Rational Fuzzy Cone Contractions on Fuzzy Cone Metric Spaces with an Application to Fredholm Integral Equations

Rehman, Saif Ur; Aydi, Hassen

doi:https://doi.org/10.1155/2021/5527864

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Volume 2021 | Article ID 5527864 | https://doi.org/10.1155/2021/5527864

Rational Fuzzy Cone Contractions on Fuzzy Cone Metric Spaces with an Application to Fredholm Integral Equations

Saif Ur Rehman¹and Hassen Aydi^2,3,4

Academic Editor: Liliana Guran

Received28 Feb 2021

Accepted22 Apr 2021

Published10 May 2021

Abstract

This paper is aimed at proving some common fixed point theorems for mappings involving generalized rational-type fuzzy cone-contraction conditions in fuzzy cone metric spaces. Some illustrative examples are presented to support our work. Moreover, as an application, we ensure the existence of a common solution of the Fredholm integral equations: and , for all , , and , where is the space of all -valued continuous functions on the interval and .

1. Introduction

In 1922, Banach [1] proved a “Banach contraction principle,” which is stated as follows: “A self-mapping on a complete metric space verifying the contraction condition has a unique fixed point.” This principle plays a very important role in the fixed point theory. A number of researches have generalized it in many directions for single-valued and multivalued mappings in the context of metric spaces. Some of the findings can be found in [2–13] and the references therein. Currently, the fixed point theory is one of the most interested research areas in the field of mathematics. In the last decades, it has been investigated in many fields, such as game theory, graph theory, economics, computer sciences, and engineering.

The theory of fuzzy sets was introduced by Zadeh [14], while the concept of a fuzzy metric space (FM space) was given by Kramosil and Michalek [15]. After that, the stronger form of the metric fuzziness was presented by George and Veeramani in [16]. Later on, in [17], Gregori and Sapena proved some contractive-type fixed point results in complete FM spaces. Some more fixed point results in FM spaces can be found in [18–27] and the references therein.

Initially, in 2007, the concept of a cone metric space was reintroduced by Huang and Zhang [28]. They proved some nonlinear contractive-type fixed point results in cone metric spaces. After the publication of this article, a number of researchers have contributed their ideas in cone metric spaces. Some of such works can be found in [29–34] and the references therein.

In 2015, the basic concept of a fuzzy cone metric space (FCM space) was given by Öner et al. [35]. They presented some key attributes and a “fuzzy cone Banach contraction theorem” in FCM spaces. Later, Rehman and Li [36] extended and improved a “fuzzy cone Banach contraction theorem” and proved some generalized fixed point theorems in FCM spaces. Some more properties and related fixed point results can be found in [37–47].

The aim of this research work is to establish some rational-type fuzzy cone-contraction results in FCM spaces. We use the concept of [36, 39] and prove some common fixed theorems under generalized rational-type fuzzy cone-contraction conditions in FCM spaces. Some illustrative examples are presented. In the last section, we give an application of two Fredholm integral equations (FIEs).

2. Preliminaries

Definition 1 [47]. An operation is called a continuous -norm if (i) is commutative, associative, and continuous(ii) and , whenever and , for all The basic -norms: the minimum, the product, and the Lukasiewicz continuous -norms are defined by [47]

Definition 2 [35]. A 3-tuple is said to be a FCM space if is a cone of , is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions: (1) and (2)(3)(4) is continuousfor all .

Definition 3 [35]. Let be a FCM space and and be a sequence in . (i) converges to if for and there is such that , for . We may write this or as (ii) is Cauchy if for and there is such that , for (iii) is complete if every Cauchy sequence is convergent in (iv) is fuzzy cone contractive if there is so that

Lemma 4 [35]. Let be a FCM space and let be sequence in converging to a point iff as for each .

Definition 5 [36]. Let be a FCM space. The fuzzy cone metric is triangular if

Definition 6 [35]. Let be a FCM space and . Then, is said to be fuzzy cone contractive if there is such that A “fuzzy cone Banach contraction theorem” [35] is stated as follows: “Let be a complete FCM space in which fuzzy cone contractive sequences are Cauchy and be a fuzzy cone contractive mapping. Then, has a unique fixed point.”

In this paper, we present some rational-type fuzzy cone-contraction theorems in FCM spaces by using the concept of [36, 39]. Namely, we prove some common fixed theorems under generalized rational-type fuzzy cone-contraction conditions in FCM spaces without the assumption that the fuzzy cone contractive sequences are Cauchy. We use “the triangular property of the fuzzy cone metric.” We also present some illustrative examples to support our work. In the last section, an application of Fredholm integral equations is provided.

3. Main Results

In this section, we prove some common fixed point theorems via generalized rational-type fuzzy cone-contraction conditions in FCM spaces.

Theorem 7. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a common fixed point in .

Proof. Fix and construct a sequence of points in such that Then, by (5), for ,

By Definition 2 (3), , for . One writes

After simplification, we get that

where since . Similarly,

Again, by Definition 2 (3), , for . We have

After simplification, we have

where the value of is the same as in (9). Now, from (9) and (12) and by induction, we have

which shows that is a fuzzy cone-contractive sequence in , and we get that

Note that is triangular; then, for all ,

which yields that is a Cauchy sequence in . Since is complete, there is such that

Now, we prove that . Since is triangular,

By (5), (14), and (16), for ,

Again, by Definition 2 (3), , for . It follows that

Then,

This together with (17) and (16) implies

Note that because . Then, , that is, . Similarly, we can show that because is triangular. Therefore,

Now, again by (5), (14), and (16), one writes for

Again, by Definition 2 (3), , for It follows that

Then,

This together with (22) and (16) implies

Note that since . Then, , that is, .

Hence, is a common fixed point of and .

Example 1. Let , be a continuous -norm and be written as Then, easily one can verify that is triangular and is a complete FCM space. Now, we define by Then, for , we have Hence, the pair of self-mapping is a fuzzy cone-contraction. Now, from Definition 2 (3), and , for . We get the following: Hence, from the above, we conclude that all the conditions of Theorem 7 are satisfied with , , and . The mappings and have a common fixed point, i.e., .

Putting in Theorem 7, we get the following corollary.

Corollary 8. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a common fixed point in .

In the following corollary, we prove that the mappings and have a unique common fixed point in by using the constant in Theorem 7.

Corollary 9. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Hence, and have a unique common fixed point in .

Proof. It follows from the proof of Theorem 7 that is a common fixed point of and in . For uniqueness, let be another common fixed point of and in such that and . Then, by view of (32), By Definition 2 (3), It follows that Since , one writes , i.e., for .

Corollary 10. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a unique common fixed point in .

Example 2. As in Example 1, let be defined by Then, easily one can verify that is triangular and is a complete FCM space. Now, we define self-mappings by Then, from (36), for , we have Hence, all the conditions of Corollary 10 are satisfied with and . The mappings and have a common fixed point, i.e., .

Theorem 11. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a common fixed point in .

Proof. The proof is similar as the proof of Theorem 7.

Corollary 12. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a common fixed point in .

Corollary 13. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that , , , and with . Then, and have a unique common fixed point in .

Proof. It is as the proof of Theorem 7. Let be a common fixed point of and in . Let be another common fixed point of and in such that and . Then, by view of (42), By Definition 2 (3), It follows that Since , , i.e., .

Corollary 14. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a unique common fixed point in .

Example 3. Let . As in Example 2, we define self-mappings by Now, from (46), for , we have Hence, all the conditions of Corollary 14 are satisfied with and . The mappings and have a common fixed point, i.e., .

4. Application

In this section, we present an application on Fredholm integral equations. Let be the space of all -valued continuous functions on the interval , where . The Fredholm integral equations are

where and . The induced metric be defined as

The binary operation is defined by for all . A standard fuzzy metric is given as

Then, easily one can verify that is triangular and is a complete FCM space.

Theorem 15. The two FIEs are where and . Assume that are such that for every , where If there exists such that for all , where Then, the two FIEs defined in (49) have a common solution in .

Proof. Define the mappings by The FIEs in (49) have a common solution if and only if and have a common fixed point in . Now, we have to show that Theorem 7 is applied to the integral operators and . Then, for all , we have the following four cases. (a)If in (55), then from (51) and (54), we haveThis implies that for all such that . It is obvious that the inequality (58) holds if . Thus, the integral operators and satisfy all the conditions of Theorem 7 with and in (5). The integral operators and have a common fixed point, i.e., (49) has a common solution in . (b)If in (55), then from (51) and (54), we haveIt yields that for all and for . Here, we simplify the term by using Definition 2 (3) and (51), for , we have This implies that for all and for . Now, from (60) and (62), we have for all such that . It is obvious that the inequality (63) holds if . Thus, the integral operators and satisfy all the conditions of Theorem 7 with and in (5). The integral operators and have a common fixed point, i.e., (49) has a common solution in . (c)If in (55), then from (51) and (54), we haveThis implies for all and for . Here, we simplify the term , by Definition 2 (3). For , we have In view of (51) and after routine calculation, we get for . Now, from (65) and (67), we have for all such that . It is obvious that the inequality (68) holds if . Thus, the integral operators and satisfy all the conditions of Theorem 7 with and in (5). The integral operators and have a common fixed point, i.e., (49) has a common solution in . (d)If in (55), then from (51) and (54), we haveThis implies that for all such that . It is obvious that the inequality (70) holds if . Thus, the integral operators and satisfy all the conditions of Theorem 7 with and in (5). The integral operators and have a common fixed point, i.e., (49) has a common solution in .

5. Conclusion

In this paper, we presented the concept of rational-type fuzzy cone contractions in FCM spaces and some common fixed point results under generalized rational-type fuzzy cone-contraction conditions in complete FCM spaces by using the “triangular property of fuzzy cone metric” as a basic tool. Moreover, we resolved some Fredholm integral equations as an application. So, one can use this concept to prove more rational-type fuzzy cone-contraction results in complete FCM spaces with different types of applications.

Data Availability

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

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

The authors have equally contributed to the final manuscript.

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Copyright © 2021 Saif Ur Rehman and Hassen Aydi. 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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