# Common Fixed Points of Generalized Cocyclic Mappings in Complex Valued Metric Spaces

Authors: Mujahid Abbas; Manuel De la Sen; Talat Nazir
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

We present fixed point results of mappings satisfying generalized contractive conditions in complex valued metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of generalized contractive-type mappings involved in cocyclic representation of a nonempty subset of a complex valued metric space are also obtained. Some examples are also presented to support the results proved herein. These results extend and generalize many results in the existing literature.


---

## Body

## 1. Introduction and Preliminaries

Fixed point theory is one of the famous and traditional theories in mathematics that provide important and traditional tools for proving the existence of solutions of many problems in both applied and pure mathematics. In metric fixed point theory, the interplay between contractive condition and the existence and uniqueness of a fixed point has been very strong and fruitful. There are many applications to the study of fixed points of mappings which satisfy contractive conditions related to the solutions of differential and integral equations [1–3]. Fixed point results for mappings inspired and motivated the development of many other important kinds of points like equilibrium points, coincidence points, periodic points, sectional points, intersection points, and so forth.Banach’s fixed point theorem is one of the basic and most widely applied fixed point theorems in mathematical analysis. Kirk et al. [4] established fixed point theorems for mappings satisfying cyclical contractive conditions. The cyclic contraction mappings need not be continuous, while the Banach contraction mappings are continuous mappings. Păcurar and Rus [5] proved some fixed point results for cyclic weakly contractive mappings. Karapinar [6] obtained fixed point results for cyclic weakly contractive mappings. Derafshpour and Rezapour [7] proved some results on the existence of best proximity points of cyclic contractions. Recently, Abbas et al. [8] established some common fixed point results of rational-type cocyclic mappings in the setup of multiplicative metric spaces. For further interesting results in this direction, we refer readers to [9–17]. There are many generalizations of metric spaces such as partial metric spaces, generalized metric spaces, cone metric spaces, and quasimetric spaces. Recently, Azam et al. [18] obtained the generalization of Banach’s contraction principal by introducing the concept 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, some other useful results appeared in [19–24]. Abbas et al. [25] obtained the well-posedness and periodic point property of mappings satisfying a rational inequality in an ordered complex valued metric space. Recently, Nashine et al. [26] proved common fixed point theorems under rational contractions in complex valued metric spaces.In this paper, we establish fixed point result of a generalized contractive mapping in the setup of complex valued metric spaces. Further, by employing this result, a common fixed point of a pair of weakly compatible mappings is obtained. We study the sufficient conditions for the existence of common fixed points of pair of rational contractive-type mappings involved in cocyclic representation of a nonempty subset of a complex valued metric space. Our results generalize and extend comparable results.ByR
,
R
+
,
C
,
C
+, and N we denote the set of all real numbers, the set of all nonnegative real numbers, the set of all complex numbers, the set of all nonnegative complex numbers, and the set of all natural numbers, respectively.Consistent with Azam et al. [18], the following definitions and results will be needed in the sequel.Letz
1
,
z
2
∈
C. Define a partial order ⪯ on C as follows:z
1
⪯
z
2
if and only ifR
e
(
z
1
)
≤
R
e
(
z
2
), I
m
(
z
1
)
≤
I
m
(
z
2
).It follows that z
1
⪯
z
2 if one of the following conditions is satisfied:(1)
R
e
z
1
=
R
e
z
2
,
I
m
(
z
1
)
<
I
m
(
z
2
).
(2)
R
e
(
z
1
)
<
R
e
(
z
2
)
,
I
m
(
z
1
)
=
I
m
(
z
2
).
(3)
R
e
(
z
1
)
<
R
e
(
z
2
)
,
I
m
(
z
1
)
<
I
m
(
z
2
).
(4)
R
e
(
z
1
)
=
R
e
(
z
2
)
,
I
m
(
z
1
)
=
I
m
(
z
2
).In particular, we will writez
1
⪯
z
2 if one of (1), (2), and (3) is satisfied and we will write z
1
≺
z
2 if only (3) is satisfied.The partial order⪯ on C satisfies the following properties:(i)
Ifθ
⪯
z
1
⪯
z
2, then z
1
≤
z
2, where θ is the zero of complex number.
(ii)
z
1
⪯
z
2 is equivalent to z
1
-
z
2
⪯
θ
.
(iii)
Ifz
1
⪯
z
2 and r
≥
0 is a real number, then r
z
1
⪯
r
z
2
.
(iv)
Ifθ
⪯
z
1 and θ
⪯
z
2 with z
1
+
z
2
≠
θ, then z
1
2
/
z
1
+
z
2
⪯
z
1
.
(v)
θ
⪯
z
1 and θ
⪯
z
2 do not imply θ
⪯
z
1
z
2
.
(vi)
θ
⪯
z
1 does not imply θ
⪯
1
/
z
1. Moreover, if θ
⪯
z
1 and θ
⪯
1
/
z
1, then I
m
(
z
1
)
=
0
.Definition 1.
LetX be a nonempty set. Suppose that the mapping d
:
X
×
X
→
C satisfies the conditions(1)
θ
⪯
d
(
x
,
y
) for all x
,
y
∈
X and d
(
x
,
y
)
=
θ if and only if x
=
y;
(2)
d
(
x
,
y
)
=
d
(
y
,
x
) for all x
,
y
∈
X;
(3)
d
(
x
,
y
)
⪯
d
(
x
,
z
)
+
d
(
z
,
y
) for all x
,
y
,
z
∈
X.
Thend is called acomplex valued metric on X and (
X
,
d
) is called a complex valued metric space.
A pointx
∈
X is called aninterior point of a set A
⊆
X whenever there exists θ
≺
r
∈
C such that B
(
x
,
r
)
=
{
y
∈
X
:
d
(
x
,
y
)
≺
r
}
⊆
A. A subset A in X is calledopen whenever each point of A is an interior point of A. The family F
=
B
x
,
r
:
x
∈
X
,
θ
≺
r is a subbasis for a Hausdorff topology τ on X.
A pointx
∈
X is called alimit point of A whenever, for every θ
≺
r
∈
C
,
B
(
x
,
r
)
∩
A
∖
x
≠
ϕ. A subset B
⊆
X is calledclosed whenever each limit point of B belongs to B.
Let{
x
n
} be a sequence in X and x
∈
X
. If, for every c
∈
C, with θ
≺
c there is n
0
∈
N such that, for all n
>
n
0, d
x
n
,
x
≺
c, then x is called thelimit of x
n and we write lim
n
→
∞
⁡
x
n
=
x or x
n
→
x as n
→
∞
. If for every c
∈
C, with θ
≺
c, there is n
0
∈
N such that, for all n
>
n
0
,
d
(
x
n
,
x
n
+
m
)
≺
c, then {
x
n
} is called aCauchy sequence in (
X
,
d
)
. If every Cauchy sequence is convergent in X
,
d, then X
,
d is called acomplete complex valued metric space.Lemma 2 (see [18]).
Let(
X
,
d
) be a complex valued metric space and let x
n be a sequence in X. Then x
n converges to x if and only if d
x
n
,
x
→
0 as n
→
∞.Lemma 3.
Let(
X
,
d
) be a complex valued metric space and let x
n be a sequence in X. Then x
n is a Cauchy sequence if and only if sup
m
≥
1
⁡
d
x
n
,
x
n
+
m
→
0 as n
→
∞.Recall that iff and g are two self-maps on a set X and u
=
f
x
=
g
x for some x in X, then x is called acoincidence pointoff and g and u is called apoint of coincidence of f and g
.Definition 4 (see [27]).
Two self-maps,f and g, on a nonempty set X are called weakly compatible if they commute at their coincidence point.We will also need the following proposition from [27].Proposition 5.
Letf and g be weakly compatible self-maps on a set X
. If f and g have a unique point of coincidence—w
=
f
x
=
g
x—then w is the unique common fixed point of f and g
.Definition 6 (see [12]).
Let{
X
i
:
i
=
1,2
,
…
,
p
} be a finite collection of nonempty subsets of a set X, where p is some positive integer and f
,
g
:
X
→
X. The set X is said to have a cocyclic representation with respect to the collection {
X
i
:
i
=
1,2
,
…
,
p
} and a pair (
f
,
g
) if(1)
X
=
∪
i
=
1
p
X
i,
(2)
f
(
X
1
)
⊆
g
(
X
2
)
,
…
,
f
(
X
p
-
1
)
⊆
g
(
X
p
)
,
f
(
X
p
)
⊆
g
(
X
1
)
.Recall that a functionf
:
C
→
C is said to be lower semicontinuous at z
0 in C if for every ε
≻
θ there exists a neighborhood U of z
0 such that f
(
z
)
⪰
f
(
z
0
)
-
ε for all z in U. This can be expressed as liminf
z
→
z
0
⁡
f
z
⪰
f
z
0
. Also f is upper semicontinuous at z
0 in C if for every ε
≻
θ there exists a neighborhood U of z
0 such that f
(
z
)
⪯
f
(
z
0
)
+
ε for all z in U. This can be expressed as limsup
z
→
z
0
⁡
f
z
⪯
f
z
0.Definition 7.
The control functionsψ and φ are defined as follows:(i)
ψ
:
C
+
→
C
+ is a continuous nondecreasing function with ψ
(
z
)
=
θ if and only if z
=
θ.
(ii)
φ
:
C
+
→
C
+ is a lower semicontinuous function with φ
(
z
)
=
θ if and only if z
=
θ
.


## 2. Main Results

Now, we prove fixed and common fixed point theorems in the setup of complex valued metric space. Our first result is the following.Theorem 8.
LetX
,
d be a complete complex valued metric space and f
:
X
→
X. Suppose that there exist control functions ψ and φ such that, for x
,
y
∈
X,(1)
ψ
d
f
x
,
f
y
⪯
ψ
U
f
x
,
y
-
φ
U
f
x
,
yholds, where(2)
U
f
x
,
y
∈
d
x
,
y
,
d
f
x
,
x
d
f
y
,
y
1
+
d
x
,
y
,
d
f
x
,
y
d
f
y
,
y
1
+
d
x
,
y
,
d
f
x
,
y
d
f
y
,
x
1
+
d
x
,
y
.Then f has almost one fixed point.Proof.
Letx
0 be an arbitrary element in X
. Construct a sequence {
x
n
} in X as x
n
+
1
=
f
x
n or equivalently as x
n
=
f
n
x
0 for n
∈
N
. If, for some n
∈
N
,
x
n
=
x
n
+
1 holds, then we have f
x
n
=
x
n
+
1
=
x
n and x
n is the fixed point of f. So we assume that x
n
+
1
≠
x
n, for all n
∈
N; that is, d
(
x
n
+
1
,
x
n
)
≻
θ for all n
∈
N. From (1), we have(3)
ψ
d
x
n
+
1
,
x
n
+
2
=
ψ
d
f
x
n
,
f
x
n
+
1
⪯
ψ
U
f
x
n
,
x
n
+
1
-
φ
U
f
x
n
,
x
n
+
1
,where(4)
U
f
x
n
,
x
n
+
1
∈
d
x
n
,
x
n
+
1
,
d
f
x
n
,
x
n
d
f
x
n
+
1
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
,
d
f
x
n
,
x
n
+
1
d
f
x
n
+
1
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
,
d
f
x
n
,
x
n
+
1
d
f
x
n
+
1
,
x
n
1
+
d
x
n
,
x
n
+
1
=
d
x
n
,
x
n
+
1
,
d
x
n
+
1
,
x
n
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
,
d
x
n
+
1
,
x
n
+
1
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
,
d
x
n
+
1
,
x
n
+
1
d
x
n
+
2
,
x
n
1
+
d
x
n
+
1
,
x
n
=
d
x
n
,
x
n
+
1
,
d
x
n
+
1
,
x
n
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
,
θ
,
θ
.If U
f
(
x
n
,
x
n
+
1
)
=
d
(
x
n
,
x
n
+
1
), then(5)
ψ
d
x
n
+
1
,
x
n
+
2
⪯
ψ
d
x
n
,
x
n
+
1
-
φ
d
x
n
,
x
n
+
1
⪯
ψ
d
x
n
,
x
n
+
1
,and, by the property of ψ, we obtain d
(
x
n
+
1
,
x
n
+
2
)
⪯
d
(
x
n
,
x
n
+
1
)
.
In caseU
f
(
x
n
,
x
n
+
1
)
=
d
(
x
n
+
1
,
x
n
)
d
(
x
n
+
2
,
x
n
+
1
)
/
1
+
d
(
x
n
,
x
n
+
1
), then(6)
ψ
d
x
n
+
1
,
x
n
+
2
⪯
ψ
d
x
n
+
1
,
x
n
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
-
φ
d
x
n
+
1
,
x
n
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
⪯
ψ
d
x
n
+
2
,
x
n
+
1
-
φ
d
x
n
+
1
,
x
n
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1implies φ
(
d
(
x
n
+
1
,
x
n
)
d
(
x
n
+
2
,
x
n
+
1
)
/
1
+
d
(
x
n
,
x
n
+
1
)
)
⪯
θ, a contradiction.
Finally ifU
f
(
x
n
,
x
n
+
1
)
=
θ, then ψ
(
d
(
x
n
+
1
,
x
n
+
2
)
)
⪯
θ gives a contradiction.
Thus(7)
d
x
n
+
1
,
x
n
+
2
⪯
d
x
n
,
x
n
+
1for all n
∈
N and {
d
(
x
n
,
x
n
+
1
)
} is the decreasing of positive real numbers. Consequently, there exists θ
⪯
c
∈
C
+ such that {
d
(
x
n
,
x
n
+
1
)
} converges to c. Suppose that c
≻
θ. Now(8)
ψ
d
x
n
+
1
,
x
n
+
2
⪯
ψ
U
f
x
n
,
x
n
+
1
-
φ
U
f
x
n
,
x
n
+
1
,where U
f
(
x
n
,
x
n
+
1
)
∈
{
d
(
x
n
,
x
n
+
1
)
,
d
(
x
n
+
1
,
x
n
)
d
(
x
n
+
2
,
x
n
+
1
)
/
1
+
d
(
x
n
,
x
n
+
1
)
,
θ
}
.
IfU
f
(
x
n
,
x
n
+
1
)
=
d
(
x
n
,
x
n
+
1
), then lower semicontinuity of φ gives that(9)
l
i
m
s
u
p
n
→
∞
ψ
d
x
n
+
1
,
x
n
+
2
⪯
l
i
m
s
u
p
n
→
∞
ψ
d
x
n
,
x
n
+
1
-
l
i
m
i
n
f
n
→
∞
φ
d
x
n
,
x
n
+
1
,which implies ψ
(
c
)
⪯
ψ
(
c
)
-
φ
(
c
)
≺
ψ
(
c
), a contradiction as φ
(
c
)
≻
θ.
In caseU
f
(
x
n
,
x
n
+
1
)
=
d
(
x
n
+
1
,
x
n
)
d
(
x
n
+
2
,
x
n
+
1
)
/
1
+
d
(
x
n
,
x
n
+
1
), then we obtain(10)
l
i
m
s
u
p
n
→
∞
ψ
d
x
n
+
1
,
x
n
+
2
⪯
l
i
m
s
u
p
n
→
∞
ψ
d
x
n
+
1
,
x
n
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
-
l
i
m
i
n
f
n
→
∞
φ
d
x
n
+
1
,
x
n
d
x
n
+
2
,
x
n
+
1
1
+
d
x
n
,
x
n
+
1
,which implies that ψ
(
c
)
⪯
ψ
(
c
2
/
1
+
c
)
-
φ
(
c
2
/
1
+
c
)
≺
ψ
(
c
2
/
1
+
c
), and so c
⪯
c
2
/
1
+
c which implies 1
+
c
⪯
c, a contradiction.
IfU
f
(
x
n
,
x
n
+
1
)
=
θ, then we have limsup
n
→
∞
⁡
ψ
(
d
(
x
n
+
1
,
x
n
+
2
)
)
⪯
θ, a contradiction.
Thereforec
=
θ; that is, lim
n
→
∞
⁡
d
(
x
n
,
x
n
+
1
)
=
θ.
Now, we claim thatlim
n
,
m
→
∞
⁡
d
(
x
n
,
x
m
)
=
θ. If not, then there exist θ
≺
ε
∈
C
+ and sequences {
n
k
}, {
m
k
} in N, with n
k
>
m
k
≥
k, such that d
(
x
n
k
,
x
m
k
)
⪰
ε for all k
∈
N
. Without any loss of generality, we can assume that d
(
x
n
k
,
x
m
k
-
1
)
≺
ε
. Since {
d
(
x
m
k
-
1
,
x
m
k
)
} is a subsequence of convergent sequence {
d
(
x
n
-
1
,
x
n
)
}
→
θ as n
→
∞, therefore, {
d
(
x
m
k
-
1
,
x
m
k
)
}
→
θ as k
→
∞. Now(11)
ε
⪯
d
x
m
k
,
x
n
k
⪯
d
x
n
k
,
x
m
k
-
1
+
d
x
m
k
-
1
,
x
m
kimplies that(12)
lim
k
→
∞
⁡
d
x
m
k
,
x
n
k
=
ε
.From (12) and inequality d
(
x
m
k
,
x
n
k
)
⪯
d
(
x
m
k
,
x
m
k
-
1
)
+
d
(
x
m
k
-
1
,
x
n
k
), it follows that ε
⪯
lim
k
→
∞
⁡
d
(
x
m
k
-
1
,
x
n
k
). Also, the inequality d
(
x
m
k
-
1
,
x
n
k
)
⪯
d
(
x
m
k
-
1
,
x
m
k
)
+
d
(
x
m
k
,
x
n
k
) and (12) give that lim
k
→
∞
⁡
d
(
x
m
k
-
1
,
x
n
k
)
⪯
ε, and hence we have(13)
lim
k
→
∞
⁡
d
x
m
k
-
1
,
x
n
k
=
ε
.From (13) and inequality d
(
x
m
k
-
1
,
x
n
k
)
⪯
d
(
x
m
k
-
1
,
x
n
k
+
1
)
+
d
(
x
n
k
+
1
,
x
n
k
), it is implied that ε
⪯
lim
k
→
∞
⁡
d
(
x
m
k
-
1
,
x
n
k
+
1
), while inequality d
(
x
m
k
-
1
,
x
n
k
+
1
)
⪯
d
(
x
m
k
-
1
,
x
n
k
)
+
d
(
x
n
k
,
x
n
k
+
1
) and (13) imply that lim
k
→
∞
⁡
d
(
x
m
k
-
1
,
x
n
k
+
1
)
⪯
ε, and hence we have(14)
lim
k
→
∞
⁡
d
x
m
k
-
1
,
x
n
k
+
1
=
ε
.From (12) and d
(
x
m
k
,
x
n
k
)
⪯
d
(
x
m
k
,
x
n
k
+
1
)
+
d
(
x
n
k
+
1
,
x
n
k
), we have ε
⪯
lim
k
→
∞
⁡
d
(
x
m
k
,
x
n
k
+
1
), and the inequality d
(
x
m
k
,
x
n
k
+
1
)
⪯
d
(
x
m
k
,
x
n
k
)
+
d
(
x
n
k
,
x
n
k
+
1
) and (12) give that lim
k
→
∞
⁡
d
(
x
m
k
,
x
n
k
+
1
)
⪯
ε. So(15)
lim
k
→
∞
⁡
d
x
m
k
,
x
n
k
+
1
=
ε
.As(16)
M
f
x
n
k
,
x
m
k
-
1
∈
d
x
n
k
,
x
m
k
-
1
,
d
f
x
n
k
,
x
n
k
d
f
x
m
k
-
1
,
x
m
k
-
1
1
+
d
x
n
k
,
x
m
k
-
1
,
d
f
x
n
k
,
x
m
k
-
1
d
f
x
m
k
-
1
,
x
m
k
-
1
1
+
d
x
n
k
,
x
m
k
-
1
,
d
f
x
n
k
,
x
m
k
-
1
d
f
x
m
k
-
1
,
x
n
k
1
+
d
x
n
k
,
x
m
k
-
1
=
d
x
n
k
,
x
m
k
-
1
,
d
x
n
k
+
1
,
x
n
k
d
x
m
k
,
x
m
k
-
1
1
+
d
x
n
k
,
x
m
k
-
1
,
d
x
n
k
+
1
,
x
m
k
-
1
d
x
m
k
,
x
m
k
-
1
1
+
d
x
n
k
,
x
m
k
-
1
,
d
x
n
k
+
1
,
x
m
k
-
1
d
x
m
k
,
x
n
k
1
+
d
x
n
k
,
x
m
k
-
1
,so we have(17)
lim
k
→
∞
⁡
U
f
x
n
k
,
x
m
k
-
1
∈
ε
,
θ
,
θ
,
ε
2
1
+
ε
.Now, if lim
k
→
∞
⁡
U
f
(
x
n
k
,
x
m
k
-
1
)
=
ε, then, from (1), it follows that(18)
ψ
d
x
n
k
+
1
,
x
m
k
=
ψ
d
f
x
n
k
,
f
x
m
k
-
1
⪯
ψ
U
f
x
n
k
,
x
m
k
-
1
-
φ
U
f
x
n
k
,
x
m
k
-
1and taking upper limit as k
→
∞ implies that ψ
(
ε
)
⪯
ψ
(
ε
)
-
φ
(
ε
)
≺
ψ
(
ε
), a contradiction as ε
≻
θ.
In caselim
k
→
∞
⁡
U
f
(
x
n
k
,
x
m
k
-
1
)
=
θ, then, from (1), it follows that limsup
k
→
∞
⁡
ψ
(
d
(
x
n
k
+
1
,
x
m
k
)
)
⪯
θ, a contradiction.
Whenlim
k
→
∞
⁡
U
f
(
x
n
k
,
x
m
k
-
1
)
=
ε
2
/
1
+
ε, then, from (1), it follows that(19)
ψ
ε
=
limsup
k
→
∞
⁡
ψ
d
x
n
k
+
1
,
x
m
k
=
limsup
k
→
∞
⁡
ψ
d
f
x
n
k
,
f
x
m
k
-
1
⪯
limsup
k
→
∞
⁡
ψ
U
f
x
n
k
,
x
m
k
-
1
-
liminf
k
→
∞
⁡
φ
U
f
x
n
k
,
x
m
k
-
1
=
ψ
ε
2
1
+
ε
-
φ
ε
2
1
+
ε
⪯
ψ
ε
2
1
+
ε
,which implies ε
⪯
ε
2
/
1
+
ε; that is, 1
+
ε
⪯
ε, a contradiction.
Thus, we obtain thatlim
n
,
m
→
∞
⁡
d
(
x
n
,
x
m
)
=
θ, and hence {
x
n
} is a Cauchy sequence in the complex valued metric space (
X
,
d
)
.
Next, we assume there exists a pointu
∈
X such that lim
n
→
∞
⁡
d
(
u
,
x
n
)
=
θ.
Thus,(20)
U
f
u
,
x
n
∈
d
u
,
x
n
,
d
f
u
,
u
d
f
x
n
,
x
n
1
+
d
u
,
x
n
,
d
f
u
,
x
n
d
f
x
n
,
x
n
1
+
d
u
,
x
n
,
d
f
u
,
x
n
d
f
x
n
,
u
1
+
d
u
,
x
n
=
d
u
,
x
n
,
d
f
u
,
u
d
x
n
+
1
,
x
n
1
+
d
u
,
x
n
,
d
f
u
,
x
n
d
x
n
+
1
,
x
n
1
+
d
u
,
x
n
,
d
f
u
,
x
n
d
x
n
+
1
,
u
1
+
d
u
,
x
n
.Taking limit as n
→
∞, we conclude that lim
n
→
∞
⁡
M
f
(
u
,
x
n
)
=
θ
. Hence(21)
ψ
d
f
u
,
f
x
n
+
1
⪯
ψ
U
f
u
,
x
n
-
φ
U
f
u
,
x
nand taking upper limit as n
→
∞ implies that(22)
ψ
d
f
u
,
u
⪯
θ
,which further implies that f
u
=
u
.
To prove the uniqueness of fixed point off, assume on the contrary that f
v
=
v and f
w
=
w with v
≠
w. Note that(23)
ψ
d
v
,
w
=
ψ
d
f
v
,
f
w
⪯
ψ
U
f
v
,
w
-
φ
U
f
v
,
w
,where(24)
U
f
v
,
w
∈
d
v
,
w
,
d
f
v
,
v
d
f
w
,
w
1
+
d
v
,
w
,
d
f
v
,
w
d
f
w
,
w
1
+
d
v
,
w
,
d
f
v
,
w
d
f
w
,
v
1
+
d
v
,
w
=
d
v
,
w
,
d
v
,
v
d
w
,
w
1
+
d
v
,
w
,
d
v
,
w
d
w
,
w
1
+
d
v
,
w
,
d
v
,
w
d
w
,
v
1
+
d
v
,
w
=
d
v
,
w
,
θ
,
θ
,
d
v
,
w
2
1
+
d
v
,
w
.Now, if U
f
(
v
,
w
)
=
d
(
v
,
w
), then from (23) it follows that(25)
ψ
d
v
,
w
=
ψ
d
f
v
,
f
w
⪯
ψ
d
v
,
w
-
φ
d
v
,
w
≺
ψ
d
v
,
w
,a contradiction as d
(
v
,
w
)
≻
θ
. Hence v
=
w.
In caseU
f
(
v
,
w
)
=
θ, then (23) gives a contradiction.
Finally, ifU
f
(
v
,
w
)
=
d
(
v
,
w
)
2
/
1
+
d
(
v
,
w
), then, from (23),(26)
ψ
d
v
,
w
⪯
ψ
d
v
,
w
2
1
+
d
v
,
w
-
φ
d
v
,
w
2
1
+
d
v
,
w
≺
ψ
d
v
,
w
2
1
+
d
v
,
w
,which gives 1
+
d
(
v
,
w
)
⪯
d
(
v
,
w
), a contradiction. Hence for all cases we have v
=
w
.Corollary 9.
LetX
,
d be a complete complex valued metric space and f
:
X
→
X. Suppose that there exist control functions ψ and φ such that for x
,
y
∈
X
(27)
ψ
d
f
n
x
,
f
n
y
⪯
ψ
U
f
n
x
,
y
φ
U
f
n
x
,
yholds for n
∈
N, where(28)
U
f
n
x
,
y
∈
d
x
,
y
,
d
f
n
x
,
x
d
f
n
y
,
y
1
+
d
x
,
y
,
d
f
n
x
,
y
d
f
n
y
,
y
1
+
d
x
,
y
,
d
f
n
x
,
y
d
f
n
y
,
x
1
+
d
x
,
y
.Then f has almost one fixed point u in X and f
n
u
=
f
u.Proof.
TakingT
=
f
n
. It follows from Theorem 8 that T has a unique fixed point u. Also f
(
u
)
=
f
(
T
u
)
=
f
n
+
1
(
u
)
=
f
n
(
f
u
)
=
T
f
u implies that f
u is also the fixed point of T. From the uniqueness of fixed point of T, it is implied that f
u
=
u
.Recently, Haghi et al. [28] proved the following lemma.Lemma 10.
LetX be a nonempty set and let f be a self-map on X. Then there exists a subset E
⊆
X such that f
:
E
→
X is one-to-one and f
(
E
)
=
f
(
X
).Theorem 11.
LetX
,
d be a complex valued metric space and f
,
g
:
X
→
X. Suppose that there exist control functions ψ and φ such that for any x
,
y
∈
X
(29)
ψ
d
f
x
,
f
y
⪯
ψ
U
g
x
,
g
y
-
φ
U
g
x
,
g
yholds, where(30)
U
x
,
y
∈
d
g
x
,
g
y
,
d
f
x
,
g
x
d
f
y
,
g
y
1
+
d
g
x
,
g
y
,
d
f
x
,
g
y
d
f
y
,
g
y
1
+
d
g
x
,
g
y
,
d
f
x
,
g
y
d
f
y
,
g
x
1
+
d
g
x
,
g
y
.If g
X is a complete subspace of X, then f and g have at most one coincidence point in X. Furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point in X.Proof.
It follows from Lemma10 that there exists E
⊆
X such that g
:
E
→
X is one-to-one and g
(
E
)
=
g
(
X
). Define a map h
:
g
(
E
)
→
g
(
E
) by h
(
g
x
)
=
f
x. Since g is one-to-one on E, h is well-defined. Note that(31)
ψ
d
h
g
x
,
h
g
y
⪯
ψ
U
g
x
,
g
y
-
φ
U
g
x
,
g
y
,for all g
x
,
g
y
∈
g
(
E
), where(32)
U
g
x
,
g
y
∈
d
g
x
,
g
y
,
d
h
g
x
,
g
x
d
h
g
y
,
g
y
1
+
d
g
x
,
g
y
,
d
h
g
x
,
g
y
d
h
g
y
,
g
y
1
+
d
g
x
,
g
y
,
d
h
g
x
,
g
y
d
h
g
y
,
g
x
1
+
d
g
x
,
g
y
.Since g
(
E
)
=
g
(
X
) is complete, by Theorem 8, there exists x
∗
∈
X such that h
(
g
x
∗
)
=
g
x
∗
=
f
x
∗. Hence, f and g have at most one point of coincidence in X. By Proposition 5, f and g have at most one common fixed point.


## 3. Cocyclic Contraction Mappings

In this section, we prove common fixed point results for self-maps satisfying cocyclic contractions defined on a complex valued metric space. Our first result is the following.Theorem 12.
LetX
,
d be a complex valued metric space, let A
1
,
A
2
,
…
,
A
p be nonempty closed subsets of X, and let Y
=
∪
i
=
1
p
A
i. Suppose that f
,
g
:
Y
→
Y are such that(a)
Y has a cocyclic representation with respect to pair (
f
,
g
) and to the collection {
A
i
:
i
=
1,2
,
…
,
p
};
(b)
there exist control functionsψ and φ such that, for any x
,
y
∈
A
i
×
A
i
+
1,  i
=
1,2
,
…
,
p,(33)
ψ
d
f
x
,
f
y
⪯
ψ
U
f
,
g
x
,
y
-
φ
U
f
,
g
x
,
y
holds where(34)
U
f
,
g
x
,
y
∈
d
g
x
,
g
y
,
d
f
x
,
g
x
d
f
y
,
g
y
1
+
d
g
x
,
g
y
,
d
f
x
,
g
y
d
f
y
,
g
y
1
+
d
g
x
,
g
y
,
d
f
x
,
g
y
d
f
y
,
g
x
1
+
d
g
x
,
g
y
,
withA
p
+
1
=
A
1.
Ifg
A
i is complete subspace of X for each i
∈
{
1,2
,
…
,
p
}, then f and g have a unique coincidence point z
∈
∩
i
=
1
p
g
A
i
=
K provided that f
K
⊆
g
K
⊆
K and g
K is closed. Furthermore, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
Letx
0
∈
∪
i
=
1
p
A
i be a given element. By the given condition, there exists i
0
∈
{
1,2
,
…
,
p
} such that x
0
∈
A
i
0. Since f
(
A
i
0
)
⊆
g
(
A
i
0
+
1
), we can choose a point x
1
∈
A
i
0
+
1 such that f
(
x
0
)
=
g
(
x
1
)
. Continuing this process, for n
>
0, there exists i
n
∈
{
1,2
,
…
,
p
} such that, having chosen x
n
∈
A
i
n, we obtain x
n
+
1
∈
A
i
n
+
1 such that f
(
x
n
)
=
g
(
x
n
+
1
). If for some n
0
≥
0,   f
x
n
0
=
f
x
n
0
+
1; that is, f
x
n
0
=
g
x
n
0 and x
n
0 is the coincidence point of f and g. Taking d
(
f
x
n
,
f
x
n
+
1
)
≻
θ for all n, we have(35)
ψ
d
g
x
n
+
1
,
g
x
n
+
2
=
ψ
d
f
x
n
,
f
x
n
+
1
⪯
ψ
U
f
,
g
x
n
,
x
n
+
1
-
φ
U
f
,
g
x
n
,
x
n
+
1
⪯
ψ
U
f
,
g
x
n
,
x
n
+
1
,where(36)
U
f
,
g
x
n
,
x
n
+
1
∈
d
g
x
n
,
g
x
n
+
1
,
d
f
x
n
,
g
x
n
d
f
x
n
+
1
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
,
d
f
x
n
,
g
x
n
+
1
d
f
x
n
+
1
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
,
d
f
x
n
,
g
x
n
+
1
d
f
x
n
+
1
,
g
x
n
1
+
d
g
x
n
,
g
x
n
+
1
=
d
g
x
n
,
g
x
n
+
1
,
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
,
d
g
x
n
+
1
,
g
x
n
+
1
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
2
,
d
g
x
n
+
1
,
g
x
n
+
1
d
g
x
n
+
2
,
g
x
n
1
+
d
g
x
n
,
g
x
n
+
1
=
d
g
x
n
,
g
x
n
+
1
,
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
,
θ
,
θ
.If, for some n,  U
f
,
g
(
x
n
,
x
n
+
1
)
=
d
(
g
x
n
,
g
x
n
+
1
), then(37)
ψ
d
g
x
n
+
1
,
g
x
n
+
2
⪯
ψ
d
g
x
n
,
g
x
n
+
1
-
φ
d
g
x
n
,
g
x
n
+
1implies that d
(
g
x
n
+
2
,
g
x
n
+
1
)
⪯
d
(
g
x
n
,
g
x
n
+
1
)
.
In caseU
f
,
g
(
x
n
,
x
n
+
1
)
=
d
(
g
x
n
+
1
,
g
x
n
)
d
(
g
x
n
+
2
,
g
x
n
+
1
)
/
1
+
d
g
x
n
,
g
x
n
+
1, then(38)
ψ
d
g
x
n
+
1
,
g
x
n
+
2
⪯
ψ
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
-
φ
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
⪯
ψ
d
g
x
n
+
2
,
g
x
n
+
1
-
φ
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1implies φ
(
d
(
g
x
n
+
1
,
g
x
n
)
d
(
g
x
n
+
2
,
g
x
n
+
1
)
/
1
+
d
(
g
x
n
,
g
x
n
+
1
)
)
⪯
θ, a contradiction.
WhenU
f
,
g
(
x
n
,
x
n
+
1
)
=
θ, then ψ
(
d
(
g
x
n
+
1
,
g
x
n
+
2
)
)
⪯
θ gives a contradiction.
Thus(39)
d
g
x
n
+
1
,
g
x
n
+
2
⪯
d
g
x
n
,
g
x
n
+
1for all n
∈
N and {
d
(
g
x
n
,
g
x
n
+
1
)
} is the decreasing of positive real numbers. Consequently, there exists θ
⪯
δ
∈
C
+ such that {
d
(
x
n
,
x
n
+
1
)
} converges to δ
. Suppose that δ
≻
θ
. Now(40)
ψ
d
g
x
n
+
1
,
g
x
n
+
2
⪯
ψ
U
f
,
g
x
n
,
x
n
+
1
-
φ
U
f
,
g
x
n
,
x
n
+
1
,where(41)
U
f
,
g
x
n
,
x
n
+
1
∈
d
g
x
n
,
g
x
n
+
1
,
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
,
θ
.If  U
f
(
x
n
,
x
n
+
1
)
=
d
(
g
x
n
,
g
x
n
+
1
), then lower semicontinuity of φ gives that(42)
limsup
n
→
∞
⁡
ψ
d
g
x
n
+
1
,
g
x
n
+
2
⪯
limsup
n
→
∞
⁡
ψ
d
g
x
n
,
g
x
n
+
1
-
liminf
n
→
∞
⁡
φ
d
g
x
n
,
g
x
n
+
1
,which implies that ψ
(
δ
)
⪯
ψ
(
δ
)
-
φ
(
δ
)
≺
ψ
(
δ
), a contradiction as φ
(
δ
)
≻
θ.
In caseU
f
,
g
(
x
n
,
x
n
+
1
)
=
d
(
g
x
n
+
1
,
g
x
n
)
d
(
g
x
n
+
2
,
g
x
n
+
1
)
/
1
+
d
(
g
x
n
,
g
x
n
+
1
), then we obtain(43)
limsup
n
→
∞
⁡
ψ
d
g
x
n
+
1
,
g
x
n
+
2
⪯
limsup
n
→
∞
⁡
ψ
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
-
liminf
n
→
∞
⁡
φ
d
g
x
n
+
1
,
g
x
n
d
g
x
n
+
2
,
g
x
n
+
1
1
+
d
g
x
n
,
g
x
n
+
1
,which implies that ψ
(
δ
)
⪯
ψ
(
δ
2
/
1
+
δ
)
-
φ
(
δ
2
/
1
+
δ
)
≺
ψ
(
δ
2
/
1
+
δ
), and so δ
⪯
δ
2
/
1
+
δ which implies 1
+
δ
⪯
δ, a contradiction.
IfU
f
,
g
(
x
n
,
x
n
+
1
)
=
θ, then we have ψ
(
δ
)
=
limsup
n
→
∞
⁡
ψ
(
d
(
g
x
n
+
1
,
g
x
n
+
2
)
)
⪯
θ, a contradiction. Therefore δ
=
θ; that is,(44)
lim
n
→
∞
⁡
d
g
x
n
+
1
,
g
x
n
=
θ
.Now we claim that {
g
x
n
} is a complex valued Cauchy sequence. If not, there are ε
≻
θ and even integers n
k and m
k with m
k
>
n
k
>
k such that(45)
d
g
x
m
k
,
g
x
n
k
⪰
εand d
(
g
x
m
k
-
1
,
g
x
n
k
)
≺
ε
. Note that(46)
ε
⪯
d
g
x
m
k
,
g
x
n
k
⪯
d
g
x
n
k
,
g
x
m
k
-
1
+
d
g
x
m
k
-
1
,
g
x
m
k
.From (44) and (45), it follows that(47)
lim
k
→
∞
⁡
d
g
x
m
k
,
g
x
n
k
=
ε
.By (47) and(48)
d
g
x
n
k
+
1
,
g
x
m
k
⪯
d
g
x
n
k
+
1
,
g
x
n
k
+
d
g
x
n
k
,
g
x
m
k
,we have lim
k
→
∞
⁡
d
(
g
x
n
k
+
1
,
g
x
m
k
)
⪯
ε. Also, by (47) and(49)
d
g
x
m
k
,
g
x
n
k
⪯
d
g
x
n
k
+
1
,
g
x
n
k
+
d
g
x
n
k
+
1
,
g
x
m
k
,we obtain that ε
⪯
lim
k
→
∞
⁡
d
(
g
x
n
k
+
1
,
g
x
m
k
). Hence(50)
lim
k
→
∞
⁡
d
g
x
n
k
+
1
,
g
x
m
k
=
ε
.From (44) and(51)
d
g
x
n
k
+
1
,
g
x
m
k
-
1
⪯
d
g
x
n
k
+
1
,
g
x
n
k
+
d
g
x
n
k
,
g
x
m
k
-
1
,we have lim
k
→
∞
⁡
d
(
g
x
n
k
+
1
,
g
x
m
k
-
1
)
⪯
ε. By (50) and the inequality(52)
d
g
x
n
k
+
1
,
g
x
m
k
⪯
d
g
x
n
k
+
1
,
g
x
m
k
-
1
+
d
g
x
m
k
-
1
,
g
x
m
k
,we have ε
⪯
lim
k
→
∞
⁡
d
(
g
x
n
k
+
1
,
g
x
m
k
-
1
). Thus(53)
lim
k
→
∞
⁡
d
g
x
n
k
+
1
,
g
x
m
k
-
1
=
ε
.As(54)
U
f
,
g
x
n
k
,
x
m
k
-
1
∈
d
g
x
n
k
,
g
x
m
k
-
1
,
d
f
x
n
k
,
g
x
n
k
d
f
x
m
k
-
1
,
g
x
m
k
-
1
1
+
d
g
x
n
k
,
g
x
m
k
-
1
,
d
f
x
n
k
,
g
x
m
k
-
1
d
f
x
m
k
-
1
,
g
x
m
k
-
1
1
+
d
g
x
n
k
,
g
x
m
k
-
1
,
d
f
x
n
k
,
g
x
m
k
-
1
d
f
x
m
k
-
1
,
g
x
n
k
1
+
d
g
x
n
k
,
g
x
m
k
-
1
=
d
g
x
n
k
,
g
x
m
k
-
1
,
d
g
x
n
k
+
1
,
g
x
n
k
d
g
x
m
k
,
g
x
m
k
-
1
1
+
d
g
x
n
k
,
g
x
m
k
-
1
,
d
g
x
n
k
+
1
,
g
x
m
k
-
1
d
g
x
m
k
,
g
x
m
k
-
1
1
+
d
g
x
n
k
,
g
x
m
k
-
1
,
d
g
x
n
k
+
1
,
g
x
m
k
-
1
d
g
x
m
k
,
g
x
n
k
1
+
d
g
x
n
k
,
g
x
m
k
-
1
,so we have lim
k
→
∞
⁡
U
f
,
g
(
x
n
k
,
x
m
k
-
1
)
∈
{
ε
,
θ
,
θ
,
ε
2
/
1
+
ε
}.
Now, iflim
k
→
∞
⁡
U
f
,
g
(
x
n
k
,
x
m
k
-
1
)
=
ε, then, from (b), it follows that(55)
ψ
d
g
x
n
k
+
1
,
g
x
m
k
=
ψ
d
f
x
n
k
,
f
x
m
k
-
1
⪯
ψ
M
f
,
g
x
n
k
,
x
m
k
-
1
-
φ
M
f
,
g
x
n
k
,
x
m
k
-
1and taking upper limit as k
→
∞ implies that ψ
(
ε
)
⪯
ψ
(
ε
)
-
φ
(
ε
)
≺
ψ
(
ε
), a contradiction as ε
≻
θ.
In caselim
k
→
∞
⁡
M
f
,
g
(
x
n
k
,
x
m
k
-
1
)
=
θ, then, from (b), it follows that limsup
k
→
∞
⁡
ψ
(
d
(
g
x
n
k
+
1
,
g
x
m
k
)
)
⪯
θ, a contradiction.
Whenlim
k
→
∞
⁡
M
f
,
g
(
x
n
k
,
x
m
k
-
1
)
=
ε
2
/
1
+
ε, then, from (b), it follows that(56)
ψ
ε
=
limsup
k
→
∞
⁡
ψ
d
g
x
n
k
+
1
,
g
x
m
k
=
limsup
k
→
∞
⁡
ψ
d
f
x
n
k
,
f
x
m
k
-
1
⪯
limsup
k
→
∞
⁡
ψ
U
f
,
g
x
n
k
,
x
m
k
-
1
-
liminf
k
→
∞
⁡
φ
U
f
,
g
x
n
k
,
x
m
k
-
1
=
ψ
ε
2
1
+
ε
-
φ
ε
2
1
+
ε
⪯
ψ
ε
2
1
+
ε
,which implies ε
⪯
ε
2
/
1
+
ε; that is, 1
+
ε
⪯
ε, a contradiction.
Hence{
g
x
n
} is a Cauchy sequence in ∪
i
=
1
p
g
A
i. Since ∪
i
=
1
p
g
A
i is complete, there exists y
∈
∪
i
=
1
p
g
A
i such that(57)
lim
n
→
∞
⁡
d
g
x
n
,
y
=
θ
.Consequently, we can find a point z in Y such that g
(
z
)
=
y
.
Now we show thaty
∈
∩
i
=
1
p
g
A
i
=
K
. From condition (a) and x
0
∈
A
i
0 for some i
0
∈
{
1,2
,
…
,
p
}, we can choose a subsequence {
x
n
k
} in A
i
0 out of the sequence {
x
n
}
. Obviously, {
g
x
n
k
}
⊆
g
(
A
i
0
). As g
(
A
i
0
) is closed, so y
∈
g
(
A
i
0
). Similarly, we can choose a subsequence {
x
n
k
+
1
} in A
i
0
+
1 out of the sequence {
x
n
}
. Obviously, {
g
x
n
k
+
1
}
⊆
g
(
A
i
0
+
1
). As g
(
A
i
0
+
1
) is closed, so y
∈
g
(
A
i
0
+
1
). Continuing this way, we obtain that y
∈
∩
i
=
1
p
g
A
i and hence ∩
i
=
1
p
g
(
A
i
)
≠
∅.
Now we show thatf
(
z
)
=
g
(
z
). Since z
∈
Y, there exists some i in {
1,2
,
…
,
p
} such that z
∈
A
i. Choose a subsequence {
x
n
k
} of {
x
n
} with x
n
k
∈
A
i
-
1. From (b), we have(58)
ψ
d
g
x
n
k
+
1
,
f
z
=
ψ
d
f
x
n
k
,
f
z
⪯
ψ
U
f
,
g
x
n
k
,
z
-
φ
U
f
,
g
x
n
k
,
z
,where(59)
U
f
,
g
x
n
k
,
z
∈
d
g
x
n
k
,
g
z
,
d
f
x
n
k
,
g
x
n
k
d
f
z
,
g
z
1
+
d
g
x
n
k
,
g
z
,
d
f
x
n
k
,
g
z
d
f
z
,
g
z
1
+
d
g
x
n
k
,
g
z
,
d
f
x
n
k
,
g
z
d
f
z
,
g
x
n
k
1
+
d
g
x
n
k
,
g
z
=
d
g
x
n
k
,
g
z
,
d
g
x
n
k
+
1
,
g
x
n
k
d
f
z
,
g
z
1
+
d
g
x
n
k
,
g
z
,
d
g
x
n
k
+
1
,
g
z
d
f
z
,
g
z
1
+
d
g
x
n
k
,
g
z
,
d
g
x
n
k
+
1
,
g
z
d
f
z
,
g
x
n
k
1
+
d
g
x
n
k
,
g
z
,which gives lim
k
→
∞
⁡
U
f
,
g
(
x
n
k
,
z
)
=
θ. Thus ψ
d
(
g
z
,
f
z
)
⪯
θ and hence f
(
z
)
=
g
(
z
)
. Thus z is the coincidence point of f and g.
ThusK being a finite intersection of closed sets is closed and hence complete. Consider the restrictions mappings f
|
K
,
g
|
K
:
K
→
K. Clearly, f
|
K
K
⊆
g
|
K
(
K
)
⊆
K
. Also g
|
Z
(
K
) is closed and hence complete. From Theorem 11, it follows that f
|
K and g
|
K have at most one coincidence point in K. As f and g are weakly compatible, from Proposition 5, it follows that f
|
K and g
|
K have at most one common fixed point.Corollary 13.
LetX
,
d be a complex valued metric space, let A
1
,
A
2
,
…
,
A
p be nonempty closed subsets of X, and let Y
=
∪
i
=
1
p
A
i. Suppose that f
,
g
:
Y
→
Y are such that(a)
Y has a cocyclic representation with respect to pair (
f
m
,
g
n
) and to the collection {
A
i
:
i
=
1,2
,
…
,
p
} for some m
,
n
∈
N;
(b)
there exist two control functions,ψ and φ, such that, for any x
,
y
∈
A
i
×
A
i
+
1
,
i
=
1,2
,
…
,
p,(60)
ψ
d
f
m
x
,
f
m
y
⪯
ψ
U
f
,
g
x
,
y
-
φ
U
f
,
g
x
,
y
holds for somem
,
n
∈
N, where(61)
U
f
x
,
y
∈
d
g
n
x
,
g
n
y
,
d
f
m
x
,
g
n
x
d
f
m
y
,
g
n
y
1
+
d
g
n
x
,
g
n
y
,
d
f
m
x
,
g
n
y
d
f
m
y
,
g
n
y
1
+
d
g
n
x
,
g
n
y
2
,
d
f
m
x
,
g
n
y
d
f
m
y
,
g
n
x
1
+
d
g
n
x
,
g
n
y
,
withA
p
+
1
=
A
1.
Ifg
n
A
i is a complete subspace of X for each i
∈
{
1,2
,
…
,
p
}, then f
m and g
n have a unique coincidence point z
∈
∩
i
=
1
p
g
n
A
i
=
K provided that f
m
K
⊆
g
n
K
⊆
K and g
n
K is closed. Moreover, if f
m and g
n are weakly compatible, then f
m and g
n have a unique common fixed point. Furthermore, f and g have at most one common fixed point provided that f and g commute.Proof.
SetF
=
f
m and G
=
g
n
. From Theorem 12, F and G have a unique common fixed point u. Now F
(
f
u
)
=
f
F
u
=
f
u and G
f
u
=
f
G
u
=
f
u imply that f
u is the common fixed point of F and G. Also, G
(
g
u
)
=
g
G
u
=
g
u and F
g
u
=
g
F
u
=
g
u imply that g
u is also the common fixed point of F and G
. By the uniqueness of common fixed point of F and G, we have f
u
=
g
u
=
u.Example 14.
LetX
=
R and let d
:
X
×
X
→
C
+ be the complex valued metric defined by d
(
x
,
y
)
=
x
-
y
e
i
ϑ, where ϑ
∈
0
,
π
/
2
. Suppose A
1
=
[
-
c
,
0
]
,
A
2
=
[
-
k
c
,
k
c
] and A
3
=
[
-
k
c
/
2
,
c
] and A
4
=
A
1, where c
>
0 and k
∈
[
0,1
). Define f
,
g
:
∪
i
=
1
3
A
i
→
∪
i
=
1
3
A
i by(62)
f
x
=
-
α
x
c
,
if
x
>
0
,
0
,
otherwise
,
g
x
=
β
x
c
,where α
,
β
≥
1 with α
<
2
β
≤
c
. Note that f
A
1
=
0
⊆
-
k
β
,
k
β
=
g
A
2,  f
A
2
=
-
k
α
,
0
⊆
[
-
k
β
/
2
,
β
]
=
g
(
A
3
), and f
A
3
=
-
α
,
0
⊆
[
-
β
,
0
]
=
g
(
A
1
)
. Hence Y
=
A
1
∪
A
2
∪
A
3 is a cocyclic representation with respect to pair (
f
,
g
) and the collection {
A
1
,
A
2
,
A
3
}. Define ψ
,
φ
:
C
+
→
C
+ by(63)
ψ
z
=
β
z
,
φ
z
=
β
z
4
,
i
f
z
∈
θ
,
2
β
e
i
ϑ
β
z
,
if
t
≻
2
β
e
i
ϑ
.Clearly ψ is continuous and nondecreasing and φ is lower semicontinuous with ψ
(
z
)
=
φ
(
z
)
=
θ if and only if z
=
θ
.
We show that condition (b) is satisfied:(I)
Letx
∈
A
1 and y
∈
A
2
. If y
∈
[
-
k
c
,
0
], then d
(
f
x
,
f
y
)
=
θ and hence (b) holds. If y
∈
(
0
,
k
c
], then we have(64)
ψ
d
f
x
,
f
y
=
ψ
d
0
,
-
α
y
c
=
α
β
c
y
e
i
ϑ
⪯
3
β
2
4
c
y
-
x
e
i
ϑ
=
3
β
4
d
β
x
c
,
β
y
c
=
3
β
4
d
g
x
,
g
y
⪯
3
β
4
U
f
,
g
x
,
y
=
β
U
f
,
g
x
,
y
-
β
4
U
f
,
g
x
,
y
=
ψ
U
f
,
g
x
,
y
-
φ
U
f
,
g
x
,
y
.
(II)
Letx
∈
A
2
,
y
∈
A
3
. If x
∈
[
-
k
c
,
0
] and y
∈
[
-
k
c
/
2,0
], then d
(
f
x
,
f
y
)
=
θ and hence (b) is satisfied. If x
∈
[
-
k
c
,
0
] and y
∈
(
0
,
c
], then we obtain that(65)
ψ
d
f
x
,
f
y
=
ψ
d
0
,
-
β
y
c
=
β
2
c
y
e
i
ϑ
≺
3
β
α
+
β
4
c
y
e
i
ϑ
=
3
β
4
d
-
α
y
c
,
β
y
c
=
3
β
4
d
f
y
,
g
y
⪯
3
β
4
U
f
,
g
x
,
y
=
β
U
f
,
g
x
,
y
-
β
4
U
f
,
g
x
,
y
=
ψ
U
f
,
g
x
,
y
-
φ
U
f
,
g
x
,
y
.
Whenx
∈
(
0
,
k
c
] and y
∈
[
-
k
c
/
2,0
], we have(66)
ψ
d
f
x
,
f
y
=
ψ
d
-
α
x
c
,
0
=
α
β
c
x
e
i
ϑ
≺
3
β
α
+
β
4
c
x
e
i
ϑ
=
3
β
4
d
-
α
x
c
,
β
x
c
=
3
β
4
d
f
x
,
g
x
⪯
3
β
4
U
f
,
g
x
,
y
=
β
U
f
,
g
x
,
y
-
β
4
U
f
,
g
x
,
y
=
ψ
U
f
,
g
x
,
y
-
φ
U
f
,
g
x
,
y
.
Forx
∈
(
0
,
k
c
] and y
∈
(
0
,
c
], we obtain that(67)
ψ
d
f
x
,
f
y
=
ψ
d
-
α
x
c
,
-
α
y
c
=
α
β
c
x
-
y
e
i
ϑ
≺
3
β
2
4
c
x
-
y
e
i
ϑ
=
3
β
4
d
β
x
c
,
β
y
c
=
3
β
4
d
g
x
,
g
y
⪯
3
β
4
U
f
,
g
x
,
y
=
β
U
f
,
g
x
,
y
-
β
4
U
f
,
g
x
,
y
=
ψ
U
f
,
g
x
,
y
-
φ
U
f
,
g
x
,
y
.
(III)
Letx
∈
A
3
,
y
∈
A
1
. If x
∈
[
-
k
c
/
2,0
], then d
(
f
x
,
f
y
)
=
θ and (b) is satisfied. If x
∈
(
0
,
c
], then we have(68)
ψ
d
f
x
,
f
y
=
ψ
d
-
α
x
c
,
0
=
α
β
c
x
e
i
ϑ
≺
3
β
α
+
β
4
c
x
e
i
ϑ
=
3
β
4
d
-
α
x
c
,
β
x
c
=
3
β
4
d
f
x
,
g
x
⪯
3
β
4
U
f
,
g
x
,
y
=
β
U
f
,
g
x
,
y
-
β
4
U
f
,
g
x
,
y
=
ψ
U
f
,
g
x
,
y
-
φ
U
f
,
g
x
,
y
.Hence f and g satisfy all the conditions of Theorem 12. Moreover, 0 is a unique common fixed point of f and g in ∩
i
=
1
3
A
i.We denoteΥ
=
{
ϕ
:
C
+
→
C
+
:
ϕ is Lebesgue integrable with finite integral on each compact subset of C
+ and, for each θ
≺
ε
∈
C
+,   ∫
θ
ε
‍
φ
(
z
)
d
z
≻
θ
}. As a consequence of Theorem 12, we obtain the following common fixed point result for mappings satisfying cocyclic contractive condition of integral type in a complex valued metric space.Corollary 15.
LetX
,
d be a complex valued metric space, let A
1
,
A
2
,
…
,
A
p, be nonempty closed subsets of X, and let Y
=
∪
i
=
1
p
A
i. Suppose that f
,
g
:
Y
→
Y are mappings such that(a)
Y has a cocyclic representation with respect to pair (
f
,
g
) and to the collection {
A
i
:
i
=
1,2
,
…
,
p
};
(b)
there exist control functionsψ and φ such that, for any x
,
y
∈
A
i
×
A
i
+
1
,
i
=
1,2
,
…
,
p,(69)
∫
θ
ψ
d
f
x
,
f
y
ϕ
w
d
w
⪯
∫
θ
ψ
U
f
,
g
x
,
y
ϕ
w
d
w
-
∫
θ
φ
U
f
,
g
x
,
y
ϕ
w
d
w
is satisfied, whereϕ
∈
Υ and(70)
U
f
,
g
x
,
y
∈
d
g
x
,
g
y
,
d
f
x
,
g
x
d
f
y
,
g
y
1
+
d
g
x
,
g
y
,
d
f
x
,
g
y
d
f
y
,
g
y
1
+
d
g
x
,
g
y
2
,
d
f
x
,
g
y
d
f
y
,
g
x
1
+
d
g
x
,
g
y
withA
p
+
1
=
A
1.If g
A
i is a complete subspace of X for each i
∈
{
1,2
,
…
,
p
}, then f and g have a unique coincidence point z
∈
∩
i
=
1
p
g
A
i
=
K provided that f
K
⊆
g
K
⊆
K and g
K is closed. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.Proof.
DefineΨ
:
C
+
→
C
+ by Ψ
(
z
)
=
∫
θ
z
‍
ϕ
(
w
)
d
w
. From (69), it follows that(71)
Ψ
ψ
d
f
x
,
f
y
⪯
Ψ
ψ
U
f
,
g
x
,
y
-
Ψ
φ
U
f
,
g
x
,
y
,which can be written as(72)
ψ
∗
d
f
x
,
f
y
⪯
ψ
∗
U
f
,
g
x
,
y
-
φ
∗
U
x
,
y
,where ψ
∗
=
Ψ
∘
ψ and φ
∗
=
Ψ
∘
φ. Clearly, ψ
∗
,
φ
∗
:
C
+
→
C
+
,
ψ
∗ is continuous and nondecreasing, φ
∗ is lower semicontinuous, and ψ
∗
(
z
)
=
φ
∗
(
z
)
=
θ if and only if z
=
θ
. Hence, by Theorem 12, f and g have a unique common fixed point.Remark 16.
It is worth mentioning that the results obtained in this paper are distinct from the results in [8] in the following sense:(1)
The results in this paper are proved in the setup of complex valued metric spaces in which the distance between any two elements is a complex number but the results proved in [8] are in the setup of multiplicative metric spaces, where the distance between any two elements is a real number greater than or equal to 1.
(2)
The contractive conditions imposed on the mappings established in this paper are substantially different from contractions used in [8].



---

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