Abstract
In this paper, we prove some common fixed point theorems for rational contraction mapping on complex partial -metric space. The presented results generalize and expand some of the literature’s well-known results. We also explore some of the applications of our key results.
1. Introduction
Introduced in 1989 by Bakhtin [1] and Czerwick [2], the concept of -metric spaces provided a framework to extend the results already known in the classical setting of metric spaces. The concept of complex valued metric spaces was introduced in 2011 by Azam et al. [3] and given some common fixed point theorems under the condition of contraction. Rao et al. [4] introduced the definition of complex valued -metric spaces in 2013 and provided a scheme to expand the results, as well as proved a common fixed point theorem under contraction. In 2017, Dhivya and Marudai [5] introduced the concept of complex partial metric space and suggested a plan to expand the results, as well as proved common fixed point theorems under the rational expression contraction condition. Gunaseelan [6, 7] presented the concept of complex partial -metric space in 2019, as well as proved the fixed point theorem under the contractive condition. Many researchers have studied some intriguing concepts and applications and have shown significant results [7–23]. In this paper, we prove some common fixed point theorems for rational contraction mapping 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 and .
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 (i), (ii), and (iii) is satisfied and we write if only (iii) is satisfied. Notice that (a)If , then (b)If and , then (c)If and , then for all
Here, denote the set of nonnegative complex numbers.
Now, let us recall some basic concepts and notations, which will be used in the sequel.
Definition 1 [6]. A complex partial -metric on a nonvoid set is a function such that for all , (i)(ii)(iii)(iv) a real number and is an independent of such that A complex partial -metric space is a pair such that is a nonvoid set and is the complex partial -metric on . The number is called the coefficient of .
Definition 2 [6]. Let be a complex partial -metric space with coefficient . Let be any sequence in and . Then, (1)The sequence is said to converge to , if (2)The sequence is said to be Cauchy sequence in if exists and is finite(3) is said to be a complete complex partial -metric space if for every Cauchy sequence in , there exists such that .
Definition 3. Let and be self-mappings of nonempty set . A point is called a common fixed point of and if .
In 2019, Gunaseelan [6] proved some fixed point theorems on complex partial -metric space as follows.
Theorem 4. Let be any complete complex partial -metric space with coefficient and be a mapping satisfying for all , where . Then, has a unique fixed point and .
Inspired by the study made by Gunaseelan [6], here, we prove some common fixed point theorems for rational contraction mapping on complex partial -metric space with an application.
3. Main Results
In this section, we will give our main result of this paper, where some common fixed point theorems for rational contraction mapping on complex partial -metric space are given.
Theorem 5. Let be a complete complex partial -metric space with the coefficient and be mappings satisfying for all , where are nonnegative reals with . Then, and have a unique common fixed point in .
Proof. Let be arbitrary point in , and define a sequence in such that
Next, show that the sequence is Cauchy. By using (3), we get
so that
By the hypothesis of theorem, we get
Hence,
Similarly,
since . Therefore, with and for all , consequently, we have
That is,
For any , we have
From (10), we get
Hence,
and hence,
Thus, is a Cauchy sequence in . Since is complete, there exists some such that as and
Assume on the contrary that there exists such that
By using the triangular inequality and (2), we obtain
which implies that
As in (18), we obtain that , a contradiction with (16). Therefore . Hence, . Similarly, we obtain .
Assume that is another common fixed point of and . Then,
so that
Hence, , which proves the uniqueness. This completes the proof of the theorem.☐
Theorem 6. Let be a complete complex partial -metric space with the coefficient and be mappings satisfying for all , where , and are nonnegative reals with . Then, and have a unique common fixed point in .
Proof. Let be arbitrary point in , and define a sequence in such that
Next, show that the sequence is Cauchy. By using (22), we get
so that
By the notion of complex partial -metric space, we get
Hence,
Similarly,
Set . Since and for all , consequently, we have
That is,
For any , we have
From (29), we get
Hence,
and hence,
Thus, is a Cauchy sequence in . Since is complete, there exists some such that as and
Assume on the contrary that there exists such that
By using the triangular inequality and (21), we obtain
which implies that
As in (37), we obtain that , a contradiction with (35). Therefore, . Hence, . Similarly, we obtain .
Assume that is another common fixed point of and . Then,
which implies that , a contradiction. So , which proves the uniqueness.☐
Example 1. Let be endowed with the order if and only if . Then, is a partial order in . Define the complex partial -metric space as follows (Table 1):
It is easy to verify that is a complete complex partial -metric space with the coefficient for . Define by ,
Let and ; we consider the following cases:
(1)If and , then and the conditions of Theorem 5 are satisfied(2)If and , then , ,(3)If and , then , (4)If and , then , (5)If and , then , Moreover for , the conditions of Theorem 5 are satisfied. Therefore, is the unique common fixed point of and .
4. Application
Consider the following systems of integral equations: where (1)[](2) is an unknown variable for each , (3) is the deterministic free term defined for (4) and are deterministic kernels defined for
In this section, we present an existence theorem for a common solution to (44) and (45) that belongs to (the set of continuous functions defined on ) by using the obtained result in Theorem 5. We consider the continuous mappings given by
Then, the existence of a common solution to the integral equations (44) and (45) is equivalent to the existence of a common fixed point of and . It is well known that , endowed with the metric defined by for all , is a complete complex partial -metric space. can also be equipped with the partial order given by
Further, let us consider that a system of integral equation as (44) and (45) under the following condition holds: (1) are continuous functions satisfying
Theorem 7. Let be a complete complex partial -metric space; then, the systems (44) and (45) under condition (3) have a unique common solution.
Proof. For and , we define the continuous mappings by Then, we have Hence, all the conditions of Theorem 5 are satisfied for , with . Therefore, the system of integral equations (44) and (45) has a unique common solution.
5. Conclusion
In this paper, we proved some common fixed point theorems for rational contraction mapping on complex partial -metric space. An illustrative example and application on complex partial -metric space is given.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
The fourth-named author extends appreciation to the Deanship of Scientific Research at King Khalid University for funding work through research groups program under grant R.G.P.1/15/42.