Unique Fixed-Point Results in Fuzzy Metric Spaces with an Application to Fredholm Integral Equations

Shamas, Iqra; Rehman, Saif Ur; Aydi, Hassen; Mahmood, Tayyab; Ameer, Eskandar

doi:https://doi.org/10.1155/2021/4429173

Journal of Function Spaces

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Unique and Non-Unique Fixed Points and their Applications

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Volume 2021 | Article ID 4429173 | https://doi.org/10.1155/2021/4429173

Unique Fixed-Point Results in Fuzzy Metric Spaces with an Application to Fredholm Integral Equations

Iqra Shamas,¹Saif Ur Rehman,¹Hassen Aydi,^2,3,4Tayyab Mahmood,⁵and Eskandar Ameer⁶

Academic Editor: Anita Tomar

Received21 Apr 2021

Accepted01 Sept 2021

Published23 Sept 2021

Abstract

This paper aims at proving some unique fixed-point results for different contractive-type self-mappings in fuzzy metric spaces by using the “triangular property of the fuzzy metric”. Some illustrative examples are presented to support our results. Moreover, we present an application by resolving a particular case of a Fredholm integral equation of the second kind.

1. Introduction

In 1922, Banach [1] proved a “Banach contraction principle (BCP),” which is stated as “a self-mapping in a complete metric space satisfying a contraction condition has a unique fixed point”. This theorem plays a very important role in the theory of fixed points. Many researchers gave generalization and improved the BCP in many directions for single-valued and multivalued mappings in the context of metric spaces by ensuring the existence of fixed point, common fixed point, and coincidence point results with different types of applications, such as differential-type applications, integral-type applications, functional-type applications. In 2004, Ran and Reurings [2] proved a fixed-point theorem in a metric space by using partially ordered sets and they present some applications to matrix equations. While in [3], Nieto and Rodrguez-López extended and improved the result of Ran and Reurings [2] by using increasing mappings and applied the result to get a unique solution for the first-order ordinary differential equation with periodic boundary equations. In 2017, Priskillal and Thangavelu [4] established some fixed-point theorems in complete metric spaces by using -contractive fuzzy mappings with an application to fuzzy differential equations. Some more fixed-point results in the context of metric spaces can be found in [5–16].

In 1965, the theory of fuzzy sets was introduced by Zadeh [17]. Lately, this theory is improved, investigated, and applied in many directions. Among them, we state the theory of fuzzy logic, which is based on the notion of relative graded memberships, as inspired by the processing of human perceptions and cognitions. Fuzzy logic can deal with information arising from computational perceptions and cognitions, that is, uncertain, obscure, imprecise, partly true, or without sharp limits. A fuzzy logic permits the inclusion of vague human assessments in computing problems. The fuzzy logic is extremely useful for many people associated with innovative work including engineering (electrical, chemical, civil, environmental, mechanical, industrial, geological, etc.), mathematics, computer software, earth science, and physics. Some of their findings can be found in [18–25].

The other direction of fuzzy sets is used in topology and analysis by many mathematicians. Subsequently, several authors have applied various forms of general topologies and developed the concept of fuzzy spaces. Kramosil and Michalek [26] developed the concept of a fuzzy metric space (FM-space). Later on, Grabeic [27] extended the BCP and proved a fixed-point result in FM-spaces in the sense of Kramosil and Michalek. George and Veeramani [28] modified the concept of FM-spaces with the help of continuous -norms and proved some basic properties in this direction. In 2002, Gregori and Sapena [29] proved some contractive-type fixed-point theorems in complete FM-spaces in the sense of Kramosil and Michalek [26] and in the sense of George and Veeramani [28]. Rana et al. [30] established some fixed-point theorems in FM-spaces by using implicit relations. Many authors have introduced the number of fixed-point theorems in FM-spaces by using the concept of compatible maps, implicit relations, weakly compatible maps, and R-weakly compatible maps (see [31–38] and the references therein). Furthermore, Beg and Abbas [39], Popa [40], and Imad et al. [41, 42] obtained some fixed-point and invariant approximation results in FM-spaces. Recently, Li et al. [43] proved some strong coupled fixed-point theorems in FM-spaces with an integral-type application. Later on, Rehman et al. [44] proved some rational fuzzy-contraction theorems in FM-spaces with nonlinear integral-type application.

The purpose of this paper is at obtaining some extended unique fixed-point theorems in FM-spaces without the “assumption that all the sequences are Cauchy” by using the concept of Li et al. [43] and Rehman et al. [44]. We present some illustrative examples and an integral-type application to support our work. By using this concept, one can prove more generalized contractive-type fixed-point and common fixed-point results in FM-spaces with different types of integral equations. Our paper is organized as follows: Section 2 consists of preliminary concepts. In Section 3, we prove some generalized fixed-point results without continuity in FM-spaces and we presented some examples in the support of our obtained results. In Section 4, we consider some generalized Ćirić fuzzy contraction results in complete FM-spaces. In Section 5, we present an application of a particular case of the Fredholm integral equation of the second kind by ensuring the existence of a solution.

2. Preliminaries

The concept of a continuous -norm is given by Schweizer and Sklar [45].

Definition 1 (see [45]). An operation is known as a continuous -norm if it satisfies the following: (1) is commutative, associative, and continuous(2) and , whenever and , for all

The basic continuous -norms: the minimum, the Lukasiewicz, and the product -norms are defined, respectively, as follows:

Definition 2 (see [28]). A 3-tuple is said to be a FM-space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following: (i) and (ii)(iii)(iv) is continuous for all and

Definition 3 (see [28, 29]). Let be a FM-space, , and be a sequence in . Then, (i)A sequence in is said to be convergent to a point if for (ii) is said to be a Cauchy sequence, if for each and , there is such that , (iii) is complete, if every Cauchy sequence is convergent in (iv) is known as a fuzzy contractive, if there is so that

Definition 4 (see [46]). Let be a FM-space. The fuzzy metric is triangular if for all and .

Lemma 5. A fuzzy metric is triangular.

Proof. Let be a fuzzy metric defined by

Now, we have

This implies that

Hence, it is proved that a fuzzy metric is triangular.

Lemma 6 (see [46]). Let be a FM-space. Let and be a sequence in . Then, iff for .

Definition 7 (see [29]). Let be a FM-space and . Then, is known as a fuzzy contraction, if there is so that for all and .

3. Generalized Fixed-Point Results in FM-Spaces

In this section, we consider some generalized contraction theorems on FM-spaces for fixed points (by using the “triangular property of the fuzzy metric”).

Theorem 8. Let be a complete FM-space so that the fuzzy metric is triangular. Let satisfy for all , , with . Then, has a unique fixed point.

Proof. Fix . Take an iterative sequence such that for all Now, by view of (8), we have

Then, we have for ,

Three possibilities arise: (i)If is the minimum term in , then, after simplification, (10) can be written as(ii)If is the minimum term in , then again, (10) can be written as(iii)If is the minimum term in , then again, (10) becomes

Let . Then, from all cases, we get

Similarly,

Now, from (14) and (15) and by induction, for ,

This yields that

Since is triangular, we have

Thus, is a Cauchy sequence. Since is complete, there is so that

We shall show that . By the triangular property of , we have that

Now, by using (8), (17), and (19), we have

Hence,

Equation (22) together with (20) and (19) implies that

As , one has . This implies that .

The uniqueness is as follows: let be such that . Then, in view of (8), we have for

Hence, we get that , so . Thus, has a unique fixed point in .

Corollary 9. Let be a complete FM-space so that the fuzzy metric is triangular. Let verify that for all , , and with . Then, has a unique fixed point.

Proof. It follows by putting in Theorem (8).

Example 10. Let be equipped with a continuous -norm. Let be a fuzzy metric defined by for all and . Then, is a complete FM-space. Now, we define the mapping by

Then,

Hence, all the conditions of Corollary 9 are satisfied with . Hence, the self-mapping has a unique fixed point, that is,

Corollary 11. Let be a complete FM-space so that the fuzzy metric is triangular. Let verify that for all , , and with . Then, has a unique fixed point.

Proof. It follows by putting in Theorem (8).

Example 12. Let be endowed with a continuous -norm. Let be a fuzzy metric defined by for all and . Then, is a complete FM-space. Now, we define a mapping by

We have

Hence, all the conditions of Corollary 11 are satisfied with . Then, the self-mapping has a unique fixed point, that is,

4. Ćirić-Type Fuzzy Contraction Results in FM-Spaces

In this section, we define Ćirić-type fuzzy contraction mappings and we present a unique related fixed-point theorem on a complete FM-space.

Definition 13. Let be a complete FM-space. A self-mapping is said to be a Ćirić contraction if there is such that for all and . Here, is called the contractive constant of .

Theorem 14. Let a self-mapping be a Ćirić contraction in a complete FM-space so that is triangular and (33) satisfies with . Then, has a unique fixed point.

Proof. Fix . Take an iterative sequence such that for all Now, by using (33), we have

After simplification, for , we get

Now, there are three possibilities: (i)If is the maximum in , then, from (35), we have(ii)If is the maximum in , then, from (35), we have (iii)If is the maximum in , then, from (35), we have

Let . Using (36) and (38), we have

Similarly,

Now, from (39) and (40) and by induction, for

This yields that

Since is triangular and , we have

Hence, is a Cauchy sequence. Since is complete, there is so that

Now, we have to show that . Since is triangular, one writes

In view of (33), (42), and (44), we have for

Hence,

This together with (45) and (44), we have

Since , one gets . This implies that .

The uniqueness is as follows: let be such that . Using (33), we have

We get that for . This implies that . Thus, has a unique fixed point in .

Example 15. Let be endowed with a continuous -norm. Let a fuzzy metric be defined by for all and . Then, is a complete FM-space. Now, we define a mapping as

We have for . Hence, all the conditions of Theorem 14 are satisfied with and has a unique fixed point, that is,

5. Application

In this section, we present an integral-type equation. Let be the space of all real-valued continuous functions on the interval , where . Now, we present a particular case of a Fredholm integral equation (FIE) of the second kind given as follows: where and . The induced metric is defined by

The binary operation , being a continuous -norm, is defined by for all . The standard fuzzy metric can be expressed as

Then easily, we can show that is triangular and is a complete space.

Theorem 16. Assume that there is so that where

Then, the FIE (53) has a unique solution.

Proof. Give as

Notice that is well defined and (53) has a unique solution if and only if has a unique fixed point in . Now, we have to show that Theorem 8 is applied to the integral operator . Then, for all , we have the following six cases: (1)If is the maximum term in (57), then . Therefore, in view of (55) and (56), we have

This implies that for all such that . The inequality (60) holds if . Thus, the integral operator satisfies all the conditions of Theorem 8 with and in (8). Then, the integral operator has a unique fixed point, i.e., (53) has a solution in (2)If is the maximum term in (57), then, . Therefore, using (55) and (56), we have

It yields that for all such that (3)If is the maximum term in (57), then . Therefore, by (55) and (56), we have

That is, for all such that (4)If is the maximum term in (57), then, . Therefore, due to (55) and (56), we have

Hence, for all such that (5)If is the maximum term in (57), then, . Using (55) and (56), we have

It implies that for all such that

The inequalities (62), (64), (66), and (68) hold if . Thus, the integral operator satisfies all the conditions of Theorem 8 with and in (8). The integral operator has a unique fixed point, i.e., (53) has a solution in . (6)If is the maximum term in (57), then, . Therefore, from (55) and (56), we have

That is, for all such that . The inequality (70) holds if . Thus, the integral operator satisfies all the conditions of Theorem 8 with and in (8). The integral operator has a unique fixed point, i.e., (53) has a solution in .

Now, we present a special type of example for a particular case of an FIE of a second kind.

Example 17. Take . If we put in (53), then, we have the following integral equation:

Equation (71) is a special kind of the integral equation (53), where . Then, where . Now, we have to show that . From equation (58), we have

Hence, all conditions of Theorem 16 hold with . The integral equation (71) has a unique solution by using Theorem 16.

6. Conclusion

In this paper, we proved variant unique fixed-point results for some generalized contraction-type self-mappings in complete FM-spaces, without continuity and by using the “triangular property of the fuzzy metric” as a basic tool. We presented illustrative examples. Moreover, we provided an application about a particular case of Fredholm integral equation of second kind. In this direction, researchers can prove more fixed-point results in complete FM-spaces without using continuity via different types of applications.

Data Availability

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

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors have equally contributed to the final manuscript.

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