𝑛-Tuplet Coincidence Point Theorems in Intuitionistic Fuzzy Normed Spaces

Abstract

For an arbitrary n positive integer, we investigate the existence of n-tuplet coincidence points in intuitionistic fuzzy normed space. Results of the paper are more general than those of the coupled and the tripled fixed point works in intuitionistic fuzzy normed space.

1. Introduction

One of the most important fields in mathematics is the fixed point theory. This theory is used to solve a variety of problems in many areas such as economics, chemistry, computer science, and engineering as well as many branches of mathematics. One of the main theorems in the fixed point theory is the Banach contraction theorem [1]. This theorem states that contraction map in complete metric space has a unique fixed point. Many authors studied the generalization of the Banach contraction theorem. Generalization on the complete partial ordered metric space was given by Ran and Reurings [2] with a weaker condition. In their theorem, contraction condition is provided only for comparable elements with respect to partial order relation in complete metric space. Some fixed point theorems have been obtained by many authors based on [2]. Bhaskar and Lakshmikantham [3] defined the concept of coupled fixed point and used a theorem associated with it for existence and uniqueness of solution of the periodic boundary value problem. Later on, Lakshmikantham and Ćirić [4] introduced the concept of coincidence point which is a generalization of fixed point. By inspiring these works, coupled fixed point theorems have been studied for different type contraction mappings (see [5–12]). The interest on coupled fixed point theorem has motivated the authors to generalize it as tripled fixed point theorem in [13, 14], afterwards as quadruple fixed point theorem in [15–17], and as 𝑛-tuplet fixed point theorem in [18–21].

On the other hand, fuzzy theory was introduced by Zadeh [22] and it was generalized by Atanassov [23] as intuitionistic fuzzy theory. While fuzzy theory assigns degree of membership for each element, intuitionistic fuzzy theory assigns degree of membership and nonmembership for each element. Both of them were applied in many fields of sciences.

Introduction of the intuitionistic fuzzy metric space by Park [24] and of the intuitionistic fuzzy normed space by Saadati and Park [25] has enabled many subjects in functional analysis to be studied in intuitionstic fuzzy normed (metric) spaces. Fixed point theory is one of these subjects. Many fixed point theorems have been studied in intuitionistic fuzzy normed (metric) space. Numerous works have been produced since richness of fixed point theory and intuitionistic fuzzy functional analysis. Some of the articles concerning these fields can be found in the literature (such as [26–33]).

Coupled and tripled fixed point theorems in intuitionistic fuzzy normed space were proved via 𝑛-property in [29, 30], respectively. The purpose of our paper is to study 𝑛-tuplet coincidence point theorem without 𝑛-property in intuitionistic fuzzy normed space, which is the generalization of coupled fixed point theorem [29] and tripled fixed point theorem [30] in intuitionistic fuzzy normed space.

Let us start by recalling some of the concepts used in this paper.

Definition 1 (see [34]). A binary operation ∗∶[0,1]×[0,1] is a continuous 𝑡-norm if it satisfies the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous; (iii) 𝑎∗1=𝑎 for all 𝑎∈[0,1]; (iv) 𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for each 𝑎,𝑏,𝑐,𝑑∈[0,1].

Definition 2 (see [34]). A binary operation ⋄∶[0,1]×[0,1] is a continuous 𝑡-conorm if it satisfies the following conditions: (i) ⋄ is associative and commutative; (ii) ⋄ is continuous; (iii) 𝑎⋄0=𝑎 for all 𝑎∈[0,1]; (iv) 𝑎⋄𝑏≤𝑐⋄𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for each 𝑎,𝑏,𝑐,𝑑∈[0,1].

Definition 3 (see [25]). Let ∗ be a continuous 𝑡-norm, let ⋄ be a continuous 𝑡-conorm, and let 𝑋 be a linear space over the field 𝐼𝐹(ℝ or ℂ). If 𝜇 and 𝜈 are fuzzy sets on 𝑋×(0,∞) satisfying the following conditions, the five-tuple (𝑋,𝜇,𝜈,∗,⋄) is said to be an intuitionistic fuzzy normed space and (𝜇,𝜈) is called an intuitionistic fuzzy norm. For every 𝑥,𝑦∈𝑋 and 𝑠,𝑡>0, one has the following:(i)𝜇(𝑥,𝑡)+𝜈(𝑥,𝑡)≤1,(ii)𝜇(𝑥,𝑡)>0,(iii)𝜇(𝑥,𝑡)=1⇔𝑥=0,(iv)𝜇(𝑎𝑥,𝑡)=𝜇(𝑥,𝑡/|𝑎|) for each 𝑎≠0,(v)𝜇(𝑥,𝑡)∗𝜇(𝑦,𝑠)≤𝜇(𝑥+𝑦,𝑡+𝑠),(vi)𝜇(𝑥,⋅)∶(0,∞)→[0,1] is continuous,(vii)lim_𝑡→∞𝜇(𝑥,𝑡)=1 and lim_𝑡→0𝜇(𝑥,𝑡)=0,(viii)𝜈(𝑥,𝑡)<1,(ix)𝜈(𝑥,𝑡)=0⇔𝑥=0,(x)𝜈(𝑎𝑥,𝑡)=𝜈(𝑥,𝑡/|𝑎|) for each 𝑎≠0,(xi)𝜈(𝑥,𝑡)⋄𝜈(𝑦,𝑠)≥𝜈(𝑥+𝑦,𝑡+𝑠),(xii)𝜈(𝑥,⋅)∶(0,∞)→[0,1] is continuous,(xiii)lim_𝑡→∞𝜈(𝑥,𝑡)=0 and lim_𝑡→0𝜈(𝑥,𝑡)=1;

we further assume that (𝑋,𝜇,𝜈,∗,⋄) satisfies the following axiom:(xiv)𝑎⋄𝑎=𝑎 and 𝑎∗𝑎=𝑎 for all 𝑎∈[0,1].

Throughout this paper, expression “intuitionistic fuzzy normed space” will be denoted by IFNS for short.

Similar to Definition  4.12 in [25], Definition 4 can be given as the following:

Definition 4 (see [25]). Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS. (𝑋^𝑛,Φ,Ψ,∗,⋄) is called a cartesian product of intuitionistic fuzzy normed spaces if (Φ,Ψ) is a cartesian product of intuitionistic fuzzy norms defined Φ(𝑥,𝑡)=^𝑛_𝑖=1𝜇𝑥_𝑖,Ψ()=,𝑡𝑥,𝑡^𝑛_𝑖=1𝜈𝑥_𝑖,𝑡, (1)
where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛),∏^𝑛_𝑖=1𝑎_𝑖=𝑎₁∗𝑎₂∗⋯∗𝑎_𝑛,∐^𝑛_𝑖=1𝑎_𝑖=𝑎₁⋄𝑎₂⋄⋯⋄𝑎_𝑛 and 𝑡>0.

Definition 5 (see [25]). Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS. Then a sequence (𝑥_𝑘) in 𝑋 is said to be Cauchy sequence if for each 𝜖>0 and 𝑡>0 there exists 𝑘₀∈ℕ such that 𝜇𝑥_𝑘−𝑥_𝑚,𝑡>1−𝜖,𝜈𝑥_𝑘−𝑥_𝑚,𝑡<𝜖 (2) for each 𝑘,𝑚>𝑘₀.

Definition 6 (see [25]). Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS. (𝑋,𝜇,𝜈,∗,⋄) is said to be complete if every Cauchy sequence in (𝑋,𝜇,𝜈,∗,⋄) is convergent.

Definition 7 (see [35]). Let 𝑋 and 𝑌 be two IFNSs. 𝑓∶𝑋→𝑌 is continuous at 𝑥₀∈𝑋 if (𝑓(𝑥_𝑘)) in 𝑌 convergences to 𝑓(𝑥₀) for any (𝑥_𝑘) in 𝑋 converging to 𝑥₀. If 𝑓∶𝑋→𝑌 is continuous at each element of 𝑋, then 𝑓∶𝑋→𝑌 is said to be continuous on 𝑋.

Definition 8 (see [18]). Let (𝑋,⪯) be partially ordered set and 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋. It is said that 𝐹 has the mixed 𝑔-monotone property if 𝐹(𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛) is monotone 𝑔-nondecreasing in its odd argument and it is monotone 𝑔-nonincreasing in its even argument. That is, for any 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛∈𝑋, 𝑦₁,𝑧₁∈𝑋,𝑔𝑦₁⪯𝑔𝑧₁⟹𝐹𝑦₁,𝑥₂,𝑥₃,…,𝑥_𝑛⪯𝐹𝑧₁,𝑥₂,𝑥₃,…,𝑥_𝑛,𝑦₂,𝑧₂∈𝑋,𝑔𝑦₂⪯𝑔𝑧₂⟹𝐹𝑥₁,𝑦₂,𝑥₃,…,𝑥_𝑛≽𝐹𝑥₁,𝑧₂,𝑥₃,…,𝑥_𝑛,⋮𝑦_𝑛,𝑧_𝑛∈𝑋,𝑔𝑦_𝑛⪯𝑔𝑧_𝑛⟹𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑦_𝑛⪯𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑧_𝑛(if𝑛isodd),𝑦_𝑛,𝑧_𝑛∈𝑋,𝑔𝑦_𝑛⪯𝑔𝑧_𝑛⟹𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑦_𝑛≽𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑧_𝑛(if𝑛iseven). (3) Note that if 𝑔 is the identity mapping, this definition is reduced to Definition 1 in [18].

Definition 9 (see [18]). Let 𝑋 be a nonempty set and let 𝐹∶𝑋^𝑛→𝑋 be a given mapping. An element (𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛)∈𝑋^𝑛 is called an 𝑛-tuplet coincidence point of 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 if 𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛=𝑔𝑥₁,𝐹𝑥₂,𝑥₃,…,𝑥_𝑛,𝑥₁=𝑔𝑥₂,⋮𝐹𝑥_𝑛,𝑥₁,𝑥₂,…,𝑥_𝑛−1=𝑔𝑥_𝑛. (4) Note that if 𝑔 is the identity mapping, this definition is reduced to Definition 2 in [18].

Definition 10 (see [18]). Let 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 be two mappings. 𝐹 and 𝑔 are called commutative if 𝑔𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛=𝐹𝑔𝑥₁,𝑔𝑥₂,𝑔𝑥₃,…,𝑔𝑥_𝑛 (5) for all 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛∈𝑋.

2. Main Results

Theorem 11. Let 𝐹∶𝑋^𝑛→𝑋 be continuous map having mixed 𝑔-monotone property on the complete (𝑋,𝜇,𝜈,∗,⋄) having partial order relation denoted by ⪯. Also 𝐹(𝑋^𝑛)⊂𝑔(𝑋); 𝑔 is continuous and commutes with 𝐹. Suppose that 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 hold the following conditions, for all 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛,𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑛∈𝑋 and 𝛼∈(0,1): 𝜇𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛−𝐹𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑛𝑔𝑥,𝛼𝑡≥𝜇₁−𝑔𝑦₁𝑔𝑥,𝑡∗𝜇₂−𝑔𝑦₂,𝑡∗⋯∗𝜇𝑔𝑥_𝑛−𝑔𝑦_𝑛,𝑡,𝜈𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛−𝐹𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑛𝑔𝑥,𝛼𝑡≤𝜈₁−𝑔𝑦₁𝑔𝑥,𝑡⋄𝜈₂−𝑔𝑦₂,𝑡⋄⋯⋄𝜈𝑔𝑥_𝑛−𝑔𝑦_𝑛,𝑡, (6)(7) where 𝑔(𝑥_2𝑖−1)⪯𝑔(𝑦_2𝑖−1),𝑖∈{1,2,…,(𝑛+1)/2}, and 𝑔(𝑥_2𝑖)≽𝑔(𝑦_2𝑖),𝑖∈{1,2,…,𝑛/2}. If there exist 𝑥¹₀,𝑥²₀,𝑥³₀,…,𝑥^𝑛₀∈𝑋 such that 𝑔𝑥¹₀⪯𝐹𝑥¹₀,𝑥²₀,𝑥³₀,…,𝑥^𝑛₀,𝑔𝑥²₀≽𝐹𝑥²₀,𝑥³₀,…,𝑥^𝑛₀,𝑥¹₀,⋮𝑔𝑥^𝑛₀⪯𝐹𝑥^𝑛₀,𝑥¹₀,𝑥²₀,…,𝑥₀^𝑛−1,𝑔𝑥𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑_𝑛0≽𝐹𝑥^𝑛₀,𝑥¹₀,𝑥²₀,…,𝑥₀^𝑛−1𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛, (8) then there exist 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛∈𝑋 such that 𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛=𝑔𝑥₁,𝐹𝑥₂,𝑥₃,…,𝑥_𝑛,𝑥₁=𝑔𝑥₂,⋮𝐹𝑥_𝑛,𝑥₁,𝑥₂,…,𝑥_𝑛−1=𝑔𝑥_𝑛; (9) that is, 𝐹 and 𝑔 have an 𝑛-tuplet coincidence point.

Proof. Proof of this theorem consists of four steps.
Step 1. In first step, let us define (𝑥¹_𝑘),(𝑥²_𝑘),…,(𝑥^𝑛_𝑘). Let 𝑥¹₀,𝑥²₀,𝑥³₀,…,𝑥^𝑛₀∈𝑋 be as in (8). Since 𝐹(𝑋^𝑛)⊂𝑔(𝑋), we construct the sequence (𝑥¹_𝑘), (𝑥²_𝑘),…,(𝑥^𝑛_𝑘) as in [18]: 𝑔𝑥¹_𝑘=𝐹𝑥¹_𝑘−1,𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1,𝑔𝑥²_𝑘=𝐹𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1,⋮𝑔𝑥^𝑛_𝑘=𝐹𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1,…,𝑥^𝑛−1_𝑘−1 (10) for 𝑘=1,2,3,….
Step 2. We prove that the following inequalities hold: 𝑔𝑥¹_𝑘−1⪯𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘−1≽𝑔𝑥²_𝑘,⋮𝑔𝑥^𝑛_𝑘−1⪯𝑔𝑥^𝑛_𝑘(if𝑛isodd),𝑔𝑥^𝑛_𝑘−1≽𝑔𝑥^𝑛_𝑘(if𝑛iseven) (11) for all 𝑘≥1. This step is similar to a part of proof of Theorem 1 in [18]. So, we omit it. However, since we use it in the proof, we express it again for the integrity of our proof. We can write (12) from (11) as follows: ⋯≽𝑔𝑥¹_𝑘≽𝑔𝑥¹_𝑘−1≽⋯≽𝑔𝑥¹₁≽𝑔𝑥¹₀,⋯⪯𝑔𝑥²_𝑘⪯𝑔𝑥²_𝑘−1⪯⋯⪯𝑔𝑥²₁⪯𝑔𝑥²₀,⋮⋯≽𝑔𝑥^𝑛_𝑘≽𝑔𝑥^𝑛_𝑘−1≽⋯≽𝑔𝑥^𝑛₁≽𝑔𝑥^𝑛₀(if𝑛isodd),⋯⪯𝑔𝑥^𝑛_𝑘⪯𝑔𝑥^𝑛_𝑘−1⪯⋯⪯𝑔𝑥^𝑛₁⪯𝑔𝑥^𝑛₀(if𝑛iseven). (12)
Step 3. In this step, we show that 𝑔(𝑥¹_𝑘),𝑔(𝑥²_𝑘),…,𝑔(𝑥^𝑛_𝑘) are Cauchy sequences in (𝑋,𝜇,𝜈,∗,⋄).
To do this, we define 𝛿¹_𝑘(𝑡)=𝜇𝑔𝑥¹_𝑘−𝑔𝑥¹_𝑘+1𝑔𝑥,𝑡∗𝜇²_𝑘−𝑔𝑥²_𝑘+1𝑔𝑥,𝑡∗⋯∗𝜇^𝑛_𝑘−𝑔𝑥^𝑛_𝑘+1,𝛿,𝑡_2𝑘(𝑡)=𝑣𝑔𝑥¹_𝑘−𝑔𝑥¹_𝑘+1𝑔𝑥,𝑡⋄𝑣²_𝑘−𝑔𝑥²_𝑘+1𝑔𝑥,𝑡⋄⋯⋄𝑣^𝑛_𝑘−𝑔𝑥^𝑛_𝑘+1,𝑡. (13)
By considering (6) and (12), we have in the following inequalities: 𝜇𝑔𝑥¹_𝑘−𝑔𝑥¹_𝑘+1𝐹𝑥,𝑡=𝜇¹_𝑘−1,𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1−𝐹𝑥¹_𝑘,𝑥²_𝑘,…,𝑥^𝑛_𝑘𝑡,𝛼𝑎≥𝜇𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼∗𝜇𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝛼∗⋯∗𝜇𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼=𝛿_𝑘−1𝑡𝛼,𝜇𝑔𝑥²_𝑘−𝑔𝑥²_𝑘+1𝐹𝑥,𝑡=𝜇²_𝑘−1,…,𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1−𝐹𝑥²_𝑘,…,𝑥^𝑛_𝑘,𝑥¹_𝑘𝑡,𝛼𝛼≥𝜇𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝛼∗⋯∗𝜇𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼∗𝜇𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼=𝛿¹_𝑘−1𝑡𝛼⋮𝜇𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛_𝑘+1𝐹𝑥,𝑡=𝜇^𝑛_𝑘−1,𝑥¹_𝑘−1,…,𝑥^𝑛−1_𝑘−1−𝐹𝑥^𝑛_𝑘,𝑥¹_𝑘,…,𝑥_𝑘^𝑛−1𝑡,𝛼𝛼≥𝜇𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼∗𝜇𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼∗⋯∗𝜇𝑔𝑥^𝑛−1_𝑘−1−𝑔𝑥_𝑘^𝑛−1,𝑡𝛼=𝛿¹_𝑘−1𝑡𝛼. (14) Using the property (iv) of 𝑡-norm and property (xiv) in Definition 3 together with (14), we get 𝛿¹_𝑘(𝑡)≥𝛿¹_𝑘−1𝑡𝛼∗𝛿¹_𝑘−1𝑡𝛼∗⋯∗𝛿¹_𝑘−1𝑡𝛼=𝛿¹_𝑘−1𝑡𝛼. (15)
Again, by (6) and (12), 𝜇𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼=𝜇𝐹𝑥¹_𝑘−2,𝑥²_𝑘−2,…,𝑥^𝑛_𝑘−2−𝐹𝑥¹_𝑘−1,𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1𝑡,𝛼𝛼²≥𝜇𝑔𝑥¹_𝑘−2−𝑔𝑥¹_𝑘−1,𝑡𝛼²∗𝜇𝑔𝑥²_𝑘−2−𝑔𝑥²_𝑘−1,𝑡𝛼²∗⋯∗𝜇𝑔𝑥^𝑛_𝑘−2−𝑔𝑥^𝑛_𝑘−1,𝑡𝛼²=𝛿¹_𝑘−2𝑡𝛼²,𝜇𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝛼=𝜇𝐹𝑥²_𝑘−2,…,𝑥^𝑛_𝑘−2,𝑥¹_𝑘−2−𝐹𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1𝑡,𝛼𝛼²≥𝜇𝑔𝑥²_𝑘−2−𝑔𝑥²_𝑘−1,𝑡𝛼²∗⋯∗𝜇𝑔𝑥^𝑛_𝑘−2−𝑔𝑥^𝑛_𝑘−1,𝑡𝛼²∗𝜇𝑔𝑥¹_𝑘−2−𝑔𝑥¹_𝑘−1,𝑡𝛼²=𝛿¹_𝑘−2𝑡𝛼²⋮𝜇𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼=𝜇𝐹𝑥^𝑛_𝑘−2,𝑥¹_𝑘−2,…,𝑥^𝑛−1_𝑘−2−𝐹𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1,…,𝑥^𝑛−1_𝑘−1𝑡,𝛼𝛼²≥𝜇𝑔𝑥^𝑛_𝑘−2−𝑔𝑥^𝑛_𝑘−1,𝑡𝛼²∗𝜇𝑔𝑥¹_𝑘−2−𝑔𝑥¹_𝑘−1,𝑡𝛼²∗⋯∗𝜇𝑔𝑥^𝑛−1_𝑘−2−𝑔𝑥^𝑛−1_𝑘−1,𝑡𝛼²=𝛿¹_𝑘−2𝑡𝛼². (16)
By the property (iv) of 𝑡-norm together with (16), we get 𝛿¹_𝑘−1𝑡𝛼≥𝛿¹_𝑘−2𝑡𝛼². (17) Thus, if we continue this process in this way, we have 𝛿¹_𝑘(𝑡)≥𝛿¹_𝑘−1𝑡𝛼≥𝛿¹_𝑘−2𝑡𝛼²≥⋯≥𝛿¹₀𝑡𝛼^𝑘. (18)
Now we will do the same calculations for 𝛿²_𝑘(𝑡). By using (7) and (12), 𝑣𝑔𝑥¹_𝑘−𝑔𝑥¹_𝑘+1𝐹𝑥,𝑡=𝑣¹_𝑘−1,𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1−𝐹𝑥¹_𝑘,𝑥²_𝑘,…,𝑥^𝑛_𝑘𝑡,𝛼𝛼≤𝑣𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼⋄𝑣𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝛼⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼=𝛿²_𝑘−1𝑡𝛼,𝑣𝑔𝑥²_𝑘−𝑔𝑥²_𝑘+1𝐹𝑥,𝑡=𝑣²_𝑘−1,…,𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1−𝐹𝑥²_𝑘,…,𝑥^𝑛_𝑘,𝑥¹_𝑘𝑡,𝛼𝛼≤𝑣𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝛼⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼⋄𝑣𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼=𝛿²_𝑘−1𝑡𝛼⋮𝑣𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛_𝑘+1𝐹𝑥,𝑡=𝑣^𝑛_𝑘−1,𝑥¹_𝑘−1,…,𝑥^𝑛−1_𝑘−1−𝐹𝑥^𝑛_𝑘,𝑥¹_𝑘,…,𝑥_𝑘^𝑛−1𝑡,𝛼𝛼≤𝑣𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼⋄𝑣𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼⋄⋯⋄𝑣𝑔𝑥^𝑛−1_𝑘−1−𝑔𝑥_𝑘^𝑛−1,𝑡𝛼=𝛿²_𝑘−1𝑡𝛼. (19) Using the property (iv) of 𝑡-norm and property (xiv) in Definition 3 together with (19), we get 𝛿²_𝑘(𝑡)≤𝛿²_𝑘−1𝑡𝛼⋄𝛿²_𝑘−1𝑡𝛼⋄⋯⋄𝛿²_𝑘−1𝑡𝛼=𝛿²_𝑘−1𝑡𝛼. (20)
Again, by (7) and (12), 𝑣𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝛼=𝑣𝐹𝑥¹_𝑘−2,𝑥²_𝑘−2,…,𝑥^𝑛_𝑘−2−𝐹𝑥¹_𝑘−1,𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1𝑡,𝛼𝛼²≤𝑣𝑔𝑥¹_𝑘−2−𝑔𝑥¹_𝑘−1,𝑡𝛼²⋄𝑣𝑔𝑥²_𝑘−2−𝑔𝑥²_𝑘−1,𝑡𝛼²⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘−2−𝑔𝑥^𝑛_𝑘−1,𝑡𝛼²=𝛿²_𝑘−2𝑡𝛼²,𝑣𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝛼=𝑣𝐹𝑥²_𝑘−2,…,𝑥^𝑛_𝑘−2,𝑥¹_𝑘−2−𝐹𝑥²_𝑘−1,…,𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1𝑡,𝛼𝛼²≤𝑣𝑔𝑥²_𝑘−2−𝑔𝑥²_𝑘−1,𝑡𝛼²⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘−2−𝑔𝑥^𝑛_𝑘−1,𝑡𝛼²⋄𝑣𝑔𝑥¹_𝑘−2−𝑔𝑥¹_𝑘−1,𝑡𝛼²=𝛿²_𝑘−2𝑡𝛼²⋮𝑣𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝛼=𝑣𝐹𝑥^𝑛_𝑘−2,𝑥¹_𝑘−2,…,𝑥^𝑛−1_𝑘−2−𝐹𝑥^𝑛_𝑘−1,𝑥¹_𝑘−1,…,𝑥^𝑛−1_𝑘−1𝑡,𝛼𝛼²≤𝑣𝑔𝑥^𝑛_𝑘−2−𝑔𝑥^𝑛_𝑘−1,𝑡𝛼²⋄𝑣𝑔𝑥¹_𝑘−2−𝑔𝑥¹_𝑘−1,𝑡𝛼²⋄⋯⋄𝑣𝑔𝑥^𝑛−1_𝑘−2−𝑔𝑥^𝑛−1_𝑘−1,𝑡𝛼²=𝛿²_𝑘−2𝑡𝛼². (21)
Hence, from (21),𝛿²_𝑘−1𝑡𝛼≤𝛿²_𝑘−2𝑡𝛼². (22) Thus, if we continue this process in this way, we have 𝛿²_𝑘(𝑡)≤𝛿²_𝑘−1𝑡𝛼≤𝛿²_𝑘−2𝑡𝛼²≤⋯≤𝛿²₀𝑡𝛼^𝑘. (23)
Now, we can show 𝑔(𝑥¹_𝑘),𝑔(𝑥²_𝑘),…,𝑔(𝑥^𝑛_𝑘) are Cauchy sequences in (𝑋,𝜇,𝜈,∗,⋄) by means of (18) and (23). For each 𝑡>0 and 𝑝>0, 𝜇𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_𝑘,𝑡∗𝜇𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_𝑘𝑔𝑥,𝑡∗⋯∗𝜇^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_𝑘𝑔𝑥,𝑡=𝜇¹_𝑘+𝑝−𝑔𝑥¹_{𝑘+𝑝−1}+𝑔𝑥¹_{𝑘+𝑝−1}−𝑔𝑥¹_{𝑘+𝑝−2}+𝑔𝑥¹_{𝑘+𝑝−2}−⋯−𝑔𝑥¹_𝑘−1+𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝑝+𝑡𝑝𝑡+⋯+𝑝∗𝜇𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_{𝑘+𝑝−1}+𝑔𝑥²_{𝑘+𝑝−1}−𝑔𝑥²_{𝑘+𝑝−2}+𝑔𝑥²_{𝑘+𝑝−2}−⋯−𝑔𝑥²_𝑘−1+𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝑝+𝑡𝑝𝑡+⋯+𝑝∗⋯∗𝜇𝑔𝑥^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_{𝑘+𝑝−1}+𝑔𝑥^𝑛_{𝑘+𝑝−1}−𝑔𝑥^𝑛_{𝑘+𝑝−2}+𝑔𝑥^𝑛_{𝑘+𝑝−2}−⋯−𝑔𝑥^𝑛_𝑘−1+𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝑝+𝑡𝑝𝑡+⋯+𝑝≥𝜇𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_{𝑘+𝑝−1},𝑡𝑝∗𝜇𝑔𝑥¹_{𝑘+𝑝−1}−𝑔𝑥¹_{𝑘+𝑝−2},𝑡𝑝∗⋯∗𝜇𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝑝∗⋯∗𝜇𝑔𝑥^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_{𝑘+𝑝−1},𝑡𝑝∗𝜇𝑔𝑥^𝑛_{𝑘+𝑝−1}−𝑔𝑥^𝑛_{𝑘+𝑝−2},𝑡𝑝∗⋯∗𝜇𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝑝=𝜇𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_{𝑘+𝑝−1},𝑡𝑝∗𝜇𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_{𝑘+𝑝−1},𝑡𝑝∗⋯∗𝜇𝑔𝑥^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_{𝑘+𝑝−1},𝑡𝑝∗𝜇𝑔𝑥¹_{𝑘+𝑝−1}−𝑔𝑥¹_{𝑘+𝑝−2},𝑡𝑝∗𝜇𝑔𝑥²_{𝑘+𝑝−1}−𝑔𝑥²_{𝑘+𝑝−2},𝑡𝑝∗⋯∗𝜇𝑔𝑥^𝑛_{𝑘+𝑝−1}−𝑔𝑥^𝑛_{𝑘+𝑝−2},𝑡𝑝∗⋯∗𝜇𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝑝∗𝜇𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝑝∗⋯∗𝜇𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝑝≥𝜇𝑔𝑥¹₀−𝑔𝑥¹₁,𝑡𝑝𝛼^{𝑘+𝑝−1}∗𝜇𝑔𝑥²₀−𝑔𝑥²₁,𝑡𝑝𝛼^{𝑘+𝑝−1}∗⋯∗𝜇𝑔𝑥^𝑛₀−𝑔𝑥^𝑛₁,𝑡𝑝𝛼^{𝑘+𝑝−1}∗𝜇𝑔𝑥¹₀−𝑔𝑥¹₁,𝑡𝑝𝛼^{𝑘+𝑝−2}∗𝜇𝑔𝑥²₀−𝑔𝑥²₁,𝑡𝑝𝛼^{𝑘+𝑝−2}∗⋯∗𝜇𝑔𝑥^𝑛₀−𝑔𝑥^𝑛₁,𝑡𝑝𝛼^{𝑘+𝑝−2}∗⋯∗𝜇𝑔𝑥¹₀−𝑔𝑥¹₁,𝑡𝑝𝛼^𝑘∗𝜇𝑔𝑥²₀−𝑔𝑥²₁,𝑡𝑝𝛼^𝑘∗⋯∗𝜇𝑔𝑥^𝑛₀−𝑔𝑥^𝑛₁,𝑡𝑝𝛼^𝑘,𝑣𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_𝑘,𝑡⋄𝑣𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_𝑘𝑔𝑥,𝑡⋄⋯⋄𝑣^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_𝑘𝑔𝑥,𝑡=𝑣¹_𝑘+𝑝−𝑔𝑥¹_{𝑘+𝑝−1}+𝑔𝑥¹_{𝑘+𝑝−1}−𝑔𝑥¹_{𝑘+𝑝−2}+𝑔𝑥¹_{𝑘+𝑝−2}−⋯−𝑔𝑥¹_𝑘−1+𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝑝+𝑡𝑝𝑡+⋯+𝑝⋄𝑣𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_{𝑘+𝑝−1}+𝑔𝑥²_{𝑘+𝑝−1}−𝑔𝑥²_{𝑘+𝑝−2}+𝑔𝑥²_{𝑘+𝑝−2}−⋯−𝑔𝑥²_𝑘−1+𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝑝+𝑡𝑝𝑡+⋯+𝑝⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_{𝑘+𝑝−1}+𝑔𝑥^𝑛_{𝑘+𝑝−1}−𝑔𝑥^𝑛_{𝑘+𝑝−2}+𝑔𝑥^𝑛_{𝑘+𝑝−2}−⋯−𝑔𝑥^𝑛_𝑘−1+𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝑝+𝑡𝑝𝑡+⋯+𝑝≤𝑣𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_{𝑘+𝑝−1},𝑡𝑝⋄𝑣𝑔𝑥¹_{𝑘+𝑝−1}−𝑔𝑥¹_{𝑘+𝑝−2},𝑡𝑝⋄⋯⋄𝑣𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝑝⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_{𝑘+𝑝−1},𝑡𝑝⋄𝑣𝑔𝑥^𝑛_{𝑘+𝑝−1}−𝑔𝑥^𝑛_{𝑘+𝑝−2},𝑡𝑝⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝑝=𝑣𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_{𝑘+𝑝−1},𝑡𝑝⋄𝑣𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_{𝑘+𝑝−1},𝑡𝑝⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_{𝑘+𝑝−1},𝑡𝑝⋄𝑣𝑔𝑥¹_{𝑘+𝑝−1}−𝑔𝑥¹_{𝑘+𝑝−2},𝑡𝑝⋄𝑔𝑥²_{𝑘+𝑝−1}−𝑔𝑥²_{𝑘+𝑝−2},𝑡𝑝⋄⋯⋄𝑣𝑔𝑥^𝑛_{𝑘+𝑝−1}−𝑔𝑥^𝑛_{𝑘+𝑝−2},𝑡𝑝⋄⋯⋄𝑣𝑔𝑥¹_𝑘−1−𝑔𝑥¹_𝑘,𝑡𝑝⋄𝑣𝑔𝑥²_𝑘−1−𝑔𝑥²_𝑘,𝑡𝑝⋄⋯⋄𝑣𝑔𝑥^𝑛_𝑘−1−𝑔𝑥^𝑛_𝑘,𝑡𝑝≤𝑣𝑔𝑥¹₀−𝑔𝑥¹₁,𝑡𝑝𝛼^{𝑘+𝑝−1}⋄𝑣𝑔𝑥²₀−𝑔𝑥²₁,𝑡𝑝𝛼^{𝑘+𝑝−1}⋄⋯⋄𝑣𝑔𝑥^𝑛₀−𝑔𝑥^𝑛₁,𝑡𝑝𝛼^{𝑘+𝑝−1}⋄𝑣𝑔𝑥¹₀−𝑔𝑥¹₁,𝑡𝑝𝛼^{𝑘+𝑝−2}⋄𝑣𝑔𝑥²₀−𝑔𝑥²₁,𝑡𝑝𝛼^{𝑘+𝑝−2}⋄⋯⋄𝑣𝑔𝑥^𝑛₀−𝑔𝑥^𝑛₁,𝑡𝑝𝛼^{𝑘+𝑝−2}⋄⋯⋄𝑣𝑔𝑥¹₀−𝑔𝑥¹₁,𝑡𝑝𝛼^𝑘⋄𝑣𝑔𝑥²₀−𝑔𝑥²₁,𝑡𝑝𝛼^𝑘⋄⋯⋄𝑣𝑔𝑥^𝑛₀−𝑔𝑥^𝑛₁,𝑡𝑝𝛼^𝑘, (24)𝑡/𝑝𝛼^{𝑘+𝑝−1},𝑡/𝑝𝛼^{𝑘+𝑝−2},…,𝑡/𝑝𝛼^𝑘 tend to infinity when 𝑘→∞ in (24). So, we get (25) with properties (vii) and (xiii) of intuitionistic fuzzy norm: lim_𝑘→∞𝜇𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_𝑘,𝑡∗𝜇𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_𝑘𝑔𝑥,𝑡∗⋯∗𝜇^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_𝑘,𝑡≥1∗1∗⋯∗1=1,lim_𝑘→∞𝑣𝑔𝑥¹_𝑘+𝑝−𝑔𝑥¹_𝑘,𝑡⋄𝑣𝑔𝑥²_𝑘+𝑝−𝑔𝑥²_𝑘𝑔𝑥,𝑡⋄⋯⋄𝑣^𝑛_𝑘+𝑝−𝑔𝑥^𝑛_𝑘,𝑡≤0⋄0⋄⋯⋄0=0. (25)
Hence, 𝑔(𝑥¹_𝑘),𝑔(𝑥²_𝑘),…,𝑔(𝑥^𝑛_𝑘) are Cauchy sequences in (𝑋,𝜇,𝜈,∗,⋄).
Step 4. In final step, we prove that 𝑔 and 𝐹 have an 𝑛-tuplet coincidence point. Since 𝑋 is complete, then there exist 𝑥¹,𝑥²,…,𝑥^𝑛 such that lim_𝑘→∞𝜇𝑔𝑥¹_𝑘−𝑥¹,𝑡=1,lim_𝑘→∞𝜈𝑔𝑥¹_𝑘−𝑥¹,𝑡=0,lim_𝑘→∞𝜇𝑔𝑥²_𝑘−𝑥²,𝑡=1,lim_𝑘→∞𝜈𝑔𝑥²_𝑘−𝑥²⋮,𝑡=0,lim_𝑘→∞𝜇𝑔𝑥^𝑛_𝑘−𝑥^𝑛,𝑡=1,lim_𝑘→∞𝜈𝑔𝑥^𝑛_𝑘−𝑥^𝑛,𝑡=0. (26)
By using intuitionistic fuzzy continuity of 𝑔, we write lim_𝑘→∞𝜇𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡=1,lim_𝑘→∞𝜈𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡=0,lim_𝑘→∞𝜇𝑔𝑔𝑥²_𝑘−𝑔𝑥²,𝑡=1,lim_𝑘→∞𝜈𝑔𝑔𝑥²_𝑘−𝑔𝑥²⋮,𝑡=0,lim_𝑘→∞𝜇𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡=1,lim_𝑘→∞𝜈𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡=0. (27) Since commutativity of 𝐹 and 𝑔, it follows that 𝑔𝑔𝑥¹_𝑘+1=𝑔𝐹𝑥¹_𝑘,𝑥²_𝑘,…,𝑥^𝑛_𝑘=𝐹𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘,𝑔𝑔𝑥²_𝑘+1=𝑔𝐹𝑥²_𝑘,…,𝑥^𝑛_𝑘,𝑥¹_𝑘=𝐹𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘⋮𝑔𝑔𝑥^𝑛_𝑘+1=𝑔𝐹𝑥^𝑛_𝑘,𝑥¹_𝑘,…,𝑥_𝑘^𝑛−1=𝐹𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘,…,𝑔𝑥_𝑘^𝑛−1. (28)
Using continuous of 𝑡-norm and 𝑡-conorm, we get lim_𝑘→∞Φ𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘−𝑥¹,𝑥²,…,𝑥^𝑛,𝑡=lim_𝑘→∞Φ𝑔𝑥¹_𝑘−𝑥¹,𝑔𝑥²_𝑘−𝑥²,…,𝑔𝑥^𝑛_𝑘−𝑥^𝑛,𝑡=lim_𝑘→∞𝜇𝑔𝑥¹_𝑘−𝑥¹,𝑡∗𝜇𝑔𝑥²_𝑘−𝑥²𝑔𝑥,𝑡∗⋯∗𝜇^𝑛_𝑘−𝑥^𝑛,𝑡=lim_𝑘→∞𝜇𝑔𝑥¹_𝑘−𝑥¹,𝑡∗lim_𝑘→∞𝜇𝑔𝑥²_𝑘−𝑥²,𝑡∗⋯∗lim_𝑘→∞𝜇𝑔𝑥^𝑛_𝑘−𝑥^𝑛,𝑡=1∗1∗⋯∗1=1,lim_𝑘→∞Ψ𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘−𝑥¹,𝑥²,…,𝑥^𝑛,𝑡=lim_𝑘→∞Ψ𝑔𝑥¹_𝑘−𝑥¹,𝑔𝑥²_𝑘−𝑥²,…,𝑔𝑥^𝑛_𝑘−𝑥^𝑛,𝑡=lim_𝑘→∞𝜈𝑔𝑥¹_𝑘−𝑥¹,𝑡⋄𝜈𝑔𝑥²_𝑘−𝑥²𝑔𝑥,𝑡⋄⋯⋄𝜈^𝑛_𝑘−𝑥^𝑛,𝑡=lim_𝑘→∞𝜈𝑔𝑥¹_𝑘−𝑥¹,𝑡⋄lim_𝑘→∞𝜈𝑔𝑥²_𝑘−𝑥²,𝑡⋄⋯⋄lim_𝑘→∞𝜈𝑔𝑥^𝑛_𝑘−𝑥^𝑛,𝑡=0⋄0⋄⋯⋄0=0. (29)
That is (𝑔(𝑥¹_𝑘),𝑔(𝑥²_𝑘),…,𝑔(𝑥^𝑛_𝑘))−−−−−→^(Φ,Ψ)(𝑥¹,𝑥²,…,𝑥^𝑛). Similarly (𝑔(𝑥²_𝑘),…,𝑔(𝑥^𝑛_𝑘),𝑔(𝑥¹_𝑘))−−−−−→^(Φ,Ψ)(𝑥²,…,𝑥^𝑛,𝑥¹),…,(𝑔(𝑥^𝑛_𝑘),𝑔(𝑥¹_𝑘),…,𝑔(𝑥_𝑘^𝑛−1))−−−−−→^(Φ,Ψ)(𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1). From the intuitionistic fuzzy continuous assumption of 𝐹, we write in the followings 𝐹𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥−−−−^𝑛_𝑘^𝜇,𝜈→𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝐹𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘,𝑔𝑥−−−−¹_𝑘^𝜇,𝜈→𝐹𝑥²,…,𝑥^𝑛,𝑥¹⋮𝐹𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘,…,𝑔𝑥−−−−_𝑘^{𝑛−1𝜇,𝜈}→𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1. (30)
Considering (28) and (29), we have 𝜇𝑔𝑥¹−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝑡=𝜇¹−𝑔𝑔𝑥¹_𝑘+1+𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝑡≥𝜇¹−𝑔𝑔𝑥¹_𝑘+1,𝑡2∗𝜇𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝑡2=𝜇𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝑡2∗𝜇𝑔𝐹𝑥¹_𝑘,𝑥²_𝑘,…,𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝑡2=𝜇𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝑡2∗𝜇𝐹𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝑡2𝜈𝑔𝑥¹−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝑡=𝜈¹−𝑔𝑔𝑥¹_𝑘+1+𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝑡≤𝜈¹−𝑔𝑔𝑥¹_𝑘+1,𝑡2⋄𝜈𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝑡2=𝜈𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝑡2⋄𝜈𝑔𝐹𝑥¹_𝑘,𝑥²_𝑘,…,𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝑡2=𝜈𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝑡2⋄𝜈𝐹𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝑡2,𝜇𝑔𝑥²−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝑡=𝜇²−𝑔𝑔𝑥²_𝑘+1+𝑔𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝑡≥𝜇²−𝑔𝑔𝑥²_𝑘+1,𝑡2∗𝜇𝑔𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝑡2=𝜇𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝑡2∗𝜇𝑔𝐹𝑥²_𝑘,…,𝑥^𝑛_𝑘,𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝑡2=𝜇𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝑡2∗𝜇𝐹𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝑡2,𝜈𝑔𝑥²−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝑡=𝜈²−𝑔𝑔𝑥²_𝑘+1+𝑔𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝑡≤𝜈²−𝑔𝑔𝑥²_𝑘+1,𝑡2⋄𝜈𝑔𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝑡2=𝜈𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝑡2⋄𝜈𝑔𝐹𝑥²_𝑘,…,𝑥^𝑛_𝑘,𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝑡2=𝜈𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝑡2⋄𝜈𝐹𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝑡2,⋮𝜇𝑔𝑥^𝑛−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1𝑔𝑥,𝑡=𝜇^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1+𝑔𝑔𝑥^𝑛_𝑘+1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1𝑔𝑥,𝑡≥𝜇^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝑡2∗𝜇𝑔𝑔𝑥^𝑛_𝑘+1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1,𝑡2=𝜇𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝑡2∗𝜇𝑔𝐹𝑥^𝑛_𝑘,𝑥¹_𝑘,…,𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1,𝑡2=𝜇𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝑡2∗𝜇𝐹𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘,…,𝑔𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1,𝑡2,𝜈𝑔𝑥^𝑛−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1𝑔𝑥,𝑡=𝜈^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1+𝑔𝑔𝑥^𝑛_𝑘+1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1𝑔𝑥,𝑡≤𝜈^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝑡2⋄𝜈𝑔𝑔𝑥^𝑛_𝑘+1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1,𝑡2=𝜈𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝑡2⋄𝜈𝑔𝐹𝑥^𝑛_𝑘,𝑥¹_𝑘,…,𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1,𝑡2=𝜈𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝑡2⋄𝜈𝐹𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘,…,𝑔𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,…,𝑥^𝑛−1,𝑡2. (31)

By taking the limit as 𝑘→∞ in (31), we get 𝐹𝑥¹,𝑥²,𝑥³,…,𝑥^𝑛=𝑔𝑥¹,𝐹𝑥²,𝑥³,…,𝑥^𝑛,𝑥¹=𝑔𝑥²,⋮𝐹𝑥^𝑛,𝑥¹,𝑥²,…,𝑥^𝑛−1=𝑔𝑥^𝑛. (32)

Theorem 12. Let 𝐹∶𝑋^𝑛→𝑋 be map having mixed 𝑔-monotone property on the complete (𝑋,𝜇,𝜈,∗,⋄) having partial order relation denoted by ⪯. Also 𝐹(𝑋^𝑛)⊂𝑔(𝑋); 𝑔 is continuous and commutes with 𝐹. Suppose that 𝑋 has the following property: (a)if non decreasing sequence 𝑥−−−−→_𝑘^(𝜇,𝜈)𝑥, then 𝑥_𝑘⪯𝑥 for all 𝑘,(b)if non increasing sequence 𝑦−−−−→_𝑘^(𝜇,𝜈)𝑦, then 𝑦_𝑘≽𝑦 for all 𝑘.
Also, suppose that 𝐹∶𝑋^𝑛→𝑋 and 𝑔∶𝑋→𝑋 hold the following conditions, for all 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛,𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑛∈𝑋 and 𝛼∈(0,1): 𝜇𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛−𝐹𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑛𝑔𝑥,𝛼𝑡≥𝜇₁−𝑔𝑦₁,𝑡∗𝜇𝑔𝑥₂−𝑔𝑦₂𝑔𝑥,𝑡∗⋯∗𝜇_𝑛−𝑔𝑦_𝑛,𝜈𝐹𝑥,𝑡₁,𝑥₂,𝑥₃,…,𝑥_𝑛−𝐹𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑛𝑔𝑥,𝛼𝑡≤𝜈₁−𝑔𝑦₁,𝑡⋄𝜈𝑔𝑥₂−𝑔𝑦₂𝑔𝑥,𝑡⋄⋯⋄𝜈_𝑛−𝑔𝑦_𝑛,𝑡, (33)where 𝑔(𝑥_2𝑖−1)⪯𝑔(𝑦_2𝑖−1),  𝑖∈{1,2,…,(𝑛+1)/2}, and 𝑔(𝑥_2𝑖)≽𝑔(𝑦_2𝑖),  𝑖∈{1,2,…,𝑛/2}. If there exist 𝑥¹₀,𝑥²₀,𝑥³₀,…,𝑥^𝑛₀∈𝑋 such that𝑔𝑥¹₀⪯𝐹𝑥¹₀,𝑥²₀,𝑥³₀,…,𝑥^𝑛₀,𝑔𝑥²₀≽𝐹𝑥²₀,𝑥³₀,…,𝑥^𝑛₀,𝑥¹₀,⋮𝑔𝑥^𝑛₀⪯𝐹𝑥^𝑛₀,𝑥¹₀,𝑥²₀,…,𝑥₀^𝑛−1,𝑔𝑥𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑_𝑛0≽𝐹𝑥^𝑛₀,𝑥¹₀,𝑥²₀,…,𝑥₀^𝑛−1𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛, (34) then there exist 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛∈𝑋 such that 𝐹𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛=𝑔𝑥₁,𝐹𝑥₂,𝑥₃,…𝑥_𝑛,𝑥₁=𝑔𝑥₂,⋮𝐹𝑥_𝑛,𝑥₁,𝑥₂,…,𝑥_𝑛−1=𝑔𝑥_𝑛; (35) that is, 𝐹 and 𝑔 have an 𝑛-tuplet coincidence point.

Proof . Proof of the present theorem is also in four steps. However, three steps of poof are similar to Theorem 11. We now prove the last step. Considering the hypotheses (a)-(b) given in the theorem and 𝑔(𝑥¹_𝑘)−−−−→^(𝜇,𝜈)𝑥¹,𝑔(𝑥²_𝑘)−−−−→^(𝜇,𝜈)𝑥²,…,𝑔(𝑥^𝑛_𝑘)−−−−→^(𝜇,𝜈)𝑥^𝑛, we have 𝑔𝑥¹_𝑘⪯𝑥¹,𝑔𝑥²_𝑘≽𝑥²,⋮𝑔𝑥^𝑛_𝑘⪯𝑥^𝑛(if𝑛isodd),𝑔𝑥^𝑛_𝑘≽𝑥^𝑛(if𝑛iseven) (36) with (12) for all 𝑘. Due to intuitionistic fuzzy continuity of 𝑔, we write 𝑔𝑔𝑥−−−−¹_𝑘^𝜇,𝜈→𝑔𝑥¹,𝑔𝑔𝑥−−−−²_𝑘^𝜇,𝜈→𝑔𝑥²,⋮𝑔𝑔𝑥−−−−^𝑛_𝑘^𝜇,𝜈→𝑔𝑥^𝑛. (37) Then, by (6) and (7), we have 𝜇𝑔𝑥¹−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝛼𝑡=𝜇¹−𝑔𝑔𝑥¹_𝑘+1+𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝛼𝑡≥𝜇¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2∗𝜇𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝛼𝑡2=𝜇𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2∗𝜇𝑔𝐹𝑥¹_𝑘,𝑥²_𝑘,…,𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝛼𝑡2=𝜇𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2∗𝜇𝐹𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝛼𝑡2≥𝜇𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2∗𝜇𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡2∗𝜇𝑔𝑔𝑥²_𝑘−𝑔𝑥²,𝑡2⋯∗𝜇𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡2,𝜈𝑔𝑥¹−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝛼𝑡=𝜈¹−𝑔𝑔𝑥¹_𝑘+1+𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝑔𝑥,𝛼𝑡≤𝜈¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝛼𝑡2=𝜈𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝐹𝑥¹_𝑘,𝑥²_𝑘,…,𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝛼𝑡2=𝜈𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2⋄𝜈𝐹𝑔𝑥¹_𝑘,𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝛼𝑡2≤𝜈𝑔𝑥¹−𝑔𝑔𝑥¹_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡2⋄𝜈𝑔𝑔𝑥²_𝑘−𝑔𝑥²,𝑡2⋄⋯⋄𝜈𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡2 (38) by taking limit as 𝑘→∞ and using (37), we obtain the following results from calculation mentioned above: 𝜇𝑔𝑥¹−𝐹𝑥¹,𝑥²,…,𝑥^𝑛𝜈𝑔𝑥,𝛼𝑡=1,₁−𝐹𝑥¹,𝑥²,…,𝑥^𝑛,𝛼𝑡=0. (39) Hence, 𝑔(𝑥¹)=𝐹(𝑥¹,𝑥²,…,𝑥^𝑛).
In a similar way to the previous calculations, 𝜇𝑔𝑥²−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝛼𝑡=𝜇²−𝑔𝑔𝑥²_𝑘+1+𝑔𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝛼𝑡≥𝜇²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2∗𝜇𝑔𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝛼𝑡2=𝜇𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2∗𝜇𝑔𝐹𝑥²_𝑘,…,𝑥^𝑛_𝑘,𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝛼𝑡2=𝜇𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2∗𝜇𝐹𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝛼𝑡2≥𝜇𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2∗𝜇𝑔𝑔𝑥²_𝑘−𝑔𝑥²,𝑡2∗⋯∗𝜇𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡2∗𝜇𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡2,𝜈𝑔𝑥²−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝛼𝑡=𝜈²−𝑔𝑔𝑥²_𝑘+1+𝑔𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝑔𝑥,𝛼𝑡≤𝜈²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝑥²_𝑘+1−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝛼𝑡2=𝜈𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝐹𝑥²_𝑘,…,𝑥^𝑛_𝑘,𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝛼𝑡2=𝜈𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2⋄𝜈𝐹𝑔𝑥²_𝑘,…,𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝛼𝑡2≤𝜈𝑔𝑥²−𝑔𝑔𝑥²_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝑔𝑥²_𝑘−𝑔𝑥²,𝑡2⋄⋯⋄𝜈𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡2⋄𝜈𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡2 (40) by taking limit as 𝑘→∞ and using (37), we get the following equalities: 𝜇𝑔𝑥²−𝐹𝑥²,…,𝑥^𝑛,𝑥¹𝜈𝑔𝑥,𝛼𝑡=1,₂−𝐹𝑥²,…,𝑥^𝑛,𝑥¹,𝛼𝑡=0. (41) Hence, 𝐹(𝑥²,𝑥³,…,𝑥^𝑛,𝑥¹)=𝑔(𝑥²),….
We continue process 𝜇𝑔𝑥^𝑛−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1𝑔𝑥,𝛼𝑡=𝜇^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1+𝑔𝑔𝑥^𝑛_𝑘+1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1𝑔𝑥,𝛼𝑡≥𝜇^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2∗𝜇𝑔𝑔𝑥^𝑛_𝑘+1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1,𝛼𝑡2=𝜇𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2∗𝜇𝑔𝐹𝑥^𝑛_𝑘,𝑥¹_𝑘,…,𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1,𝛼𝑡2=𝜇𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2∗𝜇𝐹𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘,…,𝑔𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1,𝛼𝑡2≥𝜇𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2∗𝜇𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡2∗𝜇𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡2∗⋯∗𝜇𝑔𝑔𝑥_𝑘^𝑛−1−𝑔𝑥^𝑛−1,𝑡2,𝜈𝑔𝑥^𝑛−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1𝑔𝑥,𝛼𝑡=𝜈^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1+𝑔𝑔𝑥¹_𝑘+1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1𝑔𝑥,𝛼𝑡≤𝜈^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝑔𝑥^𝑛_𝑘+1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1,𝛼𝑡2=𝜈𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝐹𝑥^𝑛_𝑘,𝑥¹_𝑘,…,𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1,𝛼𝑡2=𝜈𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2⋄𝜈𝐹𝑔𝑥^𝑛_𝑘,𝑔𝑥¹_𝑘,…,𝑔𝑥_𝑘^𝑛−1−𝐹𝑥^𝑛,𝑥¹,𝑥²⋯,𝑥^𝑛−1,𝛼𝑡2≤𝜈𝑔𝑥^𝑛−𝑔𝑔𝑥^𝑛_𝑘+1,𝛼𝑡2⋄𝜈𝑔𝑔𝑥^𝑛_𝑘−𝑔𝑥^𝑛,𝑡2⋄𝜈𝑔𝑔𝑥¹_𝑘−𝑔𝑥¹,𝑡2⋄⋯⋄𝜈𝑔𝑔𝑥_𝑘^𝑛−1−𝑔𝑥^𝑛−1,𝑡2 (42) by taking limit as 𝑘→∞ and using (37), we obtain the last equalities. That is, 𝜇𝑔𝑥^𝑛−𝐹𝑥^𝑛,𝑥¹,𝑥²,…,𝑥^𝑛−1𝜈𝑔𝑥,𝛼𝑡=1,_𝑛−𝐹𝑥^𝑛,𝑥¹,𝑥²,…,𝑥^𝑛−1,𝛼𝑡=0. (43) Hence, 𝐹(𝑥^𝑛,𝑥¹,𝑥²,…,𝑥^𝑛−1)=𝑔(𝑥^𝑛).
Thus, we proved that 𝐹 and 𝑔 have an 𝑛-tuplet coincidence point.

Remark 13. Theorems 11 and 12 are restricted to Theorem 2.5 in [29] for 𝑛=2; it is restricted to Theorem 3.1 in [36] for 𝑛=2 and 𝑔=𝐼. For 𝑛=3, Theorems 11 and 12 are restricted to Theorem 2.1 in [30].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Yildiz Technical University Scientific Research Projects Coordination Unit under the Project no. BAPK 2012-07-03-DOP03.