Multidimensional Fixed-Point Theorems in Partially Ordered Complete Partial Metric Spaces under (𝜓,𝜑)-Contractivity Conditions

Abstract

We study the existence and uniqueness of coincidence point for nonlinear mappings of any number of arguments under a weak (𝜓,𝜑)-contractivity condition in partial metric spaces. The results we obtain generalize, extend, and unify several classical and very recent related results in the literature in metric spaces (see Aydi et al. (2011), Berinde and Borcut (2011), Gnana Bhaskar and Lakshmikantham (2006), Berzig and Samet (2012), Borcut and Berinde (2012), Choudhury et al. (2011), Karapınar and Luong (2012), Lakshmikantham and Ćirić (2009), Luong and Thuan (2011), and Roldán et al. (2012)) and in partial metric spaces (see Shatanawi et al. (2012)).

1. Introduction

The notion of coupled fixed point was introduced by Guo and Lakshmikantham [1] in 1987. In a recent paper, Gnana Bhaskar and Lakshmikantham [2] introduced the concept mixed monotone property for contractive operators of the form 𝐹∶𝑋×𝑋→𝑋, where 𝑋 is a partially ordered metric space, and then established some coupled fixed-point theorems. After that, many results appeared on coupled fixed-point theory in different contexts (see, e.g., [3–6]). Later, Berinde and Borcut [7] introduced the concept of tripled fixed point and proved tripled fixed-point theorems using mixed monotone mappings (see also [8–10]).

Very recently, Roldán et al. [11] proposed the notion of coincidence point between mappings in any number of variables and showed some existence and uniqueness theorems that extended the mentioned previous results for this kind of nonlinear mappings, not necessarily permuted or ordered, in the framework of partially ordered complete metric spaces, using a weaker contraction condition, that also generalized other works by Berzig and Samet [12], Karapınar and Berinde [13].

Partial metric spaces were firstly introduced by Matthews in [14] as an attempt to generalize the metric spaces by establishing the condition that the distance between a point to itself (which is not necessarily zero) is less or equal than the distance between that point and another point of the space. In the mentioned papers, Matthews studied topological properties of partial metric spaces and stated a modified version of a Banach contraction mapping principle on this kind of spaces. After Matthews' pioneering work, the theory of partial metric spaces and particularly the field of fixed-point theorems have expansively been developed due to the increasing interest in this area and motivated by its possible applications (see [15, 16] and references therein).

In this paper, our main aim is to study a weaker contractivity condition for nonlinear mappings of any number of arguments. This condition can be particularized in a variety of forms that let us extend the previously mentioned results and other recent ones in this field (see [2, 5, 7, 9, 11, 12, 16–20]). We also notice that our results cannot be obtained by the very recent paper of Haghi et al. [21] (for more details see Remark 26).

2. Preliminaries

Preliminaries and notation about coincidence points can also be found in [11]. Let 𝑛 be a positive integer. Henceforth, 𝑋 will denote a nonempty set, and 𝑋^𝑛 will denote the product space 𝑋^𝑛=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}”. Let ℝ⁺₀=[0,∞[.

A metric on 𝑋 is a mapping 𝑑∶𝑋×𝑋→ℝ satisfying, for all 𝑥,𝑦,𝑧∈𝑋: (i)𝑑(𝑥,𝑦)=0 if, and only if, 𝑥=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 [22]). A triple (𝑋,𝑑,≤) is called a partially ordered metric space if (𝑋,𝑑) is a MS and ≤ is a partial order on 𝑋.

Definition 2 (see [2]). An ordered MS (𝑋,𝑑,≤) is said to have the sequential 𝑔-monotone property if it verifies(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 two non-empty subsets of Λ_𝑛={1,2,…,𝑛}; that is, 𝐴∪𝐵=Λ_𝑛 and 𝐴∩𝐵=∅. We will denote Ω_𝐴,𝐵=𝜎∶Λ_𝑛→Λ_𝑛∶𝜎(𝐴)⊆𝐴and𝜎(𝐵),Ω⊆𝐵_′𝐴,𝐵=𝜎∶Λ_𝑛→Λ_𝑛∶𝜎(𝐴)⊆𝐵and𝜎(𝐵)⊆𝐴. (1) 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. (2)

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

Definition 3 (see [11]). One says that 𝐹 and 𝑔 are commuting if 𝑔𝐹(𝑥₁,…,𝑥_𝑛)=𝐹(𝑔𝑥₁,…,𝑔𝑥_𝑛) for all 𝑥₁,…,𝑥_𝑛∈𝑋.

Definition 4 (see [11]). Let (𝑋,≤) be a partially ordered space. One says that 𝐹 has the mixed 𝑔-monotone property (with respect to {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, 𝑥𝑔𝑦≤𝑔𝑧⟹𝐹₁,…,𝑥_𝑖−1,𝑦,𝑥_𝑖+1,…,𝑥_𝑛≤_𝑖𝐹𝑥₁,…,𝑥_𝑖−1,𝑧,𝑥_𝑖+1,…,𝑥_𝑛. (3)
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 [11]). A point (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 is called a Φ-coincidence point of the mappings 𝐹and 𝑔 if 𝐹𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)}=𝑔𝑥_𝜏(𝑖)∀𝑖. (4) 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.

Definition 7 (see [14]). A partial metric on 𝑋 is a mapping 𝑝∶𝑋×𝑋→ℝ⁺₀ verifying, for all 𝑥,𝑦,𝑧∈𝑋:  (𝑃1)  𝑝(𝑥,𝑥)≤𝑝(𝑥,𝑦);  (𝑃2)  𝑝(𝑥,𝑥)=𝑝(𝑥,𝑦)=𝑝(𝑦,𝑦)⇒𝑥=𝑦;  (𝑃3)  𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥);  (𝑃4)  𝑝(𝑥,𝑧)+𝑝(𝑦,𝑦)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧).
In this case, (𝑋,𝑝) is a partial metric space (for short, a PMS).

Example 8 (see, e.g., [14]). Let 𝑋=ℝ⁺₀, and define 𝑝 on 𝑋 by 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈𝑋. Then, (𝑋,𝑝) is a partial metric space.

Example 9 (see [14]). Let 𝑋={[𝑎,𝑏]∶𝑎,𝑏∈ℝ,𝑎≤𝑏}, and define 𝑝([𝑎,𝑏],[𝑐,𝑑]) = max{𝑏,𝑑}−min{𝑎,𝑐}. Then, (𝑋,𝑝) is a partial metric space.

Example 10 (see [14]). Let 𝑋=[0,1]∪[2,3], and define 𝑝∶𝑋×𝑋→ℝ⁺₀ by 𝑝=,𝑥,𝑦max𝑥,𝑦if∩[],𝑥,𝑦2,3≠∅,𝑥−𝑦if𝑥,𝑦⊂[0,1]. (5) Then, (𝑋,𝑝) is a partial metric space.

Example 11 (see, e.g., [23, 24]). Let (𝑋,𝑑) and (𝑋,𝑝) be a metric space and a partial metric space, respectively. Functions 𝜌_𝑖∶𝑋×𝑋→ℝ⁺₀(𝑖∈{1,2,3}) given by 𝜌₁,𝜌𝑥,𝑦=𝑑𝑥,𝑦+𝑝𝑥,𝑦₂𝑢(𝑥)𝑦,𝜌𝑥,𝑦=𝑑𝑥,𝑦+max,𝑢₃𝑥,𝑦=𝑑𝑥,𝑦+𝑎, (6) define partial metrics on 𝑋, where 𝑢∶𝑋→ℝ⁺₀ is an arbitrary function and 𝑎≥0.
Obviously, if (𝑋,𝑑) is a MS and we define 𝑝=𝑑, then (𝑋,𝑝) is a PMS. Indeed, a partial metric 𝑝 on 𝑋 verifies(i)𝑝(𝑥,𝑦)=0⇒𝑥=𝑦; (ii)𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥); (iii)𝑝(𝑥,𝑧)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧),
but the condition 𝑝(𝑥,𝑥)=0 does not necessarily hold. For a partial metric 𝑝 on 𝑋, the mappings 𝑑_𝑝,𝑑_𝑚∶𝑋×𝑋→ℝ⁺₀ given by 𝑑_𝑝(),𝑑𝑥,𝑦=2𝑝𝑥,𝑦−𝑝𝑥,𝑥−𝑝𝑦,𝑦_𝑚𝑝()𝑝(),𝑥,𝑦=max𝑥,𝑦−𝑝𝑥,𝑥,𝑝𝑦,𝑦−𝑝𝑦,𝑦=𝑝𝑥,𝑦−min𝑥,𝑥,𝑝𝑦,𝑦 (7) for all 𝑥,𝑦∈𝑋, are (usual) metrics on 𝑋. On a PMS, the concepts of convergence, Cauchy sequences, completeness, and continuity are defined as follows.

Definition 12 (see [14, 25, 26]). Let {𝑥_𝑚} be a sequence on a PMS (𝑋,𝑝).(i){𝑥_𝑚}  𝑝-converges to 𝑥∈𝑋 (and one will write {𝑥_𝑚}−→^𝑝𝑥) if 𝑝(𝑥,𝑥)=lim_𝑚→∞𝑝(𝑥,𝑥_𝑚).(ii){𝑥_𝑚} is called 𝑝-Cauchy if lim_{𝑚,𝑚′→∞}𝑝(𝑥_𝑚,𝑥_𝑚′) exists (and it is finite).(iii)(𝑋,𝑝) is said to be 𝑝-complete if every 𝑝-Cauchy sequence {𝑥_𝑚} in 𝑋  𝑝-converges to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=lim_{𝑚,𝑚′→∞}𝑝(𝑥_𝑚,𝑥_𝑚′).(iv)A mapping 𝑓∶𝑋→𝑋 is said to be 𝑝-continuous at 𝑥₀∈𝑋 if, for every 𝜀>0, there exists 𝛿>0 such that 𝑓(𝐵_𝑝(𝑥₀,𝛿))⊆𝐵_𝑝(𝑓(𝑥₀),𝜀).

We have used the previous notation because we need to distinguish between 𝑝-convergence and 𝑑_𝑝-convergence on 𝑋 and usual convergence for real sequences.

Lemma 13 (see [14, 25, 26]). Let {𝑥_𝑚} be a sequence on a PMS (𝑋,𝑝).(1){𝑥_𝑚} is 𝑝-Cauchy if, and only if, it is 𝑑_𝑝-Cauchy.(2){𝑥_𝑚}−−→^𝑑𝑝𝑥 if, and only if, {𝑥_𝑚}−→^𝑝𝑥 and 𝑝(𝑥,𝑥)=lim_{𝑚,𝑚′→∞}𝑝(𝑥_𝑚,𝑥_𝑚′); that is, 𝑑_𝑝𝑥_𝑚,𝑥⟶0⟺𝑝(𝑥,𝑥)=lim_𝑚→∞𝑝𝑥,𝑥_𝑚=lim_{𝑚,𝑚′→∞}𝑝𝑥_𝑚,𝑥_𝑚′. (8)(3)(𝑋,𝑝) is complete if, and only if, the MS (𝑋,𝑑_𝑝) is complete.(4)If {𝑥_𝑚}−→^𝑝𝑥 and 𝑝(𝑥,𝑥)=0, then lim_𝑚→∞𝑝(𝑥_𝑚,𝑦)=𝑝(𝑥,𝑦) for all 𝑦∈𝑋.

3. Auxiliary Results

We will use the following results about real sequences in the proof of our main theorems.

Lemma 14. Let {𝑎¹_𝑚}_𝑚∈ℕ,…,{𝑎^𝑛_𝑚}_𝑚∈ℕ be 𝑛 real lower bounded sequences such that {max(𝑎¹_𝑚,…,𝑎^𝑛_𝑚)}_𝑚∈ℕ→𝛿. Then, there exists 𝑖₀∈{1,2,…,𝑛} and a subsequence {𝑎^𝑖0_𝑚(𝑘)}_𝑘∈ℕ such that {𝑎^𝑖0_𝑚(𝑘)}_𝑘∈ℕ→𝛿.

Proof. Let 𝑏_𝑚=max(𝑎¹_𝑚,𝑎²_𝑚,…,𝑎^𝑛_𝑚) for all 𝑚. As {𝑏_𝑚} is convergent, it is bounded. As 𝑎^𝑖_𝑚≤𝑏_𝑚 for all 𝑚 and 𝑖, then every {𝑎^𝑖_𝑚} is bounded. As {𝑎¹_𝑚}_𝑚∈ℕ is a real bounded sequence, it has a convergent subsequence {𝑎¹_𝜎1(𝑚)}_𝑚∈ℕ→𝑎₁. Consider the subsequences {𝑎²_𝜎1(𝑚)}_𝑚∈ℕ,{𝑎³_𝜎1(𝑚)}_𝑚∈ℕ,…,{𝑎^𝑛_𝜎1(𝑚)}_𝑚∈ℕ; that are 𝑛−1 real bounded sequences and the sequence {𝑏_𝜎1(𝑚)}_𝑚∈ℕ that also converges to 𝛿. As {𝑎²_𝜎1(𝑚)}_𝑚∈ℕ is a real bounded sequence, it has a convergent subsequence {𝑎²_{𝜎2𝜎1(𝑚)}}_𝑚∈ℕ→𝑎₂. Then, the sequences {𝑎³_{𝜎2𝜎1(𝑚)}}_𝑚∈ℕ, {𝑎⁴_{𝜎2𝜎1(𝑚)}}_𝑚∈ℕ,…, {𝑎^𝑛_{𝜎2𝜎1(𝑚)}}_𝑚∈ℕ also are 𝑛−2 real bounded sequences, {𝑎¹_{𝜎2𝜎1(𝑚)}}_𝑚∈ℕ→𝑎₁, and {𝑏_{𝜎2𝜎1(𝑚)}}_𝑚∈ℕ→𝛿. Repeating this process 𝑛 times, we can find 𝑛 subsequences {𝑎¹_𝜎(𝑚)}_𝑚∈ℕ, {𝑎²_𝜎(𝑚)}_𝑚∈ℕ,…, {𝑎^𝑛_𝜎(𝑚)}_𝑚∈ℕ (where 𝜎=𝜎_𝑛⋯𝜎₁) such that {𝑎^𝑖_𝜎(𝑚)}_𝑚∈ℕ→𝑎_𝑖 for all 𝑖. And {𝑏_𝜎(𝑚)}_𝑚∈ℕ→𝛿. But 𝑏_{𝜎(𝑚)𝑚∈ℕ}=max𝑎^𝑛_𝜎(𝑚),…,𝑎^𝑛_{𝜎(𝑚)𝑚∈ℕ}⟶max𝑎₁,…,𝑎_𝑛, (9) so 𝛿=max(𝑎₁,…,𝑎_𝑛), and there exists 𝑖₀∈{1,2,…,𝑛} such that 𝑎_𝑖0=𝛿. Therefore, there exists 𝑖₀∈{1,2,…,𝑛} and a subsequence {𝑎^𝑖0_𝜎(𝑚)}_𝑚∈ℕ such that {𝑎^𝑖0_𝜎(𝑚)}_𝑚∈ℕ→𝑎_𝑖0=𝛿.

Lemma 15. Let {𝑎_𝑚}_𝑚∈ℕ be a sequence of nonnegative real numbers which has not any subsequence converging to zero. Then, for all 𝜀>0, there exist 𝛿∈]0,𝜀[ and 𝑚₀∈ℕ such that 𝑎_𝑚≥𝛿 for all 𝑚≥𝑚₀.

Proof. Suppose that the conclusion is not true. Then, there exists 𝜀₀>0 such that, for all 𝛿∈]0,𝜀₀[, there exists 𝑚₀∈ℕ verifying 𝑎_𝑚0<𝛿. Let 𝑘₀∈ℕ be such that 1/𝑘₀<𝜀₀. For all 𝑘∈ℕ, take 𝛿_𝑘=1/(𝑘+𝑘₀)∈]0,𝜀₀[. Then, there exists 𝑚(𝑘)∈ℕ verifying 0≤𝑎_𝑚(𝑘)<𝛿_𝑘=1/(𝑘+𝑘₀). Taking limit when 𝑘→∞, we deduce that lim_𝑘→∞𝑎_𝑚(𝑘)=0. Then, {𝑎_𝑚} has a subsequence converging to zero (maybe, reordering {𝑎_𝑚(𝑘)}), but this is a contradiction.

Lemma 16. If {𝑥_𝑚}_𝑚∈ℕ is a sequence in a 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)(𝑘)(),𝑑𝑥𝑘<𝑚<𝑛<𝑚𝑘+1_𝑚(𝑘),𝑥_𝑛(𝑘)≥𝜀₀,𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}<𝜀₀. (10)

Proof. We know that 𝑥_𝑚isCauchy⟺∀𝜀>0,∃𝑛₀∈ℕ∶𝑚,𝑛≥𝑛₀⟹𝑑𝑥_𝑚,𝑥_𝑛<𝜀. (11) If this condition is not true, then ∃𝜀₀>0∶∀𝑛₀∈ℕ,∃𝑚,𝑛≥𝑛₀suchthat𝑑𝑥_𝑚,𝑥_𝑛≥𝜀₀. (12) Let 𝑛₀=2. Then, there exists 𝑚₁,𝑛₁∈ℕ such that 𝑚₁,𝑛₁≥𝑛₀ and 𝑑(𝑥_𝑚1,𝑥_𝑛1)≥𝜀₀. Let 𝑚(1)=min(𝑚₁,𝑛₁)≥𝑛₀=2>1, and consider the numbers 𝑑𝑥_𝑚(1),𝑥_𝑚(1)+1,𝑑𝑥_𝑚(1),𝑥_𝑚(1)+2,…,𝑑𝑥_𝑚(1),𝑥_{max(𝑚1,𝑛1)}. (13) Since 𝑑(𝑥_𝑚(1),𝑥_{max(𝑚1,𝑛1)})=𝑑(𝑥_𝑚1,𝑥_𝑛1)≥𝜀₀, between the previous numbers there exists a first nonnegative integer 𝑛(1)∈{𝑚(1)+1,𝑚(1)+2,…,max(𝑚₁,𝑛₁)} such that 𝑑(𝑥_𝑚(1),𝑥_𝑛(1))≥𝜀₀ but 𝑑(𝑥_𝑚(1),𝑥_𝑗)<𝜀₀ for all 𝑗∈{𝑚(1),𝑚(1)+1,…,𝑛(1)−1}. In particular, 𝑑(𝑥_𝑚(1),𝑥_𝑛(1)−1)<𝜀₀.
Now, let 𝑛₀=𝑛(1)+1. Then, there exists 𝑚₂,𝑛₂∈ℕ such that 𝑚₂,𝑛₂≥𝑛(1)+1 and 𝑑(𝑥_𝑚2,𝑥_𝑛2)≥𝜀₀. Let 𝑚(2)=min(𝑚₂,𝑛₂)≥𝑛₀=𝑛(1)+1>𝑛(1), and consider the numbers 𝑑𝑥_𝑚(2),𝑥_𝑚(2)+1,𝑑𝑥_𝑚(2),𝑥_𝑚(2)+2,…,𝑑𝑥_𝑚(2),𝑥_{max(𝑚2,𝑛2)}. (14) Since 𝑑(𝑥_𝑚(2),𝑥_{max(𝑚2,𝑛2)})=𝑑(𝑥_𝑚2,𝑥_𝑛2)≥𝜀₀, between the previous numbers there exists a first nonnegative integer 𝑛(2)∈{𝑚(2)+1,𝑚(2)+2,…,max(𝑚₂,𝑛₂)} such that 𝑑(𝑥_𝑚(2),𝑥_𝑛(2))≥𝜀₀ but 𝑑(𝑥_𝑚(2),𝑥_𝑗)<𝜀₀ for all 𝑗∈{𝑚(2),𝑚(2)+1,…,𝑛(2)−1}. In particular, 𝑑(𝑥_𝑚(2),𝑥_𝑛(2)−1)<𝜀₀.
Repeating this process, we can find two subsequences {𝑥_𝑚(𝑘)} and {𝑥_𝑛(𝑘)} such that, for all 𝑘∈ℕ: (𝑘)(𝑘)(),𝑑𝑥𝑘<𝑚<𝑛<𝑚𝑘+1_𝑚(𝑘),𝑥_𝑛(𝑘)≥𝜀₀,𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}<𝜀₀. (15)

Definition 17. Let Ψ be the family of all continuous, nondecreasing mappings 𝜓∶ℝ⁺₀→ℝ⁺₀ such that 𝜓(𝑡)=0 if, and only if, 𝑡=0.

These mappings are known as altering distance functions (see [27]). Note that every selected 𝜓∈Ψ commutes with max; that is, 𝜓(max(𝑠₁,𝑠₂,…,𝑠_𝑁))=max(𝜓(𝑠₁),𝜓(𝑠₂),…,𝜓(𝑠_𝑁)) for all 𝑠₁,𝑠₂,…,𝑠_𝑁∈[0,∞).

Lemma 18. If 𝜓∈Ψ and lim_𝑚→∞𝜓(𝑎_𝑚)=0, then lim_𝑚→∞𝑎_𝑚=0.

Proof. As there exists 𝜓(𝑎_𝑚), then 𝑎_𝑚∈dom𝜓=[0,∞[. If the conclusion is not true, there exists 𝜀₀>0 such that, for all 𝑛₀∈ℕ, there exists 𝑛≥𝑛₀ verifying 𝑎_𝑛≥𝜀₀. This means that {𝑎_𝑚} has a subsequence {𝑎_𝑚(𝑘)}_𝑘 such that 𝑎_𝑚(𝑘)≥𝜀₀. As 𝜓 is nondecreasing, 𝜓(𝜀₀)≤𝜓(𝑎_𝑚(𝑘)) for all 𝑘∈ℕ. Therefore, {𝜓(𝑎_𝑚)}_𝑚 has a subsequence {𝜓(𝑎_𝑚(𝑘))}_𝑘 lower bounded by 𝜓(𝜀₀)>0, but this is impossible since lim_𝑚→∞𝜓(𝑎_𝑚)=0.

With regards to coincidence points, it is possible to consider the following simplification. If 𝜏 is a permutation of Λ_𝑛, and we reorder (4), then we deduce that every coincidence point may be seen as a coincidence point associated to the identity mapping on Λ_𝑛 (see, for instance, [28]).

Lemma 19. 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. We also show some preliminary results on PMS.

Lemma 20. Let {𝑥_𝑚} be a sequence on a PMS (𝑋,𝑝), and let 𝑥∈𝑋.(1)If {𝑥_𝑚}−→^𝑝𝑥 and 𝑝(𝑥,𝑥)=0, then {𝑥_𝑚}−−→^𝑑𝑝𝑥, {𝑑_𝑝(𝑥_𝑚,𝑦)}→𝑑_𝑝(𝑥,𝑦) and {𝑝(𝑥_𝑚,𝑦)}→𝑝(𝑥,𝑦) for all 𝑦∈𝑋.(2)If {𝑥_𝑚}−−→^𝑑𝑝𝑥 and {𝑝(𝑥_𝑚,𝑥_𝑚)}→0, then 𝑝(𝑥,𝑥)=0.

Proof. (1) Since 0≤𝑝(𝑥_𝑚,𝑥_𝑚)≤𝑝(𝑥,𝑥_𝑚) and lim_𝑚→∞𝑝(𝑥,𝑥_𝑚)=𝑝(𝑥,𝑥)=0, then lim_𝑚→∞𝑝(𝑥_𝑚,𝑥_𝑚) = 0. Therefore, lim_𝑚→∞𝑑_𝑝(𝑥,𝑥_𝑚) = lim_𝑚→∞(2𝑝(𝑥,𝑥_𝑚)−𝑝(𝑥,𝑥)−𝑝(𝑥_𝑚,𝑥_𝑚)) = 0, so {𝑥_𝑚}−−→^𝑑𝑝𝑥. Since 𝑑_𝑝 is continuous, then {𝑑_𝑝(𝑥_𝑚,𝑦)}→𝑑_𝑝(𝑥,𝑦) for all 𝑦∈𝑋, and item 4 of Lemma 13 implies that {𝑝(𝑥_𝑚,𝑦)}→𝑝(𝑥,𝑦).
(2) Item 2 of Lemma 13 shows that 𝑝(𝑥,𝑥)=lim_{𝑚,𝑚′→∞}𝑝(𝑥_𝑚,𝑥_𝑚′)=lim_𝑚→∞𝑝(𝑥_𝑚,𝑥_𝑚)=0.

Remark 21. Although the limit in a MS is unique, the 𝑝-limit in a PMS is not necessarily unique. For instance, let (𝑋,𝑝) as in Example 10. Then, (𝑋,𝑝) is a complete PMS (see [14]). Consider 𝑥_𝑚=2.5−1/(2𝑚) for all 𝑚∈ℕ. Then, {𝑥_𝑚}−−→^𝑑𝑝2.5 but {𝑥_𝑚}−→^𝑝𝑥₀ whenever 𝑥₀∈[2.5,3].

Definition 22. Let 𝑁∈ℕ, let (𝑋,𝑝) be a PMS, let 𝐺∶𝑋^𝑁→𝑋 be a mapping, and let 𝑌₀=(𝑥₁,𝑥₂,…,𝑥_𝑁)∈𝑋^𝑁. We will say that 𝐺 is 𝛼_𝑝-continuous at 𝑌₀ if, for all sequences {𝑥¹_𝑚},{𝑥²_𝑚},…,{𝑥^𝑁_𝑚} on 𝑋 such that {𝑥^𝑖_𝑚}−→^𝑝𝑥_𝑖 for all 𝑖∈{1,2,…,𝑁}, 𝑝(𝑥_𝑖,𝑥_𝑖)=0 for all 𝑖∈{1,2,…,𝑁} and {𝑝(𝐺(𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚),𝐺(𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚))}→0, we have that {𝐺(𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚)}−→^𝑝𝐺(𝑌₀) and 𝑝(𝐺(𝑌₀),𝐺(𝑌₀))=0. One will say that 𝐺 is 𝛼_𝑝-continuous if it is continuous at every point 𝑌₀∈𝑋^𝑁.

Lemma 23. If (𝑋,𝑝) is a PMS, and 𝐺∶𝑋^𝑁→𝑋 is 𝑑_𝑝-continuous at 𝑌₀∈𝑋^𝑁, then 𝐺 is 𝛼_𝑝-continuous at 𝑌₀.

Proof. Let {𝑥¹_𝑚},{𝑥²_𝑚},…,{𝑥^𝑁_𝑚} sequences on 𝑋 such that {𝑥^𝑖_𝑚}−→^𝑝𝑥_𝑖 for all 𝑖∈{1,2,…,𝑁}, 𝑝(𝑥_𝑖,𝑥_𝑖)=0 for all 𝑖∈{1,2,…,𝑁}, and {𝑝(𝐺(𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚),𝐺(𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚))}→0. Item 1 of Lemma 20 implies that {𝑥^𝑖_𝑚}−−→^𝑑𝑝𝑥_𝑖 for all 𝑖∈{1,2,…,𝑁}. Since 𝐺 is 𝑑_𝑝-continuous at 𝑌₀=(𝑥₁,𝑥₂,…,𝑥_𝑁), then {𝐺(𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚)}−−→^𝑑𝑝𝐺(𝑥₁,𝑥₂,…,𝑥_𝑁). Item 2 of Lemma 13 assures us that {𝐺(𝑥¹_𝑚,…,𝑥^𝑁_𝑚)}−→^𝑝𝐺(𝑥₁,…,𝑥_𝑁) and 𝑝𝐺𝑥₁,𝑥₂,…,𝑥_𝑁,𝐺𝑥₁,𝑥₂,…,𝑥_𝑁=lim_{𝑚,𝑚′→∞}𝑝𝐺𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚,𝐺𝑥¹_𝑚′,𝑥²_𝑚′,…,𝑥^𝑁_𝑚′=lim_𝑚→∞𝑝𝐺𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚,𝐺𝑥¹_𝑚,𝑥²_𝑚,…,𝑥^𝑁_𝑚=0. (16) Then, 𝐺 is 𝛼_𝑝-continuous at 𝑌₀.

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 24. Let (𝑋,𝑝) be a complete PMS, 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 has the mixed 𝑔-monotone property on 𝑋, 𝐹(𝑋^𝑛)⊆𝑔(𝑋) and 𝑔 is 𝛼_𝑝-continuous and commuting with 𝐹. Assume that there exist 𝜓,𝜑∈Ψ such that 𝜓𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤𝜓max_{1≤𝑖≤𝑛}𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖−𝜑max_{1≤𝑖≤𝑛}𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖, (17) for which 𝑔𝑥_𝑖≤_𝑖𝑔𝑦_𝑖 for all 𝑖. Suppose either 𝐹 is 𝛼_𝑝-continuous or (𝑋,𝑑_𝑝,≤) 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 into seven steps. The first two steps are the same as in the proof of Theorem 9 in [11], since the contractivity condition does not play any role in these parts of the proof.
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 all𝑚 and all 𝑖.
Step  3. We claim that {𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)}_𝑚≥0→0 for all 𝑖 (i.e., {max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚+1)}_𝑚≥0→0).
Indeed, define 𝛿_𝑚=max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚+1) for all 𝑚. As 𝑔𝑥^𝑖_𝑚≤_𝑖𝑔𝑥^𝑖_𝑚+1for all 𝑚 and all 𝑖, then condition (17) implies that, for all 𝑚≥1 and all 𝑖:𝜓𝑝𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1=𝜓𝑝𝐹𝑥^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)≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑗)}_𝑚−1,𝑔𝑥^𝜎𝑖_𝑚^(𝑗)−𝜑max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑗)}_𝑚−1,𝑔𝑥^𝜎𝑖_𝑚^(𝑗)≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^𝑗_𝑚−1,𝑔𝑥^𝑗_𝑚=𝜓𝛿_𝑚−1. (18)Therefore, for all 𝑚≥1, 𝜓(𝛿_𝑚)=𝜓(max_{1≤𝑖≤𝑛}𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1))=max_{1≤𝑖≤𝑛}𝜓(𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1))≤𝜓(𝛿_𝑚−1). This means that the sequence {𝜓(𝛿_𝑚)}_𝑚≥1 is nonincreasing and lower bounded. Hence, it is convergent; that is, there exists Δ≥0 such that {𝜓(𝛿_𝑚)}_𝑚≥1→Δ. We are going to show that Δ=0. Since max_{1≤𝑖≤𝑛}𝜓𝑝𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1𝑚=𝜓max_{1≤𝑖≤𝑛}𝑝𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1𝑚=𝜓𝛿_𝑚𝑚⟶Δ, (19) Lemma 14 assures that there exist 𝑖₀∈{1,2,…,𝑛} and a subsequence {𝜓(𝑝(𝑔𝑥^𝑖0_𝑚(𝑘),𝑔𝑥^𝑖0_𝑚(𝑘)+1))}_𝑘 such that {𝜓(𝑝(𝑔𝑥^𝑖0_𝑚(𝑘),𝑔𝑥^𝑖0_𝑚(𝑘)+1))}_𝑘→Δ. Repeating (18), for all 𝑘≥1, 𝜓𝑝𝑔𝑥^𝑖0_𝑚(𝑘),𝑔𝑥^𝑖0_𝑚(𝑘)+1≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)−𝜑max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖0(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖0(𝑗)}_𝑚(𝑘). (20) Consider the sequence max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^{𝜎𝑖0(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖0(𝑗)}_𝑚(𝑘))_𝑘≥1. (21) Suppose that this sequence has no subsequence converging to zero. Using 𝜀=1, Lemma 15 assures us that there exists 𝛿^′∈]0,1[ and 𝑘₀∈ℕ such that max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^{𝜎𝑖0(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖0(𝑗)}_𝑚(𝑘))≥𝛿^′ for all 𝑘≥𝑘₀. It follows that −𝜑max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)≤−𝜑𝛿^′∀𝑘≥𝑘₀. (22) Then, (20) says to us 𝜓𝑝𝑔𝑥^𝑖0_𝑚(𝑘),𝑔𝑥^𝑖0_𝑚(𝑘)+1≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)−𝜑max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖0(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖0(𝑗)}_𝑚(𝑘)≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^𝜎𝑖0_𝑚(𝑘)^(𝑗)−𝜑𝛿^′≤𝜓𝛿_{𝑚(𝑘)−1}−𝜑𝛿^′. (23) Taking limit in 𝑘, we deduce that Δ≤Δ−𝜑(𝛿^′)<Δ, which is impossible. Therefore, the sequence in (21) must have a subsequence {max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^𝜎𝑖0_𝑚^(𝑗)_{′(𝑘)−1},𝑔𝑥^𝜎𝑖0_𝑚^(𝑗)_′(𝑘))}_𝑘≥1 converging to zero. Since 𝜓 and 𝜑 are continuous, taking limit when 𝑘→∞ in (20) using this subsequence, we deduce that 0≤Δ≤𝜓(0)−𝜑(0)=0, so Δ=0. Then, we have just proved that Δ=0. Therefore, {𝜓(𝛿_𝑚)}_𝑚≥1→Δ=0, and Lemma 18 assures that {𝛿_𝑚}_𝑚≥1→0, which means that {𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚+1)}→0 for all 𝑗 since 0≤𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚+1)≤𝛿_𝑚 for all 𝑚 and all 𝑗.
Step  4. {𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚)}_𝑚≥0→0 for all 𝑖 (i.e., {max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚)}_𝑚≥0→0). It is the same proof of Step  3.
Since 𝑑_𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)=2𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1)−𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚)−𝑝(𝑔𝑥^𝑖_𝑚+1,𝑔𝑥^𝑖_𝑚+1) for all 𝑚 and 𝑖, joining Steps 3 and 4, it follows that 𝑑_𝑝𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚+1⟶0∀𝑖. (24)Step  5. Every sequence {𝑔𝑥^𝑖_𝑚}_𝑚≥0 is 𝑑_𝑝-Cauchy. We reason by contradiction. Suppose that {𝑔𝑥^𝑖1_𝑚}_𝑚≥0,…,{𝑔𝑥^𝑖𝑠_𝑚}_𝑚≥0 are not 𝑑_𝑝-Cauchy (𝑠≥1) and {𝑔𝑥^𝑖𝑠+1_𝑚}_𝑚≥0,…,{𝑔𝑥^𝑖𝑛_𝑚}_𝑚≥0 are 𝑑_𝑝-Cauchy, being {𝑖₁,…,𝑖_𝑛}={1,…,𝑛}. By Lemma 16, for all 𝑟∈{1,2,…,𝑠}, there exists 𝜀_𝑟>0 and subsequences {𝑔𝑥^𝑖𝑟_{𝑚𝑟(𝑘)}}_𝑘∈ℕ and {𝑔𝑥^𝑖𝑟_{𝑛𝑟(𝑘)}}_𝑘∈ℕ such that 𝑘<𝑚_𝑟(𝑘)<𝑛_𝑟(𝑘),𝑑_𝑝𝑔𝑥^𝑖𝑟_{𝑚𝑟(𝑘)},𝑔𝑥^𝑖𝑟_{𝑛𝑟(𝑘)}≥𝜀_𝑟,𝑑_𝑝𝑔𝑥^𝑖𝑟_{𝑚𝑟(𝑘)},𝑔𝑥^𝑖𝑟_{𝑛𝑟(𝑘)−1}<𝜀_𝑟,∀𝑘∈ℕ. (25) Now, let 𝜀₀=max(𝜀₁,…,𝜀_𝑠)>0 and 𝜀^′₀=min(𝜀₁,…,𝜀_𝑠)>0. Since {𝑔𝑥^𝑖𝑠+1_𝑚}_𝑚≥0,…,{𝑔𝑥^𝑖𝑛_𝑚}_𝑚≥0 are 𝑑_𝑝-Cauchy, for all 𝑗∈{𝑖_𝑠+1,…,𝑖_𝑛}, there exists 𝑛^𝑗₁∈ℕ such that if 𝑚,𝑚^′≥𝑛^𝑗₁, then 𝑑_𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚′)<𝜀^′₀/8. Since {𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚)}→0 by Step  4, there exists 𝑛^𝑗₂∈ℕ such that if 𝑚≥𝑛^𝑗₂, then 𝑝(𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚)<𝜀^′₀/8. Define 𝑛₀=max_{𝑗∈{𝑖𝑠+1,…,𝑖𝑛}}(𝑛^𝑗₁,𝑛^𝑗₂). If 𝑚,𝑚^′≥𝑛₀, then 0≤𝑝𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚′=𝑑_𝑝𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚′+𝑝𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚+𝑝𝑔𝑥^𝑗_𝑚′,𝑔𝑥^𝑗_𝑚′2<𝜀^′₀/8+𝜀^′₀/8+𝜀^′₀/82=3𝜀^′₀<𝜀164^′0. (26) Therefore, we have proved that there exists 𝑛₀∈ℕ such that if 𝑚,𝑚^′≥𝑛₀, then 𝑑_𝑝𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚′<𝜀^′₀4,𝑝𝑔𝑥^𝑗_𝑚,𝑔𝑥^𝑗_𝑚′<𝜀^′₀4,∀𝑗∈𝑖_𝑠+1,…,𝑖_𝑛. (27)
Next, let 𝑞∈{1,2,…,𝑠} such that 𝜀_𝑞=𝜀₀=max(𝜀₁,…,𝜀_𝑠). Let 𝑘₁∈ℕ such that 𝑛₀<𝑚^′_𝑞(𝑘₁), and define 𝑚(1)=𝑚_𝑞(𝑘₁). Consider the numbers 𝑚(1)+1,𝑚(1)+2,…,𝑛_𝑞(𝑘₁) until finding the first positive integer 𝑛(1)>𝑚(1) verifying max_{1≤𝑟≤𝑠}𝑑_𝑝𝑔𝑥^𝑖𝑟_𝑚(1),𝑔𝑥^𝑖𝑟_𝑛(1)≥𝜀₀,𝑑_𝑝𝑔𝑥^𝑖𝑗_𝑚(1),𝑔𝑥^𝑖𝑗_𝑛(1)−1<𝜀₀,∀𝑗∈{1,2,…,𝑠}. (28) Now let 𝑘₂∈ℕ such that 𝑛(1)<𝑚_𝑞(𝑘₂), and define 𝑚(2)=𝑚_𝑞(𝑘₂). Consider the numbers 𝑚(2)+1,𝑚(2)+2,…,𝑛_𝑞(𝑘₂) until finding the first positive integer 𝑛(2)>𝑚(2) verifying max_{1≤𝑟≤𝑠}𝑑_𝑝𝑔𝑥^𝑖𝑟_𝑚(2),𝑔𝑥^𝑖𝑟_𝑛(2)≥𝜀₀,𝑑_𝑝𝑔𝑥^𝑖𝑗_𝑚(2),𝑔𝑥^𝑖𝑗_𝑛(2)−1<𝜀₀,∀𝑗∈{1,2,…,𝑠}. (29) 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}𝑑_𝑝𝑔𝑥^𝑖𝑟_𝑚(𝑘),𝑔𝑥^𝑖𝑟_𝑛(𝑘)≥𝜀₀,𝑑_𝑝𝑔𝑥^𝑖𝑗_𝑚(𝑘),𝑔𝑥^𝑖𝑗_{𝑛(𝑘)−1}<𝜀₀,∀𝑗∈{1,2,…,𝑠}. (30) Note that by (27), 𝑑_𝑝(𝑔𝑥^𝑖𝑟_𝑚(𝑘),𝑔𝑥^𝑖𝑟_𝑛(𝑘)),𝑑_𝑝(𝑔𝑥^𝑖𝑟_𝑚(𝑘),𝑔𝑥^𝑖𝑟_{𝑛(𝑘)−1})<𝜀^′₀/4<𝜀₀/2 for all 𝑟∈{𝑠+1,𝑠+2,…,𝑛}, so max_{1≤𝑗≤𝑛}𝑑_𝑝𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_𝑛(𝑘)=max_{1≤𝑟≤𝑠}𝑑_𝑝𝑔𝑥^𝑖𝑟_𝑚(𝑘),𝑔𝑥^𝑖𝑟_𝑛(𝑘)≥𝜀₀,𝑑_𝑝𝑔𝑥^𝑖_𝑚(𝑘),𝑔𝑥^𝑖_{𝑛(𝑘)−1}<𝜀₀, (31) for all 𝑖∈{1,2,…,𝑛} and all 𝑘≥1. Furthermore, for all 𝑗, 2𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}−𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑚(𝑘)−1}−𝑝𝑔𝑥^𝑗_{𝑛(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}=𝑑_𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}≤𝑑_𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_𝑚(𝑘)+𝑑_𝑝𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_{𝑛(𝑘)−1}≤𝑑_𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_𝑚(𝑘)+𝜀₀. (32) Therefore, for all 𝑗 and all 𝑘, 𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}≤𝜀₀+𝑑_𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_𝑚(𝑘)+𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑚(𝑘)−1}+𝑝𝑔𝑥^𝑗_{𝑛(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}×2⁻¹. (33) Next, for all 𝑘, let 𝑖(𝑘)∈{1,2,…,𝑠} be an index such that 𝑑_𝑝𝑔𝑥_𝑚(𝑘)^𝑖(𝑘),𝑔𝑥_𝑛(𝑘)^𝑖(𝑘)=max_{1≤𝑟≤𝑠}𝑑_𝑝𝑔𝑥^𝑖𝑟_𝑚(𝑘),𝑔𝑥^𝑖𝑟_𝑛(𝑘)=max_{1≤𝑗≤𝑛}𝑑_𝑝𝑔𝑥^𝑗_𝑚(𝑘),𝑔𝑥^𝑗_𝑛(𝑘)≥𝜀₀. (34) Then, for all 𝑘, 𝑝𝑔𝑥^𝑖(𝑘)_𝑚(𝑘),𝑔𝑥^𝑖(𝑘)_𝑛(𝑘)=𝑑_𝑝𝑔𝑥_𝑚(𝑘)^𝑖(𝑘),𝑔𝑥_𝑛(𝑘)^𝑖(𝑘)+𝑝𝑔𝑥_𝑚(𝑘)^𝑖(𝑘),𝑔𝑥_𝑚(𝑘)^𝑖(𝑘)+𝑝𝑔𝑥^𝑖(𝑘)_𝑛(𝑘),𝑔𝑥^𝑖(𝑘)_𝑛(𝑘)×2⁻¹≥𝜀₀2. (35) Applying the contractivity condition (17), it follows, for all 𝑘, 𝜀0<𝜓⁰2≤𝜓𝑝𝑔𝑥^𝑖(𝑘)_𝑚(𝑘),𝑔𝑥^𝑖(𝑘)_𝑛(𝑘)≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑘)}_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^{𝜎𝑖(𝑘)}_𝑛(𝑘)^(𝑗)₋₁−𝜑max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑛(𝑘)−1}. (36) Consider the sequence: max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑘)}_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^{𝜎𝑖(𝑘)}_𝑛(𝑘)^(𝑗)_−1𝑘≥1. (37) If this sequence has a subsequence that converges to zero, then we can take limit when 𝑘→∞ in (36) using this subsequence, so that we would have 0<𝜓(𝜀₀/2)≤𝜓(0)−𝜑(0)=0, which is impossible since 𝜀₀>0. Therefore, the sequence (37) has no subsequence converging to zero. In this case, taking 𝜀₀>0 in Lemma 15, there exist 𝛿∈]0,𝜀₀[ and 𝑘₀∈ℕ such that max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑛(𝑘)−1})≥𝛿, for all 𝑘≥𝑘₀. It follows that, for all 𝑘≥𝑘₀, −𝜑(max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑛(𝑘)−1}))≤−𝜑(𝛿). Thus, by (36), 𝜀0<𝜓⁰2≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑘)}_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^{𝜎𝑖(𝑘)}_𝑛(𝑘)^(𝑗)₋₁−𝜑max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑚(𝑘)−1},𝑔𝑥^{𝜎𝑖(𝑘)(𝑗)}_{𝑛(𝑘)−1}≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^{𝜎𝑖(𝑘)}_𝑚(𝑘)^(𝑗)₋₁,𝑔𝑥^{𝜎𝑖(𝑘)}_𝑛(𝑘)^(𝑗)₋₁−𝜑(𝛿)≤𝜓max_{1≤𝑗≤𝑛}𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}−𝜑(𝛿). (38) Fix any 𝛾>0 and we are going to prove that 𝜓(𝜀₀/2)+𝜑(𝛿)≤𝜓(𝜀₀/2+𝛾). Indeed, by Step  3 and (24), since max_{1≤𝑖≤𝑛}𝑝𝑔𝑥^𝑖_{𝑚(𝑘)−1},𝑔𝑥^𝑖_{𝑚(𝑘)−1},max_{1≤𝑖≤𝑛}𝑝𝑔𝑥^𝑖_{𝑛(𝑘)−1},𝑔𝑥^𝑖_{𝑛(𝑘)−1},max_{1≤𝑖≤𝑛}𝑑_𝑝𝑔𝑥^𝑖_{𝑚(𝑘)−1},𝑔𝑥^𝑖_𝑚(𝑘) (39) are sequences converging to zero, we can find 𝑚₁∈ℕ such that if 𝑚(𝑘)≥𝑚₁, then max_{1≤𝑖≤𝑛}𝑝𝑔𝑥^𝑖_{𝑚(𝑘)−1},𝑔𝑥^𝑖_{𝑚(𝑘)−1}≤𝛾2,max_{1≤𝑖≤𝑛}𝑝𝑔𝑥^𝑖_{𝑛(𝑘)−1},𝑔𝑥^𝑖_{𝑛(𝑘)−1}≤𝛾2,max_{1≤𝑖≤𝑛}𝑑_𝑝𝑔𝑥^𝑖_{𝑚(𝑘)−1},𝑔𝑥^𝑖_𝑚(𝑘)≤𝛾2. (40) Therefore, (33) implies that, for all 𝑗 and for all 𝑘 such that 𝑚(𝑘)>𝑚₁, 𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}≤𝜀₀+𝑑_𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_𝑚(𝑘)+𝑝𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑚(𝑘)−1}+𝑝𝑔𝑥^𝑗_{𝑛(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}×2⁻¹≤𝜀₀+𝛾/2+𝛾/2+𝛾/22=𝜀₀2+3𝛾4<𝜀₀2+𝛾. (41) Then, (38) guarantees that 0<𝜓(𝜀₀/2)≤𝜓(max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥^𝑗_{𝑚(𝑘)−1},𝑔𝑥^𝑗_{𝑛(𝑘)−1}))−𝜑(𝛿)≤𝜓(𝜀₀/2+𝛾)−𝜑(𝛿). This means that 𝜓(𝜀₀/2)+𝜑(𝛿)≤𝜓(𝜀₀/2+𝛾) for all 𝛾>0. If we take 𝛾=1/𝑚>0 (where 𝑚∈ℕ), we deduce that 𝜓(𝜀₀/2)+𝜑(𝛿)≤𝜓(𝜀₀/2+1/𝑚) for all 𝑚∈ℕ. Since 𝜓 is continuous, we have that 𝜓(𝜀₀/2)+𝜑(𝛿)≤𝜓(𝜀₀/2), which is impossible since 𝜑(𝛿)>0. This contradiction finally proves that every sequence {𝑔𝑥^𝑖_𝑚}_𝑚≥0 is 𝑑_𝑝-Cauchy.
Since 𝑋 is 𝑝-complete, then 𝑋 is 𝑑_𝑝-complete (item 3 of Lemma 13). Then, there exist 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋 such that {𝑔𝑥^𝑖_𝑚}−−→^𝑑𝑝𝑥_𝑖 for all 𝑖. Furthermore, 𝑝(𝑥_𝑖,𝑥_𝑖) = lim_{𝑚,𝑚′→∞}𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚′) = lim_𝑚→∞𝑝(𝑔𝑥^𝑖_𝑚,𝑔𝑥^𝑖_𝑚) = 0 for all 𝑖. Since 𝑔 is 𝛼_𝑝-continuous, then {𝑔𝑔𝑥^𝑖_𝑚}−→^𝑝𝑔𝑥_𝑖 and 𝑝(𝑔𝑥_𝑖,𝑔𝑥_𝑖)=0 for all 𝑖. Item 1 of Lemma 20 shows that {𝑔𝑔𝑥^𝑖_𝑚}−−→^𝑑𝑝𝑔𝑥_𝑖 for all 𝑖. Therefore, for all 𝑖, lim_𝑚→∞𝑝(𝑔𝑔𝑥^𝑖_𝑚+1,𝑔𝑔𝑥^𝑖_𝑚+1)=lim_{𝑚,𝑚′→∞}𝑝(𝑔𝑔𝑥^𝑖_𝑚+1,𝑔𝑔𝑥^𝑖_𝑚′+1)=𝑝(𝑔𝑥_𝑖,𝑔𝑥_𝑖)=0. Moreover, for all 𝑚 and all 𝑖, 𝑔𝑔𝑥^𝑖_𝑚+1=𝑔𝐹(𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑥^𝜎𝑛_𝑚^(𝑛))=𝐹(𝑔𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑖_𝑚^(𝑛)).
Step  6. Suppose that 𝐹 is 𝛼_𝑝-continuous. In this case, we know that {𝑔𝑥^𝑖_𝑚}−→^𝑝𝑥_𝑖 and 𝑝(𝑥_𝑖,𝑥_𝑖)=0 for all 𝑖 and 𝑝𝐹𝑔𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑖_𝑚^(𝑛),𝐹𝑔𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑖_𝑚^(𝑛)=𝑝𝑔𝑔𝑥^𝑖_𝑚+1,𝑔𝑔𝑥^𝑖_𝑚+1⟶0, (42) which implies that {𝐹(𝑔𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑖_𝑚^(𝑛))}−→^𝑝𝐹(𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)}) and 𝑝(𝐹(𝑥_𝜎𝑖(1),…,𝑥_{𝜎𝑖(𝑛)}), 𝐹(𝑥_𝜎𝑖(1),…,𝑥_{𝜎𝑖(𝑛)}))=0 for all 𝑖. Item 1 of Lemma 20 assures us that, for all 𝑖, 𝑔𝑔𝑥^𝑖_𝑚+1=𝐹𝑔𝑥^𝜎𝑖_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑖_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑖_𝑚^(𝑛)_𝑑𝑝⟶𝐹𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)}. (43) Since the limit in a MS is unique, we deduce that 𝐹(𝑥_𝜎𝑖(1),𝑥_𝜎𝑖(2),…,𝑥_{𝜎𝑖(𝑛)})=𝑔𝑥_𝑖 for all 𝑖, so (𝑥₁,𝑥₂,…,𝑥_𝑛) is a Υ-coincidence point of 𝐹 and 𝑔.
Step  7. Suppose that (𝑋,𝑑_𝑝,≤)  has the sequential 𝑔-monotone property. In this case, by Step  2, we know that 𝑔𝑥^𝑖_𝑚≤_𝑖𝑔𝑥^𝑖_𝑚+1 for all 𝑚 and all 𝑖. This means that the sequence {𝑔𝑥^𝑖_𝑚}_𝑚≥0 is monotone. As {𝑔𝑥^𝑖_𝑚}−−→^𝑑𝑝𝑥_𝑖, we deduce that 𝑔𝑔𝑥^𝑖_𝑚≤_𝑖𝑔𝑥_𝑖 for all 𝑚and 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 (44) (the first case occurs when 𝑗∈𝐴 and the second one when 𝑗∈𝐵). Then, by (17), for all 𝑗, 𝜓𝑝𝑔𝑔𝑥^𝑗_𝑚+1,𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}=𝜓𝑝𝐹𝑔𝑥^𝜎𝑗_𝑚⁽¹⁾,𝑔𝑥^𝜎𝑗_𝑚⁽²⁾,…,𝑔𝑥^𝜎𝑗_𝑚^(𝑛),𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}≤𝜓max_{1≤𝑖≤𝑛}𝑝𝑔𝑔𝑥^𝜎𝑗_𝑚^(𝑖),𝑔𝑥_{𝜎𝑗(𝑖)}−𝜑max_{1≤𝑖≤𝑛}𝑝𝑔𝑔𝑥^𝜎𝑗_𝑚^(𝑖),𝑔𝑥_{𝜎𝑗(𝑖)}≤𝜓max_{1≤𝑖≤𝑛}𝑝𝑔𝑔𝑥^𝑖_𝑚,𝑔𝑥_𝑖. (45) Since {𝑔𝑔𝑥^𝑖_𝑚}−−→^𝑑𝑝𝑔𝑥_𝑖 for all 𝑖, then lim_𝑚→∞𝑝𝑔𝑔𝑥^𝑖_𝑚,𝑔𝑥_𝑖=𝑝𝑔𝑥_𝑖,𝑔𝑥_𝑖=0∀𝑖. (46) Therefore, lim_𝑚→∞(max_{1≤𝑖≤𝑛}𝑝(𝑔𝑔𝑥^𝑖_𝑚,𝑔𝑥_𝑖))=0. Taking limit when 𝑚→∞ in (45), we deduce that lim_𝑚→∞𝜓(𝑝(𝑔𝑔𝑥^𝑗_𝑚+1,𝐹(𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)})))=0 for all 𝑗. As 𝜓∈Ψ, Lemma 18 guarantees that lim_𝑚→∞𝑝𝑔𝑔𝑥^𝑗_𝑚+1,𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}=0∀𝑗. (47) Finally, for all 𝑗, 𝑑_𝑝𝑔𝑥_𝑗,𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}=2𝑝𝑔𝑥_𝑗,𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}−𝑝𝑔𝑥_𝑗,𝑔𝑥_𝑗−𝑝𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)},𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}≤2𝑝𝑔𝑥_𝑗,𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}≤2𝑝𝑔𝑔𝑥^𝑖_𝑚,𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}𝑝𝑔𝑥_𝑗,𝑔𝑔𝑥^𝑖_𝑚+𝑝𝑔𝑔𝑥^𝑖_𝑚,𝐹𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}. (48) Using (46) and (47), we conclude that 𝑑_𝑝(𝑔𝑥_𝑗,𝐹(𝑥_𝜎𝑗(1),𝑥_𝜎𝑗(2),…,𝑥_{𝜎𝑗(𝑛)}))=0 for all 𝑗.

Remark 25. In the previous theorem, if the image Im𝑑 of the metric 𝑑 is not the whole set [0,∞[, then 𝜓 and 𝜑 can only be defined on Im𝑑, and we can consider a wider range of mappings since it is only necessary to impose that they are continuous and nondecreasing on Im𝑑.

Remark 26. We notice also that our paper cannot be deduced from the recent interesting paper of Haghi et al. [21] on partial metric space. In fact, we use a partial order ≤. Then, we only suppose (17) for which 𝑔𝑥_𝑖≤_𝑖𝑔𝑦_𝑖 for all 𝑖 (not necessarily on points which are not comparable). Further, we use a self-map 𝑔∶𝑋→𝑋 which implies that 𝑃(𝐴,𝐵)=max_{1≤𝑖≤𝑛}𝑝𝑔𝑎_𝑖,𝑔𝑏_𝑖,𝐴=𝑎₁,𝑎₂,…,𝑎_𝑛,𝐵=𝑏₁,𝑏₂,…,𝑏_𝑛∈𝑋^𝑛 (49) is not necessarily a partial metric on 𝑋^𝑛. For instance, let 𝑋=ℝ⁺₀=[0,∞) provided with its usual partial order and the partial metric 𝑝(𝑥,𝑦)=max(𝑥,𝑦). Consider 𝑔𝑥=0,if0≤𝑥≤1,𝑥−1,if𝑥>1. (50) Then, 𝑔 is continuous, but 𝑃),(),(),(((0,0,…,00,0,…,0))=𝑃((1,1,…,10,0,…,0))=𝑃((1,1,…,11,1,…,1))=0, (51) but (0,0,…,0)≠(1,1,…,1). Then, 𝑃 does not verify the axiom 𝑝(𝑥,𝑥)=𝑝(𝑥,𝑦)=𝑝(𝑦,𝑦)⇒𝑥=𝑦. Therefore, we cannot apply Theorem 2.4 on Haghi et al. [21].
As a result, we cannot use Theorem 2.7 in [21] since 𝑇 has an influence in −𝜑(max{𝑝(𝑥,𝑦),𝑝(𝑦,𝑇𝑦)}), and our mapping 𝐹 has not a role in the left side of (17).

5. Consequences

Remark 27. Theorem 9 in [11] is an easy consequence of Theorem 24 if we take 𝑝=𝑑, 𝜓(𝑡)=𝑡, and 𝜑(𝑡)=(1−𝑘)𝑡 for all 𝑡∈ℝ⁺₀.
In the next result, let Γ₀ be the family of all nondecreasing on each argument, continuous mappings 𝜙∶[0,∞[^𝑛→ℝ⁺₀ verifying 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛)=0 if, and only if, 𝑥₁=𝑥₂=⋯=𝑥_𝑛=0. Examples of such mappings are the following, where 𝑘>0, 𝛼_𝑖>0 and 𝑛_𝑖∈ℕ for all 𝑖. (i)𝜙(𝑥₁,…,𝑥_𝑛)=𝑘max_{1≤𝑖≤𝑛}𝑥_𝑖.(ii)𝜙(𝑥₁,…,𝑥_𝑛)=∑^𝑛_𝑖=1𝛼_𝑖𝑥^𝑛𝑖_𝑖.(iii)𝜙(𝑥₁,…,𝑥_𝑛)=^𝑚𝛼₁𝑥²₁+⋯+𝛼_𝑛𝑥²_𝑛.

Lemma 28. Let 𝜙∈Γ₀, and define 𝜑∶ℝ⁺₀→ℝ⁺₀ as 𝜑(𝑡)=min(𝜙(𝑡𝑒₁),𝜙(𝑡𝑒₂),…,𝜙(𝑡𝑒_𝑛)) for all 𝑡≥0, where {𝑒₁,𝑒₂,…,𝑒_𝑛} is the usual basis of ℝ^𝑛. Then, 𝜑∈Ψ and 𝜑(max_{1≤𝑖≤𝑛}𝑥_𝑖)≤𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ⁺₀.

Proof. First part is clear. If 𝑥_𝑖0=max_{1≤𝑖≤𝑛}𝑥_𝑖, then 𝜙(𝑥_𝑖0𝑒_𝑖0) = 𝜙(0,0,…,0,𝑥_𝑖0,0,…,0) ≤ 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑖0−1,𝑥_𝑖0,𝑥_𝑖0+1,…,𝑥_𝑛) = 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛). Therefore, 𝜑(max_{1≤𝑖≤𝑛}𝑥_𝑖) = 𝜑(𝑥_𝑖0)≤𝜙(𝑥_𝑖0𝑒_𝑖0)≤𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛).

Corollary 29. Thesis of Theorem 24 also holds if one replaces the contractivity condition (17) by any of the following list (for which 𝑔𝑥_𝑖≤_𝑖𝑔𝑦_𝑖 for all 𝑖).(A) This condition can be found in [11] and [12], there exist 𝜓∈Ψ and 𝜙∈Γ₀ such that 𝜓𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤𝜓max_{1≤𝑖≤𝑛}𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖−𝜙𝑝𝑔𝑥₁,𝑔𝑦₁,…,𝑝𝑔𝑥_𝑛,𝑔𝑦_𝑛. (52)(B)In [17], there exist 𝜓,𝜑∈Ψ and 𝛽₁,𝛽₂,…,𝛽_𝑛∈[0,1] such that 𝛽₁+𝛽₂+⋯+𝛽_𝑛≤1 and 𝜓𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝑝𝑦₁,𝑦₂,…,𝑦_𝑛≤𝜓^𝑛_𝑖=1𝛽_𝑖𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖−𝜑max_{1≤𝑖≤𝑛}𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖. (53)(C) There exist 𝜓,𝜑∈Ψ and 𝛼₁,𝛼₂,…,𝛼_𝑛>0 such that 𝜓𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤𝜓max_{1≤𝑖≤𝑛}𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖−𝜑^𝑛_𝑖=1𝛼_𝑖𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖. (54)(D)In [2, 7, 9], there exist 𝜓∈Ψ, 𝛼₁,𝛼₂,…,𝛼_𝑛>0, and 𝛽₁,𝛽₂,…,𝛽_𝑛≥0 such that 𝛽₁+𝛽₂+⋯+𝛽_𝑛≤1 and 𝜓𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤𝜓^𝑛_𝑖=1𝛽_𝑖𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖−^𝑛_𝑖=1𝛼_𝑖𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖. (55)(E)In [5], there exist 𝜓∈Ψ and 𝛼₁,𝛼₂,…,𝛼_𝑛>0 such that 𝜓𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛1≤𝜓𝑛^𝑛_𝑖=1𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖−^𝑛_𝑖=1𝛼_𝑖𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖. (56)(F)In [19, 20], there exist 𝜓,𝜑∈Ψ such that 𝜓 is subadditive (𝜓(𝑠+𝑡)≤𝜓(𝑠)+𝜓(𝑡) for all 𝑡,𝑠∈[0,∞)) and 𝜓𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤1𝑛𝜓^𝑛_𝑖=1𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖−𝜑max_{1≤𝑖≤𝑛}𝑝𝑔𝑥_𝑖,𝑔𝑦_𝑖. (57)

Of course, it is also interesting to particularize all the previous items to the following cases: 𝜓(𝑡)=𝜆𝑡 (where 𝜆>0), 𝜑(𝑡)=𝜇𝑡 (where 𝜇>0), or 𝑔𝑥=𝑥 for all 𝑥∈𝑋.

Proof. (A) By Lemma 28, there exists 𝜑∈Ψ such that −𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛)≤−𝜑(max_{1≤𝑖≤𝑛}𝑥_𝑖) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈[0,∞[, so (52) implies (17). (B) It is obvious that ∑^𝑛_𝑖=1𝛽_𝑖𝑝(𝑔𝑥_𝑖,𝑔𝑦_𝑖) ≤ (∑^𝑛_𝑖=1𝛽_𝑖)max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥_𝑗,𝑔𝑦_𝑗)≤max_{1≤𝑗≤𝑛}𝑝(𝑔𝑥_𝑗,𝑔𝑦_𝑗), so (53) implies (17). (C) We only take 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛)=𝜑(∑^𝑛_𝑖=1𝛼_𝑖𝑥_𝑖) in item (A). (D) It is a mixture of (B) and (C). (E) It is a particular case of (D) where 𝛽_𝑖=1/𝑛 for all 𝑖. (F) If 𝜓 is subadditive, then (1/𝑛)𝜓(𝑡)≤𝜓(𝑡/𝑛) for all 𝑡≥0, so we may choose 𝛽_𝑖=1/𝑛 for all 𝑖 in (B).

6. Uniqueness of Υ-Coincidence Points

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. (58) We say that (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).

Theorem 30. Under the hypothesis of Theorem 24, 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.

Proof. From Theorem 24, the set of Υ-coincidence points of 𝐹 and 𝑔 is nonempty. The proof is divided into two steps.
Step  1. We claim that if (𝑥₁,𝑥₂,…,𝑥_𝑛),(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑋^𝑛 are two Υ-coincidence points of 𝐹 and 𝑔, then 𝑔𝑥_𝑖=𝑔𝑦_𝑖∀𝑖. (59) 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 24, 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^𝑑𝑝_𝑚→0𝑔𝑢^𝑖_𝑚=𝑔𝑦_𝑖 for all 𝑖, so (59) will be true.
Firstly, we reason with (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) and (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛), and the same argument holds 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 24, 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≥1either𝑔𝑢^𝜎𝑗_𝑚^(𝑖)≤_𝑖𝑔𝑥_{𝜎𝑗(𝑖)}∀𝑖or𝑔𝑥_{𝜎𝑗(𝑖)}≤_𝑖𝑔𝑢^𝜎𝑗_𝑚^(𝑖)∀𝑖. (60) Define 𝛽_𝑚=max_{1≤𝑖≤𝑛}𝑝(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖) for all 𝑚. Reasoning as in Theorem 24, it is not difficult to prove that {𝛽_𝑚}_𝑚≥1→0 which means that lim_𝑚→∞𝛽_𝑚=lim_𝑚→∞(max_{1≤𝑖≤𝑛}𝑝(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖))=0. As 0≤𝑝(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖)≤𝛽_𝑚 for all 𝑚 and all 𝑖, we deduce that {𝑝(𝑔𝑢^𝑖_𝑚,𝑔𝑥_𝑖)}_𝑚≥1→0=𝑝(𝑔𝑥_𝑖,𝑔𝑥_𝑖) for all 𝑖; that is, {𝑔𝑢^𝑖_𝑚}−→^𝑝𝑔𝑥_𝑖 for all 𝑖. Item 1 of Lemma 20 shows that 𝑔𝑢⟶^𝑖_𝑚^𝑑𝑝𝑔𝑥_𝑖∀𝑖. (61) If we had supposed that (𝑔𝑥₁,𝑔𝑥₂,…,𝑔𝑥_𝑛)≤(𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛), we would have obtained the same property (61). And as (𝑔𝑢₁,𝑔𝑢₂,…,𝑔𝑢_𝑛) also is comparable to (𝑔𝑦₁,𝑔𝑦₂,…,𝑔𝑦_𝑛), we can reason in the same way to prove that {𝑔𝑢^𝑖_𝑚}−−→^𝑑𝑝𝑔𝑦_𝑖 for all 𝑖. Since the limit in a MS is unique, 𝑔𝑥_𝑖=𝑔𝑦_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 Υ-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 [11].

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 is 𝑔 is injective on 𝑋.

Corollary 31. In addition to the hypotheses of Theorem 30, 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 (59) that 𝑔𝑥_𝑖=𝑔𝑦_𝑖 for all 𝑖. As 𝑔 is injective on these points, then, 𝑥_𝑖=𝑦_𝑖 for all 𝑖.

Corollary 32. In addition to the hypotheses of Theorem 30, 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}𝑝(𝑧_𝑖,𝑧_𝑗), let 𝑗₀,𝑠₀∈{1,2,…,𝑛} such that 𝑝(𝑧_𝑗0,𝑧_𝑠0)=𝑀, and let Λ=max_{1≤𝑖≤𝑛}𝑝𝑧_{𝜎𝑗0(𝑖)},𝑧_{𝜎𝑠0(𝑖)}≤𝑀. (62) Fix 𝑗,𝑠∈{1,2,…,𝑛}. As (𝑧_𝜎𝑗(1),𝑧_𝜎𝑗(2),…,𝑧_{𝜎𝑗(𝑛)}) is comparable to (𝑧_𝜎𝑠(1),𝑧_𝜎𝑠(2),…,𝑧_{𝜎𝑠(𝑛)}), then either 𝑧_{𝜎𝑗(𝑖)}≤_𝑖𝑧_{𝜎𝑠(𝑖)} for all 𝑖 or 𝑧_{𝜎𝑠(𝑖)}≤_𝑖𝑧_{𝜎𝑗(𝑖)} for all 𝑖. Since 𝑔𝑧_𝑖=𝑧_𝑖 for all 𝑖, we know that either 𝑔𝑧_{𝜎𝑗(𝑖)}≤_𝑖𝑔𝑧_{𝜎𝑠(𝑖)} for all 𝑖 or 𝑔𝑧_{𝜎𝑠(𝑖)}≤_𝑖𝑔𝑧_{𝜎𝑗(𝑖)} for all 𝑖. In any case, applying (17), 𝜓(𝑀)=𝜓𝑝𝑧_𝑗0,𝑧_𝑠0=𝜓𝑝𝑔𝑧_𝑗0,𝑔𝑧_𝑠0=𝜓𝑝𝐹𝑧_𝜎𝑗0(1),𝑧_𝜎𝑗0(2),…,𝑧_{𝜎𝑗0(𝑛)},𝐹𝑧_𝜎𝑠0(1),𝑧_𝜎𝑠0(2),…,𝑧_{𝜎𝑠0(𝑛)}≤𝜓max_{1≤𝑖≤𝑛}𝑝𝑔𝑧_{𝜎𝑗0(𝑖)},𝑔𝑧_{𝜎𝑠0(𝑖)}−𝜑max_{1≤𝑖≤𝑛}𝑝𝑔𝑧_{𝜎𝑗0(𝑖)},𝑔𝑧_{𝜎𝑠0(𝑖)}=𝜓(Λ)−𝜑(Λ)≤𝜓(𝑀)−𝜑(Λ). (63) If Λ>0, then 𝜑(Λ)>0, so 𝜓(𝑀)≤𝜓(𝑀)−𝜑(Λ)<𝜓(𝑀), which is impossible. Then, Λ=0, and (63) implies that 𝜓(𝑀)≤𝜓(Λ)−𝜑(Λ)=𝜓(0)−𝜑(0)=0, so 𝜓(𝑀)=0. Therefore, 𝑝(𝑧_𝑖,𝑧_𝑗)=0 for all 𝑖 and 𝑗.

Example 33. Let 𝑋=ℝ provided with its usual partial order ≤ and the partial metric 𝑝(𝑥,𝑦)=max(|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), it follows that 𝐹𝑥₁,𝑥₂,…,𝑥_𝑛≤𝑎₁𝑥₁+𝑎₂𝑥₂+⋯+𝑎_𝑛𝑥_𝑛𝑁≤𝑎₁+𝑎₂+⋯+𝑎_𝑛𝑁max𝑥₁,𝑥₂,…,𝑥_𝑛=𝑘max𝑥₁,𝑥₂,…,𝑥_𝑛. (64) Therefore, 𝑝𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛=max𝐹𝑥₁,𝑥₂,…,𝑥_𝑛,𝐹𝑦₁,𝑦₂,…,𝑦_𝑛≤max𝑘max𝑥₁,𝑥₂,…,𝑥_𝑛,𝑘max𝑦₁,𝑦₂,…,𝑦_𝑛=𝑘max_𝑖𝑝𝑥_𝑖,𝑦_𝑖. (65) If 𝜓(𝑡)=𝑡 and 𝜑(𝑡)=(1−𝑘)𝑡, all conditions of Theorems 24 and 30 (and Corollaries 31 and 32) are satisfied. Indeed, it is clear that (0,0,…,0) is the unique fixed point of 𝐹.

The following example is based on Examples 1.9 and 2.2 in [29].

Example 34. Let 𝑋={0,1,2,3,4}, and let 𝑝 be the partial metric on 𝑋 given by 𝑝(𝑥,𝑦)=max(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. Then, (𝑋,𝑝) is complete, and 𝑝 generates the discrete topology on 𝑋 (indeed, 𝑑_𝑝 is the Euclidean metric on 𝑋). Consider on 𝑋 the following partial order: 𝑥,𝑦∈𝑋,𝑥≤𝑦⟺𝑥=𝑦or𝑥,𝑦=(0,2). (66) Consider 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 defined by 𝐹𝑥₁,𝑥₂,…,𝑥_𝑛=0,if𝑥₁,𝑥₂,…,𝑥_𝑛∈{0,1,2},1,otherwise,𝑔𝑥=0,if𝑥=0,2,if𝑥∈{0.5,1},3,if𝑥∈{1.5,2}. (67) It is not difficult to prove the following statements.(1)𝐹 and 𝑔 are 𝛼_𝑝-continuous mappings (since 𝑑_𝑝 generates the discrete topology on 𝑋).(2)𝐹 and 𝑔 are commuting.(3) If 𝑦,𝑧∈𝑋 verify 𝑔𝑦≤𝑔𝑧, then either 𝑦,𝑧∈{0,1,2} or 𝑦,𝑧∈{3,4}. Then, 𝐹 has the mixed (𝑔,≤)-monotone property on 𝑋.(4) If 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛∈𝑋 verify 𝑔𝑥_𝑖≤_𝑖𝑔𝑦_𝑖 for all 𝑖, then 𝐹(𝑥₁,𝑥₂,…,𝑥_𝑛) = 𝐹(𝑦₁,𝑦₂,…,𝑦_𝑛). In particular, (17) holds (whatever 𝜓 and 𝜑; for instance, 𝜓(𝑡)=2𝑡 and 𝜑(𝑡)=log(𝑡+1) for all 𝑡≥0). For simplicity, henceforth, suppose that 𝑛 is even, and let 𝐴 (resp., 𝐵) be the set of all odd (resp., even) numbers in {1,2,…,𝑛}.(5) For a mapping 𝜎∶Λ_𝑛→Λ_𝑛, we use the notation 𝜎≡(𝜎(1),𝜎(2),…,𝜎(𝑛)) and consider 𝜎_𝑖≡(𝑖,𝑖+1,…,𝑛−1,𝑛,1,2,…,𝑖−1)∀𝑖. (68) Then, 𝜎_𝑖∈Ω_𝐴,𝐵 if 𝑖 is odd, and 𝜎_𝑖∈Ω^′_𝐴,𝐵 if 𝑖 is even. Let Υ=(𝜎₁,𝜎₂,…,𝜎_𝑛).(6) Take 𝑥^𝑖₀=0 if 𝑖 is odd and 𝑥^𝑖₀=2 if 𝑖 is even. Then, 𝑔𝑥^𝑖₀≤_𝑖𝐹(𝑥^𝜎𝑖₀⁽¹⁾,𝑥^𝜎𝑖₀⁽²⁾,…,𝑥^𝜎𝑖₀^(𝑛)) for all 𝑖.(7)(𝑋,𝑑_𝑝,≤) has the sequential 𝑔-monotone property.
Therefore, we can apply Theorems 24 and 30, and Corollaries 31 and 32, to conclude that 𝐹 and 𝑔 have a unique Υ-coincidence point, which is (0,0,…,0).

Acknowledgments

This work has been partially supported by Junta de Andaluca, by projects FQM-268, FQM-178 and FQM-235 of the Andalusian CICYE.