Some New Coupled Fixed-Point Findings Depending on Another Function in Fuzzy Cone Metric Spaces with Application

Waheed, Muhammad Talha; Rehman, Saif Ur; Jan, Naeem; Gumaei, Abdu; Al-Rakhami, Mabrook

doi:https://doi.org/10.1155/2021/4144966

Mathematical Problems in Engineering

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Generalised Fuzzy Models Applied to Logical Algebras and Intelligent Systems in Engineering

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Research Article | Open Access

Volume 2021 | Article ID 4144966 | https://doi.org/10.1155/2021/4144966

Some New Coupled Fixed-Point Findings Depending on Another Function in Fuzzy Cone Metric Spaces with Application

Muhammad Talha Waheed,¹Saif Ur Rehman,¹Naeem Jan,¹Abdu Gumaei,²and Mabrook Al-Rakhami³

Academic Editor: G. Muhiuddin

Received01 May 2021

Accepted16 Jun 2021

Published08 Jul 2021

Abstract

In this paper, we introduce the new concept of coupled fixed-point (FP) results depending on another function in fuzzy cone metric spaces (FCM-spaces) and prove some unique coupled FP theorems under the modified contractive type conditions by using “the triangular property of fuzzy cone metric.” Another function is self-mapping continuous, one-one, and subsequently convergent in FCM-spaces. In support of our results, we present illustrative examples. Moreover, as an application, we ensure the existence of a common solution of the two Volterra integral equations to uplift our work.

1. Introduction

Fixed-point theory is one of the most interesting areas of research. In 1922, Banach [1] proved a “Banach contraction principle” stated as follows: “a single-valued contractive type mapping in a complete metric space has a unique FP.” After the publication of this principle, many researchers have contributed their ideas to the problems on fixed points in the context of metric spaces for single-valued and multivalued mappings with different types of applications. Kannan [2] and Chatterjea [3] proved some fixed-point theorems, while Reich [4, 5] presented some remarks concerning contractive type mappings in complete metric spaces. Covitz and Nadler [6] and Daffer and Kaneko [7] proved some multivalued fixed-point theorems, while Kaewkhao and Neammanee [8] established fixed-point theorems for multivalued Zamfirescu mapping in complete metric spaces. In 2007, Huang and Zhang [9] introduced the notion of cone metric space in which they extended and modified the concept of metric spaces. They proved the convergence properties and some fixed-point results by using the concept of the underlying cone are normal. Meanwhile, in 2008, Rezapour and Hamlbarani [10] proved fixed-point theorems without the assumption of normality of cone. After that, many others contributed their ideas to the problems on fixed-point results in cone metric spaces. Some of their contributions to the problems on cone metric spaces for fixed points can be found in [11–14].

Initially, the concept of fuzzy set theory was given by Zadeh [15]. Recently, the fuzzy set theory has been investigated, applied, and modified in many directions, in which the one direction of this theory is fuzzy logic, which has a wide range of applications again in many directions such as in engineering fields, business, and education. In education, fuzzy logic is used for the student results evaluation, which can be directly monitored by the teacher. Some of the references related to an education system based on fuzzy logic can be found in [16–19]. The other direction of the fuzzy set is the fuzzy metric theory. The notion of FM-space was introduced by Kramosil and Michalek [20]; they used the concept of a fuzzy set on metric space and proved some basic properties of the FM-space. After that, the stronger form of the metric fuzziness was given by George and Veeramani [21]. Later on, Gregori and Sapena [22] proved some contractive type FP theorems in FM-spaces. Recently, in 2020, Li et al. [23] proved some strongly coupled FP theorems by using cyclic contractive type mappings in complete FM-spaces. Meanwhile Rehman et al. [24] presented the concept of rational type contraction mappings and proved some FP theorems in complete FM-spaces with an application.

In 2015, Oner et al. [25] introduced the concept of fuzzy cone metric spaces (FCM-spaces) and proved some basic properties and “a single-valued Banach contraction theorem for FP with the assumption that all the sequences are Cauchy.” Later on, Rehman and Li [26] established some generalized fuzzy cone-contractive type results for FP without the assumption that “all the sequences are Cauchy.” After that, Jabeen et al. [27] proved common FP theorems for quasi-contraction by using the concept of compatible and weakly compatible for three self-mappings with an integral type application. In 2020, Chen et al. [28] introduced the concept of coupled contractive type mappings in FCM-spaces and proved some coupled FP results with application to nonlinear integral type application. Recently, in 2021, Rehman and Aydi [29] proved some rational type common FP theorems in FCM-spaces with an application.

In [30], Guo and Lakshmikantham introduced the coupled FP results for the nonlinear operator with applications. After that, some coupled FP theorems in partially ordered metric spaces were proved by Bhaskar and Lakshmikantham [31] and Lakshmikantham and Ciric [32]. In 2010, Sedghi et al. [33] proved common coupled FP theorems for commuting mappings in FM-spaces. Meanwhile Moradi [34] presented some results on “Kannan FP on complete and generalized metric spaces which depends on another function” by using the concept of subsequence convergence and continuity.

In this paper, we use the above concepts together and prove some unique coupled FP theorems depending on another function in FCM-spaces. Moreover, we present an application of the two Volterra integral equations for a common solution to support our results. This new concept will play an important role in the theory of fixed point to prove more coupled FP and strongly coupled FP results in complete FCM-spaces with the application of different types of differential equations. This paper is organized as follows: Section 2 gives preliminary concepts. In Section 3, we use the concepts of Guo and Lakshmikantham [30], Moradi [34], Chen et al. [28], and Jabeen et al. [27] all together and establish some unique coupled FP results depending on another continuous function which is one-one and subsequently convergent in FCM-spaces. In Section 4, we present an application of the two Volterra integral equations for the existence of a common solution to support our main work. In the last section (Section 5), we present the conclusion of our work.

2. Preliminaries

Definition 1. Let G be any set. A fuzzy set in G is a function whose domain is G and the range is .

Definition 2 (see [35]). A binary operation would be a continuous -norm if fulfils the following conditions:(i) is associative and commutative(ii) is continuous(iii)(iv) whenever and

Definition 3 (see [9]). Let be a real Banach space, and is a subset of . Then, is called a cone if(i) is closed and nonempty and (ii)If and , then (iii)If both and , then A partial ordering on a given cone is defined by . stands for and , while stands for . In this paper, all cones have a nonempty interior.

Definition 4 (see [21]). A 3-tuple is said to be an FM-space if is any set, is continuous -norm, and is a fuzzy set on satisfying(i)(ii) if and only if (iii)(iv)(v) is continuous, for and

Definition 5 (see [25]). A 3-tuple is said to be an FCM-space if is a cone of , is an arbitrary set, is continuous -norm, and is a fuzzy set on satisfying(i)(ii) if and only if (iii)(iv)(v) is continuous, for , and

Definition 6 (see [25]). Let a 3-tuple be an FCM-space, , and a sequence in is(i)Converging to if and and there is such that , for . We may write this or as ;(ii)Cauchy sequence if and and there is such that , for ;(iii) complete if every Cauchy sequence is convergent in ;(iv)Fuzzy cone contractive if , satisfying

Lemma 1. (see [25]). Let be an FCM-space and let a sequence in converge to a point if as , for .

Definition 7. (see [26]). Let be an FCM-space. The fuzzy cone metric is triangular if

Definition 8. (see [25]). Let be an FCM-space and . Then is said to be fuzzy cone contractive if there exists such that

Definition 9 (see [31]). An element is called coupled fixed point of a mapping ifNow, in the following main results, we shall prove some unique coupled FP theorems depending on another function which is continuous, one-one, and subsequently convergent in FCM-spaces. We present some illustrative examples in support of our results. As a further study, we shall present two Volterra integral equations to ensure the existence of common solution to support our work.

3. Main Results

Now, we are in the position to present our first main result.

Theorem 1. Let be a mapping in a complete FCM-space in which is triangular and is a continuous, one-one, and subsequently convergent self-mapping on , that is, , satisfyingfor all , , and with . Then has a unique coupled FP. Also, if converges sequently, then for every the iterative sequence converges to this coupled FP.

Proof. Consider any ; we define sequences and in such thatNow, from (5) for , we haveAfter simplification, we get thatwhere . Similarly, from (5), for , we havewhere is the same as in (8). Now, from (8) and (9) and by induction, for , we have thatIt is shown that is a fuzzy cone contractive sequence; therefore,Now, for and for , we haveHence, proving that is a Cauchy sequence, we have thatNow, for sequence and from (5), for , we haveAfter simplification, we get thatwhere is the same as in (8). Similarly, from (5) for , we havewhere is the same as in (8). Now, from (15) and (16) and by induction, for , we have thatIt is shown that is a fuzzy cone contractive sequence.Now, for and for , we haveHence, proving that is a Cauchy sequence, we have thatSince is complete, and are Cauchy sequences in ; therefore, and as ; that is, and . Since is subsequently convergent, has a convergent subsequence. So there exist and in such that . Since is continuous, , and . Now, from (5), for , we haveAfter simplification, for , we haveHence, we get that ; this implies that . Since is one-one, . Next, we have to prove that . Then, from (5), for , we haveAfter simplification, for , we haveHence, we get that ; this implies that . Since is one-one, we get .
For uniqueness, suppose that and are coupled fixed-point pairs in such that and . Now, from (5), for , we haveHence, we get that for ; this implies that . Similarly, again from (5), for , we haveHence, we get that for , and this implies that .

Corollary 1. Let be a mapping in a complete FCM-space in which is triangular and is a continuous, one-one, and subsequently convergent self-mapping on , that is, , satisfyingfor all , , and with . Then has a unique coupled FP. Also, if converges sequently, then, for every , the iterative sequence converges to this coupled FP.

Corollary 2. Let be a mapping in a complete FCM-space in which is triangular and is a continuous, one-one, and subsequently convergent self-mapping on , that is, , satisfyingfor all , , and with . Then has a unique coupled FP. Also, if converges sequently, then, for every , the iterative sequence converges to this coupled FP.
If we use as an identity self-mapping, that is, in Theorem 1, then we get the following corollary.

Corollary 3. Let be a mapping in a complete FCM-space in which is triangular, satisfyingfor all , , and with . Then has a unique coupled FP in .

Example 1. Let , and let a fuzzy metric be defined by and . Then easily one can verify that is triangular and is a complete FCM-space. We define by and for all ; and a mapping is defined as and . Then, by using (30), for , we haveLet . Then, from (31), we haveHence, we get thatThus, inequality (33) satisfies all the conditions of Theorem 1 with and and has a unique coupled fixed point; that is, .

Theorem 2. Let be a mapping in a complete FCM-space in which is triangular and is a continuous, one-one, and subsequently convergent self-mapping on , that is, , satisfyingfor all , , and with . Then has a unique coupled FP. Also, if converges sequently, then, for each , the iterative sequence converges to this coupled FP.

Proof. Let any , and we define sequence byNow, from (34), for , we haveAfter simplification, we get thatwhere . Similarly, again from (34), for , we havewhere is the same as in (37). Now, from (37) and (38) and by induction, for , we haveHence, we get that is a fuzzy cone contractive sequence; therefore,Now, for and for , we haveHence, proving that is a Cauchy sequence, we have thatNow, again from (34), for , we haveAfter simplification, we get thatwhere value is the same as in (37). Similarly, again from (34), for , we havewhere value is the same as in (37). Now, from (44) and (45) and by induction, for , we have thatHence, proving that is a fuzzy cone contractive sequence,Now, for and for , we haveHence, proving that is a Cauchy sequence,Since is complete, and are Cauchy sequences in ; therefore and as ; that is, and . Since is subsequently convergent, has a convergent subsequence. So there exist and such that . Since is continuous, and . Now, from (34) and (39), for , we haveAfter simplification, for , we have thatHence, we get that , and this implies that . Since is one-one, . Now, again from (34) and (46), for , we haveAfter simplification, for , we have thatHence, we get that for , and this implies that . Since is one-one, .
For uniqueness, let and be coupled fixed-point pairs in such that and . Now, from (34), for , we haveHence, we get that , and this implies that . Similarly, again from (34), for , we haveHence, we get that for , and this implies that .

Corollary 4. Let be a mapping in a complete FCM-space in which is triangular and is a continuous, one-one, and subsequently convergent self-mapping on , that is, , satisfyingfor all , , and with . Then has a unique coupled FP. Also, if converges sequently, then, for each , the iterative sequence converges to this coupled FP.
If we use as an identity self-mapping, that is, in Theorem 2, then we get the following corollary.

Corollary 5. Let be a mapping in a complete FCM-space in which is triangular, satisfyingfor all , , and with . Then has a unique coupled FP in .

Example 2. Let , and let be defined asfor all and . Then, it could be verified that is triangular and is a complete FCM-space. We define by and for all ; and a mapping is defined as and . Then, by using (58), for , we haveLet . Then, from (??), for , we haveWe haveTherefore,This implies thatThus, inequality (63) satisfies all the conditions of Theorem 2 with and and has a unique coupled FP; that is, and .

4. Application

In this section, we present an application on Volterra integral equations (VIEs) to support our main work. We prove that the solution of the two Volterra integral equations has a common FP of the integral operators of and which are defined in (71) to support our result, that is, Theorem 1. Now we shall define the following terms by using supremum norm to justify our work. Let be the space of all continuous real-valued on . The supremum norm on is defined as

Now, we define a metric by

As is continuous -norm, for all , we have , and a fuzzy metric is defined asfor and . Then, one can easily prove that is triangular and is a complete fuzzy cone metric space.

Now we present the two VIEs for a common solution to uphold our result.

Theorem 3. The two VIEs arewhere and . Assume that such that for , and and . Therefore, we defineIf there exist such thatwherewhere , then the two VIEs (67) have a unique common solution in .

Proof. Define operators and :Then,where and . Now, we may have the three following cases:(1)If , then, from (66), (71) and (72), we have for , and . Hence, operators and satisfy all the conditions of Theorem 1 with and in (5). Thus, the two VIEs in (67) have a unique common solution in . Then integral equations have a unique common solution.(2)If , then, from (66), (71) and (72), we have for , and . Hence, operators and satisfy all the conditions of Theorem 1 with and in (5). Thus, the two VIEs in (67) have a unique common solution in . Then integral equations have a unique common solution.(3)If , then, from (66), (71) and (72), we havefor , and . Hence, operators and satisfy all the conditions of Theorem 1 with and in (5). Thus, the two VIEs in (67) have a unique common solution in . Then integral equations have a unique common solution.

5. Conclusion

We presented the concept of coupled FP results depending on another function in FCM-spaces and prove some unique coupled FP-theorems by using “the triangular property of fuzzy cone metric” by using different contractive type conditions. The other function is a self-mapping that is continuous, one-one, and subsequently convergent in FCM-spaces. Further, we presented two Volterra integral equations to uplift our main work. By using this new concept, one can prove more different contractive type coupled FP results depending on another function in complete FCM-spaces. Instead of Volterra integral equations, researchers can use different types of applications such as Lebesgue integral equations, Riemann integral equations, and nonlinear integral equations to support their findings.

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors are grateful to the Deanship of Scientific Research, King Saud University, for funding through Vice Deanship of Scientific Research Chairs.

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Copyright © 2021 Muhammad Talha Waheed 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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