# Common Fixed Point Theorems Satisfying Contractive Type Conditions in Complex Valued Metric Spaces

Authors: Mahpeyker Öztürk
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Some common fixed point theorems satisfying contractive conditions involving rational expressions and product for four mappings that satisfy property (E.A) along with weak compatibility of pairs are proved and further some results using (CLR)-property are obtained in complex valued metric spaces which generalize various results of ordinary metric spaces.


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## Body

## 1. Introduction

Fixed point theory is one of the fundamental theories in nonlinear analysis which has various applications in different branches of mathematics. In this theory, to prove the existence and the uniqueness of a fixed point of operators or mappings has been a valuable research area by using the Banach contraction principle. There are many generalizations of the Banach contraction principle particularly in metric spaces. So, many researches attempted various generalizations of the concept of metric spaces such as 2-metric spaces, D-metric spaces, G-metric spaces, K-metric spaces, cone metric spaces, and probabilistic metric space.Recently, Huang and Zhang [1] generalized the concept of metric spaces, replacing the set of real numbers by an ordered Banach space; hence they have defined the cone metric spaces. They also described the convergence of sequences and introduced the notion of completeness in cone metric spaces. They have proved some fixed point theorems of contractive mappings on complete cone metric space with the assumption of normality condition of a cone. Subsequently, various authors have generalized the results of Huang and Zhang and have studied fixed point theorems in cone metric spaces over normal and nonnormal cones.Many results of analysis cannot be generalized to cone metric since the definition of these spaces is based on a Banach space which is not a division ring. So, in a recent time, Azam et al. [2] introduced and studied the notion of complex valued metric space and established some common fixed point theorems for mappings involving rational expressions which are not meaningful in cone metric spaces. Later, several authors have studied the problem of existence of uniqueness of a fixed point for mappings satisfying different type contractive conditions in the framework of complex valued metric spaces.In 2002, Aamri and Moutawakil [3] introduced the property (E.A) and pointed out that this property buys containment of ranges without any continuity requirements besides minimizing the commutativity conditions of the maps to the commutativity at their points of coincidence. Further, property (E.A) allows replacing the completeness condition of the space with a natural condition of closeness of the range. Subsequently, there are numerous papers which contain fixed point results related to property (E.A) in various metric spaces. Most recently, Sintunavarat and Kumam [4] defined the notion of the (CLR)-property (or common limit in the range property) which does not impose either completeness of the whole space or any of the range spaces or continuity of maps. The importance of this property ensures that one does not require the closeness of the range of subspaces. Various fixed point theorems have been proved by using the notion of (CLR)-property (see [4–15]).The aim of this paper is to establish common fixed point theorems for two pairs of weakly compatible self-mappings of a complex valued metric space satisfying contractive condition involving product and rational expressions using (E.A) property. Moreover, we give some results using the property common limit in the range of one of the mappings.


## 2. Basic Facts and Definitions

We recall some notations and definitions which will be utilized in our subsequent discussion.LetC be a 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).
It follows that z1⪯z2 if one of the following conditions is satisfied: (i)
Re(z1)=Re(z2),Im⁡(z1)<Im⁡(z2),(ii)
Re(z1)<Re(z2),Im⁡(z1)=Im⁡(z2),(iii)
Re(z1)<Re(z2),Im⁡(z1)<Im⁡(z2),(iv)
Re(z1)=Re(z2),Im⁡(z1)=Im⁡(z2).In (i), (ii), and (iii), we have|z1|<|z2|. In (iv), we have |z1|=|z2|. So |z1|≤|z2|. In particular, we will write z1⪯z2 if z1≠z2 and one of (i), (ii), and (iii) is satisfied. In this case |z1|<|z2|. We will write z1≺z2 if and only if (iii) is satisfied.Take into account some fundamental properties of the partial order⪯ on C as follows.(i)
If0⪯z1⪯z2, then |z1|<|z2|.(ii)
Ifz1⪯z2,z2≺z3, then z1≺z3.(iii)
Ifz1⪯z2 and λ≥0 is a real number, then λz1⪯λz2.Definition 1 (see [12]).
The “max” function for the partial order relation “⪯” is defined by the following. (i)
max⁡{z1,z2}=z2 if and only if z1⪯z2.(ii)
Ifz1⪯max⁡{z2,z3}, then z1⪯z2, or z1⪯z3.(iii)
max⁡{z1,z2}=z2 if and only if z1⪯z2 or |z1|≤|z2|.Using Definition1 one can have the following lemma.Lemma 2 (see [12]).
Letz1,z2,z3,…∈C and the partial order relation ⪯ is defined on C. Then the following statements are easy to prove. (i)
Ifz1⪯max⁡{z2,z3}, then z1⪯z2 if z3⪯z2.(ii)
Ifz1⪯max⁡{z2,z3,z4}, then z1≤z2 if max⁡{z3,z4}⪯z2.(iii)
Ifz1⪯max⁡{z2,z3,z4,z5}, then z1≤z2 if max⁡{z3,z4,z5}⪯z2.Now we give the definition of complex valued metric space which has been introduced by Azam et al. [2].Definition 3.
LetX be a nonempty set. If a mapping d:X×X→C satisfies (C1) 0⪯d(x,y) for all x,y∈X and d(x,y)=0⇔x=y,(C2) d(x,y)=d(y,x) for all x,y∈X,(C3) d(x,y)⪯d(x,z)+d(z,y), for all x,y,z∈X,
thend is called a complex valued metric on X and the pair (X,d) is called a complex valued metric space.Let{xn} be a sequence in a complex valued metric space X and x∈X. If for every ε∈C with ε≻0 there is N∈N such that, for all n>N,  d(xn,x)≺ε, then x is called the limit of {xn} and is written as limn→∞xn=x as n→∞. If for every ε∈C with ε≻0 there is N∈N such that, for all n,m>N,  d(xn,xm)≺ε, then {xn} is called a Cauchy sequence in X. If every Cauchy sequence is convergent in X, then X is called a complete complex valued metric space.The following lemma has been given in [2] that we utilize to prove the theorems.Lemma 4.
LetX be a complex valued metric space and {xn} a sequence in X. Then (i)
{xn} converges to x∈X if and only if  |d(xn,x)|→0 as n→∞;(ii)
{xn} is a Cauchy sequence if and only if  |d(xn,xm)|→0 as n,m→∞.Definition 5 (see [14]).
A pair of self-mappingsS,T:X→X is called weakly compatible if they commute at their coincidence point; that is, if there is a point z∈X such that Sz=Tz, then STz=TSz, for each z∈X.The definition of property (E.A) has been introduced by Aamri and Moutawakil in [3] and redefined by Verma and Pathak [12] in complex valued metric spaces.Definition 6.
LetS,T:X→X be two self-mappings of a complex valued metric space (X,d). The pair (S,T) is said to satisfy property (E.A), if there exists a sequence {xn} in X such that
(2)lim⁡n→∞⁡d(Sxn,u)=lim⁡n→∞⁡d(Txn,u)=0,
for some u∈X.Example 7.
LetX=C be endowed with the complex valued metric d:C×C→C as
(3)d(z1,z2)=|x1-x2|+i|y1-y2|,
where z1=x1+iy1 and x2+iy2. Then (C,d) is a complete complex valued metric space. Define the mappings T,S:X→X as Tz=2z-1,  Sz=z2 for all z∈X and consider the sequence {zn}={1+i/2n}. Thus we obtain
(4)lim⁡n→∞d(Tzn,z)=lim⁡n→∞d(Szn,z),lim⁡n→∞d(2(1+i12n)-1,1)=lim⁡n→∞d((1+i12n)2,1)=0,
where z=1 is the limit of sequence {zn}. Hence the pair (S,T) satisfies property (E.A).Definition 8 (see [15]).
LetS and T be two self-mappings of complex valued metric space X. S and T are said to satisfy the common limit in the range of S property if
(5)lim⁡n→∞⁡d(Sxn,Sx)=lim⁡n→∞⁡d(Txn,Sx)=0,
for some x∈X.Example 9.
LetX and d be given as in Example 7. Define S,T:X→X by Tz=5x+i5y and Sz=x+4+iy for all x,y∈R. Consider the sequence {zn}={xn+iyn}={1+i/n}. Then for z=1, with an easy calculation, we see that
(6)lim⁡n→∞d(Tzn,Sz)=lim⁡n→∞d(Szn,Sz)lim⁡n→∞d(5+i5n,5)=lim⁡n→∞d(5+i1n,5)=0.
HenceT,S satisfy the common limit in the range of S property ((CLRS)-property).


## 3. Main Results

In this section, initially, some common fixed point results for the pairs, which are weakly compatible and satisfy property (E.A), have been proved, by reconstructing the contractive conditions given in [16].Theorem 10.
Let(X,d) be a complex valued metric space and let S,T,I,J:X→X be four self-mappings satisfying the following: (i)
T(X)⊆I(X),  S(X)⊆J(X);(ii)
(7)[d(Sx,Ty)]3⪯αd(Ix,Jy)d(Ix,Sx)d(Jy,Ty),
for all x,y∈X, where α∈(0,1);(iii)
the pairs{S,I} and {T,J} are weakly compatible;(iv)
one of the pairs{S,I} or {T,J} satisfies property (E.A).
If the range of one of the mappingsJ(X) or I(X) is a complete subspace of X, then the mappings I,J,S, and T have a unique common fixed point in X.Proof.
Suppose that the pair{T,J} satisfies property (E.A). Then there exists a sequence {xn} in X such that
(8)lim⁡n→∞⁡Txn=lim⁡n→∞⁡Jxn=z,
for some z∈X. Further, since T(X)⊆I(X), there exists a sequence {yn} in X such that Txn=Iyn. Hence limn→∞Iyn=z. Our claim is limn→∞Syn=z. Using condition (7), we have
(9)[d(Syn,Txn)]3⪯αd(Iyn,Jxn)d(Iyn,Syn)d(Jxn,Txn)=αd(Txn,Jxn)d(Txn,Syn)d(Txn,Jxn).
By dividing two sides of the above inequality withd(Syn,Txn) we get
(10)[d(Syn,Txn)]2⪯α[d(Txn,Jxn)]2.
Thus|d(Syn,Txn)|≤α|d(Txn,Jxn)|=0 and letting n→∞ we have
(11)lim⁡n→∞⁡Syn=lim⁡n→∞⁡Txn=z.
Now, suppose thatI(X) is complete subspace of X; then z=Iu for some u∈X. Subsequently, we obtain
(12)lim⁡n→∞⁡Syn=lim⁡n→∞⁡Txn=lim⁡n→∞⁡Jxn=lim⁡n→∞⁡Iyn=z=Iu.
We claim thatSu=Iu. To prove this, in (7)
(13)[d(Su,Txn)]3⪯αd(Iu,Jxn)d(Iu,Su)d(Jxn,Txn),
and letting n→∞ and using (12) we have
(14)|[d(Su,z)]3|≤α|d(z,z)d(z,Su)d(z,z)|=0,
and consequently Su=z=Iu. Thus z is a coincidence point of {S,I}. Weak compatibility of the pair {S,I} implies that
(15)SIu=ISu=Sz=Iz.
Conversely, sinceS(X)⊆J(X), there exists v∈X such that Su=Jv. Hence Su=Iu=Jv=z. Now we show that v is a coincidence point of {T,J}; that is, Tv=Jv=z. Putting x=u, y=v in (7), we get
(16)[d(Su,Tv)]3⪯αd(Iu,Jv)d(Iu,Su)d(Jv,Tv),|[d(z,Tv)]3|≤|αd(z,Jv)d(z,z)d(Jv,Tv)|;
thus Tv=z. Hence Tv=Jv=z and v is a coincidence point of T and J. Weak compatibility of the pair {T,J} implies that
(17)TJv=JTv=Tz=Jz.
Therefore, z is a common coincidence point of S,T,I, and J.
In order to show thatz is a common fixed point of these mappings, we write in (7)
(18)[d(z,Tz)]3=[d(Su,Tz)]3⪯αd(Iu,Jz)d(Iu,Su)d(Jz,Tz)|d(z,Tz)|3≤0.
Thus,
(19)Sz=Iz=Jz=Tz=z.
A similar argument derives if we assume thatJ(X) is a complete subspace of X and also using the property (E.A) of the pair {S,J} gives us the same result.
Uniqueness. To prove that z is a unique common fixed point, let us suppose that z* is another common fixed point of I,J,S, and T. In (7) take x=z* and y=z; then
(20)[d(z*,z)]3=[d(Sz*,Tz)]3⪯αd(Iz,Jz*)d(Iz,Sz)d(Jz*,Tz*)hhhhhhhhhhhhh|[d(z*,z)]3|≤0
is a contradiction. Thus z=z*. Consequently, Sz=Iz=Tz=Jz=z and z is the unique common fixed point of I,J,S, and T.PuttingJ=I in Theorem 10 we have the following corollary.Corollary 11.
LetS,T, and I be three self-mappings of a complex valued metric space (X,d) satisfying the inequality
(21)[d(Sx,Ty)]3⪯αd(Ix,Iy)d(Ix,Sx)d(Iy,Ty)
for all x,y in X, where α∈(0,1). Suppose that the following conditions hold: (i)
I(X)⊇S(X)⋃‍T(X),(ii)
both the pairs{I,S} and {I,T} are weakly compatible,(iii)
one of the pairs{I,S} and {I,T} satisfies the property (E.A).
IfI(X) is complete subspace of X, then S,T, and I have a unique common fixed point in X.In Theorem10, if we put S=T and I=J, we have the following.Corollary 12.
Let(X,d) be a complex valued metric space and let S and T be two self-mappings satisfying the following: (i)
S(X)⊆I(X);(ii)
(22)[d(Sx,Sy)]3⪯αd(Ix,Iy)d(Ix,Sx)d(Iy,Sy);(iii)
{I,S} is a weakly compatible pair;(iv)
the pair{I,S} satisfies property (E.A).
IfI(X) is complete subspace of  X, then  I  and  S  have the unique common fixed point in X.Theorem 13.
LetS,I and T,J be four self-mappings of a complex valued metric space (X,d) satisfying the following: (i)
T(X)⊆I(X) and S(X)⊆J(X);(ii)
(23)d(Sx,Ty)⪯α(max⁡{12[d(Ix,Jy)]2,[d(Ix,Sx)]2,hhhhhhhhhh[d(Jy,Ty)]2,12[d(Ix,Ty)]2,hhhhhhhhhh12[d(Jy,Sx)]2}hhhhh×(d(Ix,Sx)+d(Jy,Ty))-1{12[d(Ix,Jy)]2,[d(Ix,Sx)]2,)
if  d(Ix,Sx)+d(Jy,Ty)≠0 where α∈(0,1) and “max⁡” is as in Definition 1, or
(24)d(Sx,Ty)=0ifd(Ix,Sx)+d(Jy,Ty)=0,
for all x,y∈X;(iii)
the pairs{S,I} and {T,J} are weakly compatible;(iv)
one of the pairs{S,I} or {T,J} satisfies property (E.A).
If the range of one of the mappingsJ(X) or I(X) is a complete subspace of  X, then the mappings  I,J,S, and  T  have a unique common fixed point in  X.Proof.
Let us suppose thatd(Ix,Sx)+d(Jy,Ty)≠0 so d(Sx,Ty)≠0 and the pair {T,J} satisfies property (E.A). Then there exists a sequence {xn} in X such that
(25)lim⁡n→∞⁡Txn=lim⁡n→∞⁡Jxn=z,
for some z∈X. Since T(X)⊆I(X), there exists a sequence {yn} in X such that Txn=Iyn. Hence limn→∞Iyn=z. We show that limn→∞Syn=z. In inequality (23), putting x=yn and y=xn we get
(26)d(Syn,Txn)⪯α(max⁡{12[d(Iyn,Jxn)]2,[d(Iyn,Syn)]2,hhhhhhhhhh[d(Jxn,Txn)]2,12[d(Iyn,Txn)]2,hhhhhhhhhh12[d(Jxn,Syn)]2}hhhhhh×(d(Iyn,Syn)+d(Jxn,Txn))-1{12[d(Iyn,Jxn)]2,[d(Iyn,Syn)]2,)=α((d(Txn,Syn)+d(Jxn,Txn))-1max⁡{12[d(Txn,Jxn)]2,[d(Txn,Syn)]2,hhhhhhhhhh0,12[d(Jxn,Syn)]2}hhhhh×(d(Txn,Syn)+d(Jxn,Txn))-1{12[d(Iyn,Jxn)]2,[d(Iyn,Syn)]2,).
Thus,(27)|d(Syn,Txn)|≤α|max⁡{12[d(Txn,Jxn)]2,[d(Txn,Syn)]2,hhhhhhhhh0,12[d(Jxn,Syn)]2}hhhh×(d(Txn,Syn)+d(Jxn,Txn))-1{12[d(Txn,Jxn)]2,[d(Txn,Syn)]2,|,
and letting n→∞ we have
(28)(1-α)|d(Syn,z)|≤0
which is a contradiction since α∈(0,1). Therefore,
(29)lim⁡n→∞⁡Syn=lim⁡n→∞⁡Txn=z.
AssumingI(X) is complete subspace of X, then z=Iu for some u∈X. Right after, we obtain
(30)lim⁡n→∞Syn=lim⁡n→∞⁡Txn=lim⁡n→∞⁡Jxn=lim⁡n→∞⁡Iyn=z=Iu.
Our aim is to proveSu=Iu and for this putting x=u and y=xn in (23) we get
(31)d(Su,Txn)⪯α((d(Iu,Su)+d(Jxn,Txn))-1max⁡{12[d(Iu,Jxn)]2,[d(Iu,Su)]2,hhhhhhhhhh[d(Jxn,Txn)]2,hhhhhhhhhh12[d(Iu,Txn)]2,12[d(Jxn,Su)]2}hhhh×(d(Iu,Su)+d(Jxn,Txn))-1{12[d(Iu,Jxn)]2,[d(Iu,Su)]2,).
Letting n→∞ and using (30)
(32)|d(Su,z)|≤α|max⁡{0,[d(z,Su)]2,(1/2)[d(z,Su)]2}d(z,Su)|,
and hence (1-α)|d(Su,z)|≤0, and Su=z since α∈(0,1). Therefore, z is a coincidence point of {S,I}. Weak compatibility of the pair {S,I} implies that SIu=ISu=Sz=Iz.
Otherwise, sinceS(X)⊆J(X), there exists v∈X such that Su=Jv. Hence, Su=Iu=Jv=z. To show that v is a coincidence point of pair {T,J}, by using similar arguments in Theorem 10 and inequality (23) we have
(33)d(Su,Tv)⪯α(max⁡{12[d(Iu,Jv)]2,[d(Iu,Su)]2,hhhhhhhhhhh[d(Jv,Tv)]2,hhhhhhhhhhh12[d(Iu,Tv)]2,12[d(Jv,Su)]2}hhhhhh×(d(Iu,Su)+d(Jv,Tv))-1{12[d(Iu,Jv)]2,[d(Iu,Su)]2,[d(Jv,Tv)]2,),|d(z,Tv)|≤α|max⁡{0,[d(z,Tv)]2,(1/2)[d(z,Tv)]2}d(z,Tv)|≤α|d(z,Tv)|,
and then Tv=z because α∈(0,1). With the same assertions as in Theorem 13 one gets that z is a common coincidence point of S,T,I, and J.
Other details of Theorem13, in which z is a unique common fixed point of the mappings I,J,S, and T, can be obtained in view of the final part of the proof of Theorem 10 with suitable modifications.
In concluding, we note that the conclusions of Theorem13 are still valid if we replace inequality (23) with the following inequality:
(34)d(Sx,Ty)⪯αmax⁡{d(Ix,Ty)[1+d(Ix,Sx)+d(Jy,Sx)]2[1+d(Ix,Jy)],hhhhhhhhhhhd(Jy,Sx)[1+d(Ix,Ty)+d(Jy,Ty)]2[1+d(Ix,Jy)]},
where α∈(0,1) and the mappings S,I and T,J are defined as in Theorem 13.Finally, at the end of the section some common fixed point theorems for weakly compatible pairs which satisfy the(CLR)-property have been proved.Theorem 14.
LetS,I and  T,J be four self-mappings of a complex valued metric space (X,d) satisfying the following: (i)
T(X)⊆I(X) and S(X)⊆J(X);(ii)
(35)d(Sx,Ty)⪯αmax⁡{d(Ix,Ty)[1+d(Ix,Sx)+d(Jy,Sx)]2[1+d(Ix,Jy)],hhhhhhhhhhhd(Jy,Sx)[1+d(Ix,Ty)+d(Jy,Ty)]2[1+d(Ix,Jy)]}
for all  x,y∈X where α∈(0,1) and “max⁡” is as in Definition 1;(iii)
{S,I} and {T,J} are weakly compatible pairs.
If the pair{S,I} satisfies (CLRS)-property, or the pair {T,J} satisfies (CLRT)-property, then the mappings  I,J,S, and  T have a unique common fixed point in  X.Proof.
Let us suppose that the pair{T,J} satisfies (CLRT)-property; then by Definition 8 there exists a sequence {xn}∈X such that
(36)lim⁡n→∞Txn=lim⁡n→∞Jxn=Tx
for some x∈X. And also, since T(X)⊆I(X), we have Tx=Iz for some z∈X. We claim that Sz=Iz=u. Then putting x=z and y=xn in inequality (35) we have
(37)d(Sz,Txn)⪯αmax⁡{d(Iz,Txn)[1+d(Iz,Sz)+d(Jxn,Sz)]2[1+d(Iz,Jxn)],hhhhhhhhhhhd(Jxn,Sz)[1+d(Iz,Txn)+d(Jxn,Txn)]2[1+d(Iz,Jxn)]}.
If n tends to infinity and with equality (36), then
(38)|d(Sz,Iz)|≤α|max⁡{0,d(Iz,Sz)2}|,(1-α2)|d(Sz,Iz)|≤0,
which is possible for Iz=Sz since α∈(0,1). Therefore, Sz=Iz=u; that is, z is a coincidence point of the pair {S,I}. Also weak compatibility of the mappings S and I implies the following equality:
(39)ISz=SIz=Iu=Su.
Besides, sinceS(X)⊆J(X), there exist some ω∈X such that Sz=Jω. We claim that Tω=u. Then from (35), we have
(40)d(Sz,Tω)⪯αmax⁡{d(Iz,Tω)[1+d(Iz,Sz)+d(Jω,Sz)]2[1+d(Iz,Jω)],hhhhhhhhhhhd(Jω,Sz)[1+d(Iz,Tω)+d(Jω,Tω)]2[1+d(Iz,Jω)]},d(u,Tω)⪯αmax⁡{d(u,Tω)[1+d(u,u)+d(Sz,Sz)]2[1+d(u,u)],hhhhhhhhhhhd(Sz,Sz)[1+d(u,Tω)+d(u,Tω)]2[1+d(u,u)]}.
Thus,(41)[d(u,Tω)]⪯αmax⁡{12d(u,Tω),0},(1-α2)|d(u,Tω)|≤0,
which implies that Tω=u, since α∈(0,1). Hence,
(42)Iz=Sz=u=Tω=Jω,
and this shows that ω is a coincidence point of the pair {T,J}. Weak compatibility of the pair {T,J} yields that TJω=JTω=Tu=Ju. In conclusion we show that u is a common fixed point of I,J,S, and T. Using (35), we get
(43)d(u,Tu)=d(Sz,Tu)⪯αmax⁡{d(Iz,Tu)[1+d(Iz,Sz)+d(Ju,Sz)]2[1+d(Iz,Ju)],hhhhhhhhhhhd(Ju,Sz)[1+d(Iz,Tu)+d(Ju,Tu)]2[1+d(Iz,Ju)]},d(u,Tu)=d(Sz,Tu)⪯αmax⁡{d(u,Tu)[1+d(u,u)+d(Tu,u)]2[1+d(u,Tu)],hhhhhhhhhhhd(Tu,u)[1+d(u,Tu)+d(Tu,Tu)]2[1+d(u,Tu)]},(1-α2)|d(u,Tu)|≤0,
and hence Tu=u which is the desired result. The uniqueness of common fixed point u follows easily. The details of the proof of this theorem can be obtained by using the argument that the pair {S,I} satisfies (CLRS)-property with suitable modifications. This completes the proof.Theorem14 is still true if we replace condition (35) with the following condition:
(44)d(Sx,Ty)≤α(max⁡{12[d(Ix,Jy)]2,[d(Ix,Sx)]2,hhhhhhhhhhhhhh[d(Jy,Ty)]2,hhhhhhhhhhhhhh12[d(Ix,Ty)]2,12[d(Jy,Sx)]2}hhhhhhh×(d(Ix,Sx)+d(Jy,Ty))-1{12[d(Ix,Jy)]2,[d(Ix,Sx)]2,),
where α∈(0,1) and the mappings S,I and T,J are defined as in Theorem 14.Theorem 15.
Let(X,d) be a complex valued metric space and let S,T,I,J:X→X be four self-mappings satisfying the following: (i)
T(X)⊆I(X), S(X)⊆J(X);(ii)
[d(Sx,Ty)]3⪯αd(Ix,Jy)d(Ix,Sx)d(Jy,Ty), for all x,y∈X, where α∈(0,1);(iii)
the pairs{S,I} and {T,J} are weakly compatible.
If the pair{S,I} satisfies (CLRS)-property, or the pair {T,J} satisfies (CLRT)-property, then the mappings I,J,S, and T have a unique common fixed point in X.Proof.
This theorem can be obtained by using a similar technique as in the above theorem. So we omit it.



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