Volume 2013, Article ID 406026,9pages http://dx.doi.org/10.1155/2013/406026
Research Article
Meir-Keeler Type Multidimensional Fixed Point Theorems in Partially Ordered Metric Spaces
Erdal Karap J nar,
1Antonio Roldán,
2Juan Martínez-Moreno,
2and Concepcion Roldán
21Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
2University of Ja´en, Campus las Lagunillas s/n, 23071 Ja´en, Spain
Correspondence should be addressed to Erdal Karapınar; [email protected] Received 20 December 2012; Accepted 19 February 2013
Academic Editor: Mohamed Amine Khamsi
Copyright © 2013 Erdal Karapınar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 as Banach 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 ofcoupled fixed pointand 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, Rold´an et al. [35] generalized this idea by introducing the notion ofΦ-fixed point, that is to say, the multidimensional fixed point.
Another remarkable generalization of Banach contrac- tion 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]. Let𝑛be a positive integer. Henceforth,𝑋 will denote a nonempty set, and𝑋𝑛will denote the product space𝑋 × 𝑋 ×. . . × 𝑋. Throughout this paper,𝑛 𝑚and𝑘will denote nonnegative integers and𝑖, 𝑗, 𝑠 ∈ {1, 2, . . . , 𝑛}. Unless otherwise stated, “for all𝑚” will mean “for all𝑚 ≥ 0” and “for all𝑖” will mean “for all𝑖 ∈ {1, 2, . . . , 𝑛}.”
Ametric on𝑋is a mapping𝑑 : 𝑋 × 𝑋 → Rsatisfying, for all𝑥,𝑦,𝑧 ∈ 𝑋,
(i) 𝑑 (𝑥, 𝑦) = 0, iff𝑥 = 𝑦;
(ii) 𝑑 (𝑥, 𝑦) ≤ 𝑑 (𝑧, 𝑥) + 𝑑 (𝑧, 𝑦) . (1) From these properties, we can easily deduce that𝑑(𝑥, 𝑦) ≥ 0 and𝑑(𝑦, 𝑥) = 𝑑(𝑥, 𝑦)for all𝑥, 𝑦 ∈ 𝑋. The last requirement is called thetriangle inequality. If𝑑is a metric on𝑋, we say that (𝑋, 𝑑)is ametric space(for short, anMS).
Definition 1(see [15]). A triple(𝑋, 𝑑, ≼)is called apartially ordered metric spaceif(𝑋, 𝑑)is an MS and≼is a partial order on𝑋.
Definition 2(see [10]). An ordered MS(𝑋, 𝑑, ≼)is said to have thesequential𝑔-monotone propertyif it verifies the following.
(i) If{𝑥𝑚}is a nondecreasing sequence and{𝑥𝑚} → 𝑥,𝑑 then𝑔𝑥𝑚≼ 𝑔𝑥for all𝑚.
(ii) If{𝑦𝑚}is a nonincreasing sequence and{𝑦𝑚} → 𝑦,𝑑 then𝑔𝑦𝑚≼ 𝑔𝑦for all𝑚.
If𝑔is the identity mapping, then𝑋 is said to have the sequential monotone property.
Henceforth, fix a partition{𝐴, 𝐵}ofΛ𝑛 = {1, 2, . . . , 𝑛};
that is,𝐴 ∪ 𝐵 = Λ𝑛and𝐴 ∩ 𝐵 = 0. We will denote that Ω𝐴,𝐵 = {𝜎 : Λ𝑛→ Λ𝑛: 𝜎 (𝐴) ⊆𝐴, 𝜎 (𝐵) ⊆𝐵} , Ω𝐴,𝐵 = {𝜎 : Λ𝑛→ Λ𝑛: 𝜎 (𝐴) ⊆𝐵, 𝜎 (𝐵) ⊆𝐴} . (2) If(𝑋, ≼)is a partially ordered space,𝑥, 𝑦 ∈ 𝑋, and𝑖 ∈ Λ𝑛, we will use the following notation:
𝑥 ≼𝑖𝑦 ⇐⇒ {𝑥 ≼ 𝑦, if𝑖 ∈ 𝐴,
𝑥 ≽ 𝑦, if𝑖 ∈ 𝐵. (3)
Let𝐹 : 𝑋𝑛 → 𝑋and𝑔 : 𝑋 → 𝑋be two mappings.
Definition 3(see [35]). We say that𝐹and𝑔arecommutingif 𝑔𝐹(𝑥1, . . . , 𝑥𝑛) = 𝐹(𝑔𝑥1, . . . , 𝑔𝑥𝑛)for all𝑥1, . . . , 𝑥𝑛∈ 𝑋.
Definition 4(see [35]). Let(𝑋, ≼)be a partially ordered space.
We say that 𝐹 has the mixed 𝑔-monotone property (w.r.t.
{𝐴, 𝐵}) if𝐹is𝑔-monotone nondecreasing in arguments of𝐴 and𝑔-monotone nonincreasing in arguments of𝐵; that is, for all𝑥1, 𝑥2, . . . , 𝑥𝑛,𝑦,𝑧 ∈ 𝑋and all𝑖,
𝑔𝑦 ≼ 𝑔𝑧 ⇒ 𝐹 (𝑥1, . . . , 𝑥𝑖−1, 𝑦, 𝑥𝑖+1, . . . , 𝑥𝑛)
≼𝑖𝐹 (𝑥1, . . . , 𝑥𝑖−1, 𝑧, 𝑥𝑖+1, . . . , 𝑥𝑛) . (4)
Henceforth, let𝜎1, 𝜎2, . . . , 𝜎𝑛, 𝜏 : Λ𝑛 → Λ𝑛 be 𝑛 + 1 mappings fromΛ𝑛into itself, and letΦbe the(𝑛 + 1)-tuple (𝜎1, 𝜎2, . . . , 𝜎𝑛, 𝜏).
Definition 5(see [35]). A point(𝑥1, 𝑥2, . . . , 𝑥𝑛) ∈ 𝑋𝑛is called aΦ-coincidence point of the mappings𝐹and𝑔if
𝐹 (𝑥𝜎𝑖(1), 𝑥𝜎𝑖(2), . . . , 𝑥𝜎𝑖(𝑛)) = 𝑔𝑥𝜏(𝑖) ∀𝑖. (5) If𝑔is the identity mapping on𝑋, then(𝑥1, 𝑥2, . . . , 𝑥𝑛) ∈ 𝑋𝑛 is called aΦ-fixed point of the mapping𝐹.
Remark 6. If𝐹and𝑔are 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 and𝑔.
With regards to coincidence points, it is possible to consider the following simplification. If𝜏is a permutation of Λ𝑛and we reorder (5), then we deduce that every coincidence point may be seen as a coincidence point associated to the identity mapping onΛ𝑛.
Lemma 7. Let 𝜏 be a permutation of Λ𝑛, and let Φ = (𝜎1, 𝜎2, . . . , 𝜎𝑛, 𝜏) andΦ = (𝜎𝜏−1(1), 𝜎𝜏−1(2), . . . , 𝜎𝜏−1(𝑛), 𝐼Λ𝑛).
Then, a point(𝑥1, 𝑥2, . . . , 𝑥𝑛) ∈ 𝑋𝑛is aΦ-coincidence point of the mappings𝐹and𝑔if and only if(𝑥1, 𝑥2, . . . , 𝑥𝑛)is aΦ- coincidence point of the mappings𝐹and𝑔.
Therefore, in the sequel, without loss of generality, we will only consider Υ-coincidence points where Υ = (𝜎1, 𝜎2, . . . , 𝜎𝑛), that is, that verify𝐹(𝑥𝜎𝑖(1), 𝑥𝜎𝑖(2), . . . , 𝑥𝜎𝑖(𝑛)) = 𝑔𝑥𝑖for all𝑖.
If one represents a mapping𝜎 : Λ𝑛 → Λ𝑛throughout its ordered image, that is,𝜎 = (𝜎(1), 𝜎(2), . . . , 𝜎(𝑛)), then
(i) Gnana-Bhaskar and Lakshmikantham’s election in 𝑛 = 2is𝜎1= 𝜏 = (1, 2)and𝜎2= (2, 1);
(ii) Berinde and Borcut’s election in𝑛 = 3is𝜎1 = 𝜏 = (1, 2, 3),𝜎2= (2, 1, 2)and𝜎3= (3, 2, 1);
(iii) Karapnar’s election in𝑛 = 4is𝜎1 = 𝜏 = (1, 2, 3, 4), 𝜎2= (2, 3, 4, 1),𝜎3= (3, 4, 1, 2), and𝜎4= (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 𝜀0 > 0 and two subsequences {𝑥𝑚(𝑘)}𝑘∈Nand{𝑥𝑛(𝑘)}𝑘∈Nsuch that, for all𝑘 ∈N, 𝑘 < 𝑚(𝑘) <
𝑛(𝑘) < 𝑚(𝑘 + 1),𝑑(𝑥𝑚(𝑘), 𝑥𝑛(𝑘)) ≥ 𝜀0, and𝑑(𝑥𝑚(𝑘), 𝑥𝑛(𝑘)−1) <
𝜀0.
Meir and Keeler generalized the Banach contraction mapping principle in the following way.
Definition 9(Meir and Keeler [34]). AMeir-Keeler mapping is a mapping𝑇 : 𝑋 → 𝑋on an MS(𝑋, 𝑑) such that for all𝜀 > 0, there exists𝛿 > 0verifying that if𝑥, 𝑦 ∈ 𝑋and 𝜀 ≤ 𝑑(𝑥, 𝑦) < 𝜀 + 𝛿, then𝑑(𝑇𝑥, 𝑇𝑦) < 𝜀.
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 anL-functionif (a)𝜙(0) = 0, (b)𝜙(𝑡) > 0for all𝑡 > 0, and (c) for all𝜀 > 0, there exists𝛿 > 0such that 𝜙(𝑡) ≤ 𝜀for all𝑡 ∈ [𝜀, 𝜀 + 𝛿].
Theorem 11 (Lim [30]). Let(𝑋, 𝑑)be an MS, and let𝑇 : 𝑋 → 𝑋. Then𝑇is a Meir-Keeler mapping if and only if there exists an (nondecreasing, right-continuous) L-map𝜙such that
𝑑 (𝑇 (𝑥) , 𝑇 (𝑦)) < 𝜙 (𝑑 (𝑥, 𝑦))
∀𝑥, 𝑦 ∈ 𝑋verifying 𝑑 (𝑥, 𝑦) > 0. (6) 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 ofgeneralized Meir-Keeler type functionas follows.
Definition 12(see [36]). Let(𝑋, 𝑑, ≼)be a partially ordered metric space and𝐹 : 𝑋 × 𝑋 → 𝑋a given mapping. We say that𝐹is ageneralized Meir-Keeler type functionif for all𝜀 > 0, there exists𝛿(𝜀) > 0such that
𝑥 ≽ 𝑢, 𝑦 ≼V, 𝜀 ≤ 1
2 [𝑑 (𝑥, 𝑢) + 𝑑 (𝑦,V)] < 𝜀 + 𝛿 (𝜀)
⇒ 𝑑 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢,V)) < 𝜀.
(7)
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 let𝐹 : 𝑋𝑛 → 𝑋and𝑔 : 𝑋 → 𝑋be two mappings.
Definition 13. We will say that𝐹is a (multidimensional)𝑔- Meir-Keeler type mapping, ((MK) mapping) if it verifies the following two properties.
(MK1) If𝑥1, 𝑥2, . . . , 𝑥𝑛, 𝑦1, 𝑦2, . . . , 𝑦𝑛 ∈ 𝑋verify𝑔𝑥𝑖 = 𝑔𝑦𝑖 for all𝑖, then𝐹(𝑥1, 𝑥2, . . . , 𝑥𝑛) = 𝐹(𝑦1, 𝑦2, . . . , 𝑦𝑛).
(MK2) For all 𝜀 > 0, there exists 𝛿 > 0 such that if 𝑥1, 𝑥2, . . . , 𝑥𝑛, 𝑦1, 𝑦2, . . . , 𝑦𝑛 ∈ 𝑋verify𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖for all𝑖and
𝜀 ≤max
1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖, 𝑔𝑦𝑖) < 𝜀 + 𝛿,
then𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀. (8)
If𝑔is the identity mapping on𝑋, we will say that𝐹is 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) for𝜀 = 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 that𝑋has a partial order. In any case, if𝑛 = 1, (𝑋, 𝑑)has a partial order and𝑔is the identity mapping on𝑋, and we can only establish that if𝐹 : 𝑋 → 𝑋is a Meir-Keeler mapping in the sense of Definition9, then𝐹is a Meir-Keeler- type mapping in the sense of Definition13, but the converse does not hold.
Remark 14. If 𝑔 is an injective mapping on 𝑋, then all mappings𝐹verify (MK1).
Lemma 15. Let𝐹 : 𝑋𝑛 → 𝑋be a mapping on a partially ordered MS(𝑋, 𝑑, ≼), and let𝑥1, 𝑥2, . . . , 𝑥𝑛, 𝑦1, 𝑦2, . . . , 𝑦𝑛 ∈ 𝑋 be such that𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖for all𝑖.
(1)If𝐹verifies (MK2), then either𝑔𝑥𝑖= 𝑔𝑦𝑖for all𝑖or 𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛)) <max
1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖, 𝑔𝑦𝑖) . (9) (2)If𝐹is a g-Meir-Keeler type mapping, then
𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛)) ≤max
1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖, 𝑔𝑦𝑖) , (10) and the equality is achieved if and only if𝑔𝑥𝑖= 𝑔𝑦𝑖for all𝑖.
Proof. (1) If the condition “𝑔𝑥𝑖 = 𝑔𝑦𝑖 for all 𝑖”
does not hold, then 𝜀 = max1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖, 𝑔𝑦𝑖) > 0.
Hence, 𝑑(𝐹(𝑥1, 𝑥2, . . . , 𝑥𝑛), 𝐹(𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀 = max1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖, 𝑔𝑦𝑖). (2)If 𝐹is a 𝑔-Meir-Keeler-type map- ping, the case “𝑔𝑥𝑖 = 𝑔𝑦𝑖for all𝑖” means that the equality is achieved.
This global contractivity condition (10) is not strong enough to ensure that𝐹has a fixed point. For instance, if 𝑛 = 1, then𝐹(𝑥) = 𝑥 + 1for all𝑥 ∈Rhas 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 continuityof𝐹is, for all𝜀 > 0,
𝛿𝑔,𝐹(𝜀) =sup({𝜆 ≥ 0 : [ 𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖 ∀𝑖, max1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖, 𝑔𝑦i) < 𝜆]
⇒ 𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀}) .
(11)
Remark 17. The identity mapping on a set𝑋will be denoted by 1𝑋 : 𝑋 → 𝑋. If 𝑔 : 𝑋 → 𝑋 is a mapping, then
𝐺 : 𝑋𝑛 → 𝑋𝑛will be defined by𝐺(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 on𝑋𝑛. A partial order
≼on 𝑋 may be induced on𝑋𝑛 by 𝑃 ≼ 𝑄if and only if 𝑝𝑖 ≼ 𝑞𝑖 for all𝑖 (notice that this partial order depends on the partition{𝐴, 𝐵}ofΛ𝑛). Then,(𝑋𝑛, 𝐷, ≼)also is a partially ordered MS. Furthermore, given any𝜔 = (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
𝐷 (𝐹𝜔(𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹𝜔(𝑦1, 𝑦2, . . . , 𝑦𝑛))
= 𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛)) (12) for all𝑥1, 𝑥2, . . . , 𝑥𝑛, 𝑦1, 𝑦2, . . . , 𝑦𝑛∈ 𝑋.
Theorem 18. Let(𝑋, 𝑑, ≼)be a partially ordered MS, and let 𝐹 : 𝑋𝑛 → 𝑋and𝑔 : 𝑋 → 𝑋be two mappings. Then, the following statements are equivalent.
(MK)𝐹is a𝑔-Meir-Keeler-type mapping.
(MK3)For all𝜀 > 0, there exists𝛿 > 0such that 𝑥1, 𝑥2, . . . , 𝑥𝑛, 𝑦1, 𝑦2, . . . , 𝑦𝑛∈ 𝑋
𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖 ∀𝑖 max1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖, 𝑔𝑦𝑖) < 𝜀 + 𝛿
}} }} }
⇒ 𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀.
(13)
(MK4)𝛿𝑔,𝐹(𝜀) > 𝜀for all𝜀 > 0.
(MK5)𝐹and𝑔verify (MK1), and there exists an (nondecreas- ing, right-continuous) L-function𝜙 : [0, ∞[ → [0, ∞[
such that
𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛))
< 𝜙 (max
1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖, 𝑔𝑦𝑖)) (14)
for all𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛 ∈ 𝑋verifying𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖 for all𝑖andmax1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖, 𝑔𝑦𝑖) > 0.
(MK6)For all𝜔 ∈ 𝑋𝑛−1, the mapping𝐹𝜔 : 𝑋𝑛 → 𝑋𝑛 is a 𝐺-Meir-Keeler-type mapping on(𝑋𝑛, 𝐷, ≼).
(MK7)There exists𝜔0 ∈ 𝑋𝑛−1such that the mapping𝐹𝜔0 : 𝑋𝑛 → 𝑋𝑛 is a 𝐺-Meir-Keeler-type mapping on (𝑋𝑛, 𝐷, ≼).
Proof. [(MK)⇒(MK3)]: Fix 𝜀 > 0, and let𝛿 > 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) <
𝜀 + 𝛿. If 𝜂 = 0, then 𝑔𝑥𝑖 = 𝑔𝑦𝑖 for all 𝑖, and so 𝑑(𝐹(𝑥1, 𝑥2, . . . , 𝑥𝑛), 𝐹(𝑦1, 𝑦2, . . . , 𝑦𝑛)) = 0 < 𝜀 by (MK1).
In another case, 𝜂 > 0. If 𝜀 ≤ 𝜂 < 𝜀 + 𝛿, then 𝑑(𝐹(𝑥1, 𝑥2, . . . , 𝑥𝑛), 𝐹(𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀 by (MK2). Now, suppose that0 < 𝜂 < 𝜀. Then,𝜂 ∈ [𝜂, 𝜂 + 𝛿𝜂[, where𝛿𝜂 > 0
is also given by (MK2), and𝑑(𝐹(𝑥1, . . . , 𝑥𝑛), 𝐹(𝑦1, . . . , 𝑦𝑛)) <
𝜂 < 𝜀. Hence, (MK3) holds.
[(MK3)⇒(MK4)]: Given𝜀 > 0, let𝛿 > 0verifying (MK3).
Then,𝛿𝑔,𝐹(𝜀) ≥ 𝜀 + 𝛿, and so𝛿𝑔,𝐹(𝜀) > 𝜀.
[(MK4)⇒(MK)]: On the one hand, if𝑔𝑥𝑖 = 𝑔𝑦𝑖for all𝑖, then𝑑(𝐹(𝑥1, 𝑥2, . . . , 𝑥𝑛), 𝐹(𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀for all𝜀 > 0, and so𝐹verify (MK1). On the other hand, let𝜀 > 0, and define𝛿 = (𝛿𝑔,𝐹(𝜀) − 𝜀)/2 > 0. Therefore,𝜀 + 𝛿 < 𝛿𝑔,𝐹(𝜀).
Since𝛿𝑔,𝐹(𝜀)is a supremum, there exists𝜆0 ∈ ]𝜀 + 𝛿, 𝛿𝑔,𝐹(𝜀)]
such that if𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖for all𝑖and max1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖, 𝑔𝑦𝑖) < 𝜆0, then𝑑(𝐹(𝑥1, 𝑥2, . . . , 𝑥𝑛), 𝐹(𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀. In particular, if𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖for all𝑖and𝜀 ≤max1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖, 𝑔𝑦𝑖) < 𝜀+𝛿 < 𝜆0, then𝑑(𝐹(𝑥1, 𝑥2, . . . , 𝑥𝑛),𝐹(𝑦1, 𝑦2, . . . , 𝑦𝑛)) < 𝜀.
[(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 all𝑡 ≥ 0.
Corollary 19. Let(𝑋, 𝑑, ≼)be a partially ordered metric space, and let𝐹 : 𝑋𝑛 → 𝑋and𝑔 : 𝑋 → 𝑋be two mappings.
Assume that there exists𝑘 ∈ (0, 1)such that 𝑑 (𝐹 (𝑥1, 𝑥2, . . . , 𝑥𝑛) , 𝐹 (𝑦1, 𝑦2, . . . , 𝑦𝑛)) ≤ 𝑘max
1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖, 𝑔𝑦𝑖) (15) for all𝑥1, . . . , 𝑥𝑛, 𝑦1, . . . , 𝑦𝑛 ∈ 𝑋verifying𝑔𝑥𝑖≼𝑖𝑔𝑦𝑖for all𝑖.
Then,𝐹is a𝑔-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 Definition13.
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 Definition13taking𝑔as the identity mapping on the MS.
Proof. Suppose that𝐹 : 𝑋 × 𝑋 → 𝑋is a generalized Meir Keeler type function in the sense of Samet. Fix𝜀 > 0 and let𝛿 > 0verifying (7). Let𝑥, 𝑦, 𝑢,V ∈ 𝑋such that𝑥 ≽ 𝑢, 𝑦 ≼V, and max(𝑑(𝑥, 𝑢), 𝑑(𝑦,V)) < 𝜀+𝛿. We have to prove that 𝑑(𝐹(𝑥, 𝑦), 𝐹(𝑢,V)) < 𝜀. If𝑥 = 𝑢and𝑦 =V, there is nothing to prove. Next, suppose that max(𝑑(𝑥, 𝑢), 𝑑(𝑦,V)) > 0. Let
𝑀 = 1
2[𝑑 (𝑥, 𝑢) + 𝑑 (𝑦,V)] . (16) If𝑀 = 0, then𝑥 = 𝑢and𝑦 =V, which is false. Then,𝑀 > 0.
On the other hand, 𝑀 = 𝑑 (𝑥, 𝑢) + 𝑑 (𝑦,V)
2 ≤max(𝑑 (𝑥, 𝑢) , 𝑑 (𝑦,V)) < 𝜀 + 𝛿.
(17) If𝜀 ≤ 𝑀 < 𝜀 + 𝛿, then𝑑(𝐹(𝑥, 𝑦), 𝐹(𝑢,V)) < 𝜀by (7).
Finally, if 0 < 𝑀 < 𝜀, taking 𝜀 = 𝑀 in (7), we have that 𝑀 ∈ [𝜀, 𝜀 + 𝛿𝜀[ (where𝛿𝜀 is taken as in (7)), and
so𝑑(𝐹(𝑥, 𝑦), 𝐹(𝑢,V)) < 𝑀 < 𝜀. This proves that 𝐹is a 2- dimensional Meir-Keeler type mapping associated to𝑔 = 𝐼𝑋.
Remark 21. Converse of Lemma 20 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 consider𝐹(𝑥, 𝑦) = 𝑘𝑥 for all𝑥, 𝑦 ∈ R. Then, 𝐹 is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition13(taking𝑔as the identity mapping onR), but, if𝑘 > 1/2, it is not a generalized Meir Keeler type function in the sense of Samet.
Indeed, we firstly prove that𝐹is a 2-dimensional Meir- Keeler-type mapping in the sense of Definition13 (taking 𝑔as the identity mapping on R). Let𝜀 > 0. Consider any 𝑟 ∈]0, 1/𝑘 − 1[(i.e.,𝑘(1 + 𝑟) < 1) and define𝛿 = 𝑟𝜀 > 0.
Consider𝑥, 𝑦, 𝑢,V ∈Rsuch that𝑥 ≥ 𝑢and𝑦 ≤Vverifying 𝜀 ≤max(𝑑(𝑥, 𝑢), 𝑑(𝑦,V)) =max(|𝑥 − 𝑢|, |𝑦 −V|) < 𝜀 + 𝛿. In particular,|𝑥 − 𝑢| < 𝜀 + 𝛿. Then,
𝑑 (𝐹 (𝑥, 𝑦) , 𝐹 (𝑢,V)) = 𝑑 (𝑘𝑥, 𝑘𝑢) = 𝑘 |𝑥 − 𝑢| < 𝑘 (𝜀 + 𝛿)
= 𝑘 (𝜀 + 𝑟𝜀) = 𝑘 (1 + 𝑟) 𝜀 < 𝜀.
(18) It follows that𝐹is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition13. Next, we claim that if𝑘 > 1/2, then𝐹is not a generalized Meir Keeler type function in the sense of Samet. Let𝜀 > 0. If𝐹was a generalized Meir Keeler type function in the sense of Samet, it would be 𝛿 > 0 verifying (7). Take𝑥 = 𝜀,𝑢 = − 𝜀, and𝑦 = V = 0. Then, 𝑥 ≥ 𝑢,𝑦 ≤Vand
𝑑 (𝑥, 𝑢) + 𝑑 (𝑦,V)
2 = |𝜀 − (− 𝜀)| + |0 − 0|
2
=2𝜀
2 = 𝜀 ∈ [𝜀, 𝜀 + 𝛿[ .
(19)
However,𝑑(𝐹(𝑥, 𝑦), 𝐹(𝑢,v)) = 𝑑(𝑘𝜀, 𝑘(−𝜀)) = 𝑘|𝜀 − (−𝜀)| = 2𝑘𝜀 > 𝜀since𝑘 > 1/2.
4. Main Results
In the following result, we show sufficient conditions to ensure the existence of Υ-coincidence points, where Υ = (𝜎1, 𝜎2, . . . , 𝜎𝑛).
Theorem 22. Let(𝑋, 𝑑)be a complete MS, and let≼a partial order on𝑋. LetΥ = (𝜎1, 𝜎2, . . . , 𝜎𝑛)be an𝑛-tuple of mappings from{1, 2, . . . , 𝑛}into itself verifying𝜎𝑖 ∈ Ω𝐴,𝐵 if𝑖 ∈ 𝐴and 𝜎𝑖 ∈ Ω𝐴,𝐵 if𝑖 ∈ 𝐵. Let𝐹 : 𝑋𝑛 → 𝑋and𝑔 : 𝑋 → 𝑋 be two mappings such that𝐹is a𝑔-Meir-Keeler-type mapping and has the mixed 𝑔-monotone property on 𝑋, 𝐹(𝑋𝑛) ⊆ 𝑔(𝑋), and 𝑔is continuous and commuting with𝐹. Suppose that either𝐹is continuous or(𝑋, 𝑑, ≼)has the sequential g- monotone property. If there exist𝑥10, 𝑥20, . . . , 𝑥𝑛0 ∈ 𝑋verifying 𝑔𝑥𝑖0≼𝑖𝐹(𝑥0𝜎𝑖(1), 𝑥0𝜎𝑖(2), . . . , 𝑥𝜎0𝑖(𝑛))for all𝑖, then𝐹and𝑔have, at least, oneΥ-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 {𝑥1𝑚}𝑚≥0, {𝑥2𝑚}𝑚≥0, . . . , {𝑥𝑛𝑚}𝑚≥0such that𝑔𝑥𝑚+1𝑖 = 𝐹(𝑥𝜎𝑚𝑖(1),𝑥𝜎𝑚𝑖(2), . . . , 𝑥𝜎𝑚𝑖(𝑛))for all 𝑚and all𝑖.
Step 2.𝑔𝑥𝑖𝑚≼𝑖𝑔𝑥𝑖𝑚+1for all𝑚and all𝑖.
Define 𝛿𝑚 = max1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖𝑚, 𝑔𝑥𝑖𝑚+1) ≥ 0 for all 𝑚.
Firstly, suppose that there exists𝑚0 ∈ Nsuch that𝛿𝑚0 = 0.
Then,𝑔𝑥𝑖𝑚0 = 𝑔𝑥𝑚𝑖0+1 = 𝐹(𝑥𝜎𝑚𝑖(1)0 , 𝑥𝜎𝑚𝑖(2)0 , . . . , 𝑥𝜎𝑚𝑖(𝑛)0 )for all𝑖, so (𝑥1𝑚0, 𝑥2𝑚0, . . . , 𝑥𝑚𝑛0)is aΥ-coincidence point of𝐹and𝑔and we have finished. Therefore, we may reduce to the case in which 𝛿𝑚 > 0for all𝑚; that is,
∀𝑚,there exists 𝑗such that𝑔𝑥𝑗𝑚 ̸= 𝑔𝑥𝑗𝑚+1. (20) Step 3. We claim that {𝑑(𝑔𝑥𝑚𝑖 , 𝑔𝑥𝑖𝑚+1)}𝑚≥0 → 0 for all 𝑖 (i.e., {max1≤𝑗≤𝑛𝑑(𝑔𝑥𝑗𝑚, 𝑔𝑥𝑚+1𝑗 )}𝑚≥0 → 0). Indeed, as 𝑔𝑥𝑖𝑚≼𝑖𝑔𝑥𝑖𝑚+1 for all 𝑚 and all 𝑖, then condition (MK2), Lemma15, and (20) imply that, for all𝑚 ≥ 1and all𝑖,
𝑑 (𝑔𝑥𝑖𝑚, 𝑔𝑥𝑖𝑚+1) = 𝑑 (𝐹 (𝑥𝜎𝑚−1𝑖(1), 𝑥𝜎𝑚−1𝑖(2), . . . , 𝑥𝜎𝑚−1𝑖(𝑛)) , 𝐹 (𝑥𝜎𝑚𝑖(1), 𝑥𝜎𝑚𝑖(2), . . . , 𝑥𝜎𝑚𝑖(𝑛)))
<max
1≤𝑗≤𝑛𝑑 (𝑔𝑥𝜎𝑚−1𝑖(𝑗), 𝑔𝑥𝜎𝑚𝑖(𝑗))
≤max
1≤𝑗≤𝑛𝑑 (𝑔𝑥𝑗𝑚−1, 𝑔𝑥𝑗𝑚) = 𝛿𝑚−1.
(21)
Taking maximum on𝑖, we deduce that the sequence{𝛿𝑚}𝑚≥1 is nonincreasing and lower bounded. Therefore, it is conver- gent; that is, there existsΔ ≥ 0such that{𝛿𝑚}𝑚≥1 → Δ(and Δ ≤ 𝛿𝑚for all𝑚). We claim thatΔ = 0. On the contrary, assume thatΔ > 0. Let𝛿 > 0be a positive number associated to𝜀 = Δ > 0by (MK2). Since
{max1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖𝑚, 𝑔𝑥𝑖𝑚+1)}
𝑚= {𝛿𝑚}𝑚↘ Δ, (22) there exists 𝑚0 ∈ N such that if 𝑚 ≥ 𝑚0, then Δ ≤ max1≤𝑖≤𝑛𝑑(𝑔𝑥𝑖𝑚, 𝑔𝑥𝑖𝑚+1) < Δ + 𝛿. By (MK2), it follows that, for all𝑖,
𝑑 (𝑔𝑥𝑖𝑚0, 𝑔𝑥𝑖𝑚0+1)
= 𝑑 (𝐹 (𝑥𝜎𝑚𝑖(1)0 , 𝑥𝜎𝑚𝑖(2)0 , . . . , 𝑥𝜎𝑚𝑖(𝑛)0 ) , 𝐹 (𝑥𝜎𝑚𝑖(1)0+1, 𝑥𝜎𝑚𝑖0(2)+1, . . . , 𝑥𝜎𝑚𝑖(𝑛)0+1)) < Δ.
(23)
Taking maximum on𝑖, we deduce that Δ ≤ 𝛿𝑚0=max
1≤𝑖≤𝑛𝑑 (𝑔𝑥𝑖𝑚0, 𝑔𝑥𝑖𝑚0+1) < Δ. (24) But this is impossible. Then, we have just proved thatΔ = 0.
Therefore,{𝛿𝑚}𝑚≥1 → Δ = 0, which means that
𝑚 → ∞lim 𝛿𝑚= lim
𝑚 → ∞(max
1≤𝑗≤𝑛𝑑 (𝑔𝑥𝑗𝑚, 𝑔𝑥𝑗𝑚+1)) = 0. (25)
As 0 ≤ 𝑑(𝑔𝑥𝑖𝑚, 𝑔𝑥𝑖𝑚+1) ≤ 𝛿𝑚 for all 𝑚 and all 𝑖, then {𝑑(𝑔𝑥𝑖𝑚, 𝑔𝑥𝑖𝑚+1)} → 0for all𝑖.
Step 4. Every sequence{𝑔𝑥𝑖𝑚}𝑚≥0is Cauchy. Suppose that {𝑔𝑥𝑖𝑚1}𝑚≥0, . . . , {𝑔𝑥𝑖𝑚𝑠}𝑚≥0 are not Cauchy and {𝑔𝑥𝑖𝑚𝑠+1}𝑚≥0, . . . , {𝑔𝑥𝑖𝑚𝑛}𝑚≥0 are Cauchy, being {𝑖1, . . . , 𝑖𝑛} = {1, . . . , 𝑛}. By Lemma8, for all 𝑟 ∈ {1, 2, . . . , 𝑠}, there exist 𝜀𝑟 > 0and subsequences{𝑔𝑥𝑖𝑚𝑟
𝑟(𝑘)}𝑘∈Nand{𝑔𝑥𝑖𝑛𝑟
𝑟(𝑘)}𝑘∈Nsuch that
𝑘 < 𝑚𝑟(𝑘) < 𝑛𝑟(𝑘) , 𝑑 (𝑔𝑥𝑖m𝑟 𝑟(𝑘), 𝑔𝑥𝑖𝑛𝑟
𝑟(𝑘)) ≥ 𝜀𝑟, 𝑑 (𝑔𝑥𝑖𝑚𝑟
𝑟(𝑘), 𝑔𝑥𝑖𝑛𝑟
𝑟(𝑘)−1) < 𝜀𝑟, ∀𝑘 ∈N.
(26)
Let𝜀0 =max(𝜀1, . . . , 𝜀𝑠)and𝜀0 =min(𝜀1, . . . , 𝜀𝑠) > 0. Since {𝑔𝑥𝑖𝑚𝑠+1}𝑚≥0, . . . , {𝑔𝑥𝑖𝑚𝑛}𝑚≥0 are Cauchy, there exists 𝑛0 ∈ N such that if 𝑛, 𝑚 ≥ 𝑛0, then 𝑑(𝑔𝑥𝑚𝑗, 𝑔𝑥𝑗𝑛) < 𝜀0/2 for all 𝑗 ∈ {𝑖𝑠+1, . . . , 𝑖𝑛}.
Let 𝑘0 ∈ 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(𝑔𝑥𝑖𝑚𝑟
𝑟(𝑘0), 𝑔𝑥𝑖𝑛𝑟
𝑟(𝑘0)) ≥ 𝜀𝑟 ≥ 𝜀0. Thus, we can consider the numbers 𝑚(1) + 1,𝑚(1) + 2, . . .until finding the first positive integer 𝑛(1) > 𝑚(1)verifying
max1≤𝑟≤𝑠𝑑 (𝑔𝑥𝑖𝑚(1)𝑟 , 𝑔𝑥𝑖𝑛(1)𝑟 ) ≥ 𝜀0,
𝑑 (𝑔𝑥𝑖𝑚(1)𝑗 , 𝑔𝑥𝑖𝑛(1)−1𝑗 ) < 𝜀0, ∀𝑗 ∈ {1, 2, . . . , 𝑠} .
(27)
Now, let𝑘1 ∈ N such that𝑛(1) < min(𝑚1(𝑘1),𝑚2(𝑘1) , . . . , 𝑚𝑠(𝑘1)) and define 𝑚(2) = min(𝑚1(𝑘1), 𝑚2(𝑘1) , . . . , 𝑚𝑠(𝑘1)). Since𝑚(2) ∈ {𝑚1(𝑘1), 𝑚2(𝑘1), . . . , 𝑚𝑠(𝑘1)}, we can consider the numbers𝑚(2) + 1, 𝑚(2) + 2, . . .until finding the first positive integer𝑛(2) > 𝑚(2)verifying
max1≤𝑟≤𝑠𝑑 (𝑔𝑥𝑖𝑚(2)𝑟 , 𝑔𝑥𝑛(2)𝑖𝑟 ) ≥ 𝜀0,
𝑑 (𝑔𝑥𝑖𝑚(2)𝑗 , 𝑔𝑥𝑛(2)−1𝑖𝑗 ) < 𝜀0, ∀𝑗 ∈ {1, 2, . . . , 𝑠} .
(28)
Repeating this process, we can find sequences such that, for all𝑘 ≥ 1,
𝑛0< 𝑚 (𝑘) < 𝑛 (𝑘) < 𝑚 (𝑘 + 1) , max1≤𝑟≤𝑠𝑑 (𝑔𝑥𝑖𝑚(𝑘)𝑟 , 𝑔𝑥𝑖𝑛(𝑘)𝑟 ) ≥ 𝜀0,
𝑑 (𝑔𝑥𝑖𝑚(𝑘)𝑗 , 𝑔𝑥𝑛(𝑘)−1𝑖𝑗 ) < 𝜀0, ∀𝑗 ∈ {1, 2, . . . , 𝑠} .
(29)
Since𝑛0 < 𝑚(𝑘) < 𝑛(𝑘), we know that 𝑑(𝑔𝑥𝑗𝑚(𝑘), 𝑔𝑥𝑗𝑛(𝑘)), 𝑑(𝑔𝑥𝑗𝑚(𝑘), 𝑔𝑥𝑗𝑛(𝑘)−1), 𝑑(𝑔𝑥𝑗𝑚(𝑘)−1, 𝑔𝑥𝑗𝑛(𝑘)−1) < 𝜀0/2for all𝑗 ∈ {𝑖𝑠+1, . . . , 𝑖𝑛}. Therefore, for all𝑘,
1≤𝑗≤𝑛max𝑑 (𝑔𝑥𝑗𝑚(𝑘), 𝑔𝑥𝑗𝑛(𝑘)) =max
1≤𝑟≤𝑠𝑑 (𝑔𝑥𝑖𝑚(𝑘)𝑟 , 𝑔𝑥𝑖𝑛(𝑘)𝑟 ) ≥ 𝜀0,
1≤𝑗≤𝑛max𝑑 (𝑔𝑥𝑗𝑚(𝑘), 𝑔𝑥𝑗𝑛(𝑘)−1) < 𝜀0. (30)
Let𝛿 > 0verifying (MK3) using𝜀0, and let𝑘1 ∈Nsuch that if𝑘 ≥ 𝑘1, then𝑑(𝑔𝑥𝑗𝑚(𝑘)−1, 𝑔𝑥𝑗𝑚(𝑘)) < 𝛿for all𝑗. Then, for all𝑗 and all𝑘 ≥ 𝑘1,
𝑑 (𝑔𝑥𝑗𝑚(𝑘)−1, 𝑔𝑥𝑗𝑛(𝑘)−1) ≤ 𝑑 (𝑔𝑥𝑗𝑚(𝑘)−1, 𝑔𝑥𝑗𝑚(𝑘))
+ 𝑑 (𝑔𝑥𝑚(𝑘)𝑗 , 𝑔𝑥𝑗𝑛(𝑘)−1) < 𝛿 + 𝜀0, (31) Applying (MK3), it follows, for all𝑘 ≥ 𝑘0and all𝑖, that
𝑑 (𝑔𝑥𝑖𝑚(𝑘), 𝑔𝑥𝑖𝑛(𝑘))
= 𝑑 (𝐹 (𝑥𝜎𝑚(𝑘)−1𝑖(1) , . . . , 𝑥𝜎𝑚(𝑘)−1𝑖(𝑛) ) , 𝐹 (𝑥𝜎𝑛(𝑘)−1𝑖(1) , . . . , 𝑥𝜎𝑛(𝑘)−1𝑖(𝑛) ))
< 𝜀0,
(32) but this contradicts (30) since max1≤𝑗≤𝑛𝑑(𝑔𝑥𝑗𝑚(𝑘), 𝑔𝑥𝑛(𝑘)𝑗 ) ≥ 𝜀0. This contradiction shows us that every sequence{𝑔𝑥𝑖𝑚}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∈ 𝑋such that𝑥𝑖=lim𝑚 → ∞𝑔𝑥𝑖𝑚for all𝑖. As𝑔 is continuous and𝐹commutes with𝑔,
𝑚 → ∞lim 𝐹 (𝑔𝑥𝜎𝑚𝑖(1), 𝑔𝑥𝜎𝑚𝑖(2), . . . , 𝑔𝑥𝜎𝑚𝑖(𝑛))
=𝑚 → ∞lim 𝑔𝐹 (𝑥𝜎𝑚𝑖(1), 𝑥𝜎𝑚𝑖(2), . . . , 𝑥𝜎𝑚𝑖(𝑛))
=𝑚 → ∞lim 𝑔𝑔𝑥𝑖𝑚+1= 𝑔𝑥𝑖 ∀𝑖.
(33)
Step 5. Suppose that 𝐹 is continuous. In this case, since {𝑔𝑥𝜎𝑚𝑖(𝑗)}𝑚 → 𝑥𝜎𝑖(𝑗)for all𝑖, 𝑗and𝐹is continuous,
𝑚 → ∞lim 𝑔𝑔𝑥𝑖𝑚+1=𝑚 → ∞lim 𝐹 (𝑔𝑥𝜎𝑚𝑖(1), 𝑔𝑥𝜎𝑚𝑖(2), . . . , 𝑔𝑥𝜎𝑚𝑛(𝑛))
= 𝐹 (𝑥𝜎𝑖(1), 𝑥𝜎𝑖(2), . . . , 𝑥𝜎𝑖(𝑛)) (34) for all𝑖. Then,𝐹(𝑥𝜎𝑖(1), 𝑥𝜎𝑖(2), . . . , 𝑥𝜎𝑖(𝑛)) = 𝑔𝑥𝑖for all𝑖; that is, (𝑥1, 𝑥2, . . . , 𝑥𝑛)is aΥ-coincidence point of𝐹and𝑔.
Step 6.Suppose that(𝑋, 𝑑, ≼)has the sequential𝑔-monotone property.In this case, by step 2, we know that𝑔𝑥𝑖𝑚≼𝑖𝑔𝑥𝑖𝑚+1 for all𝑚 and all 𝑖. This means that the sequence{𝑔𝑥𝑖𝑚}𝑚≥0 is monotone. As 𝑥𝑖 = lim𝑚 → ∞𝑔𝑥𝑖𝑚, we deduce that 𝑔𝑔𝑥𝑖𝑚≼𝑖𝑔𝑥𝑖for all𝑚and all 𝑖. This condition implies that, for all𝑚and all𝑗,
either [𝑔𝑔𝑥𝜎𝑚𝑗(𝑖)≼𝑖𝑔𝑥𝜎𝑗(𝑖) ∀𝑖]
or [𝑔𝑥𝜎𝑗(𝑖)≼𝑖𝑔𝑔𝑥𝜎𝑚𝑗(𝑖) ∀𝑖]
(35)
(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(𝑔𝑥𝜎𝑚𝑗(1), 𝑔𝑥𝜎𝑚𝑗(2), . . . , 𝑔𝑥𝜎𝑚𝑗(𝑛)) = 𝐹(𝑥𝜎𝑗(1), 𝑥𝜎𝑗(2), . . . , 𝑥𝜎𝑗(𝑛)). Indeed, let𝜀 > 0arbitrary, and let𝛿 > 0verifying
