Integral Equations Approach in Complex-Valued Generalized <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces

Mehmood, Shahid; Rehman, Saif Ur; Ullah, Ihsan; Bantan, Rashad A. R.; Elgarhy, Mohammed

doi:https://doi.org/10.1155/2022/7454498

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Volume 2022 | Article ID 7454498 | https://doi.org/10.1155/2022/7454498

Integral Equations Approach in Complex-Valued Generalized -Metric Spaces

Shahid Mehmood,¹Saif Ur Rehman,¹Ihsan Ullah,²Rashad A. R. Bantan,³and Mohammed Elgarhy^1,4

Academic Editor: Sun Young Cho

Received15 Nov 2021

Revised19 Dec 2021

Accepted27 Dec 2021

Published24 Jan 2022

Abstract

In this paper, we study a rational type common fixed-point theorem (CFP theorem) in complex-valued generalized -metric spaces (-metric spaces) by using three self-mappings under the generalized contraction conditions. We find CFP and prove its uniqueness. To justify our result, we provide an illustrative example. Furthermore, we present a supportive application of the three Urysohn type integral equations (UTIEs) for the validity of our result. The UTIEs are

where , where is the set of all real-valued continuous functions defined on and .

1. Introduction

In 1922, Banach [1] proved a “Banach contraction principle” which is stated as “a single-valued contractive type mapping in a complete metric space has a unique FP.” Later on, this principle generalized in many directions, and some researchers contributed to the theory of FP. In [2], Nazam et al. presented the idea of weakly increasing -contractions and established some results in ordered partial-metric space with an application. Hu and Gu [3] presented a new idea of Menger probabilistic S-metric space by using the concepts of probabilistic and S-metric spaces. They investigated its topological characteristics and established some FP theorems with illustrative examples.

Bakhtin [4] introduced the idea of -metric space. After that, Czerwik [5] established some FP results by using -metric spaces. In [6], Boriceanu et al. extended the concept of fractal operator theory by introducing it in b-metric spaces and presented some generalized CFP results. While, Akkouchi [7] used the concept of an implicit relation in the said spaces and established CFP results for contractive type maps. Došenović at el. [8] discussed, complemented, improved, generalized, and enriched some FP results of () contraction in ordered b-metric spaces. They made a different approach of taking Picard’s sequence which help shorten the proof comparing other previous studies in literature. They also complimented and enriched CFP results for contraction maps.

Delfani et al. [9] contributed in ordered b-metric spaces by establishing FP theorems. In these results, they introduced the generalizations of and contractions. They also provided a suitable example to verify their FP result. In [10], Karapinar et al. contributed in b-metric spaces by generalizing it to prove FP results in view of -Meir-Keeler type contractions. These results improved and unified some previous results. Abdeljawad et al. [11] extended the b-metric spaces to double controlled metric spaces by improving control functions and on the right side of b-triangle, that is, . They also provided some examples in which two functions are not comparable.

Mustafa and Sims [12] introduced the generalized idea of metric space and established some FP theorems by using Dhage’s theory. Later on, Mustafa et al. [13] proved some modified contractive-type FP results in this space. Abbas and Rhoades [14] started to investigate CFP results in the said spaces. In [15], Chugh et al. established the property P in -metric space and proved some results. In this context, Mohanta et al. [16] contributed by establishing a CFP result which improved and supplemented some of the existing results. Mustafa introduced a mapping pair and obtained many CFP results for different contraction conditions. He supported these results through examples.

In 2014, Aghajani et al. [17] presented the idea of -metric space and proved a weakly compatible CFP theorem. Aydi [18] improved, complemented, unified, and generalized some well-known existing results in said spaces and established some coupled and tripled coincidence point results. Gupta in [19] extended some existing results from literature and obtained various FP results in -metric spaces. In [20], Makran et al. proved general CFP theorem by using multivalued maps and established its application. Mebawondu et al. [21] proved FP results for a different contraction type maps which involve -class, -admissible, Suzuki-type maps in -metric spaces.

Ege [22] gave the idea of complex-valued -metric space and proved “Banach contraction principle and Kannan’s contraction theorems for FP.” Ege [23] used the idea of -series to prove CFP results in said spaces. He also introduced contraction maps to prove CFP results. Kumar and Vashistha [24] introduced the idea of coupled FP for mapping in the said space. They proved coupled FP results and supported it by providing an example satisfying their main result. Recently, Mehmood et al. [25] proved some CFP theorems by using compatible single-valued contractive type mappings on complex-valued -metric spaces with an application.

This article presents a contraction theorem in complex-valued -metric spaces by using three self-maps to establish a generalized CFP-theorem. This result improves, extends, and modifies some of the existing results (e.g, [22, 26]). We present an example to support our work. We also present an application of the three UTIEs for the existence of a common solution to support our work. This study is organized as follows: Section 2 consists of preliminary concepts. In Section 3, we present a CFP theorem with an illustrative example. While in Section 4, we present an application to support our main work. Finally, in Section 5, we discuss the conclusion of our work.

2. Preliminaries

Let the complex numbers be denoted by and . Now, we define as , iff and , where the real and imaginary parts of the complex number are denoted by and , respectively. Accordingly, , if any one of the following conditions holds:(1) and ,(2) and ,(3) and ,(4) and .

Particularly, we can write if and one of (2), (3), and (4) is satisfied.

Remark 1 (see [27]). The following given properties can be held and verified if(1) and ,(2),(3) and .

Definition 1 (see [17]). Let set and be a given real number. A mapping is called a -metric if holds the following axioms:(1) if (2) for all with (3) for all with (4), where is a permutation of (5) for all Then, a pair is called a -metric space.
Note that each G-metric space is a -metric space with .

Definition 2 (see [22]). Let set and be a given real number. A mapping is called a complex-valued -metric if holds the following axioms:(1) if (2) for all with (3) for all with (4), where is a permutation of (5) for all Then, a pair is called a complex-valued -metric space.

Example 1. Let and a mapping be defined asfor all . Then, is a complex-valued -metric with , but it is not -metric. To see this, let , and we get , , and ; thus,.
Clearly, property (5) of definition (2.3) is satisfied with ; hence,
is -metric.

Proposition 1 (see [22]). Let be a complex-valued -metric space. Then, .(1)(2)

Definition 3 (see [22]). Let be a complex-valued -metric space and be a sequence in .(1) is complex-valued -convergent to if for every , , such that .(2)A sequence is called complex-valued -Cauchy if for every , , such that .(3)If every complex-valued -Cauchy sequence is complex-valued -convergent in , then a pair is called complex-valued -complete.

Proposition 2 (see [22]). Let be a complex-valued -metric space and be a sequence in . Then, is complex-valued -convergent to if and only if as .

Theorem 1 (see [22]). Let be a complex-valued -metric space; then, for a sequence in and a point , the following are equivalent:(1) is complex-valued -convergent to (2) as (3) as (4) as

Proposition 3 (see [22]). Let be a complex-valued -metric space and be a sequence in . Then, is a complex-valued -convergent to if and only if as .Proof: Assume that is complex-valued -convergent to and let

Then, we have , and there is a natural number , such that for all . Thus, for all , and so, as .

Suppose that as . For a given with , there exists a real number , such that for ,

Considering , we have a natural number , such that for all . This means that for all , i.e., is complex-valued -convergent to .

3. Main Result

Theorem 2. Let be a complete complex-valued -metric space with coefficient and be mappings satisfyingwherefor all ,, such that ,, and . Then, , and have a unique CFP in .
Proof. Fix , and be a sequence in , such that

Now, by using (5),

This implies thatwhere

This implies that

By using Definition 2.3 (3) and after simplification, we get that

Now, there are two possibilities:(i)If is a maximum term in , then ; so, after simplification, (3.3) can be written as(ii)If is a maximum term in , then ; so, after simplification, (3.3) can be written as

Let ; then, from (13) and (14), for all , we have

Similarly,

Now, from (16) and (15) and by induction, we have that

Let and , and we have that

By Proposition 2.5 (1), we have for . If we take limit as , we obtain . It implies is a -Cauchy sequence. Since, is complete complex-valued -metric space, , such that, , as or . We have to show that ; by contrary case, let . Now by (5),

This implies thatwhere

This implies that

After simplification, we get that

Now, there are six possibilities:(i)f is a maximum term in (23), then (21) can be written as Applying on both sides, we get This implies that ; thus,(ii)If is a maximum term in (23), then (21) can be written as Applying on both sides, we get This implies that is a contradiction, where ; thus,(iii)If is a maximum term in (23), then (21) can be written as Applying on both sides, we get This implies that is a contradiction, where ; thus,(iv)If is a maximum term in (23), then (21) can be written as Applying on both sides, we get This implies that ; thus,(v)If is a maximum term in (23), then (21) can be written as Applying on both sides, we get This implies that ; thus,(vi)If is a maximum term in (23), then (21) can be written as Applying on both sides, we get This implies that ; thus,

From (26)–(41), we get that is a FP of , that is,

Next, we have to show that ; by contrary case, let . Now by (5),

This implies thatwhere

This implies that

Now, there are eight possibilities:(i)If is a maximum term in (46), then after simplification, (44) can be written as Applying on both sides, we get This implies that is a contradiction, where ; thus,(ii)If is a maximum term in (46), then (44) can be written as Applying on both sides, we get This implies that ; thus,(iii)If is a maximum term in (46), then (44) can be written as Applying on both sides, we get This implies that ; thus,(iv)If is a maximum term in (46), then (44) can be written as Applying on both sides, we get This implies that is a contradiction, where ; thus,(v)If is a maximum term in (46), then (44) can be written as Applying on both sides, we get This implies that is a contradiction, where ; thus,(vi)If is a maximum term in (46), then (44) can be written as Applying on both sides, we get This implies that ; thus,(vii)If is a maximum term in (46), then (44) can be written as Applying on both sides, we get This implies that ; thus,(viii)If is a maximum term in (46), then (44) can be written as Applying on both sides, we get This implies that ; thus,

From (49) to (70), we find that is a FP of , that is,

Now, we have to show that ; by contrary case, let . Now, by (5),

This implies thatwhere

This implies that

After simplification, we get that

Now, there are six possibilities:(i)If is a maximum term in (76), then (73) can be written as Applying on both sides, we get This implies that ; thus,(ii)If is a maximum term in (76), then (73) can be written as Applying on both sides, we get This implies that ; thus,(iii)If is a maximum term in (76), then (73) can be written as Applying on both sides, we get This implies that ; thus,(iv)If is a maximum term in (76), then (73) can be written as Applying on both sides, we get This implies that ; thus,(v)If is a maximum term in (76), then (73) can be written as Applying on both sides, we get This implies that ; thus,(vi)If is a maximum term in (76), then (73) can be written as Applying on both sides, we get This implies that ; thus,

From (79)–(94), we find that is a FP of , that is,

Hence proved that is a CFP of and , that is,

Uniqueness: assume that is another CFP of the mappings , and , such that

Then, from (5), we have that

This implies thatwhere

This implies that

Now, by using Proposition 2.5 (2) and after simplifying, we get that

Clearly, is a maximum term in (102), so (99) can be written as

This implies that is a contradiction, where . Hence proved that , and have a unique CFP in .

Corollary 1. Let be a complete complex-valued -metric space with coefficient and be mappings satisfyingwherefor all ,, such that ,, and ; then, , and have a unique CFP in .

Corollary 2. Let be a complete complex-valued -metric space with coefficient and be mappings satisfyingwherefor all ,, such that ,, and ; then, , and have a unique CFP in .

Corollary 3. Let be a complete complex-valued -metric space with coefficient and be mappings satisfyingwherefor all ,, such that ,, and ; then, , and have a unique CFP in .

Remark 2. If we put and in Theorem 2, we get (Theorem 3.7 [22]).

Example 2. Let be a complex-valued -metric space, where and , with , for all . Now, we define asNotice thatIn all regards, it is enough to show that , for all and , with , , and .For , we discuss different cases with , and . Hence, Case 1: let ; then from (112) and (113), directly we get that . Hence, (3.1) is satisfied with , and . Case 2: let ; then from (112) and (113), we find is satisfy with , i.e., Hence, (3.1) is satisfied with , and . Case 3: let ; then from (112) and (113), we find is satisfy with , i.e., Hence, (3.1) is satisfied with , and . Case 4: let ; then from (112) and (113), we find is satisfy with , i.e.,Hence, (3.1) is satisfied with , and . Thus, all the conditions of Theorem 2 are satisfied with noticing that the point , which remains fixed under mappings , and , is indeed unique.

4. Applications

In this section, we present an application of the three UTIEs to support our main work. Let be the set of all real-valued continuous functions defined on . Now, we state and prove a result based on the three UTIEs to uplift our work.

Theorem 3. Let , where and are defined asfor all and . Consider the UTIEs arewhere . Let are such that ; for every , we have thatIf there exists ,wherewithwhereThen, the three UTIEs, i.e., (120) have a unique common solution.

Proof 1. Define asThen, we have the following main two cases:(1)If is the maximum term in , then from (122), (123), and (127), we have that For all . Thus, the maps , and satisfy the hypothesis of Theorem 2 with and in (31). Then, the given three UTIEs i.e., (4.1) have a unique common solution in .(2)If is the maximum term in , then from (123), we have thatThen, there are furthermore eight subcases arising:(i)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have that For all . Thus, the maps , and satisfy the hypothesis of Theorem 2 with and in (31). Then, the given three UTIEs, i.e., (4.1) have a unique common solution in .(ii)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have that For all . Thus, the maps , and satisfy the hypothesis of Theorem 2 with and in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in .(iii)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have that For all . Thus, the maps , and satisfy the hypothesis of Theorem 2 with and in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in .(iv)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have that For all . Thus, the maps , and satisfy the hypothesis of Theorem 2 with and in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in .(v)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have that For all . Thus, the maps , and satisfy all the conditions of Theorem 2 with and in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in .(vi)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have that For all . Thus, the maps , and satisfy all the conditions of Theorem 2 with and in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in .(vii)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have that For all . Thus, the maps , and satisfy all the conditions of Theorem 2 with and in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in .(viii)If is the maximum term in (125), then ; now, from (122), (127), and (129), we have thatfor all . Thus, the maps , and satisfy all the conditions of Theorem 2 with and in (5). Then, the given three UTIEs, i.e., (120) have a unique common solution in .

5. Conclusions

We have established a generalized CFP-theorem in complex-valued -metric spaces for three self-mappings. In this result, we have used a generalized rational contraction condition and proved a unique CFP-theorem. To justify our result, we presented an illustrative example in the said space by using three self-maps. Also, we present an application of integral equations to get the existing result for a common solution to support our work. By using this concept, one can prove different contractive-type FP and CFP results for many self-mappings in complex-valued -metric spaces with different types of integral operators.

Data Availability

No datasets were generated or analyzed to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah (D-715-150-1441).

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Copyright © 2022 Shahid Mehmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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