1. Introduction
Azam et al. [
1] introduced the concept of complex-valued metric spaces and studied some fixed point theorems for mappings satisfying a rational inequality.
Two years later, in [
2], Rao et al. discussed for the first time the idea of complex-valued b-metric spaces.
In 2017, Dhivya and Marudai [
3] introduced the concept of complex partial metric space and suggested a plan to expand the results, as well as proving common fixed-point theorems under the rational expression contraction condition. This idea has been followed by Gunaseelan [
4], who introduced the concept of complex partial b-metric spaces and discussed some results of fixed-point theory for self-mappings in these new spaces.
In [
5], Prakasam and Gunaseelan proved the existence and uniqueness of a common fixed-point (with an illustrative example) theorem using CLR and E.A. properties in complex partial b-metric spaces. Their proved results generalize and extend some of the well-known results in the literature.
In [
6], Gunaseelan et al. proved a fixed-point theorem in complex partial b-metric spaces under a contraction mapping. They also gave some applications of their main results.
In this paper, we prove some common fixed-point theorems on complex partial metric space.
2. Preliminaries
Let be the set of complex numbers and . Define a partial order ⪯ on as follows:
if and only if , .
Consequently, one can infer that if one of the following conditions is satisfied:
- (i)
, ,
- (ii)
, ,
- (iii)
, ,
- (iv)
, .
In particular, we write if , and one of and is satisfied and we write if only is satisfied. Notice that
- (a)
If , then
- (b)
If and , then ,
- (c)
If and , then for all .
Here and denote the set of non-negative complex numbers and the set of non negative real numbers, respectively.
Now, let us recall some basic concepts and notations that will be used below.
Definition 1 ( [
3])
. A complex partial metric on a non-void set G is a function such that for all :- (i)
- (ii)
- (iii)
if and only if
- (iv)
.
A complex partial metric space is a pair such that G is a non-void set and is the complex partial metric on G.
Definition 2 ( [
3])
. Let be a complex partial metric space. Let be any sequence in G. Then- (i)
The sequence is said to converge to θ, if .
- (ii)
The sequence is said to be a Cauchy sequence in if
exists and is finite.
- (iii)
is said to be a complete complex partial metric space if for every Cauchy sequence in G there exists such that
.
- (iv)
A mapping is said to be continuous at if for every , there exists such that .
Definition 3 ( [
3])
. Let Π and Ψ be self-mappings of non-void set G. A point is called a common fixed point of Π and Ψ if . Theorem 1 ( [
3])
. Let be a partially ordered set and suppose that there exists a complex partial metric in G such that is a complete complex partial metric space. Let be a pair of weakly increasing mappings, and suppose that for every comparable we have eitherfor with , , orIf Π or Ψ is continuous, then Π and Ψ have a common fixed point and .
Inspired by Theorem 1, here we prove some common fixed-point theorems on complex partial metric space with an application. For complex partial metric space, we will use the CPMS notation.
3. Main Results
Theorem 2. Let be a complete CPMS and be two continuous mappings such thatfor all , where and . Then, the pair has a unique common fixed point and . Proof. Let
be arbitrary point in
G and define a sequence
as follows:
Then by (
1) and (
2), we obtain
Case I:
If
, then we have
This implies , which is a contradiction.
Case II:
If
, then we have
From the next step, we have
The following three cases arise.
Case IIa:
which implies
, which is a contradiction.
Since , we get . Therefore is a Cauchy sequence in G.
From (
3) and (
5),
, we get
For
, with
, we have
Moreover, by using (
5), we get
Hence, is a Cauchy sequence in G.
Case III:
If .
For the next step, we have
Then, we have the following three cases:
Case IIIa:
which implies
, which is a contradiction.
Then by (
6) and (
7), we get
, where
. Hence
is a Cauchy sequence in
G.
Using (
6) and (
8) yields
where
.
Then,
and we get
For
, with
, we have
Hence,
is a Cauchy sequence in
G. In all cases above discussed, we get the sequence
, which is a Cauchy sequence. Since
G is complete, there exists
such that
as
and
By the continuity of
, it follows that
as
.
Next, we have to prove that
is a fixed point of
.
As
, we obtain
. Thus,
. Hence
and
. In the same way, we have
such that
as
and
By the continuity of
, it follows
as
.
Next we have to prove that
is a fixed point of
.
As , we obtain . Thus, . Hence, and . Therefore, is a common fixed point of the pair .
To prove uniqueness, let us consider
is another common fixed point for the pair
. Then
This implies that . □
In the absence of the continuity condition for the mappings and , we get the the following theorem.
Theorem 3. Let be a complete CPMS and be two mappings such thatfor all , where and . Then, the pair has a unique common fixed point and . Proof. Following from Theorem 2, we get that the sequence is a Cauchy sequence. Since G is complete, there exists such that as .
Since and are not continuous, we have .
From the Cauchy property of
, the above limit is zero, and then,
Hence, , which is a contradiction. Then . In the same way, we obtain . Hence is a common fixed point for the pair and . Uniqueness of the common fixed point follows from Theorem 2. □
For , we get the following fixed points results on CPMS.
Theorem 4. Let be a complete CPMS and be a continuous mapping such thatfor all , where and . Then the pair Π has a unique fixed point and . Remark 1. Similarly, we get a fixed point result in the absence of continuity condition for the mapping Π.
Corollary 1. Let be a complete CPMS and be a continuous mapping such thatfor all , where , and . Then, Ψ has a unique fixed point and . Proof. By Theorem 2, we get
such that
and
. Then, we get
Hence . Then has a unique fixed point. □
Remark 2. From the above Corollary 1, similarly, we get a fixed-point result in the absence of continuity condition for the mapping Ψ.
Next, we present a new generalization of a common fixed point theorem on CPMS.
Theorem 5. Let be a complete CPMS and be two continuous mappings such thatfor all , where and . Then, the pair has a unique common fixed point and . Proof. Let
be arbitrary point in
G and define a sequence
as follows:
Then, by (
12) and (
13), we obtain
If
, then
This shows that
, which is a contradiction. Therefore
From (
14) and (
15),
, we get
For
, with
, we have
Hence,
is a Cauchy sequence in
G. Since
G is complete, there exists
such that
as
and
Since
is continuous, it yields
Similarly, by the continuity of
, we get
. Then the pair
has a common fixed point. To prove uniqueness, let us consider that
is another common fixed point for the pair
. Then
This implies that . □
In the absence of the continuity condition for the mapping and in the Theorem 5, we obtain the following result.
Theorem 6. Let be a complete CPMS and be two mappings such thatfor all , where and . Then the pair has a unique common fixed point and . Proof. Following from Theorem 5, we get that the sequence
is a Cauchy sequence. Since
G is complete, then there exists
such that
as
and
Since and are not continuous, we have .
Hence, , which is a contradiction. Then . In the same way, we obtain . Hence, is a common fixed point for the pair . For uniqueness of the common fixed point, follows from Theorem 5. □
For , we get the following fixed-points results on CPMS.
Theorem 7. Let be a complete CPMS and be a continuous mapping such thatfor all , where and . Then, Π has a unique fixed point and . Remark 3. Similarly, in the absence of continuity condition, we can get a fixed point result on Π.
Corollary 2. Let be a complete CPMS and be a continuous mapping such thatfor all , where and . Then Π has a unique fixed point and . Proof. By Theorem 5, we get
such that
and
. Then we get
Hence . Then, has a unique fixed point. □
Remark 4. From the above corollary 2, similarly, we get a fixed point result in the absence of continuity condition for the mapping Π.
Example 1. Let be endowed with the order if and only if . Then, ⪯ is a partial order in G. Define the complex partial metric space as follows: | |
(1,1), (2,2) | 0 |
(1,2),(2,1),(1,3),(3,1),(2,3),(3,2),(3,3) | |
(1,4),(4,1),(2,4),(4,2),(3,4),(4,3),(4,4) | |
Obviously, is a complete CPMS for . Define by , Clearly Π and Ψ are continuous functions. Now, for , we consider the following cases:
- (A)
If and , then and the conditions of Theorem 2 are satisfied.
- (B)
If , , then , , - (C)
If , , then , , - (D)
If , , then , , - (E)
If , , then , , Moreover, for , with , the conditions of Theorem 2 are satisfied. Therefore, 1 is the unique common fixed point of Π and Ψ.
4. Application
Consider the following systems of nonlinear integral equations:
and
where
- (i)
is a continuous mapping and is a given function in ,
- (ii)
and are unknown variables for each , ,
- (iii)
and are deterministic kernels defined for .
In this section, we present an existence theorem for a common solution to (
18) and (
19) that belongs to
(the set of continuous functions defined on
J) by using the obtained result in Theorem 2. We consider the continuous mappings
given by
and
Then, the existence of a common solution to the nonlinear integral Equations (
18) and (
19) is equivalent to the existence of a common fixed point of
and
. It is well known that
G, endowed with the metric
, defined by
for all
, is a complete CPMS.
G can also be equipped with the partial order ⪯ given by
Further, let us consider a system of nonlinear integral equation as (
18) and (
19) under the following condition hold:
- (A)
are continuous functions satisfying
where
Theorem 8. Let be a complete CPMS; then, the system (18) and (19) under condition (A) have a unique common solution.
Proof. For
and
, we define the continuous mappings
by
and
Hence, all the conditions of Theorem 2 are satisfied for
with
. Therefore the system of nonlinear integral Equations (
18) and (
19) have a unique common solution. □