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

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.

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 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 𝑋^𝑛 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}.”

A metric on 𝑋 is a mapping 𝑑∶𝑋×𝑋→ℝ satisfying, for all 𝑥, 𝑦, 𝑧∈𝑋, (i)𝑑()𝑑().𝑥,𝑦=0,iff𝑥=𝑦;ii𝑥,𝑦≤𝑑𝑧,𝑥+𝑑𝑧,𝑦 (1) 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 sequential 𝑔-monotone property if it verifies the following.   (i) If {𝑥_𝑚} is a nondecreasing sequence and {𝑥_𝑚}→−^𝑑𝑥, then 𝑔𝑥_𝑚≼𝑔𝑥 for all 𝑚.  (ii) If {𝑦_𝑚} is a nonincreasing sequence and {𝑦_𝑚}→−^𝑑𝑦, then 𝑔𝑦_𝑚≼𝑔𝑦 for all 𝑚.
If 𝑔 is the identity mapping, then 𝑋 is said to have the sequential monotone property.

Henceforth, fix a partition {𝐴,𝐵} of Λ_𝑛={1,2,…,𝑛}; that is, 𝐴∪𝐵=Λ_𝑛 and 𝐴∩𝐵=∅. We will denote that Ω_𝐴,𝐵=𝜎∶Λ_𝑛⟶Λ_𝑛(𝐴)(𝐵),Ω∶𝜎⊆𝐴,𝜎⊆𝐵_′𝐴,𝐵=𝜎∶Λ_𝑛⟶Λ_𝑛∶𝜎(𝐴)⊆𝐵,𝜎(𝐵)⊆𝐴. (2) If (𝑋,≼) is a partially ordered space, 𝑥,𝑦∈𝑋, and 𝑖∈Λ_𝑛, we will use the following notation: 𝑥≼_𝑖𝑦⟺𝑥≼𝑦,if𝑖∈𝐴,𝑥≽𝑦,if𝑖∈𝐵. (3)

Let 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 be two mappings.

Definition 3 (see [35]). We say that 𝐹 and 𝑔 are commuting if 𝑔𝐹(𝑥₁,…,𝑥_𝑛)=𝐹(𝑔𝑥₁,…,𝑔𝑥_𝑛) for all 𝑥₁,…,𝑥_𝑛∈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}) if 𝐹 is 𝑔-monotone nondecreasing in arguments of 𝐴 and 𝑔-monotone nonincreasing in arguments of 𝐵; that is, for all 𝑥₁,𝑥₂,…,𝑥_𝑛,  𝑦,  𝑧∈𝑋 and all 𝑖, 𝑔𝑦≼𝑔𝑧⇒𝐹𝑥₁,…,𝑥_𝑖−1,𝑦,𝑥_𝑖+1,…,𝑥_𝑛≼_𝑖𝐹𝑥₁,…,𝑥_𝑖−1,𝑧,𝑥_𝑖+1,…,𝑥_𝑛. (4)

Henceforth, let 𝜎₁,𝜎₂,…,𝜎_𝑛,𝜏∶Λ_𝑛→Λ_𝑛 be 𝑛+1 mappings from Λ_𝑛 into itself, and let Φ be the (𝑛+1)-tuple (𝜎₁,𝜎₂,…,𝜎_𝑛,𝜏).

Definition 5 (see [35]). A point (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 is called a Φ-coincidence point of the mappings 𝐹 and 𝑔 if 𝐹𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)}=𝑔𝑥_𝜏(𝑖)∀𝑖. (5) If 𝑔 is the identity mapping on 𝑋, then (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 is called a Φ-fixed point of the mapping 𝐹.

Remark 6. If 𝐹 and 𝑔 are commuting and (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 is a Φ-coincidence point of 𝐹 and 𝑔, then (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛) also is a Φ-coincidence point of 𝐹 and 𝑔.

With regards to coincidence points, it is possible to consider the following simplification. If 𝜏 is a permutation of Λ_𝑛 and we reorder (5), then we deduce that every coincidence point may be seen as a coincidence point associated to the identity mapping on Λ_𝑛.

Lemma 7. Let 𝜏 be a permutation of Λ_𝑛, and let Φ=(𝜎₁,𝜎₂,…,𝜎_𝑛,𝜏) and Φ^′=(𝜎_𝜏−1(1),𝜎_𝜏−1(2),…,𝜎_{𝜏−1(𝑛)},𝐼_Λ𝑛). Then, a point (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 is a Φ-coincidence point of the mappings 𝐹 and 𝑔 if and only if (𝑥₁,𝑥₂,…,𝑥_𝑛) is a Φ^′-coincidence point of the mappings 𝐹 and 𝑔.

Therefore, in the sequel, without loss of generality, we will only consider Υ-coincidence points where Υ=(𝜎₁,𝜎₂,…,𝜎_𝑛), that is, that verify 𝐹(𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)})=𝑔𝑥_𝑖 for all 𝑖.

If one represents a mapping 𝜎∶Λ_𝑛→Λ_𝑛 throughout its ordered image, that is, 𝜎=(𝜎(1),𝜎(2),…,𝜎(𝑛)), then(i)Gnana-Bhaskar and Lakshmikantham's election in 𝑛=2 is 𝜎₁=𝜏=(1,2) and 𝜎₂=(2,1);(ii) Berinde and Borcut's election in 𝑛=3 is 𝜎₁=𝜏=(1,2,3), 𝜎₂=(2,1,2) and 𝜎₃=(3,2,1);(iii) Karapnar's election in 𝑛=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 {𝑥_𝑚}_𝑚∈ℕ is a sequence in an MS (𝑋,𝑑) that is not Cauchy, then there exist 𝜀₀>0 and two subsequences {𝑥_𝑚(𝑘)}_𝑘∈ℕ and {𝑥_𝑛(𝑘)}_𝑘∈ℕ such that, for all 𝑘∈ℕ, 𝑘<𝑚(𝑘)<𝑛(𝑘)<𝑚(𝑘+1), 𝑑(𝑥_𝑚(𝑘),𝑥_𝑛(𝑘))≥𝜀₀, and 𝑑(𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1})<𝜀₀.

Meir and Keeler generalized the Banach contraction mapping principle in the following way.

Definition 9 (Meir and Keeler [34]). A Meir-Keeler mapping is a mapping 𝑇∶𝑋→𝑋 on an MS (𝑋,𝑑) such that for all 𝜀>0, there exists 𝛿>0 verifying that if 𝑥,𝑦∈𝑋 and 𝜀≤𝑑(𝑥,𝑦)<𝜀+𝛿, then 𝑑(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 function 𝜙∶[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 𝜀>0, there exists 𝛿>0 such that 𝜙(𝑡)≤𝜀 for all 𝑡∈[𝜀,𝜀+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 𝑑𝑇(𝑥),𝑇𝑦<𝜙𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋verifying𝑑𝑥,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 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 𝜀>0, there exists 𝛿(𝜀)>0 such that 1𝑥≽𝑢,𝑦≼𝑣,𝜀≤2𝑑()(𝜀)𝐹()𝑥,𝑢+𝑑𝑦,𝑣<𝜀+𝛿⇒𝑑𝑥,𝑦,𝐹𝑢,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 let 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 be two mappings.

Definition 13. We will say that 𝐹 is a (multidimensional) 𝑔-Meir-Keeler type mapping, ((MK) mapping) if it verifies the following two properties.

(MK1) If 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛∈𝑋 verify 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖, then 𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛)=𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛).

(MK2) For all 𝜀>0, there exists 𝛿>0 such that if 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛∈𝑋 verify 𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖 for all 𝑖 and 𝜀≤max_{1≤𝑖≤𝑛}𝑑𝑔𝑥_𝑖,𝑔𝑦_𝑖𝐹𝑥<𝜀+𝛿,then𝑑₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛<𝜀. (8)

If 𝑔 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) for 𝜀=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. If 𝑔 is an injective mapping on 𝑋, then all mappings 𝐹 verify (MK1).

Lemma 15. Let 𝐹∶𝑋^𝑛→𝑋 be a mapping on a partially ordered MS (𝑋,𝑑,≼), and let 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛∈𝑋 be such that 𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖 for all 𝑖. (1)If 𝐹 verifies (MK2), then either 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖 or 𝑑𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛<max_{1≤𝑖≤𝑛}𝑑𝑔𝑥_𝑖,𝑔𝑦_𝑖. (9)(2)If 𝐹 is a g-Meir-Keeler type mapping, then 𝑑𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤max_{1≤𝑖≤𝑛}𝑑𝑔𝑥_𝑖,𝑔𝑦_𝑖, (10) and the equality is achieved if and only if 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖.

Proof. (1) If the condition “𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖” does not hold, then 𝜀=max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥_𝑖,𝑔𝑦_𝑖)>0. Hence, 𝑑(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛),𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛))<𝜀=max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥_𝑖,𝑔𝑦_𝑖). (2) If 𝐹 is a 𝑔-Meir-Keeler-type mapping, the case “𝑔𝑥_𝑖=𝑔𝑦_𝑖 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. The 𝑔-modulus of uniform continuity of 𝐹 is, for all 𝜀>0, 𝛿_𝑔,𝐹(𝜀)=sup𝜆≥0∶𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖∀𝑖,max_{1≤𝑖≤𝑛}𝑑𝑔𝑥_𝑖,𝑔𝑦_i𝐹𝑥<𝜆⇒𝑑₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛<𝜀. (11)

Remark 17. The identity mapping on a set 𝑋 will be denoted by 1_𝑋∶𝑋→𝑋. If 𝑔∶𝑋→𝑋 is a mapping, then 𝐺∶𝑋^𝑛→𝑋^𝑛 will be defined by 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑛)=(𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. If (𝑋,𝑑) is a metric space, then 𝐷∶𝑋^𝑛×𝑋^𝑛→[0,∞[, given by 𝐷(𝑃,𝑄)=max_{1≤𝑖≤𝑛}𝑑(𝑝_𝑖,𝑞_𝑖) for all 𝑃=(𝑝₁,𝑝₂,…,𝑝_𝑛), 𝑄=(𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝑋^𝑛, is a metric on 𝑋^𝑛. A partial order ≼ on 𝑋 may be induced on 𝑋^𝑛 by 𝑃≼𝑄 if and only if 𝑝_𝑖≼𝑞_𝑖 for all 𝑖 (notice that this partial order depends on the partition {𝐴,𝐵} of Λ_𝑛). Then, (𝑋^𝑛,𝐷,≼) also is a partially ordered MS. Furthermore, given any 𝜔=(𝜔₂,𝜔₃,…,𝜔_𝑛)∈𝑋^𝑛−1, 𝐹_𝜔∶𝑋^𝑛→𝑋^𝑛 will denote the mapping defined by 𝐹_𝜔(𝑥₁,𝑥₂,…,𝑥_𝑛)=(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛),𝜔₂,𝜔₃,…,𝜔_𝑛) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. It is obvious that 𝐷𝐹_𝜔𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹_𝜔𝑦₁,𝑦₂,…,𝑦_𝑛=𝑑𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛 (12) for all 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛∈𝑋.

Theorem 18. Let (𝑋,𝑑,≼) be a partially ordered MS, and let 𝐹∶𝑋^𝑛→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 all 𝜀>0, there exists 𝛿>0 such that 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛∈𝑋𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖∀𝑖max_{1≤𝑖≤𝑛}𝑑𝑔𝑥_𝑖,𝑔𝑦_𝑖𝐹𝑥<𝜀+𝛿⇒𝑑₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛<𝜀. (13)

(MK4) 𝛿_𝑔,𝐹(𝜀)>𝜀 for all 𝜀>0.

(MK5) 𝐹 and 𝑔 verify (MK1), and there exists an (nondecreasing, right-continuous) L-function 𝜙∶[0,∞[→[0,∞[ such that 𝑑𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛<𝜙max_{1≤𝑖≤𝑛}𝑑𝑔𝑥_𝑖,𝑔𝑦_𝑖 (14) for all 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛∈𝑋 verifying 𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖 for all 𝑖 and max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥_𝑖,𝑔𝑦_𝑖)>0.

(MK6) For all 𝜔∈𝑋^𝑛−1, the mapping 𝐹_𝜔∶𝑋^𝑛→𝑋^𝑛 is a 𝐺-Meir-Keeler-type mapping on (𝑋^𝑛,𝐷,≼).

(MK7) There exists 𝜔₀∈𝑋^𝑛−1 such that the mapping 𝐹_𝜔0∶𝑋^𝑛→𝑋^𝑛 is a 𝐺-Meir-Keeler-type mapping on (𝑋^𝑛,𝐷,≼).

Proof. [(MK)⇒(MK3)]: Fix 𝜀>0, and let 𝛿>0 given by (MK2). Let 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛∈𝑋 be such that 𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖 for all 𝑖, and let 𝜂=max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥_𝑖,𝑔𝑦_𝑖)<𝜀+𝛿. If 𝜂=0, then 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖, and so 𝑑(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛),𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛))=0<𝜀 by (MK1). In another case, 𝜂>0. If 𝜀≤𝜂<𝜀+𝛿, then 𝑑(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛),𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛))<𝜀 by (MK2). Now, suppose that 0<𝜂<𝜀. Then, 𝜂∈[𝜂,𝜂+𝛿_𝜂[, where 𝛿_𝜂>0 is also given by (MK2), and 𝑑(𝐹(𝑥₁,…,𝑥_𝑛),𝐹(𝑦₁,…,𝑦_𝑛))<𝜂<𝜀. Hence, (MK3) holds.
[(MK3)⇒(MK4)]: Given 𝜀>0, let 𝛿>0 verifying (MK3). Then, 𝛿_𝑔,𝐹(𝜀)≥𝜀+𝛿, and so 𝛿_𝑔,𝐹(𝜀)>𝜀.
[(MK4)⇒(MK)]: On the one hand, if 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖, then 𝑑(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛),𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛))<𝜀 for all 𝜀>0, and so 𝐹 verify (MK1). On the other hand, let 𝜀>0, and define 𝛿=(𝛿_𝑔,𝐹(𝜀)−𝜀)/2>0. Therefore, 𝜀+𝛿<𝛿_𝑔,𝐹(𝜀). Since 𝛿_𝑔,𝐹(𝜀) is a supremum, there exists 𝜆₀∈]𝜀+𝛿,𝛿_𝑔,𝐹(𝜀)] such that if 𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖 for all 𝑖 and max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥_𝑖,𝑔𝑦_𝑖)<𝜆₀, then 𝑑(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛),𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛))<𝜀. In particular, if 𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖 for all 𝑖 and 𝜀≤max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥_𝑖,𝑔𝑦_𝑖)<𝜀+𝛿<𝜆₀, then 𝑑(𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛), 𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛))<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 taking 𝜙(𝑡)=𝑘𝑡 for all 𝑡≥0.

Corollary 19. Let (𝑋,𝑑,≼) be a partially ordered metric space, and let 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 be two mappings. Assume that there exists 𝑘∈(0,1) such that 𝑑𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤𝑘max_{1≤𝑖≤𝑛}𝑑𝑔𝑥_𝑖,𝑔𝑦_𝑖 (15) for all 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛∈𝑋 verifying 𝑔𝑥_𝑖≼_𝑖𝑔𝑦_𝑖 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 Definition 13 taking 𝑔 as the identity mapping on the MS.

Proof. Suppose that 𝐹∶𝑋×𝑋→𝑋 is a generalized Meir Keeler type function in the sense of Samet. Fix 𝜀>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))<𝜀+𝛿. We have to prove that 𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))<𝜀. If 𝑥=𝑢 and 𝑦=𝑣, there is nothing to prove. Next, suppose that max(𝑑(𝑥,𝑢),𝑑(𝑦,𝑣))>0. Let 1𝑀=2𝑑(𝑥,𝑢)+𝑑𝑦,𝑣. (16) If 𝑀=0, then 𝑥=𝑢 and 𝑦=𝑣, which is false. Then, 𝑀>0. On the other hand, 𝑑()𝑀=𝑥,𝑢+𝑑𝑦,𝑣2≤max𝑑(𝑥,𝑢),𝑑𝑦,𝑣<𝜀+𝛿. (17)
If 𝜀≤𝑀<𝜀+𝛿, then 𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))<𝜀 by (7). Finally, if 0<𝑀<𝜀, taking 𝜀^′=𝑀 in (7), we have that 𝑀∈[𝜀^′,𝜀^′+𝛿_𝜀′[ (where 𝛿_𝜀′ is taken as in (7)), and so 𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))<𝑀<𝜀. This proves that 𝐹 is a 2-dimensional Meir-Keeler type mapping associated to 𝑔=𝐼_𝑋.

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 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 that 𝐹 is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13 (taking 𝑔 as the identity mapping on ℝ). Let 𝜀>0. Consider any 𝑟∈]0,1/𝑘−1[ (i.e., 𝑘(1+𝑟)<1) and define 𝛿=𝑟𝜀>0. Consider 𝑥,𝑦,𝑢,𝑣∈ℝ such that 𝑥≥𝑢 and 𝑦≤𝑣 verifying 𝜀≤max(𝑑(𝑥,𝑢),𝑑(𝑦,𝑣))=max(|𝑥−𝑢|,|𝑦−𝑣|)<𝜀+𝛿. In particular, |𝑥−𝑢|<𝜀+𝛿. Then, 𝑑𝐹()()||()()()𝑥,𝑦,𝐹𝑢,𝑣=𝑑𝑘𝑥,𝑘𝑢=𝑘𝑥−𝑢<𝑘𝜀+𝛿=𝑘𝜀+𝑟𝜀=𝑘1+𝑟𝜀<𝜀. (18) 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 𝜀>0. If 𝐹 was a generalized Meir Keeler type function in the sense of Samet, it would be 𝛿>0 verifying (7). Take 𝑥=𝜀,  𝑢=−𝜀, and 𝑦=𝑣=0. Then, 𝑥≥𝑢, 𝑦≤𝑣 and 𝑑(𝑥,𝑢)+𝑑𝑦,𝑣2=|𝜀−(−𝜀)|+|0−0|2=2𝜀2=𝜀∈[𝜀,𝜀+𝛿[. (19) However, 𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,v))=𝑑(𝑘𝜀,𝑘(−𝜀))=𝑘|𝜀−(−𝜀)|=2𝑘𝜀>𝜀  since  𝑘>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,…,𝜎_𝑛).

Theorem 22. Let (𝑋,𝑑) be a complete MS, and let ≼ a partial order on 𝑋. Let Υ=(𝜎₁,𝜎₂,…,𝜎_𝑛) be an 𝑛-tuple of mappings from {1,2,…,𝑛} into itself verifying 𝜎_𝑖∈Ω_𝐴,𝐵 if 𝑖∈𝐴 and 𝜎_𝑖∈Ω^′_𝐴,𝐵 if 𝑖∈𝐵. Let 𝐹∶𝑋^𝑛→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(𝑋^𝑛)⊆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 𝑥¹₀,𝑥²₀,…,𝑥^𝑛₀∈𝑋 verifying 𝑔𝑥^𝑖₀≼_𝑖𝐹(𝑥^𝜎𝑖₀⁽¹⁾,𝑥^𝜎𝑖₀⁽²⁾,…,𝑥^𝜎𝑖₀^(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 {𝑥¹_𝑚}_𝑚≥0, {𝑥²_𝑚}_𝑚≥0,…,{𝑥^𝑛_𝑚}_𝑚≥0 such that 𝑔𝑥^𝑖_𝑚+1=𝐹(𝑥^𝜎𝑖_𝑚⁽¹⁾, 𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑥^𝜎𝑖_𝑚^(𝑛)) for all 𝑚 and all 𝑖.
Step  2. 𝑔𝑥^𝑖_𝑚≼_𝑖𝑔𝑥^𝑖_𝑚+1 for all 𝑚 and all 𝑖.
Define 𝛿_𝑚=max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)≥0 for all 𝑚. Firstly, suppose that there exists 𝑚₀∈ℕ such that 𝛿_𝑚0=0. Then, 𝑔𝑥^𝑖_𝑚0=𝑔𝑥^𝑖_𝑚0+1=𝐹(𝑥^𝜎𝑖_𝑚⁽¹⁾₀,𝑥^𝜎𝑖_𝑚⁽²⁾₀,…,𝑥^𝜎𝑖_𝑚^(𝑛)₀) for all 𝑖, so (𝑥¹_𝑚0,𝑥²_𝑚0,…,𝑥^𝑛_𝑚0) is a Υ-coincidence point of 𝐹 and 𝑔 and we have finished. Therefore, we may reduce to the case in which 𝛿_𝑚>0 for all 𝑚; that is, ∀𝑚,thereexists𝑗suchthat𝑔𝑥^𝑗_𝑚≠𝑔𝑥^𝑗_𝑚+1. (20)
Step  3. We claim that {𝑑(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)}_𝑚≥0→0 for all 𝑖 (i.e., {max_{1≤𝑗≤𝑛}𝑑(𝑔𝑥^𝑗_𝑚, 𝑔𝑥^𝑗_𝑚+1)}_𝑚≥0→0). Indeed, as 𝑔𝑥^𝑖_𝑚≼_𝑖𝑔𝑥^𝑖_𝑚+1 for all 𝑚 and all 𝑖, then condition (MK2), Lemma 15, and (20) imply that, for all 𝑚≥1 and all 𝑖, 𝑑𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1=𝑑𝐹𝑥^𝜎𝑖(1)_𝑚−1,𝑥^𝜎𝑖(2)_𝑚−1,…,𝑥^{𝜎𝑖(𝑛)}_𝑚−1,𝐹𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑥^𝜎𝑖_𝑚^(𝑛)<max_{1≤𝑗≤𝑛}𝑑𝑔𝑥^{𝜎𝑖(𝑗)}_𝑚−1,𝑔𝑥^𝜎𝑖_𝑚^(𝑗)≤max_{1≤𝑗≤𝑛}𝑑𝑔𝑥^𝑗_𝑚−1,𝑔𝑥^𝑗_𝑚=𝛿_𝑚−1. (21) Taking maximum on 𝑖, we deduce that the sequence {𝛿_𝑚}_𝑚≥1 is nonincreasing and lower bounded. Therefore, it is convergent; that is, there exists Δ≥0 such that {𝛿_𝑚}_𝑚≥1→Δ (and Δ≤𝛿_𝑚 for all 𝑚). We claim that Δ=0. On the contrary, assume that Δ>0. Let 𝛿>0 be a positive number associated to 𝜀=Δ>0 by (MK2). Since max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)_𝑚=𝛿_𝑚𝑚↘Δ, (22) there exists 𝑚₀∈ℕ such that if 𝑚≥𝑚₀, then Δ≤max_{1≤𝑖≤𝑛}𝑑(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)<Δ+𝛿. By (MK2), it follows that, for all 𝑖, 𝑑𝑔𝑥^𝑖_𝑚0,𝑔𝑥^𝑖_𝑚0+1=𝑑𝐹𝑥^𝜎𝑖_𝑚⁽¹⁾₀,𝑥^𝜎𝑖_𝑚⁽²⁾₀,…,𝑥^𝜎𝑖_𝑚^(𝑛)₀,𝐹𝑥^𝜎𝑖_𝑚⁽¹⁾₀₊₁,𝑥^𝜎𝑖_𝑚⁽²⁾₀₊₁,…,𝑥^𝜎𝑖_𝑚^(𝑛)₀₊₁<Δ. (23) Taking maximum on 𝑖, we deduce that Δ≤𝛿_𝑚0=max_{1≤𝑖≤𝑛}𝑑𝑔𝑥^𝑖_𝑚0,𝑔𝑥^𝑖_𝑚0+1<Δ. (24) But this is impossible. Then, we have just proved that Δ=0. Therefore, {𝛿_𝑚}_𝑚≥1→Δ=0, which means that lim_𝑚→∞𝛿_𝑚=lim_𝑚→∞max_{1≤𝑗≤𝑛}𝑑𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚+1=0. (25) As 0≤𝑑(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)≤𝛿_𝑚 for all 𝑚 and all 𝑖, then {𝑑(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)}→0 for all 𝑖.
Step  4. Every sequence  {𝑔𝑥^𝑖_𝑚}_𝑚≥0  is Cauchy. Suppose that {𝑔𝑥^𝑖1_𝑚}_𝑚≥0,…,{𝑔𝑥^𝑖𝑠_𝑚}_𝑚≥0 are not Cauchy and {𝑔𝑥^𝑖𝑠+1_𝑚}_𝑚≥0,…,{𝑔𝑥^𝑖𝑛_𝑚}_𝑚≥0 are Cauchy, being {𝑖₁,…,𝑖_𝑛}={1,…,𝑛}. By Lemma 8, for all 𝑟∈{1,2,…,𝑠}, there exist 𝜀_𝑟>0 and subsequences {𝑔𝑥^𝑖𝑟_{𝑚′𝑟(𝑘)}}_𝑘∈ℕ and {𝑔𝑥^𝑖𝑟_{𝑛′𝑟(𝑘)}}_𝑘∈ℕ such that 𝑘<𝑚^′_𝑟(𝑘)<𝑛^′_𝑟(𝑘),𝑑𝑔𝑥^𝑖𝑟_{m′𝑟(𝑘)},𝑔𝑥^𝑖𝑟_{𝑛′𝑟(𝑘)}≥𝜀_𝑟,𝑑𝑔𝑥^𝑖𝑟_{𝑚′𝑟(𝑘)},𝑔𝑥^𝑖𝑟_{𝑛′𝑟(𝑘)−1}<𝜀_𝑟,∀𝑘∈ℕ. (26) Let 𝜀₀=max(𝜀₁,…,𝜀_𝑠) and 𝜀^′₀=min(𝜀₁,…,𝜀_𝑠)>0. Since {𝑔𝑥^𝑖𝑠+1_𝑚}_𝑚≥0,…,{𝑔𝑥^𝑖𝑛_𝑚}_𝑚≥0 are Cauchy, there exists 𝑛₀∈ℕ such that if 𝑛,  𝑚≥𝑛₀, then 𝑑(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑛)<𝜀^′₀/2 for all 𝑗∈{𝑖_𝑠+1,…,𝑖_𝑛}.
Let 𝑘₀∈ℕ such that 𝑛₀<min(𝑚^′₁(𝑘₀), 𝑚^′₂(𝑘₀),…,𝑚^′_𝑠(𝑘₀)), and define 𝑚(1)=min(𝑚^′₁(𝑘₀), 𝑚^′₂(𝑘₀),…,𝑚^′_𝑠(𝑘₀)). As 𝑚(1)=𝑚^′_𝑟(𝑘₀) for some 𝑟∈{1,2,…,𝑠}, there exists 𝑛^′_𝑟(𝑘₀) such that 𝑑(𝑔𝑥^𝑖𝑟_{𝑚′𝑟(𝑘0)}, 𝑔𝑥^𝑖𝑟_{𝑛′𝑟(𝑘0)})≥𝜀_𝑟≥𝜀₀. Thus, we can consider the numbers 𝑚(1)+1, 𝑚(1)+2,… until finding the first positive integer 𝑛(1)>𝑚(1) verifying max_{1≤𝑟≤𝑠}𝑑𝑔𝑥^𝑖𝑟_𝑚(1),𝑔𝑥^𝑖𝑟_𝑛(1)≥𝜀₀,𝑑𝑔𝑥^𝑖𝑗_𝑚(1),𝑔𝑥^𝑖𝑗_𝑛(1)−1<𝜀₀,∀𝑗∈{1,2,…,𝑠}. (27)
Now, let 𝑘₁∈ℕ such that 𝑛(1)<min(𝑚^′₁(𝑘₁), 𝑚^′₂(𝑘₁),…,𝑚^′_𝑠(𝑘₁)) and define 𝑚(2)=min(𝑚^′₁(𝑘₁), 𝑚^′₂(𝑘₁),…,𝑚^′_𝑠(𝑘₁)). Since 𝑚(2)∈{𝑚^′₁(𝑘₁),𝑚^′₂(𝑘₁),…,𝑚^′_𝑠(𝑘₁)}, we can consider the numbers 𝑚(2)+1,𝑚(2)+2,… until finding the first positive integer 𝑛(2)>𝑚(2) verifying max_{1≤𝑟≤𝑠}𝑑𝑔𝑥^𝑖𝑟_𝑚(2),𝑔𝑥^𝑖𝑟_𝑛(2)≥𝜀₀,𝑑𝑔𝑥^𝑖𝑗_𝑚(2),𝑔𝑥^𝑖𝑗_𝑛(2)−1<𝜀₀,∀𝑗∈{1,2,…,𝑠}. (28) Repeating this process, we can find sequences such that, for all 𝑘≥1, 𝑛₀<𝑚(𝑘)<𝑛(𝑘)<𝑚(𝑘+1),max_{1≤𝑟≤𝑠}𝑑𝑔𝑥^𝑖𝑟_𝑚(𝑘),𝑔𝑥^𝑖𝑟_𝑛(𝑘)≥𝜀₀,𝑑𝑔𝑥^𝑖𝑗_𝑚(𝑘),𝑔𝑥^𝑖𝑗_{𝑛(𝑘)−1}<𝜀₀,∀𝑗∈{1,2,…,𝑠}. (29) Since 𝑛₀<𝑚(𝑘)<𝑛(𝑘), we know that 𝑑(𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_𝑛(𝑘)), 𝑑(𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_{𝑛(𝑘)−1}), 𝑑(𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1})<𝜀^′₀/2 for all 𝑗∈{𝑖_𝑠+1,…,𝑖_𝑛}. Therefore, for all 𝑘, max_{1≤𝑗≤𝑛}𝑑𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_𝑛(𝑘)=max_{1≤𝑟≤𝑠}𝑑𝑔𝑥^𝑖𝑟_𝑚(𝑘),𝑔𝑥^𝑖𝑟_𝑛(𝑘)≥𝜀₀,max_{1≤𝑗≤𝑛}𝑑𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_{𝑛(𝑘)−1}<𝜀^′₀. (30) Let 𝛿>0 verifying (MK3) using 𝜀₀, and let 𝑘₁∈ℕ such that if 𝑘≥𝑘₁, then 𝑑(𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_𝑚(𝑘))<𝛿 for all 𝑗. Then, for all 𝑗 and all 𝑘≥𝑘₁, 𝑑𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}≤𝑑𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_𝑚(𝑘)+𝑑𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_{𝑛(𝑘)−1}<𝛿+𝜀₀, (31) Applying (MK3), it follows, for all 𝑘≥𝑘₀ and all 𝑖, that 𝑑𝑔𝑥^𝑖_𝑚(𝑘),𝑔𝑥^𝑖_𝑛(𝑘)=𝑑𝐹𝑥^𝜎𝑖_𝑚(𝑘)⁽¹⁾₋₁,…,𝑥^𝜎𝑖_𝑚(𝑘)^(𝑛)₋₁,𝐹𝑥^𝜎𝑖_𝑛(𝑘)⁽¹⁾₋₁,…,𝑥^𝜎𝑖_𝑛(𝑘)^(𝑛)₋₁<𝜀₀, (32) but this contradicts (30) since max_{1≤𝑗≤𝑛}𝑑(𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_𝑛(𝑘))≥𝜀₀. This contradiction shows us that every sequence {𝑔𝑥^𝑖_𝑚}_𝑚≥0 is Cauchy.
Existence of a fixed point is derived by standard techniques. Indeed, since (𝑋,𝑑) is complete, there exist 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 such that 𝑥_𝑖=lim_𝑚→∞𝑔𝑥^𝑖_𝑚 for all 𝑖. As 𝑔 is continuous and 𝐹 commutes with 𝑔, lim_𝑚→∞𝐹𝑔𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑖_𝑚^(𝑛)=lim_𝑚→∞𝑔𝐹𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑥^𝜎𝑖_𝑚^(𝑛)=lim_𝑚→∞𝑔𝑔𝑥^𝑖_𝑚+1=𝑔𝑥_𝑖∀𝑖. (33)
Step  5. Suppose that 𝐹 is continuous. In this case, since {𝑔𝑥^𝜎𝑖_𝑚^(𝑗)}_𝑚→𝑥_{𝜎𝑖(𝑗)} for all 𝑖,𝑗 and 𝐹 is continuous, lim_𝑚→∞𝑔𝑔𝑥^𝑖_𝑚+1=lim_𝑚→∞𝐹𝑔𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑛_𝑚^(𝑛)=𝐹𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)} (34) for all 𝑖. Then, 𝐹(𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)})=𝑔𝑥_𝑖 for all 𝑖; that is, (𝑥₁,𝑥₂,…,𝑥_𝑛) 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 𝑔𝑥^𝑖_𝑚≼_𝑖𝑔𝑥^𝑖_𝑚+1 for all 𝑚and all𝑖. This means that the sequence {𝑔𝑥^𝑖_𝑚}_𝑚≥0 is monotone. As 𝑥_𝑖=lim_𝑚→∞𝑔𝑥^𝑖_𝑚, we deduce that 𝑔𝑔𝑥^𝑖_𝑚≼_𝑖𝑔𝑥_𝑖 for all 𝑚 and all𝑖. This condition implies that, for all 𝑚 and all 𝑗, either𝑔𝑔𝑥^𝜎𝑗_𝑚^(𝑖)≼_𝑖𝑔𝑥_{𝜎𝑗(𝑖)}∀𝑖or𝑔𝑥_{𝜎𝑗(𝑖)}≼_𝑖𝑔𝑔𝑥^𝜎𝑗_𝑚^(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 lim_𝑚→∞𝐹(𝑔𝑥^𝜎𝑗_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑗_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑗_𝑚^(𝑛))=𝐹(𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}). Indeed, let 𝜀>0 arbitrary, and let 𝛿>0 verifying (MK3). Since {𝑔𝑔𝑥^𝜎𝑗_𝑚^(𝑖)}_𝑚→𝑔𝑥_{𝜎𝑗(𝑖)} for all 𝑖, there exists 𝑚₂∈ℕ such that if 𝑚≥𝑚₂, then 𝑑(𝑔𝑔𝑥^𝜎𝑗_𝑚^(𝑖),𝑔𝑥_{𝜎𝑗(𝑖)})<𝜀+𝛿 for all 𝑖. Applying (MK3) and (35), 𝑑𝐹𝑔𝑥^𝜎𝑗_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑗_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑗_𝑚^(𝑛),𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}<𝜀. (36) Therefore, 𝐹(𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)})=lim_𝑚→∞𝐹(𝑔𝑥^𝜎𝑗_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑗_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑗_𝑚^(𝑛))=lim_𝑚→∞𝑔𝑔𝑥^𝑗_𝑚+1=𝑔𝑥_𝑗 for all 𝑗. In conclusion, (𝑥₁,𝑥₂,…,𝑥_𝑛) is a Υ-coincidence point of 𝐹 and 𝑔.

5. Uniqueness of Υ-Coincidence Points

For the uniqueness of a fixed point, we need the following notion. Consider on the product space 𝑋^𝑛 the following partial order: for (𝑥₁,𝑥₂,…,𝑥_𝑛),(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑋^𝑛, 𝑥₁,𝑥₂,…,𝑥_𝑛≤𝑦₁,𝑦₂,…,𝑦_𝑛⟺𝑥_𝑖≼_𝑖𝑦_𝑖,∀𝑖. (37) We say that two points (𝑥₁,𝑥₂,…,𝑥_𝑛) and (𝑦₁,𝑦₂,…,𝑦_𝑛) are comparable if (𝑥₁,𝑥₂,…,𝑥_𝑛)≤(𝑦₁,𝑦₂,…,𝑦_𝑛) or (𝑥₁,𝑥₂,…,𝑥_𝑛)≥(𝑦₁,𝑦₂,…,𝑦_𝑛).

By following the lines of Theorem 11 in [35], we prove the uniqueness of the coincidence point.

Theorem 23. Under the hypothesis of Theorem 22, assume that for all Υ-coincidence points (𝑥₁,𝑥₂,…,x_𝑛), (𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑋^𝑛 of 𝐹 and 𝑔, there exists (𝑢₁,𝑢₂,…,𝑢_𝑛)∈𝑋^𝑛 such that (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) is comparable, at the same time, to (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛) and to (𝑔𝑦₁,𝑔𝑦₂,…,𝑔𝑦_𝑛).
Then, 𝐹 and 𝑔 have a unique Υ-coincidence point (𝑧₁,𝑧₂,…,𝑧_𝑛)∈𝑋^𝑛 such that 𝑔𝑧_𝑖=𝑧_𝑖 for all 𝑖.

It is natural to say that 𝑔 is injective on the set of all Υ-coincidence points of 𝐹 and 𝑔 when 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖 implies 𝑥_𝑖=𝑦_𝑖 for all 𝑖 when (𝑥₁,𝑥₂,…,𝑥_𝑛),(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑋^𝑛 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 Theorem 23, 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,…,𝑥_𝑛) and (𝑦₁,𝑦₂,…,𝑦_𝑛) are two Υ-coincidence points of 𝐹 and 𝑔, we have proved in (A.1) that 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖. As 𝑔 is injective on these points, then 𝑥_𝑖=𝑦_𝑖 for all 𝑖.

Corollary 25. In addition to the hypotheses of Theorem 23, suppose that (𝑧_𝜎𝑖(1),𝑧_𝜎𝑖(2),…,𝑧_{𝜎𝑖(𝑛)}) is comparable to (𝑧_𝜎𝑗(1),𝑧_𝜎𝑗(2),…,𝑧_{𝜎𝑗(𝑛)}) for all 𝑖,𝑗. Then, 𝑧₁=𝑧₂=⋯=𝑧_𝑛.
In particular, there exists a unique 𝑧∈𝑋 such that 𝐹(𝑧,𝑧,…,𝑧)=𝑧, which verifies 𝑔𝑧=𝑧.

Proof. Let 𝑀=max_{1≤𝑖,𝑗≤𝑛}𝑑(𝑧_𝑖,𝑧_𝑗) 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(𝑧_𝑗0,𝑧_𝑠0)=𝑀. As (𝑧_𝜎𝑗0(1),𝑧_𝜎𝑗0(2),…,𝑧_{𝜎𝑗0(𝑛)}) is comparable to (𝑧_𝜎𝑠0(1),𝑧_𝜎𝑠0(2),…,𝑧_{𝜎𝑠0(𝑛)}), then either 𝑧_{𝜎𝑗0(𝑖)}≼_𝑖𝑧_{𝜎𝑠0(𝑖)} for all 𝑖 or 𝑧_{𝜎𝑠0(𝑖)}≼_𝑖𝑧_{𝜎𝑗0(𝑖)} for all 𝑖. Since 𝑔𝑧_𝑖=𝑧_𝑖 for all 𝑖, we know that either 𝑔𝑧_{𝜎𝑗0(𝑖)}≼_𝑖𝑔𝑧_{𝜎𝑠0(𝑖)} for all 𝑖 or 𝑔𝑧_{𝜎𝑠0(𝑖)}≼_𝑖𝑔𝑧_{𝜎𝑗0(𝑖)} for all 𝑖. Now, we have to distinguish between two cases.
If 𝑔𝑧_{𝜎𝑗0(𝑖)}=𝑔𝑧_{𝜎𝑠0(𝑖)} for all 𝑖 (i.e., 𝑧_{𝜎𝑗0(𝑖)}=𝑧_{𝜎𝑠0(𝑖)} for all 𝑖), then 𝑧_𝑗0=𝑔𝑧_𝑗0=𝐹𝑧_𝜎𝑗0(1),𝑧_𝜎𝑗0(2),…,𝑧_{𝜎𝑗0(𝑛)}=𝐹𝑧_𝜎𝑠0(1),𝑧_𝜎𝑠0(2),…,𝑧_{𝜎𝑠0(𝑛)}=𝑔𝑧_𝑠0=𝑧_𝑠0, (38) which is impossible since 𝑑(𝑧_𝑗0,𝑧_𝑠0)=𝑀>0. Now, suppose that max_{1≤𝑖≤𝑛}𝑑(𝑔𝑧_{𝜎𝑗0(𝑖)}, 𝑔𝑧_{𝜎𝑠0(𝑖)})>0. In this case, item 1 of Lemma 15 guarantees that 𝑀=𝑑𝑧_𝑗0,𝑧_𝑠0=𝑑𝑔𝑧_𝑗0,𝑔𝑧_𝑠0=𝑑𝐹𝑧_𝜎𝑗0(1),𝑧_𝜎𝑗0(2),…,𝑧_{𝜎𝑗0(𝑛)},𝐹𝑧_𝜎𝑠0(1),𝑧_𝜎𝑠0(2),…,𝑧_{𝜎𝑠0(𝑛)}<max_{1≤𝑖≤𝑛}𝑑𝑔𝑧_{𝜎𝑗0(𝑖)},𝑔𝑧_{𝜎𝑠0(𝑖)}≤max_{1≤𝑖,𝑗≤𝑛}𝑑𝑔𝑧_𝑖,𝑔𝑧_𝑗=max_{1≤𝑖,𝑗≤𝑛}𝑑𝑧_𝑖,𝑧_𝑗=𝑀, (39) which also is impossible. This contradiction proves that 𝑀=0; that is, 𝑧_𝑖=𝑧_𝑗 for all 𝑖,𝑗.

Remark 26. Notice that a mixed strict monotone mapping 𝐹∶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. Let 𝑋=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. Let 𝑋=ℝ be provided with its usual partial order ≤ and its usual metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Let 𝑛∈ℕ, and let 𝑎₁,𝑎₂,…,𝑎_𝑛∈ℝ⧵{0} real numbers such that there exist 𝑖₀,𝑗₀∈{1,2,…,𝑛} verifying 𝑎_𝑖0<0<𝑎_𝑗0. Let 𝑁>|𝑎₁|+|𝑎₂|+⋯+|𝑎_𝑛|, and consider 𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛)=(𝑎₁𝑥₁+𝑎₂𝑥₂+⋯+𝑎_𝑛𝑥_𝑛)/𝑁 and 𝑔𝑥=𝑥, for all 𝑥,𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Then, 𝐹 is monotone nondecreasing in those arguments for which 𝑎_𝑖>0 and monotone nonincreasing in those arguments for which 𝑎_𝑖<0. Furthermore, taking 𝑘=(|𝑎₁|+|𝑎₂|+⋯+|𝑎_𝑛|)/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.

Appendix

Proof of Theorem 23

Proof. From Theorem 22, the set of Υ-coincidence points of 𝐹 and 𝑔 is nonempty. The proof is divided in two steps.
Step  1. We claim that if (𝑥₁,𝑥₂,…,𝑥_𝑛),(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑋^𝑛are two Υ-coincidence points of 𝐹 and 𝑔, then 𝑔𝑥_𝑖=𝑔𝑦_𝑖∀𝑖. (A.1) Let (𝑥₁,𝑥₂,…,𝑥_𝑛),(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑋^𝑛 be two Υ-coincidence points of 𝐹 and 𝑔, and let (𝑢₁,𝑢₂,…,𝑢_𝑛)∈𝑋^𝑛 be a point such that (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) is comparable, at the same time, to (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛) and to (𝑔𝑦₁,𝑔𝑦₂,…,𝑔𝑦_𝑛). Using (𝑢₁,𝑢₂,…,𝑢_𝑛) define the following sequences. Let 𝑢^𝑖₀=𝑢_𝑖 for all 𝑖. Reasoning as in Theorem 22, we can determine sequences {𝑢¹_𝑚}_𝑚≥0, {𝑢²_𝑚}_𝑚≥0,…,{𝑢^𝑛_𝑚}_𝑚≥0 such that 𝑔𝑢^𝑖_𝑚+1=𝐹(𝑢^𝜎𝑖_𝑚⁽¹⁾,𝑢^𝜎𝑖_𝑚⁽²⁾,…,𝑢^𝜎𝑖_𝑚^(𝑛)) for all 𝑚 and all 𝑖. We are going to prove that 𝑔𝑥_𝑖=lim_𝑚→0𝑔𝑢^𝑖_𝑚=𝑔𝑦_𝑖 for all 𝑖, and so (A.1) will be true.
Firstly, we reason with (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) and (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛), and the same argument will be true for (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) and (𝑔𝑦₁,𝑔𝑦₂,…,𝑔𝑦_𝑛). As (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) and (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛) are comparable, we can suppose that (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛)≤(𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛) (the other case is similar); that is, 𝑔𝑢^𝑖₀=𝑔𝑢_𝑖≤_𝑖𝑔𝑥_𝑖 for all 𝑖. Using that 𝐹 has the mixed 𝑔-monotone property and reasoning as in Theorem 22, it is possible to prove that 𝑔𝑢^𝑖_𝑚≤_𝑖𝑔𝑥_𝑖 for all 𝑚≥1 and all 𝑖. This condition implies that, for all 𝑗 and all 𝑚≥1, either𝑔u^𝜎𝑗_𝑚^(𝑖)≤_𝑖𝑔𝑥_{𝜎𝑗(𝑖)}∀𝑖or𝑔𝑥_{𝜎𝑗(𝑖)}≤_𝑖𝑔𝑢^𝜎𝑗_𝑚^(𝑖)∀𝑖. (A.2) Define 𝛽_𝑚=max_{1≤𝑖≤𝑛}𝑑(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖) for all 𝑚. Reasoning as in Theorem 22, one can observe that {𝛽_𝑚}_𝑚≥1→0, which means that lim_𝑚→∞𝛽_𝑚=lim_𝑚→∞(max_{1≤𝑖≤𝑛}𝑑(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖))=0. As 0≤𝑑(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖)≤𝛽_𝑚 for all 𝑚 and all 𝑖, we deduce that {𝑑(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖)}_𝑚≥1→0 for all 𝑖; that is, lim_𝑚→∞𝑔𝑢^𝑖_𝑚=𝑔𝑥_𝑖∀𝑖. (A.3) If we had supposed that (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛)≤(𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛), we would have obtained the same property (A.3). And as (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) also is comparable to (𝑔𝑦₁,𝑔𝑦₂,…,𝑔𝑦_𝑛), we can reason in the same way to prove that 𝑔𝑦_𝑖=lim_𝑚→0𝑔𝑢^𝑖_𝑚=𝑔𝑥_𝑖 for all 𝑖.
Let (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 be a Υ-coincidence point of 𝐹 and 𝑔, and define 𝑧_𝑖=𝑔𝑥_𝑖 for all 𝑖. As (𝑧₁,𝑧₂,…,𝑧_𝑛)=(𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛), Remark 6 assures us that (𝑧₁,𝑧₂,…,𝑧_𝑛) also is a Υ-coincidence point of 𝐹 and 𝑔.
Step  2. We claim that (𝑧₁,𝑧₂,…,𝑧_𝑛) is the unique Φ-coincidence point of 𝐹 and 𝑔 such that 𝑔𝑧_𝑖=𝑧_𝑖 for all 𝑖. It is similar to Step  2 in Theorem 11 in [35].