# n-Tuplet Coincidence Point Theorems in Partially Ordered Probabilistic Metric Spaces

Authors: Arife Aysun Karaaslan; Vatan Karakaya
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, we investigate existence ofn-tuplet coincidence point theorems in partially ordered probabilistic metric spaces. Also, we gave uniqueness of n-tuplet fixed point theorems in this space.


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## Body

## 1. Introduction

Probabilistic metric spaces were introduced by Menger in his fundamental paper [1] in 1942. In Menger’s theory, the notion of distance between two points x and y is replaced by a distribution function Fx,y. The value of Fx,y at any point t is represented by Fx,y(t). Fx,y(t) is interpreted as probability that the distance between x and y is less than t. Sehgal first introduced the notion of contraction mapping on a probabilistic metric space in [2, 3]. In fact the study of such spaces received an impetus with the pioneering works of Schewizer and Sklar [4]. Then several authors have studied probabilistic spaces; see [5–9].Guo and Lakshmikantham initiated the concept of coupled fixed point [10] in 1987. After that, Bhaskar and Lakshmikantham introduced the notion of mixed monotone property and gave some coupled fixed point theorems in ordered metric spaces in 2006 [11]. In 2012, the extention of coupled fixed point theorems to tripled fixed point theorems for nonlinear mapping in partially ordered metric space was introduced by Berinde and Borcut [12]. Then, some coupled and tripled fixed point results were obtained by many authors [13–19]. In 2013, Ertürk and Karakaya gave the concept of n-tuplet fixed point theorems in partially ordered metric spaces [20]. Alam, Imdad, and Ali unified n-tuplet fixed point results in ordered metric space in 2016 [21]. Their survey article is recommended to someone who wants to have details about this theory.Hu and Ma studied couple coincidence point theorems in partially ordered probabilistic metric spaces in [22]. Recently, Binayak S. et al. [23] gave tripled coincidence point results in partially ordered probabilistic metric spaces.Inspired by the above studies, we introducen-tuplet fixed point theorems in partially ordered probabilistic metric spaces. This paper is organized as follows. Section 2 is devoted to giving some preliminaries. In Section 3, we obtain existence of n-tuplet fixed point theorems. Finally, Section 4 concerns the uniqueness of fixed point. These are the extensions of coupled and tripled fixed points in partially ordered probabilistic metric spaces.


## 2. Preliminaries

Definition 1 (see [4]).
A triangular normshorterΔ-norm/t-norm is a binary operation Δ on the interval [0,1], (1)Δ:0,1×0,1→0,1that satisfies the following conditions:(a)
Δa,1=a, Δ0,0=0,(b)
Δa,b=Δb,a,(c)
Δc,d≥Δa,b for c≥a, d≥b,(d)
For alla,b,c∈0,1, Δ(Δ(a,b),c)=Δ(a,Δ(b,c)).
Principal examples ofΔ-norms are (i)
ΔMa,b=min⁡a,b,(ii)
ΔPa,b=a.b,(iii)
ΔLa,b=max⁡a+b-1,0,(iv)
ΔD(a,b)=min⁡(a,b),ifmax(a,b)=10,otherwise.
AboveΔ-norms have the following relations: (2)ΔD<ΔL<ΔP<ΔM.Definition 2 (see [4]).
A functionF:R→R+ is called a distribution function if it is nondecreasing, left continuous with (3)inf⁡Ft:t∈R=0andsup⁡Ft:t∈R=1.In addition, if F(0)=0, then F is called a distance distribution function. We denote the set of all distance distribution functions by L+ and H is a specific distance distribution function (also known as Heaviside function) defined by(4)Ht=0,t≤01,t>0.Definition 3 (see [1]).
LetX be a nonempty set, F be a mapping defined on X×X into L+, and Δ be a t-norm. If the following conditions are satisfied, (X,F,Δ) is called Menger probabilistic metric spaces: (1)
Fx,y(t)=H(t) for all t>0 if and only if x=y;(2)
Fx,y(t)=Fy,x(t), for all t>0 and x,y∈X;(3)
Fx,y(t+s)≥Δ(Fx,z(t),Fz,y(s)),∀t,s≥0,x,y,z∈X.Example 4.
LetΔ be an arbitrary t-norm, X=-1,1, and(5)Fx,yt=e-x-y/t,t>00,t=0for every x,y∈X. Then (X,F,Δ) is a Menger Probabilistic Metric (for short PM) space given in [24].Definition 5 (see [4, 25]).
Let(X,F,Δ) be a Menger space.(i)
A sequence(xn) in X is said to be convergent to a point x∈X if, for every >0, limn→∞Fxn,x(t)=1.(ii)
A sequence(xn) in X is called Cauchy sequence if, for each 0<ε<1 and t>0, there exists n0∈NFxn,xm(t)≥1-ε for each n,m≥n0.(iii)
A Menger space in which every Cauchy sequence is convergent is said to be complete.Lemma 6 (see [26]).
If(X,F,Δ) be a Menger space where Δ is continuous t-norm, then for every fixed t>0, if (xn)→x,(yn)→y, then limn→∞Fxn,yn(t)=Fx,y(t).Lemma 7 (see [23]).
Ifh:Rn→R is continuous and (aα,β)α=1∞,β=1,2,…,n are such that liminfα→∞aα,γ=aα,γ for all γ≠m for some m and (aα,α)α=1∞ is bounded, then, liminfα→∞h(aα,1,aα,2,…,aα,n)=h(a1,a2,…,liminfα→∞aα,m,…,an).LetR+=[0,∞) and Φ=ϕ:ϕ:R+→R+. Each ϕ∈Φ satisfies the following conditions:(1)
ϕ is strict increasing,(2)
ϕ is upper semicontinuous from the right,(3)
∀t>0, ∑n=0∞‍ϕn(t)<∞.Ifϕ∈Φ,ϕ(t)≤t for all t>0.Lemma 8 (see [27]).
Let(un) be sequence in a Menger space (X,F,Δ), where Δ is a minimum t-norm. If there exists a function ϕ∈Φ∀t>0 and n≥1, (6)Fun,un+1ϕt≥min⁡Fun-1,unt,Fun,un+1t.Then (un) is a Cauchy sequence in X.Lemma 9 (see [27]).
Let(X,F,Δ) be a Menger space. If there exists a function ϕ∈Φ∀t>0 and x,y∈X,
F x , y ( ϕ ( t ) + 0 ) ≥ F x , y ( t ), then x=y.


## 3. Main Results

Definition 10 (see [20]).
Let(X,≤) be a partially ordered set and G:Xn→X. G is said to have mixed monotone property if G is monotone nondecreasing in its odd arguments and it is monotone nonincreasing in its even argument. That is, for any x1,x2,x3,…,xn∈X, (7)y1,z1∈X,y1≤z1⟹Gy1,x2,x3,…,xn≤Gz1,x2,x3,…,xny2,z2∈X,y2≤z2⟹Gx1,y2,x3,…,xn≥Gx1,z2,x3,…,xn⋮yn,zn∈X,yn≤zn⟹Gx1,x2,x3,…,yn≤Gx1,x2,x3,…,znifnoddyn,zn∈X,yn≤zn⟹Gx1,y2,x3,…,xn≥Gx1,x2,x3,…,znifneven.Definition 11 (see [20]).
Let(X,≤) be a partially ordered set and G:Xn→X,g:X→X. We say that G has the mixed g-monotone property if G is monotone g-nondecreasing in its odd arguments and it is monotone g-nonincreasing in its even argument. That is,
for anyx1,x2,x3,…,xn∈X(8)y1,z1∈X,gy1≤gz1⟹Gy1,x2,x3,…,xn≤Gz1,x2,x3,…,xny2,z2∈X,gy2≤gz2⟹Gx1,y2,x3,…,xn≥Gx1,z2,x3,…,xn⋮yn,zn∈X,gyn≤gzn⟹Gx1,x2,x3,…,yn≤Gx1,x2,x3,…,znifnoddyn,zn∈X,gyn≤gzn⟹Gx1,y2,x3,…,xn≥Gx1,x2,x3,…,znifneven.Ifg is taken identity mapping Definition 11 reduces to Definition 10.Definition 12 (see [28]).
LetX≠∅. An element x1,x2,x3,…,xn∈Xn is defined as a n-tuplet fixed point of the mapping G:Xn→X if (9)Gx1,x2,x3,…,xn=x1Gx2,x3,…,xn,x1=x2⋮Gxn,x1,x2,…,xn-1=xn.Definition 13 (see [28]).
LetX≠∅. An element x1,x2,x3,…,xn∈Xn is defined as a n-tuplet coincidence point of the mapping G:Xn→X and g:X→X if (10)Gx1,x2,x3,…,xn=gx1Gx2,x3,…,xn,x1=gx2⋮Gxn,x1,x2,…,xn-1=gxn.Wheng is taken identity mapping Definition 13 reduces to Definition 12.Definition 14 (see [20]).
Let(X,≤) be a partially ordered set and G:Xn→X and g:X→X is called commutative if for all x1,x2,x3,…,xn∈X(11)gGx1,x2,x3,…,xn=Ggx1,gx2,gx3,…,gxn.Definition 15.
Let(X,F,Δ) be a Menger space. G:Xn→X and g:X→X are said to be compatible if for all t>0(12)limk→∞⁡FgGxk1,…,xkn,Ggxk1,…,gxknt=1,limk→∞⁡FgGxk2,…,xkn,xk1,Ggxk1,…,gxkn,gxk1t=1,⋮limk→∞⁡FgGxkn,xk1,…,xkn-1,Ggxkn,gxk1,…,gxkn-1t=1,whenever (xk1),(xk2),…,(xkn) sequences in X such that(13)limk→∞⁡Gxk1,xk2,…,xkn=limk→∞⁡gxk1=x1,limk→∞⁡Gxk2,…,xkn,xk1=limk→∞⁡gxk2=x2,⋮limk→∞⁡Gxkn,xk1,…,xkn-1=limk→∞⁡gxkn=xn.Theorem 16.
Let(X,≤) be a partially ordered set and (X,F,Δ) be a complete Menger space, where Δ is a minimum t-norm. Let G:Xn→X,g:X→X be two mappings such that G has the mixed g-monotone property. Suppose there exist ϕ∈Φ and p≥0 such that(14)FGx1,x2,…,xn,Gy1,y2,…,ynϕt+p1-max⁡Fgx1,Gy1,y2,…,ynϕt,Fgy1,Gx1,x2,…,xnϕt≥min⁡Fgx1,gy1t,Fgx1,Gx1,x2,…,xnt,Fgy1,Gy1,y2,…,yntfor all t>0, x1,x2,…,xn,y1,y2,…,yn∈X with (15)gx1≥gy1gx2≤gy2⋮gxn≥gynifnisoddgxn≤gynifnisevenAssume that g is continuous, monotonic increasing, compatible with G such that G(Xn)⊆g(X) and suppose either(a)
G is continuous or(b)
X has the following property:
(i)
if nondecreasing sequence(xn)→x, then xn≤x for all n≥0,(ii)
if nondecreasing sequence(yn)→y, then yn≤y for all n≥0.
If there existx01,x02,…,x0n∈X such that(16)gx01≤Gx01,x02,…,x0ngx02≥Gx02,x03,…,x0n,x01⋮gx0n≤Gx0n,x01,x02,…,x0n-1ifnisoddgx0n≥Gx0n,x01,x02,…,x0n-1ifniseventhen there exist x1,x2,…,xn∈X such that (17)Gx1,x2,x3,…,xn=gx1,Gx2,x3,…,xn,x1=gx2,⋮Gxn,x1,x2,…,xn-1=gxn.That is, G and g have a n-tuplet coincidence point.Proof.
Letx01,x02,…,x0n∈X satisfy condition (16). Since G(Xn)⊂g(X), we can define (xk1),(xk2),…,(xkn)∈X sequences as follows:
fork=1,2,3,…,(18)gxk1=Gxk-11,xk-12,…,xk-1ngxk2=Gxk-12,xk-13,…,xk-1n,xk-11⋮gxkn=Gxk-1n,xk-11,…,xk-1n-1.Next step, we show that, for all k≥1,(19)gxk-11≤gxk1gxk-12≥gxk2⋮gxk-1n≤gxknifnisoddgxk-1n≥gxknifniseven.To prove this claim, we will use the inductive method for mathematics. Because of the inequalities in (16), (19) holds for k=1. We have (20)gx01≤Gx01,x02,…,x0n=gx11,gx02≥Gx02,x03,…,x0n,x01=gx12,⋮gx0n≤Gx0n,x01,x02,…,x0n-1=gx1nifnisodd,gx0n≥Gx0n,x01,x02,…,x0n-1=gx1nifniseven.Let us assume that (19) is true for k=m; that is,(21)gxm-11≤gxm1,gxm-12≥gxm2,⋮gxm-1n≤gxmnifnisoddgxm-1n≥gxmnifniseven.Since G has the mixed g-monotone property and from (21), (22)gxm1=Gxm-11,xm-12,…,xm-1n≤Gxm1,xm-12,…,xm-1n≤Gxm1,xm2,xm-13,…,xm-1n-1,xm-1n⋮≤Gxm1,xm2,…,xmn-1,xmn=gxm+11.Similar way we get (23)gxm2=Gxm-12,…,xm-1n,xm-11≥Gxm2,xm-13,…,xm-1n,xm-11≥Gxm2,xm3,xm-14,…,xm-1n,xm-11⋮≥Gxm2,xm3,…,xmn,xm1=gxm+12.⋮gxmn≤Gxmn,xm1,…,xmn-1=gxm+1nifnisoddgxmn≥Gxmn,xm1,…,xmn-1=gxm+1nifniseven.So, inequalities in (19) are true for all k≥1.(24)⋯≥gxk1≥gxk-11≥⋯≥gx11≥gx01⋯≤gxk2≤gxk-12≤⋯≤gx12≤gx02⋮⋯≥gxkn≥gxk-1n≥⋯≥gx1n≥gx0nifnisodd⋯≤gxkn≤gxk-1n≤⋯≤gx1n≤gx0nifniseven.Now, we will show that (g(xk1)),(g(xk2)),…,(g(xkn)) are Cauchy sequences.
For allt>0,n≥1, (25)Fgxk1,gxk+11ϕt=FGxk-11,xk-12,…,xk-1n,Gxk1,xk2,…,xknϕt≥min⁡Fgxk-11,gxk1t,Fgxk-11,Gxk-11,xk-12,…,xk-1nt,Fgxk1,Gxk1,xk2,…,xknt-p1-max⁡Fgxk-11,Gxk1,xk2,…,xknϕt,Fgxk1,Gxk-11,xk-12,…,xk-1nϕt=min⁡Fgxk-11,gxk1t,Fgxk-11,gxk1t,Fgxk1,gxk+11t-p1-max⁡Fgxk-11,gxk+11ϕt,Fgxk1,gxk1ϕtFgxk1,gxk+11ϕt≥min⁡Fgxk-11,gxk1t,Fgxk1,gxk+11t-p1-1Fgxk1,gxk+11ϕt≥minFgxk-11,gxk1t,Fgxk1,gxk+11t.By Lemma 8, we obtain that (g(xk1)) is a Cauchy sequence. (26)Fgxk2,gxk+12ϕt=FGxk-12,xk-13,…,xk-1n,xk-11,Gxk2,xk3,…,xkn,xk1ϕt≥min⁡Fgxk-12,gxk2t,Fgxk-12,Gxk-12,xk-13,…,xk-1n,xk-11t,Fgxk2,Gxk2,xk3,…,xkn,xk1t-p1-max⁡Fgxk-12,Gxk2,xk3,…,xkn,xk1ϕt,Fgxk2,Gxk-12,xk-13,…,xk-1n,xk-11ϕt=min⁡Fgxk-12,gxk2t,Fgxk-12,gxk2t,Fgxk2,gxk+12t-p1-max⁡Fgxk-12,gxk+12ϕt,Fgxk2,gxk2ϕtFgxk2,gxk+12ϕt≥min⁡Fgxk-12,gxk2t,Fgxk2,gxk+12t-p1-1Fgxk2,gxk+12ϕt≥minFgxk-12,gxk2t,Fgxk2,gxk+12t.We obtain that (g(xk2)) is a Cauchy sequence.
Using same way we conclude that(g(xk3)),…,(g(xkn)) are also Cauchy sequences. Since X is complete metric space, there exist x1,x2,…,xn∈X such that(27)limk→∞⁡gxk1=x1,limk→∞⁡gxk2=x2,⋮limk→∞⁡gxkn=xn.Since g is continuous, we can write(28)limk→∞⁡ggxk1=gx1,limk→∞⁡ggxk2=gx2,⋮limk→∞⁡ggxkn=gxn.As (G,g) is compatible and g is continuous,(29)gx1=limk→∞⁡ggxk+11=limk→∞⁡gGxk1,xk2,…,xkn=limk→∞⁡Ggxk1,gxk2,…,gxkn,gx2=limk→∞⁡ggxk+12=limk→∞⁡gGxk2,…,xkn,xk1=limk→∞⁡Ggxk2,…,gxkn,gxk1,⋮gxn=limk→∞⁡ggxk+1n=limk→∞⁡gGxkn,xk1,…,xkn-1=limk→∞⁡Ggxkn,gxk1,…,gxkn-1.We will indicate that (30)Gx1,x2,…,xn=gx1,Gx2,x3,…,xn,x1=gx2,⋮Gxn,x1,…,xn-1=gxn.Suppose that (a) holds. From (27), (28), and (29), (31)gx1=limk→∞⁡Ggxk1,gxk2,…,gxkn=Glimk→∞⁡gxk1,limk→∞⁡gxk2,…,limk→∞⁡gxkn=Gx1,x2,…,xngx2=limk→∞⁡Ggxk2,gxk3,…,gxkn,gxk1=Glimk→∞⁡gxk2,limk→∞⁡gxk3,…,limk→∞⁡gxkn,limk→∞⁡gxk1=Gx2,x3,…,xn,x1⋮gxn=limk→∞⁡Ggxkn,gxk1,…,gxkn-1=Glimk→∞⁡gxkn,limk→∞⁡gxk1,…,limk→∞⁡gxkn-1=Gxn,x1,…,xn-1.Now assume that (b) holds. From (19) and (29), we have for all k(32)gxk1≤x1gxk2≥x2⋮gxkn≤xnifnisoddgxkn≥xnifniseven.For all t>0,0<λ<1, we have (33)Fgx1,Gx1,x2,…,xnϕt≥ΔFgx1,ggxk+11ϕt-ϕλt,Fggxk+11,Gx1,x2,…,xnϕλt.If g(x1)=limk→∞g(g(xk+11)), limk→∞⁡Fg(x1),g(g(xk+11))(t)=1.
When we apply limit to both parts of above inequality, we get(34)Fgx1,Gx1,x2,…,xnϕt≥limk→∞⁡inf⁡ΔFgx1,ggxk+11ϕt-ϕλt,Fggxk+11,Gx1,x2,…,xnϕλt=Δlimk→∞⁡Fgx1,ggxk+11ϕt-ϕλt,limk→∞⁡inf⁡Fggxk+11,Gx1,x2,…,xnϕλt=min⁡1,limk→∞⁡inf⁡FGgxk1,gxk2,…,gxk1,Gx1,x2,…,xnϕλt=limk→∞⁡inf⁡FGgxk1,gxk2,…,gxk1,Gx1,x2,…,xnϕλt≥limk→∞⁡inf⁡minFggxk1,gx1λt,Fggxk1,Ggxk1,gxk2,…,gxknλt,Fgx1,Gx1,x2,…,xnλt-p1-maxFggxk1,Gx1,x2,…,xnϕλt,Fgx1,Ggxk1,gxk2,…,gxknϕλt=minlimk→∞⁡Fggxk1,gx1λt,limk→∞⁡inf⁡Fggxk1,Ggxk1,gxk2,…,gxknλt,Fgx1,Gx1,x2,…,xnλt-p1-maxlimk→∞⁡Fggxk1,Gx1,x2,…,xnϕλt,limk→∞⁡Fgx1,Ggxk1,gxk2,…,gxknϕλt=min⁡Fgx1,gx1λt,Fgx1,gx1λt,Fgx1,Gx1,x2,…,xnλt-p1-max⁡Fgx1,Gx1,x2,…,xnϕλt,Fgx1,gx1ϕλt≥min⁡1,Fgx1,Gx1,x2,…,xnλt-p1-1≥Fgx1,Gx1,x2,…,xnλt.Because λ is arbitrary 0<λ<1, by taking λ→1, and left continuous property of distribution function F, we have (35)Fgx1,Gx1,x2,…,xnϕt≥Fgx1,Gx1,x2,…,xnt.ϕ is increasing and also is monotone increasing. Accordingly, (36)ϕt+0≥ϕt⟹Fgx1,Gx1,x2,…,xnϕt+0≥Fgx1,Gx1,x2,…,xnϕt≥Fgx1,Gx1,x2,…,xnt.From Lemma 9, g(x1)=G(x1,x2,…,xn).
By the same way, we can find(37)gx2=Gx2,…,xn,x1,⋮gxn=Gxn,x1,…,xn-1.Consequently, g and G have n-tuplet coincidence point in X.Takingp=0 in Theorem 16, we obtain Corollary 17.Corollary 17.
Let(X,≤) be a partially ordered set and (X,F,Δ) be a complete Menger space, where Δ is minimum t-norm. Let G:Xn→X,g:X→X be two mappings such that G has the mixed g-monotone property. Let ϕ∈Φ exist such that(38)FGx1,x2,…,xn,Gy1,y2,…,ynϕt≥min⁡Fgx1,gy1t,Fgx1,Gx1,x2,…,xnt,Fgy1,Gy1,y2,…,yntfor all t>0, x1,x2,…,xn,y1,y2,…,yn∈X with (39)gx1≥gy1gx2≤gy2⋮gxn≥gynifnisoddgxn≤gynifniseven.Also g is continuous, monotonic increasing, compatible with G and G(Xn)⊆g(X). And suppose either(a)
G is continuous or(b)
X has the following properties:
(i)
if a nondecreasing sequence(xn)→x, then xn≤x∀n≥0,(ii)
if a nonincreasing sequence(yn)→y, then yn≤y∀n≥0.
If there arex01,x02,…,x0n∈X such that (40)gx01≤Gx01,x02,…,x0ngx02≥Gx02,x03,…,x0n,x01⋮gx0n≤Gx0n,x01,…,x0n-1ifnisoddgx0n≥Gx0n,x01,…,x0n-1ifniseven,
theng and G have n-tuplet coincidence point in X.
That is, there existx1,x2,…,xn∈X such that (41)Gx1,x2,x3,…,xn=gx1Gx2,x3,…,xn,x1=gx2⋮Gxn,x1,x2,…,xn-1=gxn.When we takeϕ(t)=ct, c∈(0,1) in Theorem 16, we obtain Corollary 18.Corollary 18.
Let(X,≤) be a partially ordered set and (X,F,Δ) be a complete Menger space, where Δ is minimum t-norm. Let G:Xn→X,g:X→X be two mappings such that G has the mixed g-monotone property. Let ϕ∈Φ and p≥0 exist such that(42)FGx1,x2,…,xn,Gy1,y2,…,ynct+p1-max⁡Fgx1,Gy1,y2,…,ynct,Fgy1,Gx1,x2,…,xnct≥min⁡Fgx1,gy1t,Fgx1,Gx1,x2,…,xnt,Fgy1,Gy1,y2,…,yntfor all t>0, x1,x2,…,xn,y1,y2,…,yn∈X with (43)gx1≥gy1gx2≤gy2⋮gxn≥gynifnisoddgxn≤gynifniseven.
Alsog is continuous, monotonic increasing, compatible with G and G(Xn)⊆g(X). And suppose either(a)
G is continuous or(b)
X has the following properties:
(i)
if a nondecreasing sequence(xn)→x, then xn≤x∀n≥0,(ii)
if a nonincreasing sequence(yn)→y, then yn≤y∀n≥0.
If there arex01,x02,…,x0n∈X such that (44)gx01≤Gx01,x02,…,x0ngx02≥Gx02,x03,…,x0n,x01⋮gx0n≤Gx0n,x01,…,x0n-1ifnisoddgx0n≥Gx0n,x01,…,x0n-1ifniseven,then g and G have n-tuplet coincidence point in X. That is, there exist x1,x2,…,xn∈X such that (45)Gx1,x2,x3,…,xn=gx1Gx2,x3,…,xn,x1=gx2⋮Gxn,x1,x2,…,xn-1=gxn.


## 4. Uniqueness ofn-Tuplet Fixed Point

For all(x1,x2,…,xn),(x1∗,x2∗,…,xn∗)∈Xn,(46)x1,x2,…,xn≤x1∗,x2∗,…,xn∗⇔x1≤x1∗,x2≥x2∗,⋮xn≤xn∗ifnisodd,xn≥xn∗ifniseven.(x1,x2,…,xn) is equal to (x1∗,x2∗,…,xn∗) iff x1=x1∗,x2=x2∗,…,xn=xn∗.Theorem 19.
In addition to the hypotheses of Theorem16, suppose that for all (x1,x2,…,xn),(x1∗,x2∗,…,xn∗)∈Xn there exist (y1,y2,…,yn)∈Xn such that (47)Gy1,y2,…,yn,Gy2,…,yn,y1,…,Gyn,y1,⋯,yn-1is comparable to (48)Gx1,x2,…,xn,Gx2,…,xn,x1,…,Gxn,x1,…,xn-1and (49)Gx1∗,x2∗,…,xn∗,Gx2∗,…,xn∗,x1∗,…,Gxn∗,x1∗,…,xn-1∗.Then g and G have a unique n-tuplet common fixed point; that is, there exist (x1,x2,…,xn)∈Xn such that (50)x1=gx1=Gx1,x2,x3,…,xn,x2=gx2=Gx2,x3,…,xn,x1,⋮xn=gxn=Gxn,x1,x2,…,xn-1.Proof.
The set ofn-tuplet coincidence points is nonempty due to Theorem 16. We shall show that if (x1,x2,…,xn) and (x1∗,x2∗,…,xn∗) are n-tuplet coincidence points, that is, if (51)gx1=Gx1,x2,…,xn,gx2=Gx2,…,xn,x1,⋮gxn=Gxn,x1,…,xn-1and (52)gx1∗=Gx1∗,x2∗,…,xn∗,gx2∗=Gx2∗,…,xn∗,x1∗,⋮gxn∗=Gxn∗,x1∗,…,xn-1∗,then(53)gx1=gx1∗,gx2=gx2∗,⋮gxn=gxn∗.By assumption there is (y1,y2,…,yn)∈Xn such that(54)Gy1,y2,…,yn,Gy2,…,yn,y1,…,Gyn,y1,…,yn-1is comparable to(55)Gx1,x2,…,xn,Gx2,…,xn,x1,…,Gxn,x1,…,xn-1and(56)Gx1∗,x2∗,…,xn∗,Gx2∗,…,xn∗,x1∗,…,Gxn∗,x1∗,…,xn-1∗.We can determine sequences (g(yk1)),(g(yk2)),…,(g(ykn)) such that y1=y01,y2=y02,…,yn=y0n and (57)gyk1=Gyk-11,yk-12,…,yk-1n,gyk2=Gyk-12,…,yk-1n,yk-11,⋮gykn=Gyk-1n,yk-11,…,yk-1n-1.From comparability of (56) and (55) with (54), we suppose that (58)gx1,gx2,…,gxn≥gy1,gy2,…,gyn=gy01,gy02,…,gy0n.Using (24), we have (59)gx1,gx2,…,gxn≥gyk1,gyk2,…,gyknfor  allk.Also from (46), we get(60)gx1≥gyk1,gx2≤gyk2,⋮gxn≥gyknifnisodd,gxn≤gyknifniseven.Fgx1,gyk+11ϕt=FGx1,x2,…,xn,Gyk1,yk2,…,yknϕt≥min⁡Fgx1,gyk1t,Fgx1,Gx1,x2,…,xnt,Fgyk1,Gyk1,yk2,…,yknt-p1-max⁡Fgx1,Gyk1,yk2,…,yknϕt,Fgyk1,Gx1,x2,…,xnϕt=min⁡Fgx1,gyk1t,Fgx1,gx1t,Fgyk1,gyk+11t-p1-max⁡Fgx1,gyk+11ϕt,Fgyk1,gx1ϕt≥min⁡Fgx1,gyk1t,Fgyk1,gyk+11t,Fgx2,gyk+12ϕt=FGx2,x3,…,xn,x1,Gyk2,yk3,…,ykn,yk1ϕt≥min⁡Fgx2,gyk2t,Fgx2,Gx2,x3,…,xn,x1t,Fgyk2,Gyk2,yk3,…,ykn,yk1t-p1-max⁡Fgx2,Gyk2,…,ykn,yk1ϕt,Fgyk2,Gx2,…,xn,x1ϕt=min⁡Fgx2,gyk2t,Fgx2,gx2t,Fgyk2,gyk+12t-p1-max⁡Fgx2,gyk+12ϕt,Fgyk2,gx2ϕt≥min⁡Fgx2,gyk2t,Fgyk2,gyk+12t,⋮Fgxn,gyk+1nϕt=FGxn,x1,…,xn-1,Gykn,yk1,…,ykn-1ϕt≥min⁡Fgxn,gyknt,Fgxn,Gxn,x1,…,xn-1t,Fgykn,Gykn,yk1,…,ykn-1t-p1-max⁡Fgxn,Gykn,yk1,…,ykn-1ϕt,Fgykn,Gxn,x1,…,xn-1ϕt=min⁡Fgxn,gyknt,Fgxn,gxnt,Fgykn,gyk+1nt-p1-max⁡Fgxn,gyk+1nϕt,Fgykn,gxnϕt≥min⁡Fgxn,gyknt,Fgykn,gyk+1ntfor each k≥1. If we take lower limit when k→∞ from Lemmas 8 and 9, we have(61)limk→∞⁡gyk+11=gx1,limk→∞⁡gyk+12=gx2,⋮limk→∞⁡gyk+1n=gxn.Likewise, one can show that(62)limk→∞⁡gyk+11=gx1∗,limk→∞⁡gyk+12=gx2∗,⋮limk→∞⁡gyk+1n=gxn∗.Using (61), (62), and triangle inequality property, (63)Fgx1,gx1∗t≥ΔFgx1,gyk+11t2,Fgyk+11,gx1∗t2⟶1k⟶∞,Fgx2,gx2∗t≥ΔFgx2,gyk+12t2,Fgyk+12,gx2∗t2⟶1k⟶∞,⋮Fgxn,gxn∗t≥ΔFgxn,gyk+1nt2,Fgyk+1n,gxn∗t2⟶1k⟶∞.So, we have
g ( x 1 ) = g ( x 1 ∗ ) , g ( x 2 ) = g ( x 2 ∗ ) , … , g ( x n ) = g ( x n ∗ ) . Thus, we showed (53).
Using commutativity ofG and g,(64)ggx1=gGx1,x2,…,xn=Ggx1,gx2,…,gxn,ggx2=gGx2,…,xn,x1=Ggx2,…,gxn,gx1,⋮ggxn=gGxn,x1,…,xn-1=Ggxn,gx1,…,gxn-1.Indicate g(x1)=z1,g(x2)=z2,…,g(xn)=zn. From (64),(65)gz1=Gz1,z2,…,zn,gz2=Gz2,…,zn,z1,⋮gzn=Gzn,z1,…,zn-1.Therefore, (z1,z2,…,zn) is a n-tuplet coincidence point. Then from the assumption with x1∗=z1,…,xn∗=zn it follows g(z1)=g(x1),g(z2)=g(x2),…,g(zn)=g(xn); that is,(66)gz1=z1,gz2=z2,⋮gzn=zn.By (65) and (66), (67)z1=gz1=Gz1,z2,…,zn,z2=gz2=Gz2,…,zn,z1,⋮zn=gzn=Gzn,z1,…,zn-1.Hence, (z1,z2,…,zn) is a n-tuplet common fixed point of G and g. To prove the uniqueness, assume that (q1,q2,…,qn) is another n-tuplet common fixed point. Then by assumption we have (68)q1=gq1=gz1,q2=gq2=gz2,⋮qn=gqn=gzn.



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