# Some Modified Fixed Point Results inV-Fuzzy Metric Spaces

Authors: Vishal Gupta; Manu Verma; Mohammad Saeed Khan
Journal: Advances in Fuzzy Systems
 (2019)

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## Abstract

The present research paper focuses on the existence of fixed point inV-fuzzy metric space. The presentation of V-fuzzy metric space in n-tuple encourages us to define different mapping in the symmetric V-fuzzy metric space. Here, the properties of fuzzy metric space are extended to V-fuzzy metric space. The introduction of notion for pair of mappings (f,g) on V-fuzzy metric space called V-weakly commuting of type Vf and V-R weakly commuting of type Vf is given. This proved fixed point theorem in V-fuzzy metric space employing the effectiveness of E.A. property and CLRg property. For the justification of the results, some examples are illustrated.


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## Body

## 1. Introduction

Metric space is one of the important basic areas of research for the mathematicians. Many researchers accelerated the concept of metric space either by introducing different contractions in different fields or by extending number of variables in the metric space. Different types of mappings are introduced to facilitate the fixed point in metric spaces such as weakly commuting pair of mappings [1], compatible mappings [2], and weakly compatible mappings [3]. Subsequently, Aamri and Moutawakil [4] introduced the notion of E.A. property. In 2011, Sintunavarat and Kumam[5] stamped the idea of common limit in the range of g (called CLRg property) which relaxes the requirement of completeness (or closedness) of the underlying subspace. Fixed point results are proved through the same concept in fuzzy metric spaces. Many authors [5–15] have given results about the common fixed point results in several spaces. On the basis of number of variables, there are many different generalizations, such as generalized metric space by Mustafa and Sims [16], generalized fuzzy metric spaces by Sun and Yang [17], new generalized metric space called S-metric space by Sedghi [18], and A-metric spaces by Abbas et al. [19], which is generalization of S-metric spaces. Also, V-fuzzy metric spaces were introduced by Gupta and Kanwar [20], which are based on fuzzy metric for n-tuples.The above mentioned generalizations of metric spaces are described below.Definition 1 ([16]).
LetX be a nonempty set and let G:X3→[0,+∞) be a function satisfying the following conditions for all x,y,z,a∈X: (G-1)
G(x,y,z)=0 if x=y=z,(G-2)
0≤G(x,y,z) with x≠y,(G-3)
G(x,x,y)≤G(x,y,z) with y≠z,(G-4)
G(x,y,z)=G(x,z,y)=G(y,x,z)=G(z,x,y)=G(z,y,x)=G(y,z,x),(G-5)
G(x,y,z)≤G(x,a,a)+G(a,y,z).
The functionG is called a generalized metric on X and the pair (X,G) is called a G-metric space.In 2012, Sedghi et al. [18] introduced a new generalized metric space called S-metric space.Definition 2 ([18]).
LetX be a nonempty set. Suppose a function S:X3→[0,+∞) satisfies the following conditions: (S-1)
S(x,y,z)≥0,(S-2)
S(x,y,z)=0 if and only if x=y=z=0,(S-3)
S(x,y,z)≤S(x,x,a)+S(y,y,a)+S(z,z,a) for any x,y,z,a∈X.
Then the ordered pair(X,S) is called S-metric space.Abbas et al. [19] established the notion of A-metric spaces, which is considered as generalizations of S-metric space.Definition 3 ([19]).
LetX be a nonempty set. A function A:Xn→[0,+∞) is called an A-metric on X, if for any xi,a∈X, i=1,2,3,…,n, the following conditions hold:(A-1)
A(x1,x2,x3,…,xn)≥0,(A-2)
A(x1,x2,x3,…,xn)=0 if and only if x1=x2=x3==⋯=xn=0,(A-3)
A(x1,x2,x3,…,xn)≤A(x1,x1,x1,…,(x1)n-1,a)+A(x2,x2,x2,…,(x2)n-1,a)+…+A(xn,xn,xn,…,(xn)n-1,a).
The pair(X,A) is called A-metric space.Fuzzy sets introduced by Zadeh [21] are the engender for all the research in different fields. Kramosil and Michalek [22] introduced the concept of fuzzy metric spaces.Definition 4 ([23]).
A binary operation∗:[0.1]×[0,1]→[0,1] is called continuous t-norms; it satisfies following conditions: (T-1)
∗ is commutative and associative,(T-2)
∗ is continuous,(T-3)
a∗1=a, ∀a∈[0,1],(T-4)
a∗b≤c∗d whenever a≤c and b≤d for all a,b,c,d∈[0,1].Definition 5 ([22]).
The3-tuple (X,M,∗) is called a fuzzy metric space if X is an arbitrary set, ∗ is continuous t-norm, and M is a fuzzy set in X×X×[0,∞) satisfying the following conditions:for allx,y,z∈X and s,t>0,(FM-1)
M(x,y,0)=0,(FM-2)
M(x,y,t)=1, ∀t>0 if and only if x=y,(FM-3)
M(x,y,t)=M(y,x,t),(FM-4)
M(x,y,t)∗M(y,z,s)≤M(x,z,t+s),(FM-5)
M(x,y,·):[0,∞)→[0,1] is left continuous.Note thatM(x,y,t) can be thought of as the degree of nearness between x and y with respect to t.Example 6.
Let(X,d) be a metric space. Define t-norm a∗b=ab or a∗b=min⁡{a,b}. For all x,y∈X, t>0.(1)Mx,y,t=tt+dx,y. Then (X,M,∗) is a fuzzy metric space.Lemma 7 ([24]).
Let(X,M,∗) be a fuzzy metric space. If there exists k∈(0,1) for all x,y∈X, x,t>0 such that(2)Mx,y,kt≥Mx,y,t for all x,y∈X, t>0, then x=y.In the process of generalization of fuzzy metric space, Sun and Yang [17] presented the notion of G-fuzzy metric space as follows.Definition 8 ([17]).
A3-tuple (X,G,∗) is said to be G-fuzzy metric space (denoted by GF space) if X is an arbitrary nonempty set, ∗ is continuous t-norm, and G is a fuzzy set on X×X×X×[0,∞) satisfying the following conditions:
for eachx,y,z∈X and s,t>0,(GF-1)
G(x,x,y,t)>0 with x≠y,(GF-2)
G(x,x,y,t)≥G(x,y,z,t) with y≠z,(GF-3)
G(x,y,z,t)=1 if and only if x=y=z,(GF-4)
G(x,y,z,t)=G(x,z,y,t)=G(y,x,z,t)=G(z,x,y,t)=G(y,z,x,t)=G(z,y,x,t),(GF-5)
G(x,y,z,t+s)≥G(x,a,a,t)∗G(a,y,z,s).(GF-6)
G(x,y,z,·):(0,∞)→[0,1] is left continuous.Lemma 9 ([17]).
Let(X,G,∗) be a GF space. Then G(x,y,z,t) is nondecreasing with respect to t for all x,y,z∈X.


## 2.V-Fuzzy Metric Space

These all generalizations advocateV-fuzzy metric spaces. In 2016, Gupta and Kanwar [20] stamped the move of these generalization to n-tuples as discussed below.Definition 10 ([20]).
LetX be nonempty set. A 3-tuple (X,V,∗) is said to be a V-fuzzy metric space (denoted by VF-space), where ∗ is a continuous t-norm and V is a fuzzy set on Xn×(0,∞) satisfying the following conditions for each t,s>0:(VF-1)
V(x,x,x,…,x,y,t)>0 for all x,y∈X with x≠y;(VF-2)
V(x1,x1,x1,…,x1,x2,t)≥V(x1,x2,x3,…,xn,t) for all x1,x2,x3,…,xn∈X with x2≠x3≠⋯≠xn;(VF-3)
V(x1,x2,x3,…,xn,t)=1 if x1=x2=x3=…=xn;(VF-4)
V(x1,x2,x3,…,xn,t)=V(p(x1,x2,x3,…xn),t), where p is a permutation function;(VF-5)
V(x1,x2,x3,…,xn-1,t+s)≥V(x1,x2,x3,…,xn-1,l,t)∗V(l,l,l,…,l,xn,s);(VF-6)
limt→∞V(x1,x2,x3,…,xn,t)=1;(VF-7)
V(x1,x2,x3,…,xn):(0,∞)→[0,1] is continuous.Example 11 ([20]).
Let(X,V,∗) be a V-metric space. Define t-norm a∗b=ab or a∗b=min⁡{a,b}. For all x1,x2,x3,…,xn∈X, t>0,(3)Vx1,x2,x3,…,xn,t=tt+Ax1,x2,x3,…,xn then (X,V,∗) is a V-fuzzy metric space.Lemma 12 ([20]).
Let(X,V,∗) be a V-fuzzy metric space; then V(x1,x2,x3,…,xn,t) is nondecreasing with respect to t.Lemma 13 ([20]).
Let(X,V,∗) be a V-fuzzy metric space such that(4)Vx1,x2,x3,…,xn,kt≥Vx1,x2,x3,…,xn,t, with k∈(0,1); then x1=x2=x3=…=xn.Definition 14 ([20]).
Let(X,V,∗) be a V-fuzzy metric space. A sequence {xr} is said to converge to a point x∈X if V(xr,xr,xr,…,xr,x,t)→1 as r→∞ for all t>0; that is, for each ϵ>0, there exists n∈N such that for all r≥N we have V(xr,xr,xr,…,xr,x,t)>1-ϵ and we write limt→∞xr=r.Definition 15 ([20]).
Let(X,V,∗) be a V-fuzzy metric space. A sequence {xr} is said to be a Cauchy sequence if V(xr,xr,xr,…,xr,xq,t)→1 as r,q→∞ for all t>0; that is, for each ϵ>0 there exists n0∈N such that for all r,q≥n0, we have V(xr,xr,xr,…,xr,xq,t)>1-ϵ.Definition 16 ([20]).
AV-fuzzy metric space (X,V,∗) is said to be complete if every Cauchy sequence in X is convergent.In the present research paper, topology is induced byV-fuzzy metric spaces. The introduction of concepts of V-weakly commuting of type Vf and V-R weakly commuting of type Vf in V-fuzzy metric spaces is given which helps in determining the fixed point theorem for symmetric V-fuzzy metric spaces.


## 3. Topology Induced byV-Fuzzy Metric Space

Definition 17.
Let(X,V,∗) be a V-fuzzy metric space. For t>0, the open ball BV(x,r,t) with center x∈X and radius 0<r<1 is defined as(5)BVx,r,t=y∈X:Vx,y,…,y,t>1-r.Result 1.
Every open ball is an open set.
Consider an open ballBV(x,r,t). Now(6)y∈BVx,r,t⇒Vx,y,…,y,t>1-r. Since V(x,y,…,y,t)>1-r, we can find t0, 0<t0<t, such that V(x,y,…,y,t0)>1-r.
Letr0=V(x,y,y,…,y,t0)>1-r.
Sincer0>1-r, we can find s, 0<s<1, such that r0>1-s>1-r.
Further for a givenr0 and s such that r0>1-s we can find r1, 0<r1<1 such that r0∗r1≥1-s.
Consider the Ball(7)BVy,1-r1,t-t0. We will show that(8)BVy,1-r1,t-t0⊂BVx,r,t. Now z∈BV(y,1-r1,t-t0) implies V(y,z,…,z,t-t0)>r1.
Therefore(9)Vx,z,…,z,t≥Vx,y,…,y,t0∗Vy,z,…,z,t-t0≥r0∗r1≥1-s. Therefore z∈BV(x,r,t) and hence(10)BVy,1-r,t-t0⊂BVx,r,t.Result 2.
Let(X,V,∗) be a V-fuzzy metric space. Define (11)τ=A⊂X:x∈Aifandonlyifthereexistt>0andr,0<r<1suchthatBVx,r,t⊂A. Then τ is a topology on X.Definition 18.
Let(X,V,∗) be a V-fuzzy metric space. The following condition is satisfied:(12)limni→∞Vxn1,xn2,…,xn,tn0=Vx1,x2,…,xN,t,i=0,1,2,…,N, whenever limn1→∞xn1=x1, limn2→∞xn2=x2, …, limnN→∞xnN=xN, and(13)limn0→∞Vx1,x2,…,xN,tn0=Vx1,x2,…,xN,t; then V is called continuous function on XN×(0,∞).Lemma 19.
Let(X,V,∗) be a V-fuzzy metric space. Then V is a continuous function XN×(0,∞).Proof.
Since(14)limn1→∞xn1=x1,limn2→∞xn2=x2⋮limnN→∞xnN=xN and(15)limn0→∞Vx1,x2,…,xN,tn0=Vx1,x2,…,xN,t, then there exists np∈N such that(16)t-tn0<δforn0≥npandδ<t2. As V(x1,x2,…,xn,t) is nondecreasing with respect to t, we have (17)Vxn1,xn2,…,xnN,tn≥Vxn1,xn2,…,xnN,t-δ≥Vxn1,x1,…,x1,δN∗Vx1,xn2,…,xnN,t-N+1Nδ≥Vxn1,x1,…,x1,δN∗Vxn2,x2…x2,δN∗…∗VxN-1,xN-2,…,x1,xnN,t-N+N-1Nδ≥Vxn1,x1,…,x1,δN∗Vxn2,x2…x2,δN∗…∗VxnN,xN,…,x1,xN,δN∗VxN,xN-1,…,x1,t-N-1δ and (18)Vx1,x2,…,xN,t+N-1δ≥Vx1,x2,…,xN,tn+δ≥Vx1,xn1,…,xn1,N-1Nδ∗Vxn1,x2,x3,…,xN,tn+1Nδ≥Vx1,xn1,…,xn1,δN∗Vx2,xn2,xn2,…,xn2,δN∗…∗VxN,xnN,xnN,…,xnN,tn+δN≥Vx1,xn1,xn1,…,xn1,δN∗Vx2,xn2,xn2,…,xn2,δN∗…∗VxN,xnN,xnN,…,xnN,δN∗VxN,xN-1,…,x2,x1,tn. Considering continuity of the function V with respect to t and letting n→∞, we have (19)Vx1,x2,…,xN,t+N-1δ≥VxN,xN-1,…,x1,t≥VVxN,xN-1,…,x1,t-N-1δ. Therefore, V is continuous function on XN×[0,∞).Remark 20.
In the present paper,(X,V,∗) will denote an VF-space with a continuous t-norm ∗ defined as a∗b=min⁡{a,b} for all a,b∈[0,1] and we assume that(20)limt→∞Vx1,x2,…,xN,t=1forallxi∈X,i=1,2,…,N.Define ϕ={ϕ:R+→R+}, where R+=[0,∞] and each ϕ∈Φ satisfying the following conditions: (ϕ-1)
ϕ is strict increasing,(ϕ-2)
ϕ is upper semicontinuous from the right,(ϕ-3)
∑n=1∞ϕn(t)<∞ for all t>0.Lemma 21.
Let(X,V,∗) be an VF-space. If there exists ϕ∈Φ such that (21)Vx1,x2,x3,…,xn,ϕt≥Vx1,x2,x3,…,xn,tforallt>0, then x1=x2=x3=…=xn.Proof.
Since(22)Vx1,x2,x3,…,xn,ϕt≥Vx1,x2,x3,…,xn,t and also ϕ(t)<t, by using Lemma 12, we have(23)Vx1,x2,x3,…,xn,ϕt≤Vx1,x2,x3,…,xn,t. From (22) and (23) and definition of V-fuzzy metric space, one can get x1=x2=x3=⋯=xn.Remark 22.
Letx1=w, x2=x3=…=xn-1=μ, in (VF-5); we have(24)Vw,u,u,…,u,t+s≥Vw,v,v,…,v,t∗Vv,u,u,…,u,s, which implies that(25)Vu,u,…,w,s+t≥Vv,v,…,v,w,t∗V(vu,u,…,u,v,s for all u,v,w∈X and s,t>0.AnVF-space is said to be symmetric if V(x,x,…,x,y,t)=V(x,y,y,…,y,t) for all x,y∈X and for each t>0.Lemma 23.
Let(X,V,∗) be an VF-metric space; if we define Eλ:X×X×X×⋯×X→[0,∞) by(26)Eλx1,x2,…,xn=inft>0,Vx1,x2,…,xn,t>1-λ for all λ∈(0,1] and x1,x2,…,xn∈X, then we have (i)
for eachλ∈(0,1], there exists μ∈(0,1] such that(27)Eλx1,x2,…,xn≤∑i=1n-1Eμxi,xi,…,xi+1forallx1,x2,…,xn∈X(ii)
the sequence{xn}n∈N in X is convergent if and only if Eλ(xn,xn,…,xn,x)→0 as n→∞ for all λ∈(0,1].Proof.
(i) For anyλ∈(0,1], let μ∈(0,1] and μ<λ. Therefore, by the triangular inequality (VF-5) and Remark 22(28)Vx1,x1,…,xn,∑i=1n-1Eμxi,xi,…,xi+1+n-1δ≥Vx1,x1,…,x2,Eμx1,x1,…,x2+δ∗Vx2,x2,…,x3,Eμx2,x2,…,x3+δ+⋯Vxn-1,xn-1,…,xn,Eμxn-1,xn-1,…,xn+δ≥min⁡1-μ,1-μ…1-μ≥1-λ, which gives, using (26), (29)Eλx1,x1,…,xn≤Eμx1,x1,…,x2+Eμx2,x2,…,x3+Eμxn-1,xn-1…,xn+n-1δ. Since δ>0 is arbitrary, we have (30)Eλx1,x1,x1,…,xn≤Eμx1,x1,…,x2+Eμx2,x2,…,x3+…+Eμxn-1,xn-1,…,xn.
(ii) SinceV is continuous in its (n+1)th argument (by (26)), we have(31)Vxn,xn,…,xn,x,η>1-λforallη>0. This proves the lemma.Lemma 24.
Let(X,V,∗) be a V-Fuzzy metric space and {yn} be a sequence in X. If there exists ϕ∈Φ such that(32)Vyn,yn,yn,…,yn,yn+1,ϕt≥Vyn-1,yn-1,…,yn-1,yn,t∗Vyn,yn,…,yn,yn+1,t for all t>0 and n=1,2,3,…, then {yn} is a Cauchy sequence in X.Proof.
Let{Eλ(x,y,z)}λ∈(0,1] be defined by (26).
For eachλ∈(0,1] and n∈N, put(33)an=Eλyn-1,yn-1,…,yn-1,yn. We will prove that(34)an+1≤ϕanforalln∈N.Since ϕ is upper semicontinuous from right, for given ϵ>0 and each an, there exists pn>an such that ϕ(pn)<ϕ(an)+ϵ. From (26), it follows from Pn>an=Eλ(yn-1,yn-1,…,yn) that V(yn-1,yn-1,yn-1,…,yn,Pn)>1-λ for all n∈N.
Thus, by (32), (34), and Lemma 12, we get (35)Vyn,yn,yn,…,yn+1,ϕmax⁡pn,pn+1≥Vyn-1,yn-1,…,yn,max⁡pn,pn+1∗Vyn,yn,…,yn+1,max⁡pn,pn+1≥Vyn-1,yn-1,yn-1,…,yn,pn∗Vyn,yn,…,y1,yn+1,pn+1>1-λ. Again, by (26), we get (36)Eλyn,yn,…,yn,yn+1≤ϕmax⁡pn,pn+1=max⁡ϕpn,ϕpn+1≤max⁡ϕan,ϕan+1+ϵ. By the arbitrariness of ϵ, we have(37)an+1=Eλyn,yn,…,yn+1≤max⁡ϕan,ϕan-1. So, we can interpret that an+1≤ϕ(an).
If not, then by (37), we have an+1≤ϕ(an+1)<an+1; this is a contradiction. Hence (37) implies that an+1≤ϕ(an), and (34) is proved.
By repeated application of (34), we get (38)Eλyn,yn,…,yn+1≤ϕEλyn-1,yn-1,…,yn≤⋯≤ϕnEλy0,y0,…,y1foralln∈N. By Lemma 23, for each λ∈(0,1], there exists μ∈(0,λ] such that(39)Eλyn,yn,…,yn,yn≤∑i=nm-1Eμyi,yi,…,yi,yi+1,m,n∈Nwithm>n. Since ϕ∈Φ, by condition (ϕ-3), we have(40)∑n=0∞ϕnEμy0,y0,…,y1<+∞. So, for given ϵ>0, there exists n0∈N such that ∑i=n0∞ϕi(Eμ(y0,y0,…,y1))<ϵ. Thus, it follows from (39) that (41)Eλyn,yn,…,ym≤∑ϕiEμy0,y0,y0,…,y1<ϵforalln≥n0, which implies that V(yn,yn,yn,…,yn,ϵ)>1-λ for all m,n∈N with m>n≥n0. Therefore {yn} is a Cauchy sequence in X.Definition 25.
A pair of self-mappings(f,g) of V-fuzzy metric space (X,V,∗) is said to be V-weakly commuting of type Vf if(42)Vfgx,gfx,fgx,…,ffx,t≥Vfx,gx,fx,fx,…,…,fx,t for all x∈X and t>0.Definition 26.
A pair of self-mappings(f,g) of a V-fuzzy metric space (X,V,∗) is said to be V-R weakly commuting of type Vf if there exists some positive real number R such that(43)Vfgx,gfx,fgx,…,ffxt≥Vfx,gx,fx,fx,…,fx,tr for all x∈X and t>0.Remark 27.
If we interchangef and g in above definitions, then the pair of self-mappings (f,g) of V-fuzzy metric space (X,V,∗) is said to be V-weakly commuting of type Vg and V-R weakly commuting of type Vg, respectively.For proving our main results, we use the following relation.The following example shows that a pair of mapping(f,g) that is V-weakly commuting of type Vf does not need to be V-weakly commuting of type Vg.Example 28.
Letx=(0,1] be the V-fuzzy metric space with(44)Ax1,x2,…,xn=max⁡x1-x2,x2-x3…,xn-1-xn for all x1,x2,…,xn∈X.
Definefx=x2/4, g(x)=x2.
Then we find(45)Afgx,gfx,…,ffx=1564x4 and(46)Afx,gx,gx,…,fx=34x2. Then, one can get(47)Agfx,fgx,fgx,…,ggx=1516≰Agx,fx,fx,…gx=34, which implies(48)Vfgx,gfx,…,ffx,t≥Vfx,gx,…,fx,t and(49)Vgfx,fgx,…,ggx,t≱Vgx,fx,…,gx,t. Hence the pair (f,g) is not V-weakly commuting of type Vg, but it is V-weakly commuting of type Vf.Lemma 29.
Iff and g are V-weakly commuting of type Vf or V-R-weakly commuting of type Vf, then f and g are weakly compatible.Proof.
Letx be a coincidence point of f and g; i.e., f(x)=g(x); then if pair (f,g) is V-weakly commuting of type Vf, we have (50)Vfgx,gfx,…,fgx,t=Vfgx,gfx,…,ffx,t≥Vfx,gx,…,fx,t=1. It follows that fxg=gfx. Hence f and g commute at their coincidence point.
Similarly, if pair(f,g) is V-R weakly commuting of type Vf, we have (51)Vfgx,gfx,…,fgx,t=Vfgx,gfx,…,ffx,t≥Vfx,gx,…,fx,tR=1, and thus fgx=gfx; then the pair (f,g) is weakly compatible.
The converse of the lemma need not be true.Example 30.
X = [ 0,1 ] and A(x1,x2,…,xn)=x1-x2+x2-x3.
Definef,g:X→X by f(x)=4x-1 and g(x)=3x2, x∈X; we see that x=1/3 is the only coincidence point and fg1/3=1/3 and gf1/3=g1/3, so f and g are weakly compatible.
But by easy calculation, forx=1, one can have(52)Afgx,gfx,…,ffx=36 and(53)Afx,gx,…,fx=0, i.e.,(54)Vfgx,gfx,…,ffx,t=36≰0=fx,gx,…,fx,t. Therefore, f and g are not V-weakly commuting of type Vf.Definition 31 ([4]).
A pair of self-mappings(f,g) on X is said to satisfy the property E.A. if there exists a sequence {xn} such that(55)limn→∞fxn=limn→∞gxn=zforallz∈X.Definition 32 ([25]).
A pair of self-mappings(f,g) on X is said to satisfy the CLRg property if there exists a sequence {xn} such that(56)limn→∞fxn=limn→∞gxn=gzforallz∈X.Now, we are ready to prove our results for symmetric V-fuzzy metric spaces.Theorem 33.
Let(X,V,∗) be a symmetric VF-space and mappings f,g:X→X satisfying the following conditions:(i)
f and g are V-weakly commuting of type Vf;(ii)
f(X)⊆g(X);(iii)
g(X) is a V-complete subspace of X;(iv)
these existsϕ∈Φ such that for all xi∈X and i=1,2,…,n, t>0(57)Vfx1,fx2,fx3,…,fxn,ϕt≥Vgx1,gx1,…,fx1,t∗Vgx2,gx2,…,fx2,t∗Vgx3,gx3,…,fx3,t∗…∗Vgxn,gxn,…,fxn,t.Then f and g have common fixed point.Proof.
Letz1∈X such that fz0=gz1 and z2∈X, where fz1=gz2, and then by induction we can define a sequence {yn}∈X as follows(58)yn=fzn=gzn+1,n∈N. We will prove that {yn} is a Cauchy sequence in X. (59)Vyn,yn,…,yn,yn+1,ϕt=Vfzn,fzn,…,fzn+1,ϕt≥Vgzn,gzn,…,fzn,t∗Vgzn,gzn,…,fzn,t∗⋯∗Vgzn+1,gzn+1,…,fzn+1,t≥Vgzn,gzn,…,gzn+1,t∗Vgzn,gzn,…,gzn+1,t∗⋯∗Vgzn+1,gzn+1,…,gzn+2,t, which gives (60)Vyn,yn,…,yn+1,ϕt≥Vyn-1,yn-1,…,yn,t∗⋯∗Vyn,yn,…,yn+1,t. By Lemma 24, the sequence {yn} is a V-Cauchy sequence. Since yn=gzn+1, {gzn+1} is a V-Cauchy sequence in g(x).
By hypothesis (iii), we know thatg(X) is V-complete; then there exists u∈g(X) such that(61)limn→∞gzn=u=limn→∞fzn. Now u∈g(x), so there exists p∈X such that u=gp. Therefore(62)limn→∞gzn=gp=limn→∞fzn. We will prove that fp=gp: (63)Vfp,fp,…,fzn,ϕt≥Vgp,gp,…,fp,t∗Vgp,gp,…,fp,t∗Vgp,gp,…,fp,t∗…∗Vgzn,gzn,…,fzn,t, taking limit as n→∞, (64)Vfp,fp,…,gp,ϕt≥Vgp,gp,…,fp,t∗Vgp,gp,…,fp,t∗Vgp,gp,…,gp,t, which implies, (65)Vfp,fp,…,gp,ϕt≥Vgp,gp,…,fp,t. Since V-fuzzy metric space is symmetric, we have (66)Vfp,fp,…,gp,ϕt≥Vgp,gp,…,fp,t=Vfp,fp,…,gp,t, which implies fp=gp (by Lemma 21).
Since pair(f,g) is V-weakly commuting of type Vf, then(67)Vfgp,gfp,fgp,…,ffp,ϕt≥Vfp,gp,fp,gp,…,fp,t=1, which implies(68)ffp=fgp=gfp=ggp. Hence fu=fgp=fgp=gu.
Eventually, we show thatu=gp is common fixed point of f and g. Suppose fu≠u; then (69)Vfu,fp,fp,…,fp,ϕt≥Vgu,gu,…,fu,t∗Vgp,gp,…,fp,t∗…∗Vgp,gp,…,fp,t≥Vfu,fu,…,fu,t∗Vfp,fp,…,fp,t∗…∗Vfp,fp,…,fp,t.Vfu,u,…u,ϕt≥1∗1∗…∗1=1,
which is the contradiction. Hence,fu=gu=u.
To prove the uniqueness, supposeu and v are such that u≠v, fv=gv=v and fu=gu=u; then again using condition (iv), we have (70)Vu,v,…,v,ϕt=Vfu,fv,…,fv,ϕt≥Vgu,gu,…,fu,t∗Vgv,gv,…,fv,t∗…∗Vgv,gv,…,fv,t=1∗1∗…∗1=1. Hence, V(u,v,v,…,v,ϕ(t))≥1, which gives a contradiction. Hence u=v. Therefore u is a unique common fixed point of f and g.Example 34.
LetX=[0,1] be a standard V-fuzzy metric space.
Letϕ(t)=t/2 and define f,g:X→X by f(x)=x/6, g(x)=(x/2)(x+1), x∈X.
We see thatx=0 is the only coincidence point and f and g are weakly compatible.
Letxn=1/n be a sequence such that(71)Vfp,fp,…,fxn,ϕt≥Vfp,fp,…,gp,t, where p is a coincidence point.
Then the pair(f,g) is V-weakly commuting of type Vf. Further f and g have a unique common fixed point of f and g.Corollary 35.
Theorem33 remains true if we replace V-weakly commuting and V-R-weakly commuting of type Vf by V-weakly commuting and V-R-weakly commuting of type Vg (considering the other conditions are the same).Theorem 36.
Let(X,V,∗) be a symmetric V-fuzzy metric space and suppose mappings f,g:X→X are V-weakly commuting of type Vf satisfying the following conditions: (i)
f and g satisfy the E.A property;(ii)
g(X) is a closed subspace of X;(iii)
there exists aϕ∈Φ such that for all xi∈X, i=1,2,…,n and t>0, (72)Vfx1,fx2,…,fxn,ϕt≥Vgx1,gx1,…,fx1,t∗Vgx2,gx2,…,fx2,t∗…∗Vgxn,gxn,…,fxn,t.Thenf and g have a unique common fixed point.Proof.
Since, the mappingsf and g satisfy the E.A. property, then there exists a sequence {zn} in X satisfying(73)limn→∞gzn=u=limn→∞fznforsomeu∈X. Since g(X) is closed subspace of X and limn→∞gzn=u, then there exists p∈X such that gp=u.
Also,limn→∞gzn=gp=limn→∞fzn.
We will prove,fp=gp: (74)Vfp,fp,…,fzn,ϕt≥Vgp,gp,…,fp,t∗Vgp,gp,…,fp,t∗…∗Vgzn,gzn,…,fzn,t, and, taking limit as n→∞, we have (75)Vfp,fp…,gp,ϕt≥Vgp,gp,…,fp,t∗⋯∗Vgp,gp,…,gp,tVfp,fp…,gp,ϕt≥Vgp,gp,…,fp,t. Since V-fuzzy metric space is symmetric, we have(76)Vfp,fp,…,gp,ϕt≥Vfp,fp,…,gp,t, which implies fp=gp=u (by Lemma 21).
Since pair(f,g) is V-weakly commuting of type Vf, then (77)Vfgp,gfp,…,ffp,ϕt≥Vfp,gp,fp,…,fp,t=1, which implies ffp=fgp=gfp=ggp.
Hencefu=fgp=gfp=gu.
Finally, we show thatu=gp is a common fixed point of f and g. Suppose fu≠u, then (78)Vfu,fp,…,fp,ϕt≥Vgu,gu,…,fu,t∗Vgp,gp,…,fp,t∗…∗Vgp,gp,…,fp,t≥Vfu,fu,…,fu,t∗Vfp,fp,…,fp,t∗…∗Vfp,fp,fp,…,fp,t=1∗1∗1∗…∗1=1.Thus,Vfu,u,u…u,ϕt≥1which is a contradiction. Hence fu=gu=u.
To prove the uniqueness, suppose thatu and v are such that u≠v, fu=gu=u, and fv=gv=v; then again using condition (iii), we have (79)Vu,v,…,v,ϕt=Vfu,fv,…,fv,ϕt≥Vgu,gu,…,fu,t∗Vgv,gv,…,fv,t…∗Vgv,gv,…,fv,t=1∗1∗1∗…∗1=1. Hence V(u,v,…,v,ϕ(t))≥1, which gives a contradiction. Hence u=v. Therefore ‘u’ is a unique common fixed point of f and g.Theorem 37.
Let(X,V,∗) be a symmetric V-fuzzy metric space and suppose mappings f,g:X→X are V-weakly commuting of type Vf satisfying the following conditions: (i)
f and g satisfy the CLRg property;(ii)
g(X) is a closed subspace of X;(iii)
there exists aϕ∈Φ such that for all xi∈X, i=1,2,…,n and t>0, (80)Vfx1,fx2,…,fxn,ϕt≥Vgx1,gx1,…,fx1,t∗Vgx2,gx2,…,fx2,t∗…∗Vgxn,gxn,…,fxn,t.Thenf and g have a unique common fixed point.Proof.
Proof follows on the same lines of Theorem33 and by definition of CLRg property.



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