# Weakly Compatible Maps Using E.A. and (CLR) Properties in Complex ValuedG-Metric Spaces

Authors: Balbir Singh; Vishal Gupta; Sanjay Kumar
Journal: Journal of Complex Systems
 (2013)

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## Abstract

We introduce the notion of complex valuedG-metric spaces and prove common fixed point theorems for weakly compatible maps along with E.A. and (CLR) properties in complex valued G-metric spaces.


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## Body

## 1. Introduction

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Recently, Mustafa and Sims [1, 2] have shown that most of the results concerning Dhage’s D-metric spaces are invalid; therefore, they introduced an improved version of the generalized metric space structure and called it G-metric spaces.In 2006, Mustafa and Sims [2] introduced the concept of G-metric spaces as follows.Definition 1.
LetX be a nonempty set, and let G:X×X×X→ℝ+ be a function satisfying the following properties:(G1)
G(x,y,z)=0 if x=y=z;(G2)
0<G(x,y,z) 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 (rectangle inequality);Then the functionG is called a generalized metric or, more specially a G-metric on X, and the pair (X,G) is called a G-metric space.The idea of complex metric space was initiated by Azam et al. [3] to exploite the idea of complex valued normed spaces and complex valued Hilbert spaces.Definition 2.
Letℂ be the set of complex numbers and z1, z2∈ℂ. Define a partial order ≾ on ℂ as follows:
(1)z1≾z2iffRe(z1)≤Re(z2),Im(z1)≤Im(z2).That is,z1≾z2 if one of the following holds:(C1)
Re(z1) = Re(z2) and Im(z1) = Im(z2);(C2)
Re(z1) < Re(z2) and Im(z1) = Im(z2);(C3)
Re(z1) = Re(z2) and Im(z1) < Im(z2);(C4)
Re(z1) < Re(z2) and Im(z1) < Im(z2).In particular, we will write thatz1⪱z2 if z1≠z2 and one of (C2), (C3), and (C4) is satisfied, and we will write z1≺z2 if only (C4) is satisfied.Remark 3.
We  noted  that the following statements hold:(i)
a,b∈R and a≤b⇒az≾bz for all z∈ℂ;(ii)
0≾z1⪱z2⇒|z1|<|z2|;(iii)
z1≾z2 and z2≺z3⇒z1≺z3.Now we introduce the notion of complex valuedG-metric space akin to the notion of complex valued metric spaces [3] as follows.Definition 4.
LetX be a non-empty set. Let G:X×X×X→ℂ be a function satisfying the following properties:(CG1)
G(x,y,z)=0 if x=y=z;(CG2)
0≺G(x,y,z) for all x,y∈X with x≠y;(CG3)
G(x,x,y)≾G(x,y,z)for all x,y,z∈X with y≠z;(CG4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯ (symmetry in all three variables);(CG5)
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 complex valued generalized metric or more specially, a complex valued G-metric on X, and the pair (X,G) is called a complex valued G-metric space.


## 2. The Complex ValuedG-Metric Topology

A pointx∈X is called interior point of  a set A⊆X, whenever there exists 0≺r∈ℂ such that
(2)BG(x,r)={y∈X:G(x,y,y)≺r}⊆A.A pointx∈X is called limit point of  a set A, whenever there exists 0≺r∈ℂ:
(3)BG(x,r)∩(AX)≠∅.The setA is called open, whenever each element of A is an interior point of A. A subset B⊆X is called closed, whenever each limit point of B belongs to B.Proposition 5.
Let (X,G) be complex valued G-metric space, then for any x0∈X and r> 0, one has the following:(1)
ifG(x0,x,y)≺r,
thenx,y∈BG(x0,r);(2)
ify∈BG(x0,r),  then there existsaδ>0such thatBG(y,δ)⊆BG(x0,r).Proposition 6.
Let (X,G) be complex valued G-metric space; then for all x0∈X and r>0, one has
(4)BG(x0,13r)⊆BdG(x0,r)⊆BG(x0,r),
where dG(x,y)=G(x,y,y)+G(x,x,y).Proposition 7.
Let (X,G) be a complex valued G-metric space. Then for any x,y,z, and a in X it follows that: (i)
ifG(x,y,z)=0andx=y=z;(ii)
G(x,y,z)≾G(x,x,y)+G(x,x,z);(iii)
G(x,y,y)≾2G(y,x,x);(iv)
G(x,y,z)≾G(x,a,z)+G(a,y,z);(v)
G(x,y,z)≾2/3(G(x,y,a)+G(x,a,z)+G(a,y,z));(vi)
G(x,y,z)≾(G(x,a,a)+G(y,a,a)+G(z,a,a)).Proposition 8.
Let (X,G) be a complex valued G-metric space. Then the following are equivalent:(i)
(X,G)is symmetric;(ii)
G(x,y,y)≾G(x,y,a), for allx,y,a∈X;(iii)
G(x,y,z)≾G(x,y,a)+G(z,y,b)for allx,y,z,a,b∈X.


## 3. Convergence, Continuity, and Completeness in Complex ValuedG-Metric Spaces

Now we discuss some definition regarding convergence, continuity, and completeness in complex valuedG-metric spaces.Definition 9.
Let(X,G) be a complex valued G-metric space. Let {xn} be a sequence of points of X; we say that {xn} is complex valued G-convergent to x if, for any ϵ>0, there exists k∈N such that G(x,xn,xm)≺ϵ for all n,m≥k. We refer to x as the limit of the sequence {xn}, and we write xn→(G)x.Proposition 10.
Let(X,G) be complex valued G-metric spaces. For a sequence {xn}⊆X  and point x∈X, the following are equivalent:(1)
{xn}is complex valuedG-convergent tox;(2)
|G(xn,xn,x)|→0asn→∞;(3)
|G(xn,x,x)|→0asn→∞;(4)
|G(xm,xn,x)|→0asn,m→∞.Definition 11.
Let(X,G) and (X′,G′) be two complex valued G-metric spaces. Then a function f:X→X′ is complex valued G-continuous at a point x0∈X if f-1(BG′(f(x0),r))∈τ(G) for all r>0. We say f is complex valued G-continuous if it complex valued G-continuous at all points of X; that is, continuous as a function from X with the τ(G)-topology to X′with τ(G′)-topology.Since complex valuedG-metric topologies are metric topologies, therefore we have some proposition in this regard.Proposition 12.
Let(X,G) and (X′,G′) be two complex valued G-metric spaces. Then a function f:X→X′ is complex valued G-continuous at a point x∈X if and only if it is complex valued G-sequentially continuous at x; that is, whenever {xn} is complex valued G-convergent to x, one has (f{xn}) is complex valued G-convergent to f(x).Proposition 13.
Let (X,G) be a complex valued G-metric spaces. The function G(x,y,z) is jointly continuous in all three of its variables.Proof.
Suppose that{xk}, {ym}, and {zn} are complex valued G-convergent to x,y, and z, respectively. Then by (CG5), we have
(5)G(x,y,z)≾G(y,ym,ym)+G(ym,x,z),G(z,x,ym)≾G(x,xk,xk)+G(xk,ym,z),G(z,xk,ym)≾G(z,zn,zn)+G(zn,ym,xk),
so
(6)G(x,y,z)-G(xk,ym,zn)≾G(y,ym,ym)+G(x,xk,xk)+G(z,zn,zn).
Similarly,
(7)G(xk,ym,zn)-G(x,y,z)≾G(xk,x,x)+G(ym,y,y)+G(zn,z,z).
Combining the above inequality and using (iii) of Proposition 7, we have
(8)|G(xk,ym,zn)-G(x,y,z)|≤2(G(x,xk,xk)+G(y,ym,ym)+G(z,zn,zn)),|G(xk,ym,zn)-G(x,y,z)|→ 0, as k,m,n,→∞, and the result follows by Proposition 12.Definition 14.
Let (X,G) be a complex valued G-metric space. A sequence {xn} is complex valued G-Cauchy if, given ϵ>0, there exists k∈N such that G(xn,xm,xl)≺ϵ for all n,m,l≥k.Definition 15.
A complex valuedG-metric space (X,G) is said to be complex valued G-complete if every complex valued G-Cauchy sequence is complex valued G-convergent in (X,G).Proposition 16.
Let (X,G) be a complex valued G-metric space. Then the following are equivalent:(1)
the sequence{xn}is a complex valuedG-Cauchy in X;(2)
for everyϵ>0, there existsk∈Nsuch thatG(xn,xm,xm)≺ϵfor alln,m≥k;(3)
{xn}is a Cauchy sequence in the complex valued metric space(X,
dG).Proposition 17.
Let (X,G) be a complex valued G-metric space, and let {xn} be a sequence in X. Then {xn} is complex valued G-convergent to x if and only if |G(x,xn,xm)|→0 as n,m→∞.Proof.
Suppose that{xn} is complex valued G-convergent to x. For a given real number ϵ>0, let
(9)c=ϵ2+iϵ2.
Then 0≺r∈ℂ, and there is a natural number k, such that G(x,xn,xm)≺ϵ for all n,m≥k.
Therefore,(10)|G(x,xn,xm)|<|c|=ϵ,∀n,m≥k.
It follows that |G(x,xn,xm)|→0 as n,m→∞.
Conversely, suppose that|G(x,xn,xm)|→0 as n,m→∞. Then given r∈ℂ with 0≺c, there exists a real number δ>0, such that, for z∈ℂ,
(11)|z|<δ⟹z≺c.
For this δ, there is a natural number k such that
(12)|G(x,xn,xm)|<δ,∀n,m≥k.
This means that G(x,xn,xm)≺ϵ for all n,m≥k. Hence {xn} is complex valued G-convergent to x.Proposition 18.
Let (X,G) be a complex valued G-metric space, and let {xn} be a sequence in X. Then {xn} is complex valued G-Cauchy sequence if and only if |G(xn,xm,xl)|→0 as n,m→∞.Proof.
Suppose that{xn} is complex valued G-Cauchy sequence. For a given real number ϵ>0, let
(13)c=ϵ2+iϵ2.
Then 0≺r∈ℂ, and there is a natural number k, such that G(xn,xm,xl)≺ϵ for all n,m≥k.
Therefore,(14)|G(xn,xm,xl)|<|c|=ϵ,∀n,m≥k.
It follows that |G(xn,xm,xl)|→0 as n,m→∞.
Conversely, suppose that|G(xn,xm,xl)|→0 as n,m→∞. Then given c∈ℂ with 0≺c, there exists a real number δ>0, such that, for z∈ℂ,
(15)|z|<δ⟹z≺c.
For this δ, there is a natural number k such that
(16)|G(xn,xm,xl)|<δ,∀n,m≥k.
This means that G(xn,xm,xl)≺ϵ for all n,m ≥ k. Hence {xn} is complex valued G-Cauchy sequence.


## 4. Weakly Compatible Maps

In 1996, Jungck [4] introduced the concept of weakly compatible maps as follows.Definition 19.
Two self-mapsf and g are said to be weakly compatible if they commute at coincidence points.
Now  we  prove our main result for a pair of self-mappings.Theorem 20.
Let (X,G) be a complete complex valued G-metric space. Let S,T:X→X be self-mappings satisfying the following conditions:(2.1)
S(X)⊆T(X);(2.2)
any one of the subspaceS(X) or T(X) is complete;(2.3)
G(Sx,Sy,Sz)≾k  G(Tx,Ty,Tz) for all x,y,z∈X, where 0≤k<1;(2.4)
S and T are weakly compatible self-maps.
ThenS and T have a unique common fixed point in X.Proof.
Letx0∈X be an arbitrary point in X. By (2.1), one can choose a point x1 in X such that Sx0=Tx1. In general, choose  xn+1 such that
(17)yn=Sxn=Txn+1.
Now, we  prove that {yn} is a complex valued G-Cauchy sequence in X.
Puttingx=xn,  y=xn+1,  and  z=xn+1 in (2.3), we have
(18)G(Sxn,Sxn+1,Sxn+1)≾kG(Txn,Txn+1,Txn+1)=kG(Sxn-1,Sxn,Sxn).
Continuing in the same way, we have
(19)G(Sxn,Sxn+1,Sxn+1)≾knG(Sx0,Sx1,Sx1).
This  implies that  G(yn,yn+1,yn+1)≾knG(y0,y1,y1).
Then, for alln,m∈N,  n<m, we have by (CG5)
(20)G(yn,ym,ym)≾G(yn,yn+1,yn+1)+G(yn+1,yn+2,yn+2)+G(yn+2,yn+3,yn+3)+⋯+G(ym-1,ym,ym)≾(kn+kn+1+kn+2+⋯+km-1)×G(y0,y1,y1)≾kn1-kG(y0,y1,y1).
Therefore,
(21)|G(yn,ym,ym)|≤(kn1-k)|G(y0,y1,y1)|.
Proceeding limit as n,m→∞ and since 0 ≤k<1, then  (kn/(1-k))|G(y0,y1,y1)|→0; that is, |G(yn,ym,ym)|→0.
Forn,m,l∈N, (CG5) implies that
(22)G(yn,ym,yl)≾G(yn,ym,ym)+G(yl,ym,ym).
Therefore,
(23)|G(yn,ym,yl)|≤|G(yn,ym,ym)|+|G(yl,ym,ym)|.
Taking limit as n,m,l→∞, we get |G(yn,ym,yl)|→ 0; that is, G(yn,ym,yl→0). So {yn} is complex valued G-Cauchy sequence. Since either S(X) or T(X) is complete. Without loss of generality, we assume that T(X) is complete subspace of X, and then the subsequence of {yn} must get a limit in T(X), say z. Then Tu=z for some u∈X, as {yn} is a complex valued G-Cauchy sequence containing a convergent subsequence.
Next we show thatSu=z. On setting x=u,y=xn, and  z=xn in (2.3), we have
(24)G(Su,Sxn,Sxn)≾kG(Tu,Txn,Txn).
Taking limit as n→∞, we have G(Su,z,z)≾kG(Tu,z,z).
Therefore,|G(Su,z,z)|≤k|G(Tu,z,z)| implies that Su=z.
Therefore,Su=Tu=z. That is, u is coincidence point of S and T. Since S and T are weakly compatible, it follows that STu=TSu; that is, Sz=Tz.
We now show thatSz=z. Suppose that S(z)≠z; therefore 0≺G(Sz,z,z) implies that |G(Sz,z,z)|>0.
Puttingx=z, y=u, and z=u in (2.3), we have
(25)G(Sz,Su,Su)≾kG(Tz,Tu,Tu)=kG(Sz,z,z);
That is, |G(Sz,z,z)|≤k|G(Sz,z,z)|<|G(Sz,z,z)|, which is a contradiction; therefore Sz=z. Thus Sz=Tz=z; that is, z is a common fixed point of S and T.
Uniqueness. To prove uniqueness, suppose that w≠z be another common fixed point of S and T. Then 0≺G(z,w,w) implies that |G(z,w,w)|>0.
Puttingx=z,y=u,  and  z=u in (2.3), we have
(26)G(z,w,w)=G(Sz,Sw,Sw)≾kG(Tz,Tw,Tw)=kG(z,w,w);
that is, |G(z,w,w)|≤k|G(z,w,w)|<|G(z,w,w)|, which is a contradiction; therefore z=w. Thus Sz=Tz=z; that is, z is a unique common fixed point of S and T.Example 21.
LetX=[-1,1], and let G:X×X×X→ℂ be complex valued G-metric space defined as follows:
(27)G(x,y,z)=|x-y|+|y-z|+|z-x|,∀x,y,z∈X.
Then (X,G) is complex valued G-metric space. Define  S,T:X→X as Sx=x/6  and Tx=x/2. Here we note that (2.1) S(X)⊆T(X), (2.3)  G(Sx,Sy,Sz)≾kG(Tx,Ty,Tz) holds for all x,y,z∈X,  1/3≤k<1, and (2.4)  S and T are weakly compatible because S and T commute at their coincidence point, that is, at x=0, and x=0 is the unique common fixed point of S and T and S and T also satisfy the condition (2.2).


## 5. E.A. Property and Weakly Compatible Maps

In 2002, Aamri and Moutawakil [5] introduced the notion of E.A. property as follows.Definition 22.
Two self-mappingsf and g of a metric space (X,d) are said to satisfy E.A. property if there exists a sequence {xn} in X such that limn→∞fxn=limn→∞gxn=t for some t in X.In a similar mode, we use these notions in complex valuedG-metric spaces.Example 23.
LetX=ℂ. Let G:X×X×X→ℂ be complex valued G-metric space defined as follows:
(28)G(x,y,z)=|x-y|+|y-z|+|z-x|,∀x,y,z∈X.
Then (X,G) is complex valued G-metric space. Define S, T: X→X as Sx=x+1 and Tx=2x; for all x∈X.
Consider a sequence{xn}={1-1/n},  n∈ℕ, in X; then
(29)limn→∞Sxn=limn→∞(xn+1)=limn→∞1-1n+1=2,limn→∞Txn=limn→∞2xn=limn→∞2(1-1n)=2.
Thus S and T satisfy E.A. property.Now we prove a common fixed point theorem for weakly compatible maps along with E.A. property.Theorem 24.
LetS and T be self-mappings of a complex valued G-metric space (X,G) satisfying (2.3), (2.4), and the following:(3.1)
S and T satisfy E.A. property;(3.2)
T(X) is a closed subspace of X.Then S and T have a unique common fixed point.Proof.
SinceS and T satisfy the E.A. property, therefore, there exists a sequence {xn} in X such that   limn→∞Sxn=limn→∞Txn=u∈X. Since T(X) is a closed subspace of X, therefore every convergent sequence of points of T(X) has a limit point in T(X).
Thenlimn→∞Sxn=u=Tv=limn→∞Txn for some v in X. This implies that u=Tv∈T(X).
On settingx=v,y=xn, and  z=xn, in (2.3), we have
(30)G(Sv,Sxn,Sxn)≾kG(Tv,Txn,Txn).
Taking limit as n→∞, we have G(Sv,u,u)≾kG(Tv,u,u).
Therefore,|G(Sv,u,u)|≤k|G(Tv,u,u)| implies that Sv=u.
Therefore,Sv=Tv=u. That is, v is coincidence point of S and T. Since S and T are weakly compatible, it follows that STv=TSv;thatis,Su=Tu.
Next we show thatSu=u. On setting x=u,  y=v, and  z=v in (2.3), we have
(31)G(Su,Sv,Sv)≾kG(Tu,Tv,Tv).
Therefore, |G(Su,u,u)|≤k|G(Tu,u,u)| implies that Su=u. Hence u is a  common fixed point of S and T.Uniqueness easily follows from Theorem20. Hence u is a unique common fixed point of S and T.


## 6. (CLR) Property and Weakly Compatible Maps

In 2011, Kumam and Sintunavarat [6] introduced the notion of (CLRg) property as follows.Definition 25.
Two self-mappingsf and g of a metric space (X,d) are said to satisfy (CLRg) property if there exists a sequence {xn} in X such that limn→∞fxn=limn→∞gxn=gx for some x in X.
In a similar mode, we use these notions in complex valuedG-metric spaces.Example 26.
LetX=ℂ. Let G:X×X×X→ℂ be complex valued G-metric space defined as follows:
(32)G(x,y,z)=|x-y|+|y-z|+|z-x|,∀x,y,z∈X.
Then (X,G) is complex valued G-metric space. Define S,T:X→X as Sx=x+1 and Tx=2x for all x∈X.
Consider a sequence{xn}={1-1/n},  n∈ℕ, in X; then
(33)limn→∞Sxn=limn→∞(xn+1)=limn→∞1-1n+1=2,limn→∞Txn=limn→∞2xn=limn→∞2(1-1n)=2.
Also, we have
(34)limn→∞Sxn=limn→∞Txn=2=S(1).
Thus S and T satisfy (CLRS) property
Now we prove a common fixed point theorem for weakly compatible maps along with (CLRT) property.Theorem 27.
LetS and T be self-mappings of a complex valued G-metric space (X,G) satisfying (2.1), (2.3), (2.4), and the following:(4.1)
S and T satisfy (CLRT) property.
ThenS and T have a unique common fixed point.Proof.
SinceS and T satisfy the (CLRT) property, therefore, there exists a sequence {xn} in X such that limn→∞Sxn=limn→∞Txn=u∈X. Then limn→∞Sxn=u=Tv=limn→∞Txn for some v in X. This implies that u=Tv∈T(X).
On settingx=v,  y=xn, and  z=xn in (2.3), we have
(35)G(Sv,Sxn,Sxn)≾kG(Tv,Txn,Txn).
Taking limit as n→∞,we have G(Sv,u,u)≾kG(Tv,u,u).
Therefore,|G(Sv,u,u)|≤k|G(Tv,u,u)| implies that Sv=u.
Therefore,Sv=Tv=u. That is, v is coincidence point of S and T. Since S and T are weakly compatible, it follows that STv=TSv; that is,  Su=Tu.
Next we show thatSu=u. On setting x=u,y=v, and  z=v in (2.3), we have
(36)G(Su,Sv,Sv)≾kG(Tu,Tv,Tv).
Therefore, |G(Su,u,u)|≤k|G(u,u,u)| implies that Su=u. Hence u is a  common fixed point of S and T.Uniqueness easily follows from Theorem20. Hence u is a unique common fixed point of S and T.



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