# Some Common Fixed-Point Theorems for Generalized-Contractive-Type Mappings on Complex-Valued Metric Spaces

Authors: Chakkrid Klin-eam; Cholatis Suanoom
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Fixed-point theory in complex valued metric spaces has greatly developed in
recent times. In this paper, we prove certain common fixed-point theorems for two single-valued mappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities with generalized fixed-point theorems due to Rouzkard and Imdad (2012). This extends and subsumes many results of other authors which were obtained for mappings on complex-valued metric spaces.


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## Body

## 1. Introduction

The existence and uniqueness of fixed-point theorems of operators or mappings has been a subject of great interest since the work of Banach in 1992 [1]. The Banach contraction mapping principle is widely recognized as the source of metric fixed-point theory. A mapping T:X→X, where (X,d) is a metric space, is said to be a contraction mapping if for all x,y∈X,
(1)d(Tx,Ty)≤λd(x,y),where0<λ<1.
According to the Banach contraction mapping principle, any mapping T satisfying (1) in a complete metric space will have a unique fixed point. This principle includes different directions in different spaces adopted by mathematicians; for example, metric spaces, G-metric spaces, partial metric spaces, cone metric spaces, quasimetric spaces have already been obtained.A new space called the complex-valued metric space which is more general than well-known metric spaces has been introduced by Azam et al. [2]. Azam proved some fixed-point theorems for mappings satisfying a rational inequality. Naturally, this new idea can be utilized to define complex-valued normed spaces and complex-valued inner product spaces which, in turn, offer a wide scope for further investigation. Several authors studied many common fixed point results on complex-valued metric spaces (see [3–5]).In 2012, Rouzkard and Imdad [6] extended and improved the common fixed-point theorems which are more general than the result of Azam et al. [2].Theorem 1 (see [6, Theorem 1]).
IfS and T are self-mappings defined on a complete complex-valued metric space (X,d) satisfying the condition
(2)d(Sx,Ty)≾λd(x,y)+μd(x,Sx)d(y,Ty)1+d(x,y)+γd(y,Sx)d(x,Ty)1+d(x,y)
for all x,y∈X where λ, μ, and γ are nonnegative with λ+μ+γ<1, then S and T have a unique common fixed point.Though complex-valued metric spaces from a spacial class of cone metric spaces, yet this idea is intended to define rational expressions which are not meaningful in cone metric spaces, and thus many results of analysis cannot be generalized to cone metric spaces. The aim of this paper is to establish some common fixed-point theorems for two nonlinear general contraction mappings in complex-valued metric spaces. Our results generalized Theorem1.


## 2. Preliminaries

Letℂ be the set of complex numbers and z1,z2∈ℂ, we define a partial order ≺ and ≾ on ℂ as follows:(i)
z1≺z2 if and only if Re(z1)<Re(z2) and Im(z1)<Im(z2);(ii)
z1≾z2 if and only if Re(z1)≤Re(z2) and Im(z1)≤Im(z2).Now, we briefly review the notation about complex valued metric space and some lemma for prove our main results.Definition 2.
LetX be a nonempty set. Suppose that the mapping d:X×X→ℂ satisfies the following conditions:(d1)
0≾d(x,y) for all x,y∈X;(d2)
d(x,y)=0 if and only if x=y for all x,y∈X;(d3)
d(x,y)=d(y,x) for all x,y∈X;(d4)
d(x,y)≾d(x,z)+d(z,y) for all x,y,z∈X.
Then,d is called a complex-valued metric on X, and (X,d) is called a complex valued metric space.Definition 3.
Let(X,d) be a complex-valued metric space.(i)
A pointx∈X is called interior point of a set B⊆X whenever there exists 0≺r∈ℂ such that N(x,r):={y∈X:d(x,y)≺r}⊆B.(ii)
A pointx∈X is called limit point of a set B⊆X whenever for every 0≺r∈ℂ such that N(x,y)∩(X-B)≠∅.(iii)
A subsetB⊆X is called open whenever each element of B is an interior point of B.(iv)
A subsetB⊆X is called closed whenever each limit point of B belongs to B.(v)
The familyF={N(x,r):x∈X,0≺r} is a subbasis for a topology on X. We denote this complex topology by τc. Indeed, the topology τc is Hausdorff.Definition 4 (see [2]).
Let(X,d) be a complex-valued metric space, and let {xn} be a sequence in X and x∈X.(i)
If for everyc∈ℂ, with 0≺c there is N∈ℕ such that for all n>N, d(xn,x)≺c, then {xn} is said to be convergent, {xn} converges to x and x is limit point of {xn}. We denote this by xn→x as n→∞ or limn→∞xn=x.(ii)
If for everyc∈ℂ, with 0≺c there is N∈ℕ such that for all n>N, d(xn,xn+m)≺c, where m∈ℕ, then {xn} is said to be Cauchy sequence.(iii)
If every Cauchy sequence inX is convergent, then (X,d) is said to be a complete complex-valued metric space.Lemma 5 (see [2]).
Let(X,d) be a complex-valued metric space, and let {xn} be a sequence in X. Then, {xn} converges to x if and only if |d(xn,x)|→0 as n→∞.Lemma 6 (see [2]).
Let(X,d) be a complex-valued metric space, and let {xn} be a sequence in X. Then, {xn} is a Cauchy sequence if and only if |d(xn,xn+m)|→0 as n→∞, where m∈ℕ.Definition 7.
Two families of self-mappings{Ti}i=1m and {Si}i=1n are said to be pairwise commuting if:(i)
TiTj=TjTi, i,j∈{1,2,…,m}.(ii)
SiSj=SjSi, i,j∈{1,2,…,n}.(iii)
TiSj=SjTi, i∈{1,2,…,m}, j∈{1,2,…,n}.Definition 8.
LetS and T be self-mappings of a nonempty set X.(i)
A pointx∈X is said to be a fixed point of T if Tx=x.(ii)
A pointx∈X is said to be a common fixed point of T and S if Tx=Sx=x.Remark 9.
We obtain that the following statements hold:(i)
Ifz1≾z2 and z2≾z3, then z1≾z3.(ii)
Ifz∈ℂ, a,b∈ℝ, and a≤b, then az≾bz.(iii)
If0≾z1≾z2, then |z1|≤|z2|.


## 3. Main Results

In this section, we will prove some common fixed-point theorems for the generalized contractive mappings in complex-valued metric space.Theorem 10.
IfS and T are self-mappings defined on a complete complex valued metric space (X,d) satisfying the condition
(3)d(Sx,Ty)≾Ad(x,y)+Bd(x,Sx)d(y,Ty)1+d(x,y)+Cd(y,Sx)d(x,Ty)1+d(x,y)+Dd(x,Sx)d(x,Ty)1+d(x,y)+Ed(y,Sx)d(y,Ty)1+d(x,y)
for all x,y∈X, where A, B, C, D, and E are nonnegative with A+B+C+2D+2E<1, then S and T have a unique common fixed point.Proof.
Letx0 be an arbitrary in X. Since S(X)⊆X and T(X)⊆X, we construct the sequence {xk} in X such that x2k+1=Sx2k and x2k+2=Tx2k+1 for all k≥0. From the definition of {xk} and (3), we obtain that
(4)d(x2k+1,x2k+2)=d(Sx2k,Tx2k+1)≾Ad(x2k,x2k+1)+Bd(x2k,Sx2k)d(x2k+1,Tx2k+1)1+d(x2k,x2k+1)+Cd(x2k+1,Sx2k)d(x2k,Tx2k+1)1+d(x2k,x2k+1)+Dd(x2k,Sx2k)d(x2k,Tx2k+1)1+d(x2k,x2k+1)+Ed(x2k+1,Sx2k)d(x2k+1,Tx2k+1)1+d(x2k,x2k+1).
Since x2k+1=Sx2k implies that d(x2k+1,Sx2k)=0; therefore,
(5)d(x2k+1,x2k+2)≾Ad(x2k,x2k+1)+Bd(x2k,Sx2k)d(x2k+1,Tx2k+1)1+d(x2k,x2k+1)+Dd(x2k,Sx2k)d(x2k,Tx2k+1)1+d(x2k,x2k+1),
by Remark 9 and |1+d(x2k,x2k+1)|>|d(x2k,x2k+1)|, we have
(6)|d(x2k+1,x2k+2)|≤A|d(x2k,x2k+1)|+B|d(x2k,Sx2k)||d(x2k+1,Tx2k+1)||1+d(x2k,x2k+1)|+D|d(x2k,Sx2k)||d(x2k,Tx2k+1)||1+d(x2k,x2k+1)|=A|d(x2k,x2k+1)|+B|d(x2k+1,x2k+2)||d(x2k,x2k+1)||1+d(x2k,x2k+1)|+D|d(x2k,x2k+2)||d(x2k,x2k+1)||1+d(x2k,x2k+1)|<A|d(x2k,x2k+1)|+B|d(x2k+1,x2k+2)|+D|d(x2k,x2k+2)|.
From (6) and Definition 2, we have
(7)|d(x2k+1,x2k+2)|<A|d(x2k,x2k+1)|+B|d(x2k+1,x2k+2)|+D|d(x2k,x2k+1)|+D|d(x2k+1,x2k+2)|
it follows that |d(x2k+1,x2k+2)|<((A+D)/(1-B-D))|d(x2k,x2k+1)|.
Similarly, we get(8)d(x2k+2,x2k+3)=d(x2k+3,x2k+2)=d(Sx2k+2,Tx2k+1)≾Ad(x2k+2,x2k+1)+Bd(x2k+2,Sx2k+2)d(x2k+1,Tx2k+1)1+d(x2k+2,x2k+1)+Cd(x2k+1,Sx2k+2)d(x2k+2,Tx2k+1)1+d(x2k+2,x2k+1)+Dd(x2k+2,Sx2k+2)d(x2k+2,Tx2k+1)1+d(x2k+2,x2k+1)+Ed(x2k+1,Sx2k+2)d(x2k+1,Tx2k+1)1+d(x2k+2,x2k+1).
Since x2k+2=Tx2k+1 implies that d(x2k+2Tx2k+1)=0; therefore,
(9)d(x2k+2,x2k+3)≾Ad(x2k+2,x2k+1)+Bd(x2k+2,Sx2k+2)d(x2k+1,Tx2k+1)1+d(x2k+2,x2k+1)+Ed(x2k+1,Sx2k+2)d(x2k+1,Tx2k+1)1+d(x2k+2,x2k+1);
by Remark 9 and |1+d(x2k+2,x2k+1)|>|d(x2k+2,x2k+1)|, we have
(10)|d(x2k+2,x2k+3)|≤A|d(x2k+2,x2k+1)|+B|d(x2k+2,Sx2k+2)||d(x2k+1,Tx2k+1)||1+d(x2k+2,x2k+1)|+E|d(x2k+1,Sx2k+2)||d(x2k+1,Tx2k+1)||1+d(x2k+2,x2k+1)|=A|d(x2k+2,x2k+1)|+B|d(x2k+2,x2k+3)||d(x2k+1,x2k+2)||1+d(x2k+2,x2k+1)|+E|d(x2k+1,x2k+3)||d(x2k+1,x2k+2)||1+d(x2k+2,x2k+1)|<A|d(x2k+2,x2k+1)|+B|d(x2k+2,x2k+3)|+E|d(x2k+1,x2k+3)|.
From (10) and Definition 2, we have
(11)|d(x2k+1,x2k+2)|<A|d(x2k+2,x2k+1)|+B|d(x2k+2,x2k+3)|+E|d(x2k+1,x2k+2)|+D|d(x2k+2,x2k+3)|;
it follows that |d(x2k+2,x2k+3)|<((A+E)/(1-B-E))|d(x2k+1,x2k+2)|.
Puttingk=max{((A+D)/(1-B-D)),((A+E)/(1-B-E))}, we obtain that
(12)|d(xn,xn+1)|≤k|d(xn-1,xn)|≤k2|d(xn-2,xn-1)|≤⋯≤kn|d(x0,x1)|∀n.
Thus, for any n∈ℕ, we have
(13)|d(xn,xm)|≤|d(xn,xn+1)|≤|d(xn+1,xn+2)|≤⋯≤|d(xm-1,xm)|≤(kn+kn+1+kn+2+⋯+km-1)|d(x0,x1)|≤(kn1-k)|d(x0,x1)|;
it follows that |d(xn,xm)|≤(kn/(1-k))|d(x0,x1)|→0 as n→∞.
By Lemma6, the sequence {xn} is a Cauchy. Since X is compete, there exists a point z∈X such that xn→z as n→∞.
Next, we will show thatSz=z. By the notion of a complete complex-valued metric d, we have
(14)d(z,Sz)≾d(z,x2k+2)+d(x2k+2,Sz)=d(z,x2k+2)+d(Sz,Tx2k+1)≾d(z,x2k+2)+Ad(z,x2k+1)+Bd(z,Sz)d(x2k+1,Tx2k+1)1+d(z,x2k+1)+Cd(x2k+1,Sz)d(z,Tx2k+1)1+d(z,x2k+1)+Dd(z,Sz)d(z,Tx2k+1)1+d(z,x2k+1)+Ed(x2k+1,Sz)d(x2k+1,Tx2k+1)1+d(z,x2k+1),
which implies that
(15)|d(z,Sz)|≤|d(z,x2k+2)|+A|d(z,x2k+1)|+B|d(z,Sz)||d(x2k+1,x2k+2)||1+d(z,x2k+1)|+C|d(x2k+1,Sz)||d(z,x2k+2)||1+d(z,x2k+1)|+D|d(z,Sz)||d(z,x2k+2)||1+d(z,x2k+1)|+E|d(x2k+1,Sz)||d(x2k+1,x2k+2)||1+d(z,x2k+1)|.
Taking k→∞, we have |d(z,Sz)|=0; it is obtained that d(z,Sz)=0. Thus, Sz=z. It follows that similarly Tz=z. Therefore, z is common fixed point of S and T.
Finally, to prove the uniqueness of common fixed point, letz*∈X be another common fixed point of S and T such that Sz*=Tz*=z*. Consider
(16)d(z,z*)=d(Sz,Tz*)≾Ad(z,z*)+Bd(z,Sz)d(z*,Tz*)1+d(z,z*)+Cd(z*,Sz)d(z,Tz*)1+d(z,z*)+Dd(z,Sz)d(z,Tz*)1+d(z,z*)+Ed(z*,Sz)d(z*,Tz*)1+d(z,z*)
so that
(17)|d(z,z*)|≤A|d(z,z*)|+C|d(z*,Sz)||d(z,Tz*)||1+d(z,z*)|=A|d(z,z*)|+C|d(z*,z)||d(z,z*)||1+d(z,z*)|.
Since |1+d(z,z*)|>|d(z,z*)|, therefore |d(z,z*)|<A|d(z,z*)|+C|d(z*,z)|=(A+C)|d(z,z*)|.
This is contradiction toA+C<1. Hence, z=z*. Therefore, z is a unique common fixed point of S and T.Corollary 11.
IfT is a self-mapping defined on a complete complex-valued metric space (X,d) satisfying the condition
(18)d(Tx,Ty)≾Ad(x,y)+Bd(x,Tx)d(y,Ty)1+d(x,y)+Cd(y,Tx)d(x,Ty)1+d(x,y)+Dd(x,Tx)d(x,Ty)1+d(x,y)+Ed(y,Tx)d(y,Ty)1+d(x,y)
for all x,y∈X, where A, B, C, D, and E are nonnegative with A+B+C+2D+2E<1, then T has a unique fixed point.Proof.
We can prove this result by applying Theorem10 by setting T=S.Corollary 12.
IfS and T are self-mappings defined on a complete complex valued metric space (X,d) satisfying the condition
(19)d(Sx,Ty)≾Ad(x,y)+Bd(x,Sx)d(y,Ty)1+d(x,y)+Cd(y,Sx)d(x,Ty)1+d(x,y)+Dd(x,Sx)d(x,Ty)1+d(x,y)
for all x,y∈X, where A, B, C, and D are nonnegative with A+B+C+2D<1, then S and T have a unique common fixed point.Proof.
We can prove this result by applying Theorem10 by setting E=0.Corollary 13.
IfT is a self-mapping defined on a complete complex valued metric space (X,d) satisfying the condition
(20)d(Tx,Ty)≾Ad(x,y)+Bd(x,Tx)d(y,Ty)1+d(x,y)+Cd(y,Tx)d(x,Ty)1+d(x,y)+Dd(x,Tx)d(x,Ty)1+d(x,y)
for all x,y∈X, where A, B, C, and D are nonnegative with A+B+C+2D<1, then T has a unique fixed point.Proof.
We can prove this result by applying Corollary12 by setting T=S and E=0.Corollary 14.
IfS and T are self-mappings defined on a complete complex valued metric space (X,d) satisfying the condition
(21)d(Sx,Ty)≾Ad(x,y)+Bd(x,Sx)d(y,Ty)1+d(x,y)+Cd(y,Sx)d(x,Ty)1+d(x,y)+Ed(y,Sx)d(y,Ty)1+d(x,y)
for all x,y∈X where A, B, C, and E are nonnegative with A+B+C+2E<1, then S and T have a unique common fixed point.Proof.
We can prove this result by applying Theorem10 by setting D=0.Corollary 15.
IfT is a self-mapping defined on a complete complex valued metric space (X,d) satisfying the condition
(22)d(Tx,Ty)≾Ad(x,y)+Bd(x,Tx)d(y,Ty)1+d(x,y)+Cd(y,Tx)d(x,Ty)1+d(x,y)+Dd(x,Tx)d(x,Ty)1+d(x,y)+Ed(y,Tx)d(y,Ty)1+d(x,y)
for all x,y∈X, where A, B, C, and E are nonnegative with A+B+C+2E<1, then T has a unique fixed point.Proof.
We can prove this result by applying Corollary14 by setting T=S.Remark 16.
(i) By choosingD=E=0 in Theorem 10, we get Theorem 1 of [6].
(ii) By choosingD=E=0 and S=T in Theorem 10, we get Corollary 3 of [6].
(iii) By choosingC=D=E=0 in Theorem 10, we get Theorem 4 of Azam et al. [2].
(iv) By choosingC=D=E=0 and S=T in Theorem 10, we get Corollary 5 of Azam et al. [2].Theorem 17.
If{Ti}1m and {Si}1n are two finite pairwise commuting finite families of self-mapping defined on complete complex-valued metric space (X,d) such that the mappings T and S with T=T1T2⋯Tm and S=S1S2⋯Sn satisfy condition (3), then the component maps of the two families {Ti}1m and {Si}1n have a unique common fixed point.Proof.
By Theorem10, one can infer that T and S have a unique common fixed point Z(i.e.,Tz=Sz=z). Now, we will show that z is a common fixed point of all the component maps of both families. In view of pairwise commutativity of the families {Ti}1m and {Si}1n, for every 1≤k≤m, we can write
(23)Tkz=TkSz=STkz,Tkz=TkTz=TTkz.
It implies that Tkz(∀k) is also a common fixed point of T and S. By using the uniqueness of common fixed point, we have Tkz=z(∀k). Hence, z is a common fixed point of the family {Ti}1m. Similarly, we can show that z is a common fixed point of the family {Si}1n. This completes the proof of the theorem.Corollary 18.
IfF and G are self-mappings defined on a complete complex-valued metric space (X,d) satisfying the condition
(24)d(Fmx,Gny)≾Ad(x,y)+Bd(x,Fmx)d(yGny)1+d(x,y)+Cd(y,Fmx)d(x,Gny)1+d(x,y)+Dd(x,Fmx)d(x,Gny)1+d(x,y)+Ed(y,Fmx)d(y,Gny)1+d(x,y)
for all x,y∈X, where A, B, C, D, and E are nonnegative with A+B+C+2D+2E<1, then F and G have a unique common fixed point.Proof.
We can prove this result by applying Theorem17 by setting T1=T2=⋯=Tm=F and S1=S2=⋯=Sn=G.Corollary 19.
IfT is a self-mapping defined on a complete complex valued metric space (X,d) satisfying the condition
(25)d(Tnx,Tny)≾Ad(x,y)+Bd(x,Tnx)d(y,Tny)1+d(x,y)+Cd(y,Tnx)d(x,Tny)1+d(x,y)+Dd(x,Tnx)d(x,Tny)1+d(x,y)+Ed(y,Tnx)d(y,Tny)1+d(x,y)
for all x,y∈X, where A, B, C, D, and E are nonnegative with A+B+C+2D+2E<1, then T has a unique fixed point.Proof.
We can prove this result by applying Corollary18 by setting F=G=T.Remark 20.
(i) By choosingD=E=0 in Theorem 17, we get Theorem 1 of [6].
(ii) By choosingD=E=0 in Corollary 18, we get Corollary 6 of [6].
(iii) By choosingD=E=0 in Corollary 19, we get Corollary 7 of [6].
(iv) By choosingC=D=E=0 in Corollary 19, we get Corollary 6 of Azam et al. [2].Corollary 21 (see [5]).
IfT:X→X is a mapping defined on a complete complex-valued metric space (X,d) satisfying the condition
(26)d(Tnx,Tny)≾λd(x,y)
for all x,y∈X, where λ is nonnegative reals λ<1, then T has a unique fixed point.The following example demonstrates the superiority of Bryant theorem [5] over Banach contraction theorem.Example 22.
LetX=ℂ be the set of complex numbers. Define d:ℂ×ℂ→ℂ as
(27)d(z1,z2)=|x1-x2|+i|y1-y2|,
where z1=x1+iy1 and z2=x2+iy2. Then, (ℂ,d) is a complete complex-valued metric space. Define T:ℂ→ℂ as
(28)T(x+iy)={0;x,y∈Q,2+2i;x,y∈Qc,2;x∈Qc,y∈Q,2i;x∈Q,y∈Qc.
Now, for x=1/3 and y=0, we get
(29)d(T(13),T(0))=d(2,0)=2≾λd(13,0)=λ13.
Thus, λ≥3, which is a contradiction as 0≤λ<1. However, notice that T2z=0, so that 0=d(T2z1,T2z2)≾λd(z1,z2), which shows that T2 satisfies the requirement of Bryant theorem and z=0 is the unique fixed point of T.



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