# Common Coupled Fixed-Point Theorems in Generalized Fuzzy Metric Spaces

Authors: K. P. R. Rao; I. Altun; S. Hima Bindu
Journal: Advances in Fuzzy Systems
 (2011)

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## Abstract

We prove two unique common coupled fixed-point theorems for self maps in symmetricG-fuzzy metric spaces.


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## Body

## 1. Introduction and Preliminaries

Mustafa and Sims [1–3] and Naidu et al. [4] demonstrated that most of the claims concerning the fundamental topological structure of D-metric introduced by Dhage [5–8] and hence all theorems are incorrect. Alternatively, Mustafa and Sims [1, 2] introduced a G-metric space and obtained some fixed-point theorems in it. Some interesting references in G-metric spaces are [3, 9–15]. In this paper, we prove two unique common coupled fixed-point theorems for Jungck type and for three mappings in symmetric G-fuzzy metric spaces.Before giving our main results, we recall some of the basic concepts and results inG-metric spaces and G-fuzzy metric spaces.Definition 1 (see [2]).
LetX be a nonempty set and let G:X×X×X→[0,∞) be a function satisfying the following properties:(G1)
G(x,y,z)=0 if x=y=z,(G2)
0<G(x,x,y) for all x,y∈X with x≠y,(G3)
G(x,x,y)≤G(x,y,z) for all x,y,z∈X with y≠z,(G4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯, symmetry in all three variables,(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z) for all x,y,z,a∈X.
Then, the functionG is called a generalized metric or a G-metric on X and the pair (X,G) is called a G-metric space.Definition 2 (see [2]).
TheG-metric space (X,G) is called symmetric if G(x,x,y)=G(x,y,y) for all x,y∈X.Definition 3 (see [2]).
Let(X,G) be a G-metric space and let {xn} be a sequence in X. A point x∈X is said to be limit of {xn} if and only if limn,m→∞G(x,xn,xm)=0. In this case, the sequence {xn} is said to be G-convergent to x.Definition 4 (see [2]).
Let(X,G) be a G-metric space and let {xn} be a sequence in X. {xn} is called G-Cauchy if and only if liml,n,m→∞G(xl,xn,xm)=0. (X,G) is called G-complete if every G-Cauchy sequence in (X,G) is G-convergent in (X,G).Proposition 5 (see [2]).
In aG-metric space (X,G), the following are equivalent.(i)
The sequence{xn} is G-Cauchy.(ii)
For everyϵ>0,there existsN∈N such that G(xn,xm,xm)<ϵ, for all n,m≥N.Proposition 6 (see [2]).
Let(X,G) be a G-metric space. Then, the function G(x,y,z) is jointly continuous in all three of its variables.Proposition 7 (see [2]).
Let(X,G) be a G-metric space. Then, for any x,y,z,a∈X, it follows that(i)
ifG(x,y,z)=0, then x=y=z,(ii)
G(x,y,z)≤G(x,x,y)+G(x,x,z),(iii)
G(x,y,y)≤2G(x,x,y),(iv)
G(x,y,z)≤G(x,a,z)+G(a,y,z),(v)
G(x,y,z)≤(2/3)[G(x,a,a)+G(y,a,a)+G(z,a,a)].Proposition 8 (see [2]).
Let(X,G) be a G-metric space. Then, for a sequence {xn}⊆X and a point x∈X, the following are equivalent:(i)
{xn} is G-convergent to x,(ii)
G(xn,xn,x)→0 as n→∞,(iii)
G(xn,x,x)→0 as n→∞,(iv)
G(xm,xn,x)→0 as m,n→∞.Recently, Sun and Yang [16] introduced the concept of G-fuzzy metric spaces and proved two common fixed-point theorems for four mappings.Definition 9 (see [16]).
A 3-tuple(X,G,*) is called a G-fuzzy metric space if X is an arbitrary nonempty set, * is a continuous t-norm, and G is a fuzzy set on X3×(0,∞) satisfying the following conditions for each t,s>0:(i)
G(x,x,y,t)>0 for all x,y∈X with x≠y,(ii)
G(x,x,y,t)≥G(x,y,z,t) for all x,y,z∈X with y≠z,(iii)
G(x,y,z,t)=1 if and only if x=y=z,(iv)
G(x,y,z,t)=G(p(x,y,z),t), where p is a permutation function,(v)
G(x,y,z,t+s)≥G(a,y,z,t)*G(x,a,a,s) for all x,y,z,a∈X,(vi)
G(x,y,z,·):(0,∞)→[0,1] is continuous.Definition 10 (see [16]).
AG-fuzzy metric space (X,G,*) is said to be symmetric if G(x,x,y,t)=G(x,y,y,t) for all x,y∈X and for each t>0.Example 11.
LetX be a nonempty set and let G be a G-metric on X. Denote a*b=ab for all a,b∈[0,1]. For each t>0, G(x,y,z,t)=t/(t+G(x,y,z)) is a G-fuzzy metric on X.Let(X,G,*) be a G-fuzzy metric space. For t>0,0<r<1, and x∈X, the set BG(x,r,t)={y∈X:G(x,y,y,t)>1-r} is called an open ball with center x and radius r.A subsetA of X is called an open set if for each x∈X,  there exist t>0 and 0<r<1 such that BG(x,r,t)⊆A.A sequence{xn} in G-fuzzy metric space X is said to be G-convergent to x∈X if G(xn,xn,x,t)→1 as n→∞ for each t>0. It is called a G-Cauchy sequence if G(xn,xn,xm,t)→1 as n,m→∞ for each t>0. X is called G-complete if every G-Cauchy sequence in X is G-convergent in X.Lemma 12 (see [16]).
Let(X,G,*) be a G-fuzzy metric space. Then, G(x,y,z,t) is nondecreasing with respect to t for all x,y,z∈X.Lemma 13 (see [16]).
Let(X,G,*) be a G-fuzzy metric space. Then, G is a continuous function on X3×(0,∞).Now onwards, we assume the following condition:Plim⁡t→∞G(x,y,z,t)=1∀x,y,z∈X.
Using (P), one can prove the following lemma.Lemma 14.
Let(X,G,*) be a G-fuzzy metric space. If there exists k∈(0,1) such that
(1)min⁡{G(x,y,z,kt),G(u,v,w,kt)}≥min⁡{G(x,y,z,t),G(u,v,w,t)}
for all x,y,z,u,v,w∈X and t>0, then x=y=z and u=v=w.Definition 15 (see [17]).
LetX be a nonempty set. An element (x,y)∈X×X is called a coupled fixed point of the mapping F:X×X→X if x=F(x,y) and y=F(y,x).Definition 16 (see [18]).
LetX be a nonempty set. An element (x,y)∈X×X is called(i)
a coupled coincidence point ofF:X×X→X and g:X→X if gx=F(x,y) and gy=F(y,x),(ii)
a common coupled fixed point ofF:X×X→X and g:X→X if x=gx=F(x,y) and y=gy=F(y,x).Definition 17 (see [18]).
LetX be a nonempty set. The mappings F:X×X→X and g:X→X are called W-compatible if g(F(x,y))=F(gx,gy) and g(F(y,x))=F(gy,gx) whenever gx=F(x,y) and gy=F(y,x) for some (x,y)∈X×X.Now, we give our main results.


## 2. Main Results

Theorem 18.
Let(X,G,*) be a G-fuzzy metric space with a*b=min{a,b} for all a,b∈[0,1] and S:X×X→X and let f:X→X be mappings satisfying
(2)G(S(x,y),S(u,v),S(u,v),kt)≥min⁡{G(fx,fu,fu,t),G(fy,fv,fv,t)}forallx,y,u,v∈X, where 0≤k<1,
(3)S(X×X)⊆f(X)andf(X)isacompletesubspaceofX,thepair(f,S)isW-compatible.
ThenS and f have a unique common coupled fixed point of the form (α,α) in X×X.Proof.
Letx0,y0∈X and denote zn=S(xn,yn)=fxn+1,pn=S(yn,xn)=fyn+1,n=0,1,2,…. Let dn(t)=G(zn,zn+1,zn+1,t), en(t)=G(pn,pn+1,pn+1,t). From (2), we have
(4)dn+1(kt)=G(zn+1,zn+2,zn+2,kt)=G(S(xn+1,yn+1),S(xn+2,yn+2),S(xn+2,yn+2),kt)≥min⁡{G(zn,zn+1,zn+1,t),G(pn,pn+1,pn+1,t)}≥min⁡{dn(t),en(t)}.
Also,
(5)en+1(kt)=G(pn+1,pn+2,pn+2,kt)=G(S(yn+1,xn+1),S(yn+2,xn+2),S(yn+2,xn+2),kt)≥min⁡{G(pn,pn+1,pn+1,t),G(zn,zn+1,zn+1,t)}≥min⁡{en(t),dn(t)}.
Thus, min{dn+1(kt),en+1(kt)}≥min{dn(t),en(t)}. Hence,
(6)min⁡{dn(t),en(t)}≥min⁡{dn-1(tk),en-1(tk)}≥min⁡{dn-2(tk2),en-2(tk2)}⋮≥min⁡{d0(tkn),e0(tkn)}=min⁡{G(z0,z1,z1,tkn),G(p0,p1,p1,tkn)}.
For any positive integer n and fixed positive integer p, we have
(7)G(zn,zn+p,zn+p,t)≥G(zn+p-1,zn+p,zn+p,tp)*G(zn+p-2,zn+p-1,zn+p-1,tp)*⋯*G(zn,zn+1,zn+1,tp)≥min⁡{G(z0,z1,z1,tpkn+p-1),G(p0,p1,p1,tpkn+p-1)}*min⁡⁡{G(z0,z1,z1,tpkn+p-2),G(p0,p1,p1,tpkn+p-2)}*⋯*min⁡⁡{G(z0,z1,z1,tpkn),G(p0,p1,p1,tpkn)}.
Letting n→∞ and using (P), we get
(8)lim⁡n→∞G(zn,zn+p,zn+p,t)≥1*1*⋯*1=1.
Hence, limn→∞G(zn,zn+p,zn+p,t)=1. Thus, {zn} is G-Cauchy in X. Similarly, we can show that {pn} is G-Cauchy in X. Since f(X) is G-complete, {zn} and {pn} converge to some α and β in f(X), respectively. Hence, there exist x and y in X such that α=fx,β=fy:
(9)G(zn,S(x,y),S(x,y),kt)=G(S(xn,yn),S(x,y),S(x,y),kt)≥min⁡{G(zn-1,fx,fx,t),G(pn-1,fy,fy,t)}.
Letting n→∞, we get
(10)G(fx,S(x,y),S(x,y),kt)≥min⁡{1,1}=1.
Hence, S(x,y)=fx. Similarly, it can be shown that S(y,x)=fy. Since (f,S) is W-compatible, we have
(11)fα=ffx=f(S(x,y))=S(fx,fy)=S(α,β),fβ=ffy=f(S(y,x))=S(fy,fx)=S(β,α).G(zn,fα,fα,kt)=G(S(xn,yn),S(α,β),S(α,β),kt)≥min⁡{G(zn-1,fα,fα,t),G(pn-1,fβ,fβ,t)}.
Letting n→∞, we get
(12)G(α,fα,fα,kt)≥min⁡{G(α,fα,fα,t),G(β,fβ,fβ,t)}.
Similarly, we can show that
(13)G(β,fβ,fβ,kt)≥min⁡{G(α,fα,fα,t),G(β,fβ,fβ,t)}.
Thus,
(14)min⁡{G(α,fα,fα,kt),G(β,fβ,fβ,kt)}≥min⁡{G(α,fα,fα,t),G(β,fβ,fβ,t)}.
From Lemma 14, we have fα=α and fβ=β. Thus, α=fα=S(α,β) and β=fβ=S(β,α). Hence, (α,β) is a common coupled fixed point of S and f.
Suppose(α1,β1) is another common coupled fixed point of S and f:(15)G(α,α1,α1,kt)=G(S(α,β),S(α1,β1),S(α1,β1),kt)≥min⁡{G(α,α1,α1,t),G(β,β1,β1,t)}.
Similarly,
(16)G(β,β1,β1,kt)=G(S(β,α),S(β1,α1),S(β1,α1),kt)≥min⁡{G(α,α1,α1,t),G(β,β1,β1,t)}.
Thus,
(17)min⁡{G(α,α1,α1,kt),G(β,β1,β1,kt)}≥min⁡{G(α,α1,α1,t),G(β,β1,β1,t)}.
From Lemma 14, α1=α and β1=β. Thus, (α,β) is the unique common coupled fixed point of S and f. Now, we will show that α=β:
(18)G(α,α,β,kt)=G(S(α,β),S(α,β),S(β,α),kt)≥min⁡{G(α,α,β,t),G(β,β,α,t)},G(α,β,β,kt)=G(S(α,β),S(β,α),S(β,α),kt)≥min⁡{G(α,β,β,t),G(β,α,α,t)}.
Thus,
(19)min⁡{G(α,α,β,kt),G(α,β,β,kt)}≥min⁡{G(α,α,β,t),G(α,β,β,t)}.
From Lemma 14, we have α=β. Thus, α is a common fixed point of S and f, that is, α=fα=S(α,α). Suppose α1 is another common fixed point of S and f:
(20)G(α1,α,α,t)=G(S(α1,α1),S(α,α),S(α,α),t)≥min⁡{G(α1,α,α,tk),G(α1,α,α,tk)}≥G(α1,α,α,tk2)⋮≥G(α1,α,α,tkn)⟶1asn⟶∞.
Hence, α1=α. Thus, S and f have a unique common coupled fixed point of the form (α,α).Finally, we prove a common coupled fixed-point theorem for three mappings in symmetricG-fuzzy metric spaces.Theorem 19.
Let(X,G,*) be a symmetric G-complete fuzzy metric space with a*b=min{a,b} for all a,b∈[0,1] and let S,T,R:X×X→X be mappings satisfying
(21)G(S(x,y),T(u,v),R(p,q),kt)≥min⁡{G(x,u,p,t),G(y,v,q,t),G(x,x,S(x,y),t),G(u,u,T(u,v),t),G(p,p,R(p,q),t)}forallx,y,u,v,p,q∈X, where 0≤k<1. Then, there exists (x,y)∈X×X such that
(22)x=S(x,y)=T(x,y)=R(x,y),(23)y=S(y,x)=T(y,x)=R(y,x).
Or
(24)S,T,andRhaveauniquecommoncoupledfixedpointoftheform(x,x)inX×X.Proof.
Letx0,y0∈X. Define the sequences {xn} and {yn} in X as follows: x3n+1=S(x3n,y3n),y3n+1=S(y3n,x3n); x3n+2=T(x3n+1,y3n+1), y3n+2=T(y3n+1,x3n+1); x3n+3=R(x3n+2,y3n+2), y3n+3=R(y3n+2,x3n+2), n=0,1,2,…. Suppose x3n+1=x3n for some n. Then, S(x,y)=x, where x=x3n,y=y3n. Suppose T(x,y)≠R(x,y). Then,
(25)G(x,T(x,y),R(x,y),kt)=G(S(x,y),T(x,y),R(x,y),kt)≥min⁡{1,1,1,G(x,x,T(x,y),t),G(x,x,R(x,y),t)}≥G(x,T(x,y),R(x,y),t).
It is a contradiction. Hence, T(x,y)=R(x,y). From (25) and since X is symmetric,
(26)G(x,T(x,y),T(x,y),kt)≥G(x,x,T(x,y),t)=G(x,T(x,y),T(x,y),t).
From Lemma 14, we have T(x,y)=x. Thus, S(x,y)=T(x,y)=R(x,y)=x. Similarly, if x3n+1=x3n+2 or x3n+2=x3n+3, then also we can show that S(x,y)=T(x,y)=R(x,y)=x for some x, y in X. Similarly, it can be shown that if y3n=y3n+1 or y3n+1=y3n+2 or y3n+2=y3n+3 then there exists (x,y)∈X×X such that
(27)S(y,x)=T(y,x)=R(y,x)=y.
Now, assume that xn≠xn+1 and yn≠yn+1 for all n. Write dn(t)=G(xn,xn+1,xn+2,t) and en(t)=G(yn,yn+1,yn+2,t):
(28)d3n(kt)=G(x3n,x3n+1,x3n+2,kt)=G(S(x3n,y3n),T(x3n+1,y3n+1),R(x3n-1,y3n-1),kt)≥min⁡{d3n-1(t),e3n-1(t),G(x3n,x3n,x3n+1,t),G(x3n+1,x3n+1,x3n+2,t),G(x3n-1,x3n-1,x3n,t)}≥min⁡{d3n-1(t),e3n-1(t),d3n(t),d3n(t),d3n-1(t)}.
Thus, d3n(kt)≥min{d3n-1(t),e3n-1(t)}. Similarly, we have e3n(kt)≥mind3n-1(t),e3n-1(t).
Thus,(29)min⁡{d3n(kt),e3n(kt)}≥min⁡{d3n-1(t),e3n-1(t)}.
Similarly, we can show that
(30)min⁡{d3n+1(kt),e3n+1(kt)}≥min⁡{d3n(t),e3n(t)},min⁡{d3n+2(kt),e3n+2(kt)}≥min⁡{d3n+1(t),e3n+1(t)}.
Thus,
(31)min⁡{dn+1(kt),en+1(kt)}≥min⁡{dn(t),en(t)}.
Hence
(32)min⁡{dn(t),en(t)}≥min⁡{dn(tk),en(tk)}≥min⁡{dn(tk2),en(tk2)}⋮≥min⁡{d0(tkn),e0(tkn)}=min⁡{G(x0,x1,x2,tkn),G(y0,y1,y2,tkn)}.
Thus,
(33)G(xn,xn+1,xn+2,t)≥min⁡{G(x0,x1,x2,tkn),G(y0,y1,y2,tkn)}.
From (G3), we have
(34)G(xn,xn,xn+1,t)≥G(xn,xn+1,xn+2,t)≥min⁡{G(x0,x1,x2,tkn),G(y0,y1,y2,tkn)}.
As in Theorem 18, we can show that {xn} and {yn} are G-Cauchy sequences in X. Since X is G-complete, there exist x,y∈X such that xn→x and yn→y:(35)G(S(x,y),x3n+2,x3n+3,kt)=G(S(x,y),T(x3n+1,y3n+1),R(x3n+2,y3n+2),kt)≥min⁡{G(x,x3n+1,x3n+2,t),G(y,y3n+1,y3n+2,t),G(x,x,S(x,y),t),G(x3n+1,x3n+1,x3n+2,t),G(x3n+2,x3n+2,x3n+3,t)}.
Letting n→∞,
(36)G(S(x,y),x,x,kt)≥min⁡{1,1,G(x,x,S(x,y),t),1,1}=G(x,x,S(x,y),t).
From this, we have S(x,y)=x. As in the first part of proof, we can show that S(x,y)=T(x,y)=R(x,y)=x. Similarly, it can be shown that S(y,x)=T(y,x)=R(y,x)=y. Thus, (x,y) is a common coupled fixed point of S, T, and R. Suppose (x1,y1) is another common coupled fixed point of S, T, and R. Consider
(37)G(x,x,x1,kt)=G(S(x,y),T(x,y),R(x1,y1),kt)≥min⁡{G(x,x,x1,t),G(y,y,y1,t),1,1,1}=min⁡{G(x,x,x1,t),G(y,y,y1,t)}.
Also,
(38)G(y,y,y1,kt)=G(S(y,x),T(y,x),R(y1,x1),kt)≥min⁡{G(x,x,x1,t),G(y,y,y1,t),1,1,1}=min⁡{G(x,x,x1,t),G(y,y,y1,t)}.
Thus,
(39)min⁡{G(x,x,x1,kt),G(y,y,y1,kt)}≥min⁡{G(x,x,x1,t),G(y,y,y1,t)}.
From Lemma 14, we have x1=x and y1=y. Thus, (x,y) is the unique common coupled fixed point of S, T, and R. Now, we will show that x=y. Consider
(40)G(x,x,y,kt)=G(S(x,y),T(x,y),R(y,x),kt)≥min⁡{G(x,x,y,t)G(y,y,x,t),1,1,1}=G(x,x,y,t).
Hence, x=y. Thus, S, T, and R have a unique common coupled fixed point of the form (x,x).



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*Source: 986748-2011-12-10.xml*