Common Fixed Point Theorems on <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="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 8.2614 11.6718" width="8.2614pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g></svg>-</span>Metric Spaces for Integral Type Contractions Involving Rational Terms and Application to Fractional Integral Equation

Saluja, G. S.; Nashine, Hemant Kumar; Jain, Reena; Ibrahim, Rabha W.; Nabwey, Hossam A.

doi:https://doi.org/10.1155/2024/5108481

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusion Data Availability Disclosure Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 5108481 | https://doi.org/10.1155/2024/5108481

Common Fixed Point Theorems on -Metric Spaces for Integral Type Contractions Involving Rational Terms and Application to Fractional Integral Equation

G. S. Saluja,¹Hemant Kumar Nashine,^2,3Reena Jain,²Rabha W. Ibrahim,⁴and Hossam A. Nabwey^5,6

Academic Editor: Pasquale Vetro

Received08 Dec 2023

Revised19 Jan 2024

Accepted10 Apr 2024

Published03 May 2024

Abstract

It has been shown that the findings of -metric spaces may be deduced from -metric spaces by considering . In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete -metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper’s findings generalize and expand a number of previously published conclusions. In addition, the abstract conclusions are supported by an application of the Riemann-Liouville calculus to a fractional integral problem and a supportive numerical example.

1. Introduction

The development of nonlinear analysis has been largely influenced by fixed point theory. The Banach contraction principle is generally acknowledged to be one of the most useful theorems in nonlinear analysis [1]. Its relevance derives from its wide applicability to a multitude of mathematical fields. Numerous generalizations exist for this Banach contraction concept. In contrast, a variety of generalizations of metric spaces have been accomplished, one of which is a -metric space. As a generalization of a metric, Sedghi et al. [2] established the notion of a -metric in 2012.

Definition 1 (see [2]). Let be a set and let be a function satisfying the following conditions for all :
(S1) if and only if
(S2)
Then, the function is called an -metric on , and the pair is called an -metric space (in short SMS).

The following instances are easily verifiable as SMS.

Example 1 (see [2]). Let and be a norm on ; then, is a SMS.

Example 2 (see [3]). Let be a set and be an ordinary metric on . Then, for all is a SMS.
For a set and , denote

Sedghi et al. [2] extended the well-known Banach’s contraction concept to a complete SMS. The assertion is as follows.

Theorem 2 (see [2]). Let be a complete SMS and let be a self-mapping of such that for all , where is a constant. Then, has a unique fixed point in .

In addition, Sedghi et al. [2] proved that the results of -metric spaces can be derived from SMS results if we consider . In [4, 5], Saluja discussed fixed point (FP) and common fixed point (CFP) type of results on implicit type of contraction conditions to prove CFP on SMS. Hieu et al. [6] discussed FP on Ciric quasicontractions to derive the FP for maps on SMS. In [7–9], the authors discussed FP on different types of contraction conditions on SMS. In [10], Sedghi et al. discussed CFP for two pairs of mappings on SMS.

In recent years, different contractive circumstances have been investigated using fixed point theory. Indeed, integral type contraction is among them. In 2002, Branciari [11] analyzed the existence of FP for mapping defined on a complete metric space satisfying a general contractive condition of integral type (see Theorem 2.1 of [11]). Following Branciari’s [11] finding, other studies have been conducted on generalizing integral type contractive conditions for various contractive mappings that meet a variety of known features (see [12, 13]). Rhoades [14] has done comparable work, extending the finding of Branciari [11] by substituting the following condition for the contractive condition of integral type (3) of Theorem 2.1 in [11]: for each and , where is a Lebesgue-integrable mapping which is summable on each compact subset of , nonnegative, and such that for each ,

In the framework of SMS, Rahman et al. [15] produced a CFP result of the Altman integral type for two pairs of self-mappings and provided an example to support the conclusion. Recently, Özgür and Taș [16] and Saluja [17] investigated novel integral type contractive conditions on -metric spaces, establishing certain FP theorems for various integral type contractive conditions and providing examples to illustrate their findings. They also discovered a solution to the Fredholm integral equation. Rashwan and Hammad [18] have recently proven some CFP theorems for noncommutative mappings satisfying a general contractive condition of integral type involving rational terms in the context of complete metric spaces, and they provide examples to back up their findings.

Motivated by [2, 16, 18], we demonstrate CFP results for contractive conditions of integral type using rational expressions in the context of complete -metric spaces (CSMS) in any a way that it can not convert to the results of metric spaces. To justify the work, examples are given in support. This paper’s findings extend and generalize a number of previously published findings. In addition, for practise, we recommend the fractional integral equation expressed in terms of the Riemann-Liouville calculus. Using our new fixed point theorems, we establish the necessary criteria for obtaining a unique solution. In addition, a numerical example is given in the next section.

2. Preliminaries

Now, we will review some fundamental definitions, attributes, and auxiliary results pertaining to SMS.

Definition 3 (see [2]). Let be a SMS. (1)A sequence in converges to if and only if . We denote this by or as (2)A sequence in is called a Cauchy sequence if (3)The SMS is called complete if every Cauchy sequence in is convergent in

Definition 4. Let be a set and let be two self-mappings of . Then, (i)a point is called a fixed point of mapping if (ii)a point is common fixed point of and if (iii)in [19], is called a coincidence point point of and , if for some , and is called a point of coincidence of and (iv)in [20], and are said to be commuting if for all (v)in [21], and are said to be weakly compatible if they commute at their coincidence points, i.e., if for some implies

Proposition 5. (see [19]). Let and be weakly compatible self-mappings on a set . If and have a unique point of coincidence , then is unique.

Lemma 6 (see [2], Lemma 2.5). Let be a SMS. Then, we have for all .

Lemma 6 can be considered as a symmetry condition on an -metric space.

Lemma 7 (see [2], Lemma 2.12). Let be a SMS. If and as , then as .

The link between a metric and a -metric is demonstrated in the following lemma.

Lemma 8 (see [6]). Let be a metric space. Then, the following properties are satisfied: (1) for all is a SMS(2) in if and only if in (3) is Cauchy in if and only if is Cauchy in (4) is complete if and only if is complete

The function described in Lemma 8 (1) is referred to as the -metric created by the metric . In [6, 9], there is an example of a -metric that is not derived by any metric.

3. Main Results

Our first result is the following.

Theorem 9. Let be a CSMS, and let such that for all , where , , , , , and are nonnegative reals such that and is defined in (4). Then, is a singleton set.

Proof. Let and the sequence be defined as and for . Then, from inequality (6) for and and using notion of SMS and Lemma 6, we have which implies If we take , then we find since . Using the inequality (9) again, we obtain Thus, in general, for , we have Passing limit (11), since . Condition (4) implies Next to show that the sequence is a Cauchy sequence. Assume that is not a Cauchy sequence. Then, there exists an for which we can find subsequences and of and increasing sequences of integers and such that is smallest index for which, Further corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (14). Then, Now, using (16), (S2), and Lemma 6, we have the following equation (by (16)): Passing in (17) and using (13), Owing (S2) and Lemma 6, Using the inequality (6) for and and equations (9), (15), and (20), then we obtain a contradiction to the assumption as . Thus, the sequence is a Cauchy sequence in . Thus, , i.e, , by completeness of .
Now, from the given inequality (6) for and , we find Passing in (23) and owing (S1) and Lemma 6, which implies , that is, since . This implies that . Similarly, ; hence,
Let be the another CFP. Using the inequality (6) for , , and Lemma 6, we obtain which implies , that is, since . Thus, is a singleton set.
If we take in Theorem 9; then, we obtain the following result.

Corollary 10. Let be a CSMS and such that for all , where , , , , , and are nonnegative reals such that and is defined in (4). Then, is a singleton set.

If we take and in Corollary 10, then we obtain the following result.

Corollary 11 (see [16]). Let be a CSMS and let such that for all , where is a constant and is defined in (4). Then, is a singleton set and for each .

Remark 12. (1)Corollary 11 is a generalization of Branciari [11] fixed point result from complete metric space to the setup of CSMS(2)In Corollary 11, if we set for all , we get Theorem 3.1 in [2], Theorem 3.1 in [22], and Corollary 2.5 in [8] for (3)Corollary 11 is also a generalization of Sedghi et al.’s result [2] to the case of integral type contraction condition(4)Theorem 9 and Corollary 10 are generalization of Theorem 2.4 of Özgür and Taș [16]. Indeed, if we take , , and in Theorem 9; then, we get Theorem 2.4 in [16]

Theorem 13. Let be a CSMS, and let be four mappings satisfying the following conditions: (i) and (ii)The pairs and are weakly compatiblefor all , where , , , , , and are nonnegative reals such that and is defined in (4). Then, is a singleton set.

Proof. Let and the sequence be defined as for . Then, from (27) for and and owing (28), (S1), (S2), and Lemma 6, which implies If we take , then we find since . Using the inequality (30) again, we obtain Passing in (35), since . Condition (4) implies Next to prove that the sequence is a Cauchy sequence. Let , where , from equations (27) and (28), we find

By the similar manner, for , we have following cases:

Case 1. If ,

Case 2. If ,

Passing in both the cases,

Hence, is a Cauchy sequence in a CSMS , so it is convergent to the point, say , that is, and

Since , there exists a point such that . If , then from equation (30), we get

Passing in (41) and owing (S1), (S1), (39), and Lemma 6,

or where , since . This implies a contradiction as ; therefore, , that is, . Thus, .

Hence, . Since the pair is weakly compatible, then

Similarly, , and there exists a point such that . Then, from equation (27) and using the same method as above, we can find that , so .

Hence, . Also, the pair is weakly compatible; then,

Next to prove is . Owing (27), (S1), (S2), and Lemma 6,

Passing in (47),

This implies where , since , a contradiction as ; therefore, , that is, ; also, from (44), we get

By the similar method, we can show that is a fixed point of , that is, , so from (45), we have

From equations (50) and (51), we get that . Finally, we prove uniqueness of common fixed point. To do this, be an another CFP. Using (27), (28), (S1), (S2), and Lemma 6, we have a contradiction since . Thus, we conclude that ; that is, .

Remark 14. Theorem 13 generalizes Theorem 2.4 in [16] and Banach fixed point theorem [1].

Example 3. Let be the CSMS with the -metric defined as for all . Define mapping as for all and as for all . Then, for each . Therefore, satisfies (25) of Corollary 10 for (by taking and ) (i), , and (ii), , and (iii), , and It is noted that . But does not satisfy (26) of Corollary 11 for the same values as since .

Example 4. Let and define metric on as for all . Define the mappings , , , and on by

Let the function be defined as for all ; then, for each ,

It is clear that and , so at the points and , the pairs and are weakly compatible. Now for the points and , we calculate the following:

Also,

Thus,

The inequality (61) is satisfied if for . Hence, the inequality (27) is verified, and all the conditions of Theorem 13 are hold and

4. Application on Fractional Integral Equation

The Riemann-Liouville integral is defined by where the gamma function is represented as . If is a locally integral function and is a complex integer in the half-plane , the integral is well defined. Proceeding via (62), the fractional Fredholm integral equation is formulated by where is a continuous function in a square region , such that and is a continuous function on Define the space of continuous function

Furthermore, we define the function as follows: where Thus, the function is an -metric. We then go on to show that no other metric can provide the above metric. Assume not, that is, there exists a metric, say such that

Consequently, we obtain where

Correspondingly, we have

Combining (66)–(69), we have which is not true. That is, the -metric is not generated by any metric. Consequently, is a CSMS.

Proposition 15. Suppose that is a CSMS achieving the metric (65). If then the fractional integral equation (63) has a unique solution

Proof. Define the operator by Next, we aim to prove that achieves the contraction condition. which implies As a result, based on the assumptions, the contraction requirement is met at and in Corollary 11. Hence, the Fredholm integral equation (63) has just one solution.

Example 5. Consider the fractional integral equation, which is written as on the space such that Then, the iteration solution becomes If then according to Proposition 15, for and , the integral equation (75) admits a unique solution. Note that when , we get Moreover, inequality (77) can be realized by which fits the assumption on Proposition 15.

5. Conclusion

In this paper, we have proved some unique CFP theorems for contractive conditions of integral type involving rational terms in CSMS and gave some consequences as corollaries of the main results. Also, some illustrated examples are provided to validate the results. The results of findings in this work generalize and extend several results from the existing literature. We have employed the results to get a unique solution of a fractional integral equation.

Data Availability

No underlying data was collected.

Disclosure

This article is distributed under the terms of the Creative Commons Attribution.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This research was funded by Prince Sattam bin Abdulaziz University through the project number PSAU/2024/01/822148.

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Copyright © 2024 G. S. Saluja 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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