# Meir-Keeler Type Multidimensional Fixed Point Theorems in Partially Ordered Metric Spaces

Authors: Erdal Karapınar; Antonio Roldán; Juan Martínez-Moreno; Concepcion Roldán
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the existence and uniqueness of a fixed point of the multidimensional operators which
satisfy Meir-Keeler type contraction condition. Our results extend, improve, and generalize the results mentioned
above and the recent results on these topics in the literature.


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## Body

## 1. Introduction

Fixed point theory plays a crucial role in nonlinear functional analysis. In particular, fixed point results are used to prove the existence (and also uniqueness) when solving various type of equations. On the other hand, fixed point theory has a wide application potential in almost all positive sciences, such as Economics, Computer Science, Biology, Chemistry, and Engineering. One of the initial results in this direction (given by S. Banach), which is known asBanach fixed point theorem or Banach contraction mapping principle [1] is as follows. Every contraction in a complete metric space has a unique fixed point. In fact, this principle not only guarantees the existence and uniqueness of a fixed point, but it also shows how to get the desired fixed point. Since then, this celebrated principle has attracted the attention of a number of authors (e.g., see [1–39]). Due to its importance in nonlinear functional analysis, Banach contraction mapping principle has been generalized in many ways with regards to different abstract spaces. One of the most interesting results on generalization was reported by Guo and Lakshmikantham [18] in 1987. In their paper, the authors introduced the notion of coupled fixed point and proved some related theorems for certain type mappings. After this pioneering work, Gnana Bhaskar and Lakshmikantham [10] reconsidered coupled fixed point in the context of partially ordered sets by defining the notion of mixed monotone mapping. In this outstanding paper, the authors proved the existence and uniqueness of coupled fixed points for mixed monotone mappings and they also discussed the existence and uniqueness of solution for a periodic boundary value problem. Following these initial papers, a significant number of papers on coupled fixed point theorems have been reported (e.g., see [6, 11, 13, 19, 22, 23, 29, 31–33, 36, 38, 40]).Following this trend, Berinde and Borcut [8] extended the notion of coupled fixed point to tripled fixed point. Inspired by this interesting paper, Karapınar [24] improved this idea by defining quadruple fixed point (see also [25–28]). Very recently, Roldán et al. [35] generalized this idea by introducing the notion of Φ-fixed point, that is to say, the multidimensional fixed point.Another remarkable generalization of Banach contraction mapping principle was given by Meir and Keeler [34]. In the literature of this topic, Meir-Keeler type contraction has been studied densely by many selected mathematicians (e.g., see [2–4, 9, 20, 21, 36, 39]).In this paper, we prove the existence and uniqueness of fixed point of multidimensional Meir-Keeler contraction in a complete partially ordered metric space. Our results improve, extend, and generalize the existence results on the topic in fixed point theory.


## 2. Preliminaries

Preliminaries and notation about coincidence points can also be found in [35]. Let n be a positive integer. Henceforth, X will denote a nonempty set, and Xn will denote the product space X×X×…n×X. Throughout this paper, m and k will denote nonnegative integers and i,j,s∈{1,2,…,n}. Unless otherwise stated, “for all m” will mean “for all m≥0” and “for all i”  will mean “for all i∈{1,2,…,n}.”Ametric onX is a mapping d:X×X→ℝ satisfying, for all x, y, z∈X,
(1)(i)d(x,y)=0,iffx=y;(ii)d(x,y)≤d(z,x)+d(z,y).
From these properties, we can easily deduce that d(x,y)≥0 and d(y,x)=d(x,y) for all x,y∈X. The last requirement is called the triangle inequality. If d is a metric on X, we say that (X,d) is a metric space (for short, an MS).Definition 1 (see [15]).
A triple(X,d,≼) is called a partially ordered metric space if (X,d) is an MS and ≼ is a partial order on X.Definition 2 (see [10]).
An ordered MS(X,d,≼) is said to have the sequentialg-monotone property if it verifies the following.(i) If{xm} is a nondecreasing sequence and {xm}→dx, then gxm≼gx for all m.(ii) If{ym} is a nonincreasing sequence and {ym}→dy, then gym≼gy for all m.
Ifg is the identity mapping, then X is said to have the sequential monotone property.Henceforth, fix a partition{A,B} of Λn={1,2,…,n}; that is, A∪B=Λn and A∩B=∅. We will denote that
(2)ΩA,B={σ:Λn⟶Λn:σ(A)⊆A,σ(B)⊆B},ΩA,B′={σ:Λn⟶Λn:σ(A)⊆B,σ(B)⊆A}.
If (X,≼) is a partially ordered space, x,y∈X, and i∈Λn, we will use the following notation:
(3)x≼iy⟺{x≼y,ifi∈A,x≽y,ifi∈B.LetF:Xn→X and g:X→X be two mappings.Definition 3 (see [35]).
We say thatF and g are commuting if gF(x1,…,xn)=F(gx1,…,gxn) for all x1,…,xn∈X.Definition 4 (see [35]).
Let(X,≼) be a partially ordered space. We say that F has the mixedg-monotone property(w.r.t.{A,B}) if F is g-monotone nondecreasing in arguments of A and g-monotone nonincreasing in arguments of B; that is, for all x1,x2,…,xn,  y,  z∈X and all i,
(4)gy≼gz⇒F(x1,…,xi-1,y,xi+1,…,xn)≼iF(x1,…,xi-1,z,xi+1,…,xn).Henceforth, letσ1,σ2,…,σn,τ:Λn→Λn be n+1 mappings from Λn into itself, and let Φ be the (n+1)-tuple (σ1,σ2,…,σn,τ).Definition 5 (see [35]).
A point(x1,x2,…,xn)∈Xn is called a Φ-coincidence point of the mappingsF and g if
(5)F(xσi(1),xσi(2),…,xσi(n))=gxτ(i)∀i.
If g is the identity mapping on X, then (x1,x2,…,xn)∈Xn is called a Φ-fixed point of the mappingF.Remark 6.
IfF and g are commuting and (x1,x2,…,xn)∈Xn is a Φ-coincidence point of F and g, then (gx1,gx2,…,gxn) also is a Φ-coincidence point of F and g.With regards to coincidence points, it is possible to consider the following simplification. Ifτ is a permutation of Λn and we reorder (5), then we deduce that every coincidence point may be seen as a coincidence point associated to the identity mapping on Λn.Lemma 7.
Letτ be a permutation of Λn, and let Φ=(σ1,σ2,…,σn,τ) and Φ′=(στ-1(1),στ-1(2),…,στ-1(n),IΛn). Then, a point (x1,x2,…,xn)∈Xn is a Φ-coincidence point of the mappings F and g if and only if (x1,x2,…,xn) is a Φ′-coincidence point of the mappings F and g.Therefore, in the sequel, without loss of generality, we will only considerΥ-coincidence points where Υ=(σ1,σ2,…,σn), that is, that verify F(xσi(1),xσi(2),…,xσi(n))=gxi for all i.If one represents a mappingσ:Λn→Λn throughout its ordered image, that is, σ=(σ(1),σ(2),…,σ(n)), then(i)
Gnana-Bhaskar and Lakshmikantham's election inn=2 is σ1=τ=(1,2) and σ2=(2,1);(ii)
Berinde and Borcut's election inn=3 is σ1=τ=(1,2,3), σ2=(2,1,2) and σ3=(3,2,1);(iii)
Karapnar's election inn=4 is σ1=τ=(1,2,3,4), σ2=(2,3,4,1), σ3=(3,4,1,2), and σ4=(4,1,2,3).For more details, see [35]. We will use the following result about real sequences in the proof of our main theorem.Lemma 8.
If{xm}m∈ℕ is a sequence in an MS (X,d) that is not Cauchy, then there exist ɛ0>0 and two subsequences {xm(k)}k∈ℕ and {xn(k)}k∈ℕ such that, for all k∈ℕ, k<m(k)<n(k)<m(k+1), d(xm(k),xn(k))≥ɛ0, and d(xm(k),xn(k)-1)<ɛ0.Meir and Keeler generalized the Banach contraction mapping principle in the following way.Definition 9 (Meir and Keeler [34]).
AMeir-Keeler mapping is a mapping T:X→X on an MS (X,d) such that for all ɛ>0, there exists δ>0 verifying that if x,y∈X and ɛ≤d(x,y)<ɛ+δ, then d(Tx,Ty)<ɛ.Lim characterized this kind of mappings in terms of a contractivity condition using the following class of functions.Definition 10 (Lim [30]).
A functionϕ:[0,∞[→[0,∞[ will be called an L-function if (a) ϕ(0)=0, (b) ϕ(t)>0 for all t>0, and (c) for all ɛ>0, there exists δ>0 such that ϕ(t)≤ɛ for all t∈[ɛ,ɛ+δ].Theorem 11 (Lim [30]).
Let(X,d) be an MS, and let T:X→X. Then T is a Meir-Keeler mapping if and only if there exists an (nondecreasing, right-continuous) L-map ϕ such that
(6)d(T(x),T(y))<ϕ(d(x,y))∀x,y∈Xverifyingd(x,y)>0.Using a result of Chu and Diaz [14], Meir and Keeler [34] proved that every Meir-Keeler mapping on a complete MS has a unique fixed point. Since then, many authors have developed this notion in different ways (e.g., see [2–4, 9, 20, 21, 36, 39]). For instance, in [36], Samet introduces the concept of generalized Meir-Keeler type function as follows.Definition 12 (see [36]).
Let(X,d,≼) be a partially ordered metric space and F:X×X→X a given mapping. We say that F is a generalized Meir-Keeler type function if for all ɛ>0, there exists δ(ɛ)>0 such that
(7)x≽u,y≼v,ɛ≤12[d(x,u)+d(y,v)]<ɛ+δ(ɛ)⇒d(F(x,y),F(u,v))<ɛ.Then, the author [36] proved some coupled fixed point theorems via generalized Meir Keeler type mappings. In this paper, we extend the notion of generalized Meir-Keeler type mappings in various ways and get some fixed point results by the help of these notions.


## 3. Multidimensional Meir-Keeler-Type Mappings

Henceforth, let(X,d,≼) be a partially ordered MS and let F:Xn→X and g:X→X be two mappings.Definition 13.
We will say thatF is a (multidimensional) g-Meir-Keeler type mapping, ((MK) mapping) if it verifies the following two properties.(MK1) Ifx1,x2,…,xn,y1,y2,…,yn∈X verify gxi=gyi for all i, then F(x1,x2,…,xn)=F(y1,y2,…,yn).(MK2) For allɛ>0, there exists δ>0 such that if x1,x2,…,xn,y1,y2,…,yn∈X verify gxi≼igyi for all i and
(8)ɛ≤max1≤i≤nd(gxi,gyi)<ɛ+δ,thend(F(x1,x2,…,xn),F(y1,y2,…,yn))<ɛ.Ifg is the identity mapping on X, we will say that F is a (n-dimensional)Meir-Keeler type mapping.On the one hand, notice that, in a wide sense, property (MK1) may be interpreted as property (MK2) forɛ=0. On the other hand, we observe that our definition may not be compared with the original one due to Meir and Keeler since we assume that X has a partial order. In any case, if n=1, (X,d) has a partial order and g is the identity mapping on X, and we can only establish that if F:X→X is a Meir-Keeler mapping in the sense of Definition 9, then F is a Meir-Keeler-type mapping in the sense of Definition 13, but the converse does not hold.Remark 14.
Ifg is an injective mapping on X, then all mappings F verify (MK1).Lemma 15.
LetF:Xn→X be a mapping on a partially ordered MS (X,d,≼), and let x1,x2,…,xn,y1,y2,…,yn∈X be such that gxi≼igyi for all i.   (1)
IfF verifies (MK2), then either gxi=gyi for all i or
(9)d(F(x1,x2,…,xn),F(y1,y2,…,yn))<max1≤i≤nd(gxi,gyi).(2)
IfF is a g-Meir-Keeler type mapping, then
(10)d(F(x1,x2,…,xn),F(y1,y2,…,yn))≤max1≤i≤nd(gxi,gyi),
and the equality is achieved if and only if gxi=gyi for all i.Proof.
(
1
) If the condition “gxi=gyi for all i” does not hold, then ɛ=max1≤i≤nd(gxi,gyi)>0. Hence, d(F(x1,x2,…,xn),F(y1,y2,…,yn))<ɛ=max1≤i≤nd(gxi,gyi).  (2) If F is a g-Meir-Keeler-type mapping, the case “gxi=gyi for all i” means that the equality is achieved.This global contractivity condition (10) is not strong enough to ensure that F has a fixed point. For instance, if n=1, then F(x)=x+1 for all x∈ℝ has no fixed point. In order to characterize this kind of mappings in different ways, we recall some definitions and results.Definition 16.
Theg-modulus of uniform continuity of F is, for all ɛ>0,
(11)δg,F(ɛ)=sup({λ≥0:[gxi≼igyi∀i,max1≤i≤nd(gxi,gyi)<λ]aaaaaaa⇒d(F(x1,x2,…,xn),aaaaaaaaaaaaaaF(y1,y2,…,yn))<ɛ[gxi≼igyi∀i,max1≤i≤nd(gxi,gyi)<λ]}).Remark 17.
The identity mapping on a setX will be denoted by 1X:X→X. If g:X→X is a mapping, then G:Xn→Xn will be defined by G(x1,x2,…,xn)=(gx1,gx2,…,gxn) for all x1,x2,…,xn∈X. If (X,d) is a metric space, then D:Xn×Xn→[0,∞[, given by D(P,Q)=max1≤i≤nd(pi,qi) for all P=(p1,p2,…,pn), Q=(q1,q2,…,qn)∈Xn, is a metric on Xn. A partial order ≼ on X may be induced on Xn by P≼Q if and only if pi≼qi for all i (notice that this partial order depends on the partition {A,B} of Λn). Then, (Xn,D,≼) also is a partially ordered MS. Furthermore, given any ω=(ω2,ω3,…,ωn)∈Xn-1, Fω:Xn→Xn will denote the mapping defined by Fω(x1,x2,…,xn)=(F(x1,x2,…,xn),ω2,ω3,…,ωn) for all x1,x2,…,xn∈X. It is obvious that
(12)D(Fω(x1,x2,…,xn),Fω(y1,y2,…,yn))=d(F(x1,x2,…,xn),F(y1,y2,…,yn))
for all x1,x2,…,xn,y1,y2,…,yn∈X.Theorem 18.
Let(X,d,≼) be a partially ordered MS, and let F:Xn→X and g:X→X be two mappings. Then, the following statements are equivalent.(MK)F is a g-Meir-Keeler-type mapping.(MK3) For allɛ>0, there exists δ>0 such that
(13)x1,x2,…,xn,y1,y2,…,yn∈Xgxi≼igyi∀imax1≤i≤nd(gxi,gyi)<ɛ+δ}⇒d(F(x1,x2,…,xn),F(y1,y2,…,yn))<ɛ.(MK4)δg,F(ɛ)>ɛ for all ɛ>0.(MK5)F and g verify (MK1), and there exists an (nondecreasing, right-continuous) L-function ϕ:[0,∞[→[0,∞[ such that
(14)d(F(x1,x2,…,xn),F(y1,y2,…,yn))<ϕ(max1≤i≤nd(gxi,gyi))
for all x1,…,xn,y1,…,yn∈X verifying gxi≼igyi for all i and max1≤i≤nd(gxi,gyi)>0.(MK6) For allω∈Xn-1, the mapping Fω:Xn→Xn is a G-Meir-Keeler-type mapping on (Xn,D,≼).(MK7) There existsω0∈Xn-1 such that the mapping Fω0:Xn→Xn is a G-Meir-Keeler-type mapping on (Xn,D,≼).Proof.
[(MK)⇒(MK3)]: Fix ɛ>0, and let δ>0 given by (MK2). Let x1,…,xn,y1,…,yn∈X be such that gxi≼igyi for all i, and let η=max1≤i≤nd(gxi,gyi)<ɛ+δ. If η=0, then gxi=gyi for all i, and so d(F(x1,x2,…,xn),F(y1,y2,…,yn))=0<ɛ by (MK1). In another case, η>0. If ɛ≤η<ɛ+δ, then d(F(x1,x2,…,xn),F(y1,y2,…,yn))<ɛ by (MK2). Now, suppose that 0<η<ɛ. Then, η∈[η,η+δη[, where δη>0 is also given by (MK2), and d(F(x1,…,xn),F(y1,…,yn))<η<ɛ. Hence, (MK3) holds.
[(MK3)⇒(MK4)]: Given ɛ>0, let δ>0 verifying (MK3). Then, δg,F(ɛ)≥ɛ+δ, and so δg,F(ɛ)>ɛ.
[(MK4)⇒(MK)]: On the one hand, if gxi=gyi for all i, then d(F(x1,x2,…,xn),F(y1,y2,…,yn))<ɛ for all ɛ>0, and so F verify (MK1). On the other hand, let ɛ>0, and define δ=(δg,F(ɛ)-ɛ)/2>0. Therefore, ɛ+δ<δg,F(ɛ). Since δg,F(ɛ) is a supremum, there exists λ0∈]ɛ+δ,δg,F(ɛ)] such that if gxi≼igyi for all i and max1≤i≤nd(gxi,gyi)<λ0, then d(F(x1,x2,…,xn),F(y1,y2,…,yn))<ɛ. In particular, if gxi≼igyi for all i and ɛ≤max1≤i≤nd(gxi,gyi)<ɛ+δ<λ0, then d(F(x1,x2,…,xn), F(y1,y2,…,yn))<ɛ.
[(MK)⇔(MK5)]: It is possible to follow step by step the proof of Proposition 1 in [39] with slight changes.
[(MK)⇔(MK6)⇔(MK7)]: It is apparent taking into account (12).The following result is a particular case takingϕ(t)=kt for all t≥0.Corollary 19.
Let(X,d,≼) be a partially ordered metric space, and let F:Xn→X and g:X→X be two mappings. Assume that there exists k∈(0,1) such that
(15)d(F(x1,x2,…,xn),F(y1,y2,…,yn))≤kmax1≤i≤nd(gxi,gyi)
for all x1,…,xn,y1,…,yn∈X verifying gxi≼igyi for all i. Then, F is a g-Meir-Keeler-type mapping.Next, we prove that a generalized Meir Keeler type function in the sense of Samet [36, Definition 12] is a particular case of 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13.Lemma 20.
Every generalized Meir Keeler type function in the sense of Samet is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition13 taking g as the identity mapping on the MS.Proof.
Suppose thatF:X×X→X is a generalized Meir Keeler type function in the sense of Samet. Fix ɛ>0 and let δ>0 verifying (7). Let x,y,u,v∈X such that x≽u, y≼v, and max(d(x,u),d(y,v))<ɛ+δ. We have to prove that d(F(x,y),F(u,v))<ɛ. If x=u and y=v, there is nothing to prove. Next, suppose that max(d(x,u),d(y,v))>0. Let
(16)M=12[d(x,u)+d(y,v)].
If M=0, then x=u and y=v, which is false. Then, M>0. On the other hand,
(17)M=d(x,u)+d(y,v)2≤max(d(x,u),d(y,v))<ɛ+δ.
Ifɛ≤M<ɛ+δ, then d(F(x,y),F(u,v))<ɛ by (7). Finally, if 0<M<ɛ, taking ɛ′=M in (7), we have that M∈[ɛ′,ɛ′+δɛ′[ (where δɛ′ is taken as in (7)), and so d(F(x,y),F(u,v))<M<ɛ. This proves that F is a 2-dimensional Meir-Keeler type mapping associated to g=IX.Remark 21.
Converse of Lemma20 does not hold. For instance, let X=ℝ be provided with its usual metric d(x,y)=|x-y| and partial order ≤. Take 0<k<1 and consider F(x,y)=kx for all x,y∈ℝ. Then, F is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13 (taking g as the identity mapping on ℝ), but, if k>1/2, it is not a generalized Meir Keeler type function in the sense of Samet.
Indeed, we firstly prove thatF is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13 (taking g as the identity mapping on ℝ). Let ɛ>0. Consider any r∈]0,1/k-1[ (i.e., k(1+r)<1) and define δ=rɛ>0. Consider x,y,u,v∈ℝ such that x≥u and y≤v verifying ɛ≤max(d(x,u),d(y,v))=max(|x-u|,|y-v|)<ɛ+δ. In particular, |x-u|<ɛ+δ. Then,
(18)d(F(x,y),F(u,v))=d(kx,ku)=k|x-u|<k(ɛ+δ)=k(ɛ+rɛ)=k(1+r)ɛ<ɛ.
It follows that F is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13. Next, we claim that if k>1/2, then F is not a generalized Meir Keeler type function in the sense of Samet. Let ɛ>0. If F was a generalized Meir Keeler type function in the sense of Samet, it would be δ>0 verifying (7). Take x=ɛ,  u=-ɛ, and y=v=0. Then, x≥u, y≤v and
(19)d(x,u)+d(y,v)2=|ɛ-(-ɛ)|+|0-0|2=2ɛ2=ɛ∈[ɛ,ɛ+δ[.
However, d(F(x,y),F(u,v))=d(kɛ,k(-ɛ))=k|ɛ-(-ɛ)|=2kɛ>ɛ  since  k>1/2.


## 4. Main Results

In the following result, we show sufficient conditions to ensure the existence ofΥ-coincidence points, where Υ=(σ1,σ2,…,σn).Theorem 22.
Let(X,d) be a complete MS, and let ≼ a partial order on X. Let Υ=(σ1,σ2,…,σn) be an n-tuple of mappings from {1,2,…,n} into itself verifying σi∈ΩA,B if i∈A and σi∈ΩA,B′ if i∈B. Let F:Xn→X and g:X→X be two mappings such that F is a g-Meir-Keeler-type mapping and has the mixed g-monotone property on X, F(Xn)⊆g(X), and g is continuous and commuting with F. Suppose that either F is continuous or (X,d,≼) has the sequential g-monotone property. If there exist x01,x02,…,x0n∈X verifying gx0i≼iF(x0σi(1),x0σi(2),…,x0σi(n)) for all i, then F and g have, at least, one Υ-coincidence point.Proof.
The proof is divided in six steps. We follow the strategy of Theorem 9 in [35].
Step  1. There exist n sequences {xm1}m≥0, {xm2}m≥0,…,{xmn}m≥0 such that gxm+1i=F(xmσi(1), xmσi(2),…,xmσi(n)) for all m and all i.
Step  2. gxmi≼igxm+1i for all  m and all i.
Defineδm=max1≤i≤nd(gxmi,gxm+1i)≥0 for all m. Firstly, suppose that there exists m0∈ℕ such that δm0=0. Then, gxm0i=gxm0+1i=F(xm0σi(1),xm0σi(2),…,xm0σi(n)) for all i, so (xm01,xm02,…,xm0n) is a Υ-coincidence point of F and g and we have finished. Therefore, we may reduce to the case in which δm>0 for all m; that is,
(20)∀m,thereexistsjsuchthatgxmj≠gxm+1j.
Step  3. We claim that {d(gxmi,gxm+1i)}m≥0→0 for all i (i.e., {max1≤j≤nd(gxmj, gxm+1j)}m≥0→0). Indeed, as gxmi≼igxm+1i for all m and all i, then condition (MK2), Lemma 15, and (20) imply that, for all m≥1 and all i,
(21)d(gxmi,gxm+1i)=d(F(xm-1σi(1),xm-1σi(2),…,xm-1σi(n)),aaaaF(xmσi(1),xmσi(2),…,xmσi(n)))<max1≤j≤nd(gxm-1σi(j),gxmσi(j))≤max1≤j≤nd(gxm-1j,gxmj)=δm-1.
Taking maximum on i, we deduce that the sequence {δm}m≥1 is nonincreasing and lower bounded. Therefore, it is convergent; that is, there exists Δ≥0 such that {δm}m≥1→Δ (and Δ≤δm for all m). We claim that Δ=0. On the contrary, assume that Δ>0. Let δ>0 be a positive number associated to ɛ=Δ>0 by (MK2). Since
(22){max1≤i≤nd(gxmi,gxm+1i)}m={δm}m↘Δ,
there exists m0∈ℕ such that if m≥m0, then Δ≤max1≤i≤nd(gxmi,gxm+1i)<Δ+δ. By (MK2), it follows that, for all i,
(23)d(gxm0i,gxm0+1i)=d(F(xm0σi(1),xm0σi(2),…,xm0σi(n)),aaaaaaF(xm0+1σi(1),xm0+1σi(2),…,xm0+1σi(n)))<Δ.
Taking maximum on i, we deduce that
(24)Δ≤δm0=max1≤i≤nd(gxm0i,gxm0+1i)<Δ.
But this is impossible. Then, we have just proved that Δ=0. Therefore, {δm}m≥1→Δ=0, which means that
(25)limm→∞δm=limm→∞(max1≤j≤nd(gxmj,gxm+1j))=0.
As 0≤d(gxmi,gxm+1i)≤δm for all m and all i, then {d(gxmi,gxm+1i)}→0 for all i.
Step  4. Every sequence  {gxmi}m≥0  is Cauchy. Suppose that {gxmi1}m≥0,…,{gxmis}m≥0 are not Cauchy and {gxmis+1}m≥0,…,{gxmin}m≥0 are Cauchy, being {i1,…,in}={1,…,n}. By Lemma 8, for all r∈{1,2,…,s}, there exist ɛr>0 and subsequences {gxmr′(k)ir}k∈ℕ and {gxnr′(k)ir}k∈ℕ such that
(26)k<mr′(k)<nr′(k),d(gxmr′(k)ir,gxnr′(k)ir)≥ɛr,d(gxmr′(k)ir,gxnr′(k)-1ir)<ɛr,∀k∈ℕ.
Let ɛ0=max(ɛ1,…,ɛs) and ɛ0′=min(ɛ1,…,ɛs)>0. Since {gxmis+1}m≥0,…,{gxmin}m≥0 are Cauchy, there exists n0∈ℕ such that if n,  m≥n0, then d(gxmj,gxnj)<ɛ0′/2 for all j∈{is+1,…,in}.
Letk0∈ℕ such that n0<min(m1′(k0), m2′(k0),…,ms′(k0)), and define m(1)=min(m1′(k0), m2′(k0),…,ms′(k0)). As m(1)=mr′(k0) for some r∈{1,2,…,s}, there exists nr′(k0) such that d(gxmr′(k0)ir, gxnr′(k0)ir)≥ɛr≥ɛ0. Thus, we can consider the numbers m(1)+1, m(1)+2,… until finding the first positive integer n(1)>m(1) verifying
(27)max1≤r≤sd(gxm(1)ir,gxn(1)ir)≥ɛ0,d(gxm(1)ij,gxn(1)-1ij)<ɛ0,∀j∈{1,2,…,s}.
Now, letk1∈ℕ such that n(1)<min(m1′(k1), m2′(k1),…,ms′(k1)) and define m(2)=min(m1′(k1), m2′(k1),…,ms′(k1)). Since m(2)∈{m1′(k1),m2′(k1),…,ms′(k1)}, we can consider the numbers m(2)+1,m(2)+2,… until finding the first positive integer n(2)>m(2) verifying
(28)max1≤r≤sd(gxm(2)ir,gxn(2)ir)≥ɛ0,d(gxm(2)ij,gxn(2)-1ij)<ɛ0,∀j∈{1,2,…,s}.
Repeating this process, we can find sequences such that, for all k≥1,
(29)n0<m(k)<n(k)<m(k+1),max1≤r≤sd(gxm(k)ir,gxn(k)ir)≥ɛ0,d(gxm(k)ij,gxn(k)-1ij)<ɛ0,∀j∈{1,2,…,s}.
Since n0<m(k)<n(k), we know that d(gxm(k)j,gxn(k)j), d(gxm(k)j,gxn(k)-1j), d(gxm(k)-1j,gxn(k)-1j)<ɛ0′/2 for all j∈{is+1,…,in}. Therefore, for all k,
(30)max1≤j≤nd(gxm(k)j,gxn(k)j)=max1≤r≤sd(gxm(k)ir,gxn(k)ir)≥ɛ0,max1≤j≤nd(gxm(k)j,gxn(k)-1j)<ɛ0′.
Let δ>0 verifying (MK3) using ɛ0, and let k1∈ℕ such that if k≥k1, then d(gxm(k)-1j,gxm(k)j)<δ for all j. Then, for all j and all k≥k1,
(31)d(gxm(k)-1j,gxn(k)-1j)≤d(gxm(k)-1j,gxm(k)j)+d(gxm(k)j,gxn(k)-1j)<δ+ɛ0,
Applying (MK3), it follows, for all k≥k0 and all i, that
(32)d(gxm(k)i,gxn(k)i)=d(F(xm(k)-1σi(1),…,xm(k)-1σi(n)),F(xn(k)-1σi(1),…,xn(k)-1σi(n)))<ɛ0,
but this contradicts (30) since max1≤j≤nd(gxm(k)j,gxn(k)j)≥ɛ0. This contradiction shows us that every sequence {gxmi}m≥0 is Cauchy.
Existence of a fixed point is derived by standard techniques. Indeed, since(X,d) is complete, there exist x1,x2,…,xn∈X such that xi=limm→∞gxmi for all i. As g is continuous and F commutes with g,
(33)limm→∞F(gxmσi(1),gxmσi(2),…,gxmσi(n))=limm→∞gF(xmσi(1),xmσi(2),…,xmσi(n))=limm→∞ggxm+1i=gxi∀i.
Step  5. Suppose that F is continuous. In this case, since {gxmσi(j)}m→xσi(j) for all i,j and F is continuous,
(34)limm→∞ggxm+1i=limm→∞F(gxmσi(1),gxmσi(2),…,gxmσn(n))=F(xσi(1),xσi(2),…,xσi(n))
for all i. Then, F(xσi(1),xσi(2),…,xσi(n))=gxi for all i; that is, (x1,x2,…,xn) is a Υ-coincidence point of F and g.
Step  6. Suppose that (X,d,≼) has the sequential g-monotone property. In this case, by step 2, we know that gxmi≼igxm+1i for all mand alli. This means that the sequence {gxmi}m≥0 is monotone. As xi=limm→∞gxmi, we deduce that ggxmi≼igxi for all m and alli. This condition implies that, for all m and all j,
(35)either[ggxmσj(i)≼igxσj(i)∀i]or[gxσj(i)≼iggxmσj(i)∀i]
(the first case occurs when j∈A and the second one when j∈B). Fix j∈{1,2,…,n}, and we claim that limm→∞F(gxmσj(1),gxmσj(2),…,gxmσj(n))=F(xσj(1),xσj(2),…,xσj(n)). Indeed, let ɛ>0 arbitrary, and let δ>0 verifying (MK3). Since {ggxmσj(i)}m→gxσj(i) for all i, there exists m2∈ℕ such that if m≥m2, then d(ggxmσj(i),gxσj(i))<ɛ+δ for all i. Applying (MK3) and (35),
(36)d(F(gxmσj(1),gxmσj(2),…,gxmσj(n)),aaF(xσj(1),xσj(2),…,xσj(n)))<ɛ.
Therefore, F(xσj(1),xσj(2),…,xσj(n))=limm→∞F(gxmσj(1),gxmσj(2),…,gxmσj(n))=limm→∞ggxm+1j=gxj for all j. In conclusion, (x1,x2,…,xn) is a Υ-coincidence point of F and g.


## 5. Uniqueness ofΥ-Coincidence Points

For the uniqueness of a fixed point, we need the following notion. Consider on the product spaceXn the following partial order: for (x1,x2,…,xn),(y1,y2,…,yn)∈Xn,
(37)(x1,x2,…,xn)≤(y1,y2,…,yn)⟺xi≼iyi,∀i.
We say that two points (x1,x2,…,xn) and (y1,y2,…,yn) are comparable if (x1,x2,…,xn)≤(y1,y2,…,yn) or (x1,x2,…,xn)≥(y1,y2,…,yn).By following the lines of Theorem 11 in [35], we prove the uniqueness of the coincidence point.Theorem 23.
Under the hypothesis of Theorem22, assume that for all Υ-coincidence points (x1,x2,…,xn), (y1,y2,…,yn)∈Xn of F and g, there exists (u1,u2,…,un)∈Xn such that (gu1,gu2,…,gun) is comparable, at the same time, to (gx1,gx2,…,gxn) and to (gy1,gy2,…,gyn).
Then,F and g have a unique Υ-coincidence point (z1,z2,…,zn)∈Xn such that gzi=zi for all i.It is natural to say thatg is injective on the set of all Υ-coincidence points of F and g when gxi=gyi for all i implies xi=yi for all i when (x1,x2,…,xn),(y1,y2,…,yn)∈Xn are two Υ-coincidence points of F and g. For example, this is true that g is injective on X.Corollary 24.
In addition to the hypotheses of Theorem23, suppose that g is injective on the set of all Υ-coincidence points of F and g. Then, F and g have a unique Υ-coincidence point.Proof.
If(x1,x2,…,xn) and (y1,y2,…,yn) are two Υ-coincidence points of F and g, we have proved in (A.1) that gxi=gyi for all i. As g is injective on these points, then xi=yi for all i.Corollary 25.
In addition to the hypotheses of Theorem23, suppose that (zσi(1),zσi(2),…,zσi(n)) is comparable to (zσj(1),zσj(2),…,zσj(n)) for all i,j. Then, z1=z2=⋯=zn.
In particular, there exists a uniquez∈X such that F(z,z,…,z)=z, which verifies gz=z.Proof.
LetM=max1≤i,j≤nd(zi,zj) and we are going to show that M=0 by contradiction. Assume that M>0 and let j0,s0∈{1,2,…,n} such that d(zj0,zs0)=M. As (zσj0(1),zσj0(2),…,zσj0(n)) is comparable to (zσs0(1),zσs0(2),…,zσs0(n)), then either zσj0(i)≼izσs0(i) for all i or zσs0(i)≼izσj0(i) for all i. Since gzi=zi for all i, we know that either gzσj0(i)≼igzσs0(i) for all i or gzσs0(i)≼igzσj0(i) for all i. Now, we have to distinguish between two cases.
Ifgzσj0(i)=gzσs0(i) for all i (i.e., zσj0(i)=zσs0(i) for all i), then
(38)zj0=gzj0=F(zσj0(1),zσj0(2),…,zσj0(n))=F(zσs0(1),zσs0(2),…,zσs0(n))=gzs0=zs0,
which is impossible since d(zj0,zs0)=M>0. Now, suppose that max1≤i≤nd(gzσj0(i), gzσs0(i))>0. In this case, item 1 of Lemma 15 guarantees that
(39)M=d(zj0,zs0)=d(gzj0,gzs0)=d(F(zσj0(1),zσj0(2),…,zσj0(n)),aaaaF(zσs0(1),zσs0(2),…,zσs0(n)))<max1≤i≤nd(gzσj0(i),gzσs0(i))≤max1≤i,j≤nd(gzi,gzj)=max1≤i,j≤nd(zi,zj)=M,
which also is impossible. This contradiction proves that M=0; that is, zi=zj for all i,j.Remark 26.
Notice that amixed strict monotone mappingF:X×X→X in the sense of [36, Definition 2.1] is always a mixed monotone mapping in our sense (where n=2 and g is the identity mapping on X). Then, Theorems 2.1, 2.2, 2.3, and 2.4 in [36] (and, by extension, theorems by Gnana Bhaskar and Lakshmikantham [10]) are consequence of our main results.Example 27.
LetX=ℝ and d(x,y)=|x-y| be usual metric on ℝ. Consider the mapping F(x1,x2,x3,x4,x5,x6)=(2x1-3x2+x3-x4+x5-x6)/12 and g(x)=x. It is clear that F is monotone nonincreasing in odd arguments and F is monotone nondecreasing in even arguments. All conditions of Theorems 22 and 23 are satisfied. It is clear that (0,0,0,0,0,0) is the unique fixed point.Example 28.
LetX=ℝ be provided with its usual partial order ≤ and its usual metric d(x,y)=|x-y|. Let n∈ℕ, and let a1,a2,…,an∈ℝ∖{0} real numbers such that there exist i0,j0∈{1,2,…,n} verifying ai0<0<aj0. Let N>|a1|+|a2|+⋯+|an|, and consider F(x1,x2,…,xn)=(a1x1+a2x2+⋯+anxn)/N and gx=x, for all x,x1,x2,…,xn∈X. Then, F is monotone nondecreasing in those arguments for which ai>0 and monotone nonincreasing in those arguments for which ai<0. Furthermore, taking k=(|a1|+|a2|+⋯+|an|)/N∈(0,1), Corollary 19 shows that F is a g-Meir-Keeler-type mapping. Actually, all conditions of Theorems 22 and 23 are satisfied. Indeed, it is clear that (0,0,…,0) is the unique fixed point of F.



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