding equivalent forms: Theorem 2.4. Let T be a mapping of a topological space X into itself and let X be orbital lower ordered RBS-complete. If (B) holds and if B : X → P is an orbital lower ordered RBS-continuous map, then for T there exist at least countable or finite k-forked points ξ t ∈ X.
This result is contained in Theorem 2.2 as the case for k = 1, i.e., for the case of a forked point. In this sense we obtain an immediate result for P := R 0 + and :=≤.
In this sense, we shall introduce the concept of lower BCS-convergence in a topological space X for B : X → R 0 + , i.e., a topological space X satisfies the condition of lower BCS-convergence (orbital lower BCS-convergence) if {a n (x)} n∈N is an arbitrary sequence (an arbitrary iteration sequence) in X with arbitrary x ∈ X and if B(a n (x)) → b = b(x) 0 (n → ∞) implies that {a n (x)} n∈N has a convergent subsequence {a n(k) (x)} k∈N which converges to ξ ∈ X, where In connection with this, the preceding result is an extension of our former Localization Monotone Principle of fixed point proved for the first time in Ta s k o v i ć [1985]. Annotations. From (B) and Ordering Principle it directly follows that the mapping T of a topological space X into itself on the set X B,con has at least countable or finite fixed points ξt ∈ X as minimal elements of this set.
Even more, from Theorem 2.2 it follows that T , as a mapping of a topological space X into itself where X with the property of lower BCS-convergence, has a fixed point ξ ∈ X if and only if ξ is a minimal element of the set X B,con for which we have ξ = B,con T ξ.

Annotations.
A fine illustration for Theorem 2.2 is a well known statement in 1936 which was given by Fr e u d e n t h a l and H u r e w i c z in the following form: If (X, ρ) is a compact metric space and if T is a mapping of X into itself such that ρ T (x), T (y) < ρ[x, y] for all x, y ∈ X (x = y), (4) then the mapping T has a unique fixed point ξ ∈ X.
Indeed, first, since X is a compact space it follows that the condition of lower (orbital) BCSconvergence holds. Second, let B(x) := ρ(x, T x), thus applying Theorem 2.2 we have that there exists ξ ∈ X such that (Ri). But, from (4) for ξ = T ξ we obtain i.e., we obtain a contradiction. This means that ξ = T ξ for some ξ ∈ X. The uniqueness follows immediately from (4). The proof is complete.
In connection with this statement of Fr e u d e n t h a l-H u r e w i c z [1936] there exist more extensions. An extension of this statement to give E d e l s t e i n [1962] to change the compactness with the following weak condition in the form: if {T n (x)} n∈N is an arbitrary iteration sequence in X with arbitrary x ∈ X, then he has at least one convergent subsequence in X.
On the other hand we notice that this result of Freudenthal and Hurewicz in 1936 appeared, also, at the same year independently by N i e m y t z k i [1936].
These facts are direct examples for the preceding Theorem 2.2. Also, this facts can be great for further considerations specially in the fixed point theory.
Extensions theorems of Brouwer and Schauder. This part is primarily devoted to illustraing the preceding results, thus in the forks theory we obtain extensions of Brouwer and Schauder theorems.
Proposition 2.1. Let X be a convex subset in linear topological space Y and let T be a mapping of X into itself. Then there exists a lower semicontinuous function B : X → R 0 + such that inequality (B) holds.
Proof. Let B : X → R 0 + be a convex function with the property B((x + y)/2) ≥ B(x) for all x, y ∈ X, where B is a lower semicontinuous function on the convex set X. Then, immediate (B) holds, because 0 ≤ B((x + y)/2) − B(x) ≤ B(y) − B(x) for all x, y ∈ X. Thus, B(T (x)) ≤ B(x) for every x ∈ X, i.e., inequality (B) holds. The proof is complete.
Further, in this section we apply the monotone principle of forked points (for P := R 0 + and :=≤) to the Schauder's 54th problem in Scottish Book. In this sense we have the following extension and a new solution of Schauder's problme on linear topological spaces.  Proof. (Application of Theorem 2.2 ). From Proposition 2.1 (for convex lower semicontinuous function B := D → R 0 + ) inequality (B) holds on nonempty compact convex set D, thus direct it follows that B satisfies the condition of lower BCS-convergence, also, i.e., D and B satisfy all the required hypothesis in Theorem 2.2. Applying Theorem 2.2, in this case, we obtain that T has at least countable or finite forked points ξ k ∈ D, such that B(T (ξ k )) = B(ξ k ). This implies, form the facts of B, T (ξ k ) = ξ k . The proof is complete.
We are now in a position to formulate our following known applications. In this sense we obtain three fundamental famous principles of Brouwer, Banach and Schauder.
Theorem 2.5 (General Brouwer Theorem). Suppose that C is a nonemty convex, compact subset of R n , and that T : C → C is a continuous mapping. Then T has at least countable or finite fixed points in C.
In this sense, as a direct consequence of Theorem 2.5, we obtain the following well-known Brouwer's theorem.
Theorem 2.6 (Brouwer, [1912]). Suppose that C is a nonempty convex, compact subset of R n , and that T : C → C is a continuous mapping. Then T has a fixed point in C.
We also have, as an immediate and direct consequence of Theorem 2.2, as a version of the Schauder fixed point theorem.
Theorem 2.7 (Schauder, [1930]). Let C be a nonempty, compact, convex subset of a Banach space X, and suppose T : C → C is a continuous operator. Then T has a fixed point in C.
We notice that this statement is a direct translation of the Brouwer fixed point theorem to Banach spaces.
Proof. Since C is a convex and compact subset of Banach space, from Theorem 2.1 (Ordering Principle) and Theorem 2.2, we obtain this statement.
Theorem 2.8 (General Schauder Theorem). Let C be a nonempty, compact, convex subset of a Banach space X, and suppose T : C → C is a continuous operator. Then T has at least countable or finite fixed points in C.
This statement is a direct consequence of Theorem 2.1 (Ordering Principle) and Theorem 2.2. Also, the following result is a direct consequence of Theorem 2.1 (Ordering Principle) and Theorem 2.8.