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Axiom of Infinite Choice, transversal ordered spring spaces and fixed points
Fixed Point Theory and Applicationsvolume 2018, Article number: 10 (2018)
Abstract
This paper continues the study of the Axiom of Infinite Choice on transversal ordered spring spaces in terms of fixed point and increasing inductive sets. These principles unify a number of diverse results (about three thousand papers) in fixed point theory, especially recently published. Applications in partially ordered spaces and fixed point theory are also considered.
Introduction and history
Call a poset (:= partially ordered set) P increasing inductive (increasing chain complete) when every increasing sequence has an upper bound (the least upper bound, i.e., supremum) in P.
Lemma 1
(Increasing inductiveness, Tasković [1])
If P is an increasing inductive partially ordered set, then P has at least countable or finite maximal elements.
A brief proof of a special case of this fact may be found in Tasković [1]. This statement is de facto an equivalent of Lemma of Infinite Maximality by Tasković [2].
Theorem 1
(Axiom of Infinite Choice, Tasković [2])
Let \(P:=(P,\preccurlyeq )\) be a partially ordered set. Then the following statements are equivalent:

(a)
(Lemma of Infinite Maximality, Tasković [2].) Let P be an inductive partially ordered set. Then P has at least countable or finite maximal elements.

(b)
(Local form, Tasković [1].) Let P be an increasing inductive poset and f be an increasing mapping from P into P such that
$$ a\preccurlyeq f(a)\quad \textit{for some } a\in P, $$(T)then f has at least countable or finite fixed points in P.

(c)
(Tasković [3].) Let P be an increasing inductive poset and f be a mapping from P into P such that
$$ x\preccurlyeq f(x) \quad \textit{for all } x\in P, $$(M)then f has at least countable or finite fixed points in P.
We note that a brief proof of this statement based on the Axiom of Infinite Choice may be found in Tasković [2].
Based on Lemma 1 and Theorem 1 above, we are now in a position to formulate the following new statements and facts as direct consequences of the Axiom of Infinite Choice.
Corollary 1
(Tasković [4, p. 244])
Let \(P:=(P,\preccurlyeq )\) be a partially ordered set and f be an increasing mapping from P into P. If the following set \(P(\preccurlyeq f):=\{x\in P:x\preccurlyeq f(x)\}\) is nonempty such that there exists the supremum \(s:=\sup P(\preccurlyeq f)\), then f has at least countable or finite fixed points in P.
We notice that this result is a direct consequence of an application of the Axiom of Infinite Choice on the set \(P(\preccurlyeq f)\). In 1980 Tasković proved that f has at least one fixed point for the preceding case of Corollary 1.
Corollary 2
Let \(P:=(P,\preccurlyeq )\) be a partially ordered set and f be an increasing mapping from P into P such that
where every increasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb {N}\cup\{0\}}\) in P is bounded, then f has at least countable or finite fixed points in P.
Proof
Let us consider the subset \(P(\preccurlyeq f)\) of P given by \(P(\preccurlyeq f):=\{ x\in P:x\preccurlyeq f(x)\}\). By the hypothesis, we see that \(P(\preccurlyeq f)\) is a nonempty poset. Since \(x\preccurlyeq f(x)\) implies \(f(x)\preccurlyeq f(f(x))\), we see that f maps \(P(\preccurlyeq f)\) into \(P(\preccurlyeq f)\).
Because \(a\preccurlyeq f(a)\) and f is isotone, we find \(f(a)\preccurlyeq f^{2}(a)\) and inductively that \(f^{n}(a)\preccurlyeq f^{n+1}(a)\) for each \(n\in\Bbb{N}\cup\{ 0\}\) and some \(x\in P\). Thus, \(\{f^{n}(a):n\in\Bbb{N}\cup\{0\}\}\) is an increasing sequence of iterates which is bounded, and so the least upper bound of \(\{f^{n}(a):n\in\Bbb{N}\cup\{0\}\}\) exists. The system of chain C for which
contains the nonempty chain \(\{f^{n}(a)\mid n\in\Bbb{N}\cup \{0\}\}\) and therefore contains maximal chains \(M_{k}\) by Lemma 1 (or by the Lemma of Infinite Maximality). By assumption \(\mu_{k}=\sup M_{k}\in P\) exists, where \(\sup M_{k}\) is the least upper bound of \(M_{k}\). Since \(M_{k}\) satisfies (SC), we have \(x\preccurlyeq f(x)\preccurlyeq f(\mu_{k})\) for all \(x\in M_{k}\), so that \(\mu_{k}\preccurlyeq f(\mu_{k})\).
On the other hand, if \(\mu_{k}\notin M_{k}\), then the chain \(M_{k}\cup\{f^{n}(\mu_{k})\mid n\in\Bbb{N}\cup\{0\}\}\) properly contains \(M_{k}\) and satisfies (SC) in contradiction to the maximality of \(M_{k}\). Therefore, \(\mu_{k}\in M_{k}\) and also \(f(\mu_{k})\in M_{k}\), hence \(f(\mu_{k})\preccurlyeq \mu_{k}\). This makes \(\mu_{k}\) a fixed point of f, i.e., \(f(\mu_{k})=\mu_{k}\). The proof is complete. □
Annotation
From the above it follows that \(P(\preccurlyeq f)\) is a nonempty poset with the property that each nonempty increasing chain of \(P(\preccurlyeq f)\) has an upper bound, i.e., \(P(\preccurlyeq f)\) is an increasing inductive set, and f maps \(P(\preccurlyeq f)\) into \(P(\preccurlyeq f)\), and thus, by (c) of Theorem 1, we see that f has at least countable or finite fixed points as desired.
Preliminaries and main results
Spring upper ordered spaces. Let X be a nonempty set, and in what follows, let \(P:=(P,\preccurlyeq )\) be a partially ordered set such that \(a, b\in P\) and \(a\prec b\). The set (interval) \([a, b)\) is further defined by
The function \(A: X\times X\to[a, b)\subset P\) for \(a\prec b\) is called an upper spring ordered transverse (or upper spring ordered transversal) on a nonempty set X iff \(A(x, y)=a\) if and only if \(x=y\) for all \(x, y\in X\).
An upper spring ordered transversal space is a nonempty partially ordered set X (with ordering ) together with a given upper spring ordered transverse A on X, where every decreasing sequence \(\{u_{n}\}_{n\in\Bbb{N}}\) of elements in \([a, b)\) has a unique element \(u\in[a, b)\) as limit (in notation \(u_{n}\to u\) \((n\to\infty)\)). The element \(a\in[a, b) \subset P\) is called spring of space X (cf. [5]).
In 1986 we investigated the concept of upper spring ordered TCSconvergence in a space X, i.e., an upper spring ordered transversal space \(X:=(X, A)\) satisfies the condition of upper spring ordered TCSconvergence iff \(x\in X\) and if \(A(T^{n} (x), T^{n+1}(x)) \to a\) \((n\to\infty)\) implies that \(\{T^{n} (x)\}_{n\in\Bbb{N}}\) has a convergent subsequence in X, see Tasković [6].
In connection with the above, we shall introduce the concept of upper MCSconvergence, i.e., an upper spring ordered transversal space satisfies the condition of upper MCSconvergence (i.e., upper MSCcompleteness) iff \(x\in X\) and if \(A(f^{n}(x),f^{n+1}(x))\to a\) \((n\to\infty)\) implies that \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) is bounded in X.
We notice that the sequence \(\{x_{n}\}_{n\in\Bbb{N}}\) in the upper spring ordered transversal space is convergent (or upper convergent) in notation \(x_{n}\to x\) \((n\to\infty)\) iff \(A(x_{n}, x)\to a\) as \(n\to\infty\); or equivalently, for a decreasing sequence \(\{a_{n}\}_{n\in\Bbb{N}} \in[a, b)\) which converges to a, the following inequality holds:
or for n large enough.
On the other hand, in connection with this, the sequence \(\{x_{n}\}_{n \in\Bbb{N}}\) in X will be called upper fundamental (or upper spring fundamental) if the following inequality holds:
or for n and m large enough, where the decreasing sequence \(\{a_{n}\}_{n\in\Bbb{N}}\) in \([a, b)\) converges to a (cf. [5]).
An upper spring ordered transversal space is called upper complete (or upper spring complete) if any upper fundamental sequence \(\{x_{n}\}_{n\in\Bbb{N}}\) in X is upper convergent (to a point of X, of course).
On the other hand, in this paper we shall introduce the concept of upper MBVconvergence, i.e., an upper spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) satisfies the condition of upper MBVconvergence (i.e., upper MBVcompleteness) if every increasing sequence of iterates \(\{f^{n}(x)\} _{n\in\Bbb{N}\cup\{0\}}\) is bounded in X.
An immediate direct consequence of the preceding Corollary 2 is the following new result for upper spring ordered transversal spaces.
Corollary 3
Let f be an increasing mapping of an upper spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) into itself, where X satisfies the condition of upper MBVconvergence. If
then f has at least countable or at least finite fixed points in X.
The proof of this statement is an elementary fact because the condition of upper MBVcompleteness implies that every increasing sequence of iterates is bounded in Corollary 2.
As an immediate application of Corollary 3 directly, we obtain the following new consequence on upper spring ordered transversal spaces.
Corollary 4
Let f be an increasing mapping of an upper spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) into itself, where every increasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is upper fundamental. If
then f has at least countable or at least finite fixed points in X.
Proof
Because \(a\preccurlyeq f(a)\) and f is isotone, we find \(\{f^{n}(a)\}_{n\in\Bbb {N}\cup\{0\}}\) is an increasing sequence of iterates which is upper fundamental, i.e., bounded in X. It is easy to see that X satisfies all the required hypotheses in Corollary 3. Applying Corollary 3 to this case, we obtain this statement. The proof is complete. □
Corollary 5
(Partially ordered metric spaces)
Let f be an increasing mapping of an ordered metric space \(X:=(X,\preccurlyeq ,\rho)\) into itself, where every increasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is a Cauchy sequence. If
then f has at least countable or at least finite fixed points in X.
Proof
Since an ordered metric space is an example of an upper spring ordered transversal space, thus this statement follows directly from Corollary 4. The proof is complete. □
Corollary 6
(Upper spring ordered transversal spaces)
Let f be an increasing mapping of an upper spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) into itself, where every increasing and decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is bounded. If
or
then f has at least countable or at least finite fixed points in X.
The proof of this statement is a total analogy with the preceding proofs of Corollaries 2 and 3. Thus we omit it.
Corollary 7
(Partially ordered metric spaces)
Let f be an increasing mapping of an ordered metric space \(X:=(X,\preccurlyeq ,\rho)\) into itself, where every increasing and decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is a Cauchy sequence. If (M) or (N) holds, then f has at least countable or finite fixed points in X.
This result is an example for Corollary 6. Thus we omit the proof.
Applications on partially ordered metric and other spaces
This section is mainly devoted to some applications on partially ordered metric and other spaces. Some sufficient conditions for the upper MBVcompleteness are given, which has a direct consequence on the Cauchyness of sequences in partially ordered metric spaces. On the other hand, every increasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is also a Cauchy sequence in \(X:=(X,\preccurlyeq ,A)\). This fact (in this sense) includes (unifies) at least three thousand papers published recently. We note that a partially ordered metric space directly is an example of upper spring ordered transversal spaces.
Diametral φcontraction on metric spaces. In 1980 I proved the following result of a fixed point on metric space, which is one of the most known sufficient conditions (linear and nonlinear) for the existence of a unique fixed point, cf. Tasković [7, 8], and [9]. This result generalizes a great number of known results.
Theorem 2
(Tasković [7, p. 250, Theorem 1])
Let T be a mapping of a metric space \((X,\rho)\) into itself, and let X be Torbital complete. Suppose that there exists a function \(\varphi:\Bbb{R}_{+}^{0}\to\Bbb{R}_{+}^{0}:=[0,+\infty)\) satisfying
such that the following inequality holds:
for all \(x,y\in X\). If \(\operatorname {diam}\mathcal{O}(x)\in\Bbb{R}_{+}^{0}\) for every \(x\in X\), then T has a unique fixed point \(\xi\in X\) and \(\{T^{n}(b)\}_{n\in\Bbb{N}}\) converges to ξ for arbitrary \(b\in X\).
In connection with this result, we notice that this statement is well known as “the finest theorem of nonlinear functional analysis” for metric spaces.
In the context of Theorem 2, applying Corollary 5 to this case directly, we obtain the following result for diametral φcontractions, i.e., for mappings with properties (J) and (Iφ).
Corollary 8
Let T be an increasing mapping of an ordered metric space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists a function \(\varphi:\Bbb{R}_{+}^{0}\to\Bbb{R}_{+}^{0}\) satisfying (Iφ) such that (J). If \(a\preccurlyeq T(a)\) for some \(a\in X\) and if for every \(x\in X\), then T has a unique fixed point \(\xi\in X\) and \(\{T^{n}(b)\}_{n\in\Bbb{N}\cup\{0\}}\) converges to ξ for arbitrary \(b\in X\).
Annotation
In connection with the preceding inequality (J), we notice that Tasković [10] considered a special case of this condition. Also see Kurepa [11], Ohta–Nikaido [12], and Tasković [13]. It is interesting that in 1976 Kurepa proved a geometric interpretation of condition (J). See Fig. 1.
Transversal upper edges spaces. Let X be a nonempty set. The function \(\rho: X\times X\to[a,b]\) (or \(\rho:X\times X\to[a,b)\)) for \(a< b\) (\(a, b \in\Bbb{R}_{+}^{0}:=[0,+\infty)\)) is called an upper edges transverse on X (or upper edges transversal) iff \(\rho(x, y)=\rho(y, x)\), \(\rho(x,y)=a\) if and only if \(x=y\), and if there is a function \(\psi:[a,b]^{2}\to[a,b]\) such that
for all \(x, y, z \in X\).
An upper edges transversal space (or upper edges space) is a set X together with a given upper edges transverse on X. The function \(\psi: [a,b]^{2}\to[a,b]\) in (A) is called an upper bisection function.
From (A) it follows by induction that there is a function for \(a< b\) such that the following inequality holds:
for all \(x_{0}, x_{1},\ldots, x_{n} \in X\) and for an arbitrary fixed integer \(n\ge2\).
Example 1
(Metric spaces)
A fundamental first example of upper edges space is a metric space. Indeed, if \((X, d)\) is a metric space, then for the upper bisection function \(\psi(r, t)=r+t\), we have the following upper edges transverse \(\rho:X\times X\to [a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) defined by
for all \(x, y \in X\). Thus \((X, \rho)\) is an example of an upper edges transversal space. In general, every metric space is an example of an upper edges transversal space.
Example 2
(The extended real line \(\overline{\Bbb {R}}\))
The function f defined in \(\Bbb{R}\) by \(f(x)=x/(1+x)\) is a bijection on \(\Bbb{R}\) on the open interval \((1, 1)\subset\Bbb{R}\), and the inverse mapping g being defined by \(g(x)=x/(1x)\) for \(x<1\). Let \(\overline{\Bbb{R}}\) be the set which is the union of \(\Bbb{R}\) and two new elements written +∞ and −∞ (points at infinity); then we extend f to a bijection of \(\overline{\Bbb{R}}\) onto \([1, 1]\) by putting \(f(+\infty)=1\), \(f(\infty)=1\), and write again g for the inverse mapping.
We can apply this process described to define \(\overline{\Bbb{R}}\) as an upper edges transversal space by putting for the upper edges transverse \(\rho: \overline{\Bbb{R}}\times\overline{\Bbb{R}}\to [0, 2]\), that is,
for all \(x, y \in\overline{\Bbb{R}}\). (We notice that for \(x\ge0\) is \(\rho(+\infty, x)=1/(1+x)\), and for \(x\le0\) that is \(\rho(\infty, x)=1/(1+x)\)).
Example 3
(Transversal upper edges rspaces)
A fundamental third example of a transversal upper edges space is a transversal upper edges rspace for \(r\ge1\). Indeed, in 1998 this space was known as a transversal upper space. For \(\psi(s,t):=rs+rt\) (\(r\ge1\)), we obtain directly in this case the transversal upper edges rspace. This space is well known for \(r\ge1\) as an rmetric space (or as almost metric space) introduced in 1989 by Bakhtin [14]. Also see Czerwik [15].
On the other hand, a fundamental fourth example of a transversal upper edges space is a transversal upper edges rmax space (for \(r\ge1\)), known as rmax edges space. Indeed, in 1998 this space was known as a transversal rmax edges space. For \(\psi(s,t)=r\max\{s,t\}\) for \(r\ge1\), we obtain directly in this case the transversal upper edges rspace.
Annotation
We notice that the set (class) of all wellknown bmetric spaces \((b\ge1)\) can be a proper subset (subclass) of the set (class) all transversal rmax edges space for \(r\ge1\). See Tasković [16].
For any nonempty set Y in the upper edges transversal space X, the diameter of Y is defined as \(\operatorname {diam}(Y):=\sup \{\rho(x, y): x, y \in Y \}\); it is a real number in \([a,b]\), \(A\subset B\) implies \(\operatorname {diam}(A)\le \operatorname {diam}(B)\). The relation \(\operatorname {diam}(Y)=a\) holds if and only if Y is a onepoint set.
Elements of an upper edges transversal space will usually be called points. Given an upper edges transversal space \((X, \rho)\) with the bisection function \(g:[a,b]^{2}\to[a,b]\) and a point \(z \in X\), the open ball of center z and radius \(r>0\) is the set
The convergence \(x_{n}\to x\) as \(n\to\infty\) in the upper edges transversal space \((X, \rho)\) means that \(\rho(x_{n}, x)\to a\) as \(n\to\infty\), or equivalently, for every \(\varepsilon>0\) there exists an integer \(n_{0}\) such that the relation \(n\ge n_{0}\) implies \(\rho(x_{n}, x)< a+\varepsilon\).
The sequence \(\{x_{n}\}_{n\in\Bbb{N}}\) in the upper edges transversal space \((X, \rho)\) is called a transversal sequence (or upper Cauchy sequence) iff for every \(\varepsilon>0\) there is \(n_{0}=n_{0}(\varepsilon)\) such that
Let \((X, \rho)\) be an upper edges transversal space and \(T: X\to X\). We notice, from Tasković [17], that a sequence of iterates \(\{T^{n}(x)\}_{n\in\Bbb{N}}\) in X is said to be a transversal sequence if and only if
In this sense, an upper edges transversal space is called upper complete iff every transversal sequence converges. Also, a space \((X, \rho)\) is said to be upper orbitally complete (or upper Torbitally complete) iff every transversal sequence which is contained in for some \(x \in X\) converges in X.
For further facts on upper edges transversal spaces, see Tasković [6]. A function T mapping X into the reals is Torbitally lower semicontinuous at \(p \in X\) if \(\{x_{n}\}_{n\in\Bbb{N}}\) is a sequence in and \(x_{n}\to p\) (\(n\to\infty\)) implies that \(T(p)\le \operatorname {lim.inf}T(x_{n})\).
Let \((X, \rho)\) be an upper edges transversal space. A mapping \(T:X\to X\) is said to be upper edges contraction if there exists \(0\le\lambda<1\) such that
for all points \(x, y \in X\). In addition, let \((X, \rho_{X})\) and \((Y, \rho_{Y})\) be two upper edges transversal spaces and \(T:X\to Y\). We notice, from Tasković [6], that T is upper edges continuous at \(x_{0} \in X\) iff for every \(\varepsilon>0\) there exists \(\delta>0\) such that the relation
A typical first example of an upper edges continuous mapping is the upper edges contraction on the upper edges transversal space \((X, \rho)\). For further facts on the upper edges continuous mappings, see Tasković [6].
Proposition 1
Let f be an increasing mapping of a partially ordered transversal upper edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself, where X satisfies the condition of upper MCSconvergence. Suppose that there exists a number \(0\le\lambda<1\) such that
for every \(x\in X\). If \(r\preccurlyeq f(r)\) for some \(r\in X\), then f has at least countable or finite fixed points in X.
Proof
We see that a partially ordered transversal upper edges space \(X:=(X,\preccurlyeq ,\rho)\) is an example of an upper spring ordered transversal space. From (Cu) it follows \(\rho(f^{n}(x), f^{n+1}(x))\to a\) (\(n\to\infty\)), hence by upper MCScompleteness we obtain that every increasing sequence of iterates in the form \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) is bounded in X, i.e., X satisfies the condition of upper MBVcompleteness. It is easy to see that f and X satisfy all the required hypotheses in Corollary 3, thus f has at least countable or finite fixed points in X. The proof is complete. □
Proposition 2
Let f be an increasing mapping of a partially ordered transversal upper edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists \(0\le\lambda<1\) such that
for all \(x,y\in X\). If \(r\preccurlyeq f(r)\) for some \(r\in X\), then f has a unique fixed point in X. (The space X in this statement is not obligatory complete.)
Proof
Let \(y=f(x)\) in (Lc), then it is easy to see that f and X satisfy all the required hypotheses in Proposition 1. Uniqueness follows immediately from condition (Lc). The proof is complete. □
We are now in a position to formulate the following statement, which is a roof for a great number of known results on metric spaces in the fixed point theory.
Proposition 3
Let f be an increasing mapping of a partially ordered transversal upper edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists a function \(\varphi:[a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying
such that
for all \(x,y\in X\). If \(r\preccurlyeq f(r)\) for some \(r\in X\), then f has a unique fixed point in X. (The space X in this statement is not obligatory complete.)
Proof
We begin the proof with the following lemma (as a wellknown lemma) which is essential in the following context.
Lemma 2
(Tasković [13])
Let the mapping \(\varphi:[a,b]\to[a,b]\subset\Bbb{R}^{0}_{+}\) for \(a< b\) have the property (Iu). If the sequence \((x_{n})\) of nonnegative real numbers satisfies the condition
then the sequence \((x_{n})\) tends to a. The velocity of this convergence is not necessarily geometrical.
A brief first proof of this statement may be found in Tasković [6]. We can see other brief proofs for this in Tasković [7, 8], and [13].
Proof of Proposition 3
Let x be an arbitrary point in X. We can show then that the sequence of iterates \(\{f^{n}(x)\}_{n\in \Bbb{N}\cup\{0\}}\) is a transversal sequence. It is easy to verify (cf. [7, 9], and [13]) that the sequence \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) satisfies the following inequality:
for \(n\in\Bbb{N}\), and hence applying Lemma 2 to the sequence \((\operatorname {diam}\mathcal{O}(f^{n}(x)))\), we obtain \(\lim_{n\to\infty} \operatorname {diam}\mathcal{O} (f^{n}(x)) =a\). This implies that \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) is a transversal sequence in X. Hence, \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) is bounded.
Thus f and X satisfy all the required hypotheses in Corollary 4. Uniqueness follows immediately from (B). The proof is complete. □
As an immediate consequence of the preceding Proposition 3, we obtain directly the following interesting cases of (B):
(1) There exists a nondecreasing function \(\varphi: [a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying \(\limsup_{z\to t+0}\varphi(z)< t\) for every \(t\in(a,b]\) such that
for all \(x, y \in X\).
(2) (Special affine case of condition (B) for \(\varphi(t)=\alpha t+a(1\alpha)\).) There exists a constant \(\alpha\in[0, 1)\) such that, for all \(x, y \in X\), the following inequality holds:
i.e., equivalent to
(3) (The condition of \((m+k)\)polygon.) There exists a constant \(\lambda\in[0,1)\) such that, for all \(x,y\in X\), the following inequality holds:
for arbitrary fixed integers \(m,k\ge0\). (This is a linear condition for the diameter of a finite number of points).
(4) There exists a nondecreasing function \(\varphi:[a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying \(\limsup_{z\to t+0}\varphi(z)< t\) for every \(t\in(a,b]\) such that
for arbitrary fixed integers \(m,k\ge0\) and for all \(x, y \in X\).
(5) There exists an increasing mapping, i.e., \(x_{i}\le y_{i}\) (\(i=1,\ldots, 5\)) implies \(\varphi(x_{1},\ldots, x_{5})\le \varphi(y_{1},\ldots, y_{5})\), \(\varphi: [a,b]^{5}\to[a, b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying \(\limsup_{z\to t+0} \varphi(z, z, z, z, z)< t\) for every \(t\in(a,b]\) such that
for all \(x, y \in X\).
In connection with the preceding facts, we are now in a position to formulate a localization of Proposition 3 in the following form.
Proposition 4
Let f be an increasing mapping of a partially ordered transversal upper edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists a function \(\varphi:[a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying (Iu) such that
for every \(x\in X\). If \(r\preccurlyeq f(r)\) for some \(r\in X\), then f has at least countable or finite fixed points in X. (The space X in this statement is not obligatory complete.)
The proof of this statement is a total analogy with the preceding proof of Proposition 3. Thus we omit it.
Corollary 9
(Noncomplete partially ordered metric spaces)
Let f be an increasing mapping of a partially ordered metric space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists \(\lambda\in[0,1)\) such that
for every \(x\in X\). If \(r\preccurlyeq f(r)\) for some \(r\in X\), then f has at least countable or finite fixed points in X.
Proof
From (1) it follows that the sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) is a sequence of bounded variation, i.e., it is a Cauchy sequence. Applying Corollary 5 to this case, we obtain this statement. The proof is complete. □
Corollary 10
(Noncomplete partially ordered metric spaces)
Let f be an increasing mapping of a partially ordered metric space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists \(\lambda\in[0,1)\) such that
for all \(x,y\in X\). If \(r\preccurlyeq f(r)\) for some \(r\in X\), then f has a unique fixed point in X.
Proof
Let \(y=f(x)\) in (2), then it is easy to see that f and X satisfy all the required hypotheses in Corollary 9. Uniqueness follows immediately from condition (2). The proof is complete. □
On the other hand, in this paper we shall introduce the concept of spring sup MCSconvergence, i.e., an upper spring ordered transversal space satisfies the condition of spring sup MCSconvergence iff \(x\in X\) and if \(\sup_{i, j\ge n} A(T^{i}(x), T^{j}(x))\) or \(\sup_{i, j\ge2n} A(T^{i} (x), T^{j}(x))\) or \(\sup_{i, j\ge2n+1} A(T^{i} (x), T^{j}(x))\) converges to \(u, v, c\in[a, b)\) respectively implies that \(\{T^{n} (x)\}_{n\in\Bbb{N}}\) or \(\{T^{2n}(x)\}_{n\in\Bbb{N}}\) or \(\{T^{2n+1}(x)\}_{n\in\Bbb{N}}\) is a bounded sequence in X, respectively.
Also, if \(T:X\to X\), then a function \(x\mapsto A(x,T(x))\) is ordered Torbitally lower semicontinuous at \(\xi\in X\) if \(\{x_{n}\}_{n\in\Bbb{N}}\) is a sequence in \(\sigma(x,y,\infty):= \{x,y,T(x),T(y),T^{2}(x),T^{2}(y),\ldots\}\) and \(x_{n}\to\xi(n\to\infty)\) implies that \(A(\xi,T(\xi))\preccurlyeq \lim_{n\to\infty} A(T^{n}(x), T^{n+1}(x))\).
Theorem 3
(Monotone principle of fixed point)
Let T be an increasing mapping of an upper spring ordered transversal space into itself, where X satisfies the condition of spring sup MCSconvergence. Suppose that there exists a controlling function \(B:X\times X\to[a,b)\) for \(a\prec b\) such that
for all \(x, y\in X\) or
for all \(x, y\in X\). If for some \(r\in X\), then T has at least countable or finite fixed points in X. If additionally \(A(t, t)\preccurlyeq \sup\{A(s, t), A(t, s)\}\) for all \(s, t\in X\), then T has a unique fixed point in X.
Let be a partially ordered (with ordering ) topological space and \(T:X\to X\), where \(M: X\to [a,b)\subset P\) for \(a\prec b\). In this part (by Tasković [5]), we shall introduce the concept of local sup MCSconvergence in a space X, i.e., a topological space X satisfies the condition of local spring sup MCSconvergence iff \(x\in X\) and \(\sup_{i\ge n} M(T^{i}(x))\) or \(\sup_{i\ge2n} M(T^{i} (x))\) or \(\sup_{i\ge2n+1} M(T^{i}(x))\) converges to \(u, v, c\succcurlyeq a\) respectively implies that \(\{T^{n}(x)\}_{n\in\Bbb{N}}\) or \(\{ T^{2n}(x)\}_{n\in\Bbb{N}}\) or \(\{T^{2n+1}(x)\}_{n\in\Bbb{N}}\) is a bounded sequence in X, respectively.
We are now in a position to formulate the following theorem on partially ordered (with ordering ) topological spaces with nonnumerical transverses.
Theorem 4
(Localization monotone principle)
Let T be an increasing mapping of a partially ordered (with ordering ) topological space into itself, where X satisfies the condition of local spring sup MCSconvergence. Suppose that there exists a controlling function \(N: X\to [a,b)\subset P\) for \(a\prec b\) such that
and for some \(r\in X\). Then T has at least countable or finite fixed points in X.
An immediate consequence of the preceding statement is the following result.
Corollary 11
Let T be an increasing mapping of a partially ordered with ordering topological space into itself, where X satisfies the condition of local spring sup MCSconvergence. Suppose that there exists a controlling function \(N: X\to[a,b)\subset P\) for \(a\prec b\) such that
and for some \(r\in X\). Then T has at least countable or finite fixed points in X.
The proof of this statement is an elementary fact because condition (D′) implies condition (D).
Proof of Theorem 4
Let \(x\in X\) be an arbitrary point and \(n\in\Bbb{N}\cup \{0\}\) be any nonnegative integers. From (D) for \(T^{i}(x)\) we have \(M(T^{i+1}(x)) \preccurlyeq N(T^{i} (x))\preccurlyeq \sup_{z\in\sigma(T^{i} (x),\infty)} M(z)\), and hence
i.e., we obtain that \(\{\sup_{i\ge n} M(T^{i}(x))\}_{n\in\Bbb{N}}\) is a decreasing convergent sequence in \([a,b)\subset P\). This implies (from local spring sup MCSconvergence) that its sequence of iterates \(\{T^{n}(x)\}_{n\in \Bbb{N}}\) is a bounded sequence in X, i.e., X satisfies the condition of upper MBVcompleteness.
In the cases of other two sequences, in local spring sup MCSconvergence, the proof is a total analogy. Hence we omit the proof in these cases. Also, from [5] we obtain that every partially ordered topological space is a spring ordered transversal space. It is easy to see that T and X satisfy all the required hypotheses in Corollary 3, thus T has at least countable or finite fixed points in X. The proof is complete. □
Proof of Theorem 3
Let \(M(x):=A(x,T(x))\) and \(N(x):=B(x,T(x))\), then it is easy to see that A, B, and X satisfy all the required hypotheses in Theorem 4. Uniqueness follows immediately from conditions (3) and (3′). The proof is complete. □
In 1976 Tasković proved a localization theorem on a Cartesian product of metric spaces as a solution of Kuratowski’s problem of 1932, see Brown [18], Reny George and Brian Fisher [19], Tasković [5], etc. In this context the following result holds.
Let X be an arbitrary topological space. By Tasković [10], for a mapping \(T:X^{k}\to X\) (\(k\in\Bbb{N}\) is a fixed number), we will construct the iteration sequence \(\{T^{n}(u)\}_{n\in\Bbb{N}}\) for an arbitrary point \(u:=(u_{1},\ldots,u_{k})\in X^{k}\) in the following sense. Let \(T^{0}:=\)Identical mapping and
where \(\psi:X^{k}\to X^{k}\) defined by \(\psi(u_{1},\ldots,u_{k})=(u_{2},\ldots,u_{k+1})\) for the element of the form \(u_{k+1}=T(u_{1},\ldots, u_{k})\) and \(\psi^{0}=\mbox{Identical mapping}\).
We are now in a position to formulate the following statement for mappings of Cartesian product topological spaces.
Let for \(x:=(x_{1},\ldots, x_{k})\) and \(T:X^{k}\to X\) (\(k\in\Bbb{N}\) is a fixed number). Also, . A function \(t\mapsto A(t,T(t,\ldots,t))\) is Torbital lower semicontinuous at \(p\in X\) iff \(T^{n}(x)\to p\) \((n\to\infty)\) implies that \(A(p,T(p,\ldots,p))\leqslant\liminf_{n\to\infty} A(T^{n}(x),T(T^{n}(x),\ldots,T^{n+k1}(x)))\).
In the following, we consider an increasing function \(T:X^{k}\to X\) (for fixed \(k\in\Bbb{N}\)) on the partially ordered set X with ordering , i.e., if \((i=1,\ldots, k)\) implies that .
Corollary 12
Let T be an increasing mapping of a Cartesian product of topological spaces \(X^{k}\) (\(k\in\Bbb{N}\) is a fixed number) into X which is partially ordered with ordering , where satisfies the condition of spring sup MCSconvergence. Suppose that there exists a controlling function \(B:X\times X\to [a,b)\subset P\) for \(a\prec b\) such that
for all \(u_{1}, \ldots, u_{k}, u_{k+1}\in X\) or
for all \(u_{1}, \ldots, u_{k}, u_{k+1}\in X\), where \(x:=(u_{1}, \ldots, u_{k})\). If for some \(r\in X\), then the equation \(x=T(x,\ldots, x)\) has at least countable or finite solutions on X. If additionally \(A(t, t)\preccurlyeq \sup\{A(s, t), A(t, s)\}\) for all \(s, t\in X\), then there is unique \(\zeta\in X\) such that \(T(\zeta, \ldots, \zeta)=\zeta\).
The proof of this statement is a total analogy with the former proof of Corollary 3. Thus we omit it.
Further, a mapping \(M:\Bbb{R}\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) is called an upper (distribution) function if it is nonincreasing, leftcontinuous with \(\inf M=a\), and \(\sup M=b\). We will denote by the set of all upper (distribution) functions.
The next two spaces are very interesting examples of transversal upper spaces. First, an upper statistical space is a pair \((X,{\mathcal {R}})\), where X is an abstract set and \({\mathcal {R}}\) is a mapping of \(X\times X\) into the set of all upper (distribution) functions . We shall denote the upper (distribution) function \({\mathcal {R}}(u,v)\) by \(M_{u,v}(x)\) or \(M_{u,v}\), whence the symbol \(M_{u,v}(x)\) will denote the value of \(M_{u,v}\) at \(x\in\Bbb{R}\). The functions \(M_{u,v}\) are assumed to satisfy the following conditions: \(M_{u,v}=M_{v,u}\), \(M_{u,v}(c)=b\) for some \(c\in\Bbb{R}\), and
and if \(M_{u,r}(x)=a\) and \(M_{r,v}(y)=a\) imply \(M_{u,v}(x+y)=a\) for all \(u,v,r\in X\) and for all \(x,y\in\Bbb{R}\).
In view of the condition \(M_{u,v}(c)=b\), which evidently implies that \(M_{u,v}(x)=b\) for every \(x\le c\), condition (Eq) is equivalent to the statement \(u=v\) if and only if \(M_{u,v}(x)=A(x)\), where \(A(x)=b\) if \(x\le c\) and \(A(x)=a\) if \(x>c\). See Fig. 2.
Obviously, every metric space may be regarded as an upper statistical space of a special kind. One has only to set \(M_{u,v}(x)=A(xd(u,v))\) for every pair of points \((u,v)\) in the metric space \((X,d)\). Also, \(M_{u,v}(x)\) may be interpreted as the “measure” that the distance between u and v is less than x.
A very characteristic example, for further work, of the transversal upper edges spaces is the following space in the following form.
A transversal upper edges Tspace is a pair \((X,\rho)\), where X is a transversal upper edges space and where the upper (edges) transverse \(\rho[u,v]=M_{u,v}(x)\) satisfies \(M_{u,v}=M_{v,u}\), \(M_{u,v}(c)=b\) for some \(c\in\Bbb{R}\) and (Eq).
Next, the concept of a neighborhood can be introduced and defined with the aid of the upper edges transverse. In fact, neighborhoods in transversal upper edges spaces may be defined in several nonequivalent ways. Here, we shall consider only one of these.
If \(p\in X\), \(\mu> c\) for some \(c\in\Bbb{R}\) and r is a positive real, then a \((\mu,r)\)neighborhood of p, denoted by \(U_{p}(\mu,r)\), is defined by
Corollary 13
Let T be an increasing mapping of a transversal upper edges Tspace, partially ordered with ordering , into itself, where X satisfies the condition of spring sup MCSconvergence. Suppose that there exists an upper function \(K_{x, y}(t)\), as a controlling function, such that
for all \(x, y\in X\) or
for all \(x, y\in X\), where \(\varphi: \Bbb{R}\to\Bbb{R}\) is an increasing function satisfying \(\varphi^{n}(t)\to+\infty\) \((n\to\infty)\). If for some \(r\in X\), then T has a unique fixed point in X.
The proof of this statement is a total analogy with the preceding proofs as consequences of the main statements. Thus we omit it.
In connection with the above, applying our general principle of transpose for nonnumerical transverses (see Tasković [5, p. 89]) to Corollary 13, we get an extended and generalized version of this result in the following sense.
Indeed, let be a totally ordered set. A mapping for \(a\prec b\), where \(P:=(P,\preccurlyeq )\) is a partially ordered set is called an upper ordered (distribution) function if it is nonincreasing with \(\inf M=a\) and \(\sup M=b\). We will denote by the set of all upper ordered (distribution) functions.
The next two spaces are very interesting examples of transversal ordered upper spaces. First, an upper ordered statistical space is a pair \((X,{\mathcal {R}})\), where X is an abstract set and \({\mathcal {R}}\) is a mapping of \(X\times X\) into the set of all upper ordered (distribution) functions . We shall denote the upper ordered (distribution) function \({\mathcal {R}}(u,v)\) by \(M_{u,v}(x)\) or \(M_{u,v}\), hence the symbol \(M_{u,v}(x)\) will denote the value of \(M_{u,v}\) at . The functions \(M_{u,v}\) are assumed to satisfy the following conditions: \(M_{u,v}=M_{v,u}\), \(M_{u,v}(c)=b\) for some , and
and if \(M_{u,r}(x)=a\) and \(M_{r,v}(y)=a\) imply \(M_{u,v}(x+y)=a\) for all \(u,v,r\in X\) and for all .
In view of the condition \(M_{u,v}(c)=b\), which evidently implies that \(M_{u,v}(x)=b\) for every \(x\preccurlyeq c\), condition (Eq) is equivalent to the statement \(u=v\) if and only if \(M_{u,v}(x)=A(x)\), where \(A(x)=b\) if \(x\preccurlyeq c\) and \(A(x)=a\) if \(x\succ c\). See Fig. 2.
Also, \(M_{u,v}(x)\) may be interpreted as the “measure” that the distance between u and v is less than .
A very characteristic example, for further work, of the transversal ordered upper edges spaces is the following space in the following form.
A transversal upper ordered edges Tspace is a pair \((X,\rho)\), where X is a transversal upper ordered edges space and where the upper ordered (edges) transverse \(\rho(u,v)=M_{u,v}(x)\) satisfies \(M_{u,v}=M_{v,u}\), \(M_{u,v}(c)=b\) for some , and (Eq).
If \(p\in X\), \(\mu\succ c\) for some and \(r\succ a\), then a \((\mu,r)\)neighborhood of p, denoted by \(U_{p}(\mu,r)\), is defined by
Applying our general principle of transpose for partially ordered sets to Corollary 13 and to upper spring ordered transversal space for \(A(u,v):=M_{u,v}(x)\) directly, we get an extended and generalized version of Corollary 13 in the following form.
Corollary 14
Let T be an increasing mapping of an upper transversal spring ordered space for \(A:=M_{u,v}(t)\) with the property (Eq) into itself, where X satisfies the condition of spring sup MCSconvergence. Suppose that there exists an increasing function ( is a totally ordered set) such that \(t\prec\varphi(t)\) for every and for some the following equality holds:
and there exists an upper function \(K_{x, y}(t)\), as a controlling function, such that
for all \(x, y\in X\) or
for all \(x, y\in X\). If for some \(r\in X\), then T has a unique fixed point in X.
The proof of this statement is a total analogy with the preceding proofs as consequences of the main statements. Thus we omit it.
Corollary 15
(Tasković [6, p. 549])
Let T be an increasing mapping of a transversal upper edges Tspace (partially ordered with ordering ), into itself, where X satisfies the condition of spring sup MCSconvergence. Suppose that there exists an increasing function \(\varphi:[c,+\infty)\to[c,+\infty)\) for some fixed \(c\in\Bbb{R}\) satisfying
such that :
for all \(u,v\in X\) and for every \(x>c\). If for some \(r\in X\), then T has a unique fixed point in X.
We notice that in 2001 Tasković proved a special case of this statement on a transversal upper edges Tspace \(X:=(X,M_{u,v}(t))\). Also, in 1998 Tasković introduced the concept of upper, lower, and middle transversal edges spaces.
In the context of this statement the following conditions are special cases of condition :
(1) There exists a constant \(0< k<1\) such that, for all \(p,q\in X\), for some fixed \(c\in\Bbb{R}\), and for every \(x>c\), the following inequality holds:
(2) There exists a nondecreasing function \(\varphi:[c,+\infty)\to[c,+\infty)\) for some fixed \(c\in\Bbb{R}\) satisfying (As) such that, for some fixed \(0< k<1\), for all \(u,v\in X\), and for every \(x>c\), the following inequality holds:
(3) There exists a constant \(0< k<1\) such that, for all \(u,v\in X\), for some fixed \(x\in\Bbb{R}\), and for every \(x>c\), the following inequality holds:
or in an equivalently form as
Corollary 16
(Tasković [6, p. 558])
Let T be an increasing mapping of an upper spring ordered transversal space for \(A(u,v):=M_{u,v}(t)\) with the property (Eq) into itself, where X satisfies the condition of upper MCSconvergence. Suppose that there exists an increasing function ( is a totally ordered set) such that \(t\prec\varphi(t)\) for every and for some fixed the following equality holds:
and
for all \(u,v\in X\) and for every \(x\succ c\). If for some \(r\in X\), then T has a unique fixed point in X.
Proof
Applying the general principle of transpose for posets to Corollary 15, we get a directly extended and generalized version of a result given in Corollary 15. The proof is complete. □
Annotations
The results of Corollaries 15 and 16 are two immediate consequences of Corollary 3. Uniqueness follows immediately from the conditions of given inequalities.
Spring lower ordered spaces. Let X be a nonempty set, and let \(P:=(P,\preccurlyeq )\) be a partially ordered set such that \(a, b\in P\) and \(a\prec b\). The set (interval) \((a, b]\) is defined by
The function \(A: X\times X\to(a, b]\subset P\) for \(a\prec b\) is called a lower spring ordered transverse (or lower spring ordered transversal) on a nonempty set X iff \(A(x, y)=b\) if and only if \(x=y\) for all \(x, y\in X\).
A lower spring ordered transversal space is a nonempty partially ordered set X (with ordering ) together with a given lower spring ordered transverse A on X, where every increasing sequence \(\{u_{n}\}_{n\in\Bbb{N}}\) of elements in \((a, b]\) has a unique element u in \((a, b]\) as limit (in notation \(u_{n}\to u\) \((n\to\infty)\)). The element \(b\in(a, b] \subset P\) is called spring of space X (cf. [5]).
In 1986 we investigated the concept of lower spring ordered TCSconvergence in a space X, i.e., a lower spring ordered transversal space \(X:=(X, A)\) satisfies the condition of lower spring ordered TCSconvergence iff \(x\in X\) and if \(A(T^{n} (x), T^{n+1}(x)) \to b\) \((n\to\infty)\) implies that \(\{T^{n} (x)\}_{n\in\Bbb{N}}\) has a convergent subsequence in X, see Tasković [5].
In connection with the above, we shall introduce the concept of lower MCSconvergence, i.e., a lower spring ordered transversal space satisfies the condition of lower MCSconvergence (i.e., lower MSCcompleteness) iff \(x\in X\) and if \(A(f^{n}(x),f^{n+1}(x))\to b\) \((n\to\infty)\) implies that \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) is bounded X.
We notice that the sequence \(\{x_{n}\}_{n\in\Bbb{N}}\) in the lower spring ordered transversal space \(X:=(X,A)\) is convergent (or lower convergent) in notation \(x_{n}\to x\) \((n\to\infty)\) iff \(A(x_{n}, x)\to b\) as \(n\to\infty\); or equivalently, for an increasing sequence \(\{b_{n}\}_{n\in\Bbb{N}} \in(a, b]\) which converges to b, the following inequality holds:
or for n large enough.
On the other hand, in connection with this, the sequence \(\{x_{n}\}_{n \in\Bbb{N}}\) in X will be called lower fundamental (or lower spring fundamental) if the following inequality holds:
or for n and m large enough, where the increasing sequence \(\{b_{n}\}_{n\in\Bbb{N}}\in(a, b]\) converges to b (cf. [5]).
A lower spring ordered transversal space is called lower complete (or lower spring complete) if any lower fundamental sequence \(\{x_{n}\}_{n\in\Bbb{N}}\) in X is lower convergent (to a point of X, of course).
In connection with the preceding facts, based on Lemma 1 and Theorem 1, we are now in a position to formulate the following new statements as direct consequences of the Axiom of Infinite Choice.
Corollary 17
(Tasković [4, p. 244])
Let \(P:=(P,\preccurlyeq )\) be a partially ordered set and f be an increasing mapping from P into P. If the following set \(P(f\preccurlyeq ):=\{x\in P:f(x)\preccurlyeq x\}\) is nonempty such that there exists the infimum \(I:=\inf P(f\preccurlyeq )\), then f has at least countable or finite fixed points.
For the first time, in 1980 Tasković proved that f has at least one fixed point for the preceding case of Corollary 17. Further application of the Axiom of Infinite Choice follows Corollary 17, which is a dual form of Corollary 1. The proof is a total analogy with the proof of Corollary 2.
Corollary 18
Let \(P:=(P,\preccurlyeq )\) be a partially ordered set and f be an increasing mapping from P into P such that
where every decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb {N}\cup\{0\}}\) in P is bounded, then f has at least countable or finite fixed points in P.
This statement is a dual form of Corollary 2. Thus we omit the proof of Corollary 18.
Lower spring ordered transversal spaces. In this paper we shall introduce the concept of lower MBVconvergence, i.e., a lower spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) satisfies the condition of lower MBVconvergence (i.e., lower MBVcompleteness) if every decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is bounded.
As an immediate direct consequence of the preceding Corollary 18, we give the following new result for lower spring ordered transversal spaces.
Corollary 19
Let f be an increasing mapping of a lower spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) into itself, where X satisfies the condition of lower MBVconvergence. If
then f has at least countable or at least finite fixed points in X.
The proof of this statement is an elementary fact because the condition of lower MBVcompleteness implies that every decreasing sequence of iterates is bounded in Corollary 18.
Corollary 20
Let f be an increasing mapping of a lower spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) into itself, where every decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is lower fundamental. If
then f has at least countable or at least finite fixed points in X.
Proof
Since \(f(a)\preccurlyeq a\) and f is isotone, we find \(\{f^{n}(x)\}_{n\in\Bbb {N}\cup\{0\}}\) is a decreasing sequence of iterates which is lower fundamental, i.e., bounded in X. It is easy to see that X satisfies all the required hypotheses in Corollary 19. Applying Corollary 19 to this case, we obtain this statement. The proof is complete. □
Corollary 21
(Partially ordered metric spaces)
Let f be an increasing mapping of an ordered metric space \(X:=(X,\preccurlyeq ,A)\) into itself, where every decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is a Cauchy sequence. If
then f has at least countable or at least finite fixed points in X.
Proof
Since an ordered metric space is an example of a lower spring ordered transversal space, thus this statement follows directly from Corollary 20. The proof is complete. □
Corollary 22
(Lower spring ordered transversal spaces)
Let f be an increasing mapping of a lower spring ordered transversal space \(X:=(X,\preccurlyeq ,A)\) into itself, where every increasing and decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is bounded. If
or
then f has at least countable or at least finite fixed points in X.
The proof of this statement is a total analogy with the preceding proofs of Corollaries 2 and 3 (because Corollary 22 is a dual form of Corollary 6). Thus we omit it.
Corollary 23
(Partially ordered metric spaces)
Let f be an increasing mapping of an ordered metric space \(X:=(X,\preccurlyeq ,\rho)\) into itself, where every increasing and decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is a Cauchy sequence. If (R) or (E) holds, then f has at least countable or finite fixed points in X.
This result is an example for Corollary 22. Also, Corollary 23 is a dual form of Corollary 7. Thus we omit the proof.
Transversal lower edges spaces. Let X be a nonempty set. The function \(\rho: X \times X \to[a,b] \subset\Bbb{R}_{+}^{0}\) (or \(\rho:X\times X\to(a,b]\subset \Bbb{R}_{+}^{0}\)) for \(a< b\) is called a lower edges transverse on X (or lower edges transversal) iff \(\rho(x, y)=\rho(y, x)\), \(\rho(x, y)=b\) if and only if \(x=y\), and if there is a function \(d: [a,b]^{2}\to[a,b]\) such that
for all \(x, y, z \in X\).
A lower edges transversal space (or lower edges space) is a set X together with a given lower edges transverse on X. The function \(d:[a,b]^{2}\to[a,b]\) in (Ca) is called a lower bisection function.
From (Ca) it follows by induction that there is a function for \(a< b\) such that the following inequality holds:
for all \(x_{0}, x_{1},\ldots, x_{n} \in X\) and for an arbitrary fixed integer \(n\ge2\).
Example 4
(Metric spaces)
A fundamental first example of lower edges transversal space is a metric space. Indeed, if \((X, q)\) is a metric space, then for the lower bisection function \(d(r, t)=r+t\), we have the following lower edges transverse \(\rho:X\times X\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) defined by
for all \(x, y \in X\). Thus \((X, \rho)\) is an example of a lower edges transversal space. In general, every metric space is an example of a lower edges transversal space.
Example 5
(Lower probabilistic spaces)
A mapping \(F: \Bbb{R}\to\Bbb{R}_{+}^{0}\) is called a distribution function if it is nondecreasing, leftcontinuous with \(\inf F=0\) and \(\sup F=1\). We will denote by the set of all distribution functions. We shall denote the distribution function by \(F_{p, q}(x)\), whence \(F_{p, q}(x)\) will denote the value of \(F_{p, q}\) at \(x\in\Bbb{R}\).
An example of lower edges transversal space is a lower probabilistic space which is a nonempty set X together with the functions \(F_{p, q}(x)\) with the following properties: \(F_{p, q}(x)=F_{q, p}(x)\), \(F_{p, q}(0)=0\),
and if there is a nondecreasing function \(\tau:[0, 1]^{2}\to[0, 1]\) with the property \(\tau(t, t)\ge t\) for all \(t\in[0, 1]\) such that
for all \(p, q, r \in X\) and for all \(x, y\ge0\). If we choose a lower bisection function \(d: [0,1]^{2}\to[0, 1]\) such that \(d=\tau\) (from (Nm)), then we immediately obtain that every lower probabilistic space, for \(\rho[p, q]=F_{p, q}(x): X \times X \to[0, 1]\), is a lower edges transversal space.
Example 6
(Transversal lower edges rspaces)
A fundamental example of a transversal lower edges space is a transversal lower edges rspace (for \(0< r\le1\)). In 1998 this space was known as a transversal lower space. For \(d(s,t):=r\min\{s,t\}\) and \(0< r\le1\), we obtain directly in this case the transversal lower edges rspace. This space is known for \(0< r\le1\) as a transversal lower edges rmin space or only as an rmin edges space. For this, see Tasković [16].
For any nonempty set S in the lower edges transversal space X, the diameter of S is defined as \(\operatorname {diam}(S):=\inf \{\rho(x, y): x, y \in S \}\); it is a real number in \([a,b]\), \(A\subset B\) implies \(\operatorname {diam}(B)\le \operatorname {diam}(A)\). The relation \(\operatorname {diam}(S)=b\) holds if and only if S is a onepoint set.
Elements of a lower edges transversal space will usually be called points. Given a lower edges transversal space \((X, \rho)\), with the bisection function \(d:[a,b]^{2}\to[a,b]\) and a point \(z \in X\), the open ball of center z and radius \(r>0\) is the set
The convergence \(x_{n}\to x\) as \(n\to\infty\) in the lower edges transversal space \((X, \rho)\) means that \(\rho(x_{n}, x)\to b\) as \(n\to\infty\); or equivalently, for every \(\varepsilon>0\), there exists an integer \(n_{0}\) such that the relation \(n\ge n_{0}\) implies \(\rho(x_{n}, x)>b\varepsilon\).
The sequence \(\{x_{n}\}_{n\in\Bbb{N}}\) in the lower edges transversal space \((X, \rho)\) is called a transversal sequence (or a lower Cauchy sequence) iff for every \(\varepsilon>0\) there is \(n_{0}=n_{0}(\varepsilon)\) such that
Let \((X, \rho)\) be a lower edges transversal space and \(T:X\to X\). We notice, from Tasković [16], that a sequence of iterates \(\{T^{n}(x)\}_{n\in\Bbb{N}}\) in X is said to be a transversal sequence if and only if
In this sense, a lower edges transversal space is called lower complete iff every transversal sequence converges.
Also, a space \((X, \rho)\) is said to be lower orbitally complete (or lower Torbitally complete) iff every transversal sequence which is contained in for some \(x \in X\) converges in X.
A function T mapping X into the reals is Torbitally upper semicontinuous at \(p \in X\) iff \(\{x_{n}\}_{n\in\Bbb{N}}\) is a sequence in and \(x_{n}\to p\) (\(n\to\infty\)) implies that \(T(p)\ge \operatorname {lim.sup}T(x_{n})\).
Let \((X, \rho)\) be a lower edges transversal space. A mapping \(T: X\to X\) is said to be lower edges contraction if there exists \(0\le\lambda<1\) such that
for all points \(x, y \in X\). For further facts on the lower edges contraction, see Tasković [6].
Let \((X, \rho_{X})\) and \((Y, \rho_{Y})\) be two lower edges transversal spaces, and let \(T: X\to Y\).
We notice from Tasković [6] that T is lower edges continuous at \(x_{0} \in X\) iff for every \(\varepsilon>0\), there exists \(\delta>0\) such that the relation
A typical first example of a lower edges continuous mapping is the lower edges contraction on the lower edges transversal space \((X, \rho)\). For further facts on the lower edges continuous mappings, see Tasković [6].
In connection with the preceding facts, we are now in a position to formulate a localization for lower edges contractions in the following form.
Proposition 5
Let f be an increasing mapping of a partially ordered transversal lower edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself, where X satisfies the condition of the lower MCSconvergence. Suppose that there exists \(0\le\lambda<1\) such that
for every \(x\in X\). If \(f(r)\preccurlyeq r\) for some \(r\in X\), then f has at least countable or finite fixed points in X.
Proof
We notice that a partially ordered transversal lower edges space \(X:=(X,\preccurlyeq , \rho)\) is an example of a lower spring ordered transversal space. From (Cl) it follows \(\rho(f^{n}(x), f^{n+1}(x))\to b\) \((n\to\infty)\), hence by lower MCScompleteness, we obtain that every decreasing sequence of iterates \(\{f^{n}(x)\}_{n\in\Bbb{N}\cup\{0\}}\) in X is bounded, i.e., X satisfies the condition of lower MBVcompleteness. It is easy to see that f and X satisfy all the required hypotheses in Corollary 19, thus f has at least countable or finite fixed points in X. The proof is complete. □
Proposition 6
Let f be an increasing mapping of a partially ordered transversal lower edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists \(0\le\lambda<1\) such that
for all \(x,y\in X\). If \(f(r)\preccurlyeq r\) for some \(r\in X\), then f has a unique fixed point in X. (The space X in this statement is not obligatory complete).
Proof
Let \(y=f(x)\) in (Ld), then it is easy to see that f and X satisfy all the required hypotheses in Proposition 5. Uniqueness follows immediately from condition (Ld). The proof is complete. □
Proposition 7
Let f be an increasing mapping of a partially ordered transversal lower edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists a function \(\varphi:[a,b]\to[a,b]\) satisfying
such that
for all \(x,y\in X\). If \(f(r)\preccurlyeq r\) for some \(r\in X\), then f has a unique fixed point in X. (The space X in this statement is not obligatory complete.)
The proof of this statement is a total analogy with the preceding proofs of Propositions 5 and 6. Thus we omit it.
A brief proof of this statement may be found in Tasković [6] with application of the following context.
Lemma 3
(Tasković [6])
Let the mapping \(\varphi:[a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) have the property (Id). If the sequence \((x_{n})\) of nonnegative real numbers satisfies the condition
then the sequence \((x_{n})\) tends to b. The velocity of this convergence is not necessarily geometrical.
A brief first proof of this statement may be found in Tasković [6]. We notice that Lemma 3 is a dual form of Lemma 2.
As an immediate consequence of the preceding Proposition 7, we obtain directly the following interesting cases of (D):
(6) There exists a nondecreasing function \(\varphi: [a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying \(\liminf_{z\to t0}\varphi(z)>t\) for every \(t\in[a, b)\) such that
for all \(x, y \in X\).
(7) (Special affine case of condition (D) for \(\varphi(t)=\alpha t+b(1\alpha)\).) There exists a constant \(\alpha\in[0, 1)\) such that, for all \(x, y \in X\), the following inequality holds:
i.e., equivalent to
(8) (The condition of \((m+k)\)polygon.) There exists a constant \(\lambda\in[0,1)\) such that, for all \(x,y\in X\), the following inequality holds:
for arbitrary fixed integers \(m,k\ge0\). (This is a linear condition for the diameter of a finite number of points.)
(9) There exists a nondecreasing function \(\varphi: [a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying \(\liminf_{z\to t0}\varphi(z)>t\) for every \(t\in[a, b)\) such that
for arbitrary fixed integers \(m,k\ge0\) and for all \(x, y \in X\).
(10) There exists an increasing mapping \(\psi:[a,b]^{5}\to[a, b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying \(\liminf_{z\to t0} \psi(z, z, z, z, z)>t\) for every \(t\in[a, b)\) such that
for all \(x, y \in X\).
In connection with the preceding facts, we are now in a position to formulate a localization of Proposition 7 in the following form.
Proposition 8
Let f be an increasing mapping of a partially ordered transversal lower edges space \(X:=(X,\preccurlyeq ,\rho)\) into itself. Suppose that there exists a function \(\varphi:[a,b]\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) satisfying (Id) such that
for every \(x\in X\). If \(f(r)\preccurlyeq r\) for some \(r\in X\), then f has at least countable or finite fixed points in X. (The space X in this statement is not obligatory complete.)
The proof of this statement is a total analogy with the preceding proof of Proposition 7. Thus we omit it.
A lower spring ordered transversal space is called lower complete (or lower spring complete) if any lower fundamental sequence \(\{x_{n}\}_{n\in\Bbb{N}}\) in X is lower convergent (to a point of X, of course).
On the other hand, in this paper we shall introduce the concept of spring inf MCSconvergence, i.e., a lower spring ordered transversal space satisfies the condition of spring inf MCSconvergence iff \(x\in X\) and if \(\inf_{i, j\ge n} A(T^{i}(x), T^{j}(x))\) or \(\inf_{i, j\ge2n} A(T^{i} (x), T^{j}(x))\) or \(\inf_{i, j\ge2n+1} A(T^{i} (x), T^{j}(x))\) converges to \(u, v, c\in(a, b]\) respectively implies that \(\{T^{n} (x)\}_{n\in\Bbb{N}}\) or \(\{T^{2n}(x)\}_{n\in\Bbb{N}}\) or \(\{T^{2n+1}(x)\}_{n\in\Bbb{N}}\) is a bounded sequence in X, respectively.
Theorem 5
(Monotone principle of fixed point)
Let T be an increasing mapping of a lower spring ordered transversal space into itself, where X satisfies the condition of spring inf MCSconvergence. Suppose that there exists a controlling function \(B:X\times X\to(a,b]\) for \(a\prec b\) such that
for all \(x, y\in X\) or
for all \(x, y\in X\). If for some \(r\in X\), then T has at least countable or finite fixed points in X.
If additionally \(A(t, t)\succcurlyeq \inf\{A(s, t), A(t, s)\}\) for all \(s, t\in X\), then T has a unique fixed point in X.
Let be a partially ordered with ordering topological space and \(T:X\to X\), where \(M: X\to (a,b]\subset P\) for \(a\prec b\). In this part, we shall introduce the concept of local spring inf MCSconvergence in a space X, i.e., a topological space X satisfies the condition of local spring inf MCSconvergence iff \(x\in X\) and \(\inf_{i\ge n} M(T^{i} (x))\) or \(\inf_{i\ge2n} M(T^{i} (x))\) or \(\inf_{i\ge2n+1} M(T^{i}(x))\) converges to \(u, v, c\preccurlyeq b\) respectively implies that \(\{T^{n}(x)\}_{n\in\Bbb{N}}\) or \(\{ T^{2n}(x)\}_{n\in\Bbb{N}}\) or \(\{T^{2n+1}(x)\}_{n\in\Bbb{N}}\) is a bounded sequence in X, respectively.
We are now in a position to formulate the following theorem on partially ordered (with ordering ) topological spaces with nonnumerical transverses.
Theorem 6
(Localization monotone principle)
Let T be an increasing mapping of a partially ordered with ordering topological space into itself, where X satisfies the condition of local spring inf MCSconvergence. Suppose that there exists a controlling function \(N: X\to (a,b]\subset P\) for \(a\prec b\) such that
and for some \(r\in X\). Then T has at least countable or finite fixed points in X.
An immediate consequence of the preceding statement is the following result.
Corollary 24
Let T be an increasing mapping of a partially ordered with ordering topological space into itself, where X satisfies the condition of local spring inf MCSconvergence. Suppose that there exists a controlling function \(N: X\to(a,b]\subset P\) for \(a\prec b\) such that
and for some \(r\in X\). Then T has at least countable or finite fixed points in X.
The proof of this statement is an elementary fact because condition (D′) implies condition (D).
Proof of Theorem 6
Let \(x\in X\) be an arbitrary point and \(n\in\Bbb{N}\cup \{0\}\) be any nonnegative integers. From (D) for \(T^{i}(x)\), we have \(M(T^{i+1}(x)) \succcurlyeq N(T^{i} (x))\succcurlyeq \inf_{z\in\sigma(T^{i} (x),\infty)} M(z)\), and hence
i.e., we obtain that \(\{\inf_{i\ge n} M(T^{i}(x))\}_{n\in\Bbb{N}}\) is an increasing convergent sequence in \((a,b]\subset P\). This implies (from local spring inf MCSconvergence) that its sequence of iterates \(\{T^{n}(x)\}_{n\in \Bbb{N}}\) is a bounded sequence in X, i.e., X satisfies the condition of lower MBVcompleteness.
In the cases of other two sequences, in local spring inf MCSconvergence, the proof is a total analogy. Hence we omit the proof in these cases. It is easy to see that T and X satisfy all the required hypotheses in Corollary 19, thus T has at least countable or finite fixed points in X. The proof is complete. □
Proof of Theorem 5
Let \(M(x):=A(x,T(x))\) and \(N(x):=B(x,T(x))\), then it is easy to see that A, B, and X satisfy all the required hypotheses in Theorem 6. Uniqueness follows immediately from conditions (8) and (8′). The proof is complete. □
Corollary 25
Let T be an increasing mapping of a Cartesian product of topological spaces \(X^{k}\) (\(k\in\Bbb{N}\) is a fixed number) into X which is partially ordered with ordering , where satisfies the condition of spring inf MCSconvergence. Suppose that there exists a controlling function \(B:X\times X\to (a,b]\subset P\) for \(a\prec b\) such that
for all \(u_{1}, \ldots, u_{k}, u_{k+1}\in X\) or
for all \(u_{1}, \ldots, u_{k}, u_{k+1}\in X\), where \(x:=(u_{1}, \ldots, u_{k})\). If for some \(r\in X\), then the equation \(x=T(x,\ldots, x)\) has at least countable or finite solutions on X. If additionally \(A(t, t)\succcurlyeq \inf\{A(s, t), A(t, s)\}\) for all \(s, t\in X\), then there is a unique \(\zeta\in X\) such that \(T(\zeta, \ldots, \zeta)=\zeta\).
Otherwise, a transversal edges space (or a middle transversal edges space) is an upper and a lower transversal edges space simultaneously.
As an important example of transversal lower edges spaces, we have Menger’s (probabilistic) space. Karl Menger introduced the notion of probabilistic metric space in 1942.
In this sense, a mapping \(N:\Bbb{R}\to[a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) is called a lower (distribution) function if it is nondecreasing, leftcontinuous with \(\inf N=a\) and \(\sup N=b\). We will denote by \(\mathcal{L}\) the set of all lower (distribution) functions.
A lower statistical space is a pair \((X,\mathcal{D})\), where X is an abstract set and \(\mathcal{D}\) is a mapping of \(X\times X\) into the set of all lower (distribution) functions \(\mathcal{L}\). We shall denote the lower ordered (distribution) function \(\mathcal {D}(p,q)\) by \(N_{p,q}(x)\) or \(N_{p,q}\), whence the symbol \(N_{p,q}(x)\) will denote the value of \(N_{p,q}\) at \(x\in\Bbb{R}\). The functions \(N_{p,q}\) are assumed to satisfy the following conditions: \(N_{p,q}=N_{q,p},N_{p,q}(c)=a\) for some \(c\in\Bbb{R}\), and
and if \(N_{p,q}(x)=b\) and \(N_{q,r}(y)=b\) imply \(N_{p,r}(x+y)=b\) for all \(p,q,r\in X\) and for all \(x,y\in\Bbb{R}\).
In view of the condition \(N_{p,q}(c)=a\) for some \(c\in\Bbb{R}\), which evidently implies that \(N_{p,q}(x)=a\) for all \(x\le c\), condition (Em) is equivalent to the statement: \(p=q\) if and only if \(N_{p,q}(x)=H(x)\), where \(H(x)=a\) if \(x\le c\) and \(H(x)=b\) if \(x>c\). See Fig. 3.
Every metric space may be regarded as a statistical lower space of a special kind. One has only to set \(N_{p,q}(x)=H(x\rho(p,q))\) for every pair of points \((p,q)\) in the metric space \((X,\rho)\).
In connection with the above, a transversal lower edges Tspace is a pair \((X,\rho)\), where X is a transversal lower edges space and where the lower (edges) transverse \(\rho[u,v]=N_{u,v}(x)\) satisfies \(N_{u,v}=N_{v,u}\), \(N_{u,v}(c)=a\) for some \(c\in\Bbb{R}\) and (Em). This space is a very characteristic example of transversal lower edges spaces for further work. Every Menger’s space is also a lower edges space.
The concept of a neighborhood in a lower transversal edges space X for the lower edges transverse \(\rho(p,q)=N_{p,q}(x)\) in \([a,b]\subset\Bbb{R}_{+}^{0}\) for \(a< b\) is the following. If \(p\in X\), \(\mu>c\) for some \(c\in\Bbb{R}\) and σ is a positive real, then a \((\mu,\sigma)\)neighborhood of p, denoted by \({\mathcal {O}}_{p}(\mu,\sigma)\), is defined by
Corollary 26
Let T be an increasing mapping of a transversal lower edges Tspace (partially ordered with ordering ) into itself, where X satisfies the condition of spring inf MCSconvergence. Suppose that there exists a lower function \(K_{x, y}(t)\), as a controlling function, such that
for all \(x, y\in X\) or
for all \(x, y\in X\), where \(\varphi: \Bbb{R}\to\Bbb{R}\) is an increasing function satisfying \(\varphi^{n}(t)\to+\infty\) \((n\to\infty)\). If for some \(r\in X\), then T has a unique fixed point in X.
The proof of this statement is a total analogy with the preceding proofs as consequences of the main statements. Thus we omit it.
In connection with the above, applying our general principle of transpose for nonnumerical transverses (see Tasković [5, p. 89]) to Corollary 26, we get an extended and generalized version of this result in the following sense.
Indeed, let be a totally ordered set. A mapping for \(a\prec b\), where \(P:=(P,\preccurlyeq )\) is a partially ordered set, is called a lower ordered (distribution) function if it is nonincreasing with \(\inf N=a\) and \(\sup N=b\). We will denote by the set of all lower ordered (distribution) functions.
The next two spaces are very interesting examples of transversal ordered lower spaces. First, a lower statistical space is a pair \((X,{\mathcal {R}})\), where X is an abstract set and \({\mathcal {R}}\) is a mapping of \(X\times X\) into the set of all lower ordered (distribution) functions . We shall denote the lower ordered (distribution) function \({\mathcal {R}}(u,v)\) by \(N_{u,v}(x)\) or \(N_{u,v}\), whence the symbol \(N_{u,v}(x)\) will denote the value of \(N_{u,v}\) at . The functions \(N_{u,v}\) are assumed to satisfy the following conditions: \(N_{u,v}=N_{v,u}\), \(N_{u,v}(c)=a\) for some , and
and if \(N_{u,r}(x)=b\) and \(N_{r,v}(y)=b\) imply \(N_{u,v}(x+y)=b\) for all \(u,v,r\in X\) and for all .
In view of the condition \(N_{u,v}(c)=a\), for some , which evidently implies that \(N_{u,v}(x)=a\) for every \(x\preccurlyeq c\), condition (Em) is equivalent to the statement: \(u=v\) if and only if \(N_{u,v}(x)=H(x)\), where \(H(x)=a\) if \(x\preccurlyeq c\) and \(H(x)=b\) if \(x\succ c\). See Fig. 3.
Also, \(N_{u,v}(x)\) may be interpreted as the “measure” that the distance between u and v is less than .
A very characteristic example, for further work, of the transversal lower ordered edges spaces is the following space in the following form.
A transversal lower ordered edges Tspace is a pair \((X,\rho)\), where X is a transversal lower ordered edges space and where the lower ordered (edges) transverse \(\rho(u,v)=N_{u,v}(x)\) satisfies \(N_{u,v}=N_{v,u}\), \(N_{u,v}(c)=a\) for some and (Em).
Furthermore, the concept of a neighborhood can be introduced and defined with the aid of the lower ordered edges transverse. In fact, neighborhoods in transversal lower ordered edges spaces may be defined in several nonequivalent ways. Here, we shall consider only one of these.
If \(p\in X\), \(\mu\succ c\) for some and \(r\prec b\), then a \((\mu,r)\)neighborhood of p, denoted by \(U_{p}(\mu,r)\), is defined by
Applying our general principle of transpose for partially ordered sets to Corollary 26 and to lower spring ordered transversal space for \(A(u,v):=N_{u,v}(x)\) directly, we get an extended and generalized version of Corollary 26 in the following form.
Corollary 27
Let T be an increasing mapping of a lower spring ordered transversal space for \(A:=N_{u,v}(t)\) with the property (Em) into itself, where X satisfies the condition of spring inf MCSconvergence. Suppose that there exists an increasing function ( is a totally ordered set) such that \(t\prec\varphi(t)\) for every and for some the following equality holds:
and there exists a lower function \(K_{x, y}(t)\), as a controlling function, such that
for all \(x, y\in X\) or
for all \(x, y\in X\). If for some \(r\in X\), then T has a unique fixed point in X.
The proof of this statement is a total analogy with the preceding proofs as consequences of the main statements. Thus we omit it.
Corollary 28
(Tasković [6, p. 607])
Let T be an increasing mapping of a transversal lower edges Tspace (partially ordered with ordering ), into itself, where X satisfies the condition of spring inf MCSconvergence. Suppose that there exists an increasing function \(\varphi:[c,+\infty)\to[c,+\infty)\) for some fixed \(c\in\Bbb{R}\) satisfying
such that \((F)\):
for all \(u,v\in X\) and for every \(x>c\). If for some \(r\in X\), then T has a unique fixed point in X.
We notice that in 2001 Tasković proved a special case of this statement on a transversal lower edges Tspace \(X:=(X,M_{u,v}(t))\). Also, in 1998 Tasković introduced the concept of upper, lower, and middle transversal edges spaces. Also see: [20–22] and [23].
In context of this statement, the following conditions are special cases of condition \((F)\):
(1) There exists a constant \(0< k<1\) such that, for all \(p,q\in X\), for some fixed \(c\in\Bbb{R}\), and for every \(x>c\), the following inequality holds:
(2) There exists a nondecreasing function \(\varphi:[c,+\infty)\to[c,+\infty)\) for some fixed \(c\in\Bbb{R}\) satisfying (As) such that, for some fixed \(0< k<1\), for all \(u,v\in X\), and for every \(x>c\), the following inequality holds:
(3) There exists a constant \(0< k<1\) such that, for all \(u,v\in X\), for some fixed \(x\in\Bbb{R}\), and for every \(x>c\), the following inequality holds:
or in an equivalently form as
Corollary 29
(Tasković [6, p. 616])
Let T be an increasing mapping of a lower spring ordered transversal space for \(A(u,v):=N_{u,v}(t)\) with the property (Em) into itself, where X satisfies the condition of lower MCSconvergence. Suppose that there exists an increasing function ( is a totally ordered set) such that \(t\prec\varphi(t)\) for every and for some fixed the following equality holds:
and
for all \(u,v\in X\) and for every \(x\succ c\). If for some \(r\in X\), then T has a unique fixed point in X.
Proof
Applying the general principle of transpose for posets to Corollary 28, we get directly an extended and generalized version of the result given in Corollary 28. The proof is complete. □
Annotations
The results of Corollaries 28 and 29 are two immediate consequences of Corollary 19. Uniqueness follows immediately from the conditions of given inequalities.
Conclusions
This paper presents new consequences of the Axiom of Infinite Choice in terms of ordered spring spaces and increasing mappings. Applications in nonlinear functional analysis and fixed point theory are also considered.
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MSC
 47H10
 05A15
 06A10
 04A25
 05A05
 54H25
Keywords
 Fixed points
 Axiom of Infinite Choice
 Transversal ordered spring spaces
 Spaces with the nonnumerical transversals
 Partially ordered spaces
 Increasing mappings
 Transversal edges (upper, lower, and middle) spaces
 Lower and upper (distribution) functions
 Noncomplete spaces
 Lemma of Infinite Maximality
 Increasing inductiveness
 Partially ordered metric spaces