Journal of Function Spaces

Journal of Function Spaces / 2021 / Article
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Unique and Non-Unique Fixed Points and their Applications

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Research Article | Open Access

Volume 2021 |Article ID 3856468 | https://doi.org/10.1155/2021/3856468

Arul Joseph Gnanaprakasam, Salah Mahmoud Boulaaras, Gunaseelan Mani, Mohamed Abdalla, Asma Alharbi, "Solving Integral Equations by Common Fixed Point Theorems on Complex Partial -Metric Spaces", Journal of Function Spaces, vol. 2021, Article ID 3856468, 8 pages, 2021. https://doi.org/10.1155/2021/3856468

Solving Integral Equations by Common Fixed Point Theorems on Complex Partial -Metric Spaces

Academic Editor: Liliana Guran
Received09 May 2021
Accepted09 Jun 2021
Published23 Jun 2021

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 [723]. 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 .


(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)

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