Abstract
In the theory of fuzzy fixed point, many authors have been proved different contractive type fixed point results with different types of applications. In this paper, we establish some new fuzzy cone contractive type unique coupled fixed point theorems (FP-theorems) in fuzzy cone metric spaces (FCM-spaces) by using “the triangular property of fuzzy cone metric” and present illustrative examples to support our main work. In addition, we present a Lebesgue integral type mapping application to get the existence result of a unique coupled FP in FCM-spaces to validate our work.
1. Introduction
The theory of fuzzy sets was introduced by Zadeh [1]. Later on, in 1975, Kramosil and Michalek [2] introduced the concept of fuzzy metric spaces (FM-space); they present some structural properties of FM-space. In 1988, Grabiec [3] used the concept of Kramosil and Michalek [2] and proved two fixed point theorems (FP-theorems) of “Banach and Edelstein contraction mapping theorems on complete and compact FM-spaces, respectively.” After that, the idea of FM-space given by Kramosil and Michalek [2] was modified by George and Veeramani [4], and they proved that every metric induces a fuzzy metric and also proved some fundamental properties and Baire’s theorem for FM-spaces. In 2002, Gregory and Sapena [5] proved some contractive type FP-theorems in FM-spaces. Roldan et al. [6] presented some new FP-results in FM-spaces, while Jleli et al. [7] proved some results by using cyclic -contractions in Kaleva-Seikkala’s type fuzzy metric spaces. Kiany and Harandi [8] presented the concept of set-valued fuzzy-contractive type maps and proved some FP and end point results in FM-spaces. Latterly, Rehman et al. [9] gave out the notion of rational type fuzzy contraction for FP in complete FM-spaces with an application. Some more related FP-results can be found in [10–15].
Indeed, Huang and Zhang [16] rediscovered the idea of Banach-valued metric space. Indeed, many mathematicians proposed it; but it becomes popular after Huang and Zhang’s study. By adopting the theory that the underlying cone is normal, they demonstrated the convergence properties and some FP-theorems. Rezapour and Hamlbarani [17], in 2008, proved FP-theorems without assuming the cone’s normality, while in [18] Karapinar proved some Ćirić-type nonunique FP-theorems on cone metric spaces. After that, many others contributed their ideas to the problem of FP-findings in cone metric spaces. A few of their FP-findings can be found (e.g., see [19–22]).
In 2015, Oner et al. [23] gave the idea of fuzzy cone metric space (FCM-space), and they also presented some fundamental properties and “a single-valued Banach contraction theorem for FP with the assumption that all the sequences are Cauchy.” After that, Rehman and Li [24] settled some generalized fuzzy cone contractive type FP-results neglecting that “all the sequences are Cauchy” in complete FCM-space. And later, Jabeen et al. [25] presented some common FP-theorems for three self-mappings, by taking into consideration the idea of weakly compatible in FCM-spaces with an integral type application. Chen et al. [26], in 2020, gave the idea of coupled fuzzy cone contractive-type mappings. They proved “some coupled FP-theorems in FCM-spaces with non-linear integral type application.” Latterly, Rehman and Aydi [27], in 2021, presented the concept of rational type fuzzy cone contraction mappings in FCM-spaces. They used “the triangular property of fuzzy metric” as a fundamental tool and proved some common FP-theorems and give an application.
Guo and Lakshmikantham [28] proved “coupled FP-results for the nonlinear operator with applications”. Later, Bhaskar and Lakshmikantham [29] present some coupled FP-theorems in the context of partially ordered metric spaces, and this work is also presented by Lakshmikantham and Ciric [30]. In the year 2010, Sedghi et al. [31] proved some common coupled FP-results for commuting mappings in fuzzy metric spaces.
In this paper, we present some unique coupled FP-findings in FCM-spaces by taking the idea of Guo and Lakshmikantham [28] and Chen et al. [26]. Furthermore, we have also presented an application of the two Lebesgue Integral Equations (LIE) for a common solution to uphold our work. This paper is organized as follows: Section 2 consists of preliminaries. In Section 3, we establish some unique coupled FP-results in FCM-spaces with illustrative examples. In Section 4, we present an application of Lebesgue integral mapping to get the existence result of unique coupled FP in FCM-spaces to hold up our main work. In Section 5, we discuss the conclusion of our work.
2. Preliminaries
Definition 1 [32]. 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 , for
Throughout the complete paper, -norm represents a continuous -norm.
Definition 2 [16]. Let be a real Banach space and be the zero element of , and is a subset of . Then, is called a cone if, (i) is closed and nonempty, and (ii) and , then (iii)both and and then
A partial ordering on a given cone is defined by . stands for and , while stands for . In this paper, all cones have nonempty interior.
Definition 3 [4]. A 3-tuple is said to be a FM-space if is any set, is a -norm, and is a fuzzy set on satisfying (i)(ii) (iii)(iv)(v)is continuous, and
Definition 4 [23]. A 3-tuple is said to be a FCM-space if is a cone of , is an arbitrary set,is a-norm, andis a fuzzy set onsatisfying(i)(ii) (iii)(iv)(v) is continuous, , and
Definition 5 [23]. Let a 3-tuple be a FCM-space, , which is a sequence in(i)It converges to if and ; there is such that , for , or we write it as or as (ii)It is a Cauchy sequence if and ; there is such that , for (iii) is complete if every Cauchy sequence is convergent in (iv)It is fuzzy cone contractive if and fulfilling
Lemma 6 [23]. Let be a FCM-space, and let a sequence in converge to a point which converges to 1 as , for .
Definition 7 [24]. Let be a FCM-space. The FCM is triangular, if
Definition 8 [23]. Let be a FCM-space and . Then, is said to be fuzzy cone contractive if such that
Definition 9. Let be an element in . Then, it is called coupled FP of a mapping if
Now, in the following, we prove some unique couple FP-theorems in FCM-spaces with examples to support our main work. Furthermore, we present an application of Lebesgue integral contractive type mapping to prove a unique coupled FP-theorem in FCM-spaces.
3. Main Results
Now, we present our first main result.
Theorem 10. Let be a mapping on complete FCM-spaces in which is triangular and satisfies the inequality: where , , , and with . Then, has a unique couple FP in .
Proof. Any ; we define sequences and in such that ☐
Now from (5) for , we have where
We get, after simplification, where . Similarly,
Now, from (11) and (12) and by induction, for ,
It shows that the sequence is a fuzzy cone contractive; therefore,
Now for and for , we have
Hence, the sequence is Cauchy. Now for sequence and from (5), for , we have where
We get, after simplification, where . Similarly,
Now, from (19) and (20) and by induction, for ,
It shows that the sequence is a fuzzy cone contractive; therefore,
Now for and for , we have
Hence, the sequence is Cauchy. Since is complete and are Cauchy sequences in , so such that and as or this can be written as and . Therefore,
Hence,
Similarly,
Regarding its uniqueness, suppose and are another couple fixed point pairs in such that and . Now, from (5), for , we have where
Now from (27) and for , where . Hence, we have for , .
Similarly, again from (5), for , we have where
Now from (30) and for ,
Hence, we have for , .
Corollary 11. Let be a mapping on complete FCM-spaces in which is triangular and satisfies for all , , , and with . Then, has a unique couple FP in .
Corollary 12. Let be a mapping on complete FCM-spaces in which is triangular and satisfies for all , , , and with . Then, has a unique couple FP in .
Example 1. , is a -norm, and is defined as for all and . Then, it is easy to verify that is triangular and is a complete FCM-space. We define Now from (5), for , we have It is easy to verify that conditions of Theorem 10 are satisfied with . Then, has unique coupled FP for and .
Theorem 13. Let be a mapping in a complete FCM-space in which is triangular and satisfies for all , , and with . Then, has a unique couple FP in .
Proof. Any , and we define sequence by ☐
Now, from (39), for , we have
We get, after simplification,
where . Similarly,
Now, from (42) and (43) and by induction, for , we have
Hence, the sequence is fuzzy cone contractive; therefore,
Now for and for , we have
Hence, the sequence is Cauchy. Now for sequence , again from (39), for , we have
We get, after simplification, where the value of is the same as in (42). Similarly,
Now, from (48) and (49) and by induction, for , we have that
Hence, the sequence is fuzzy cone contractive; therefore,
Now for , for , we have
Hence, the sequence is Cauchy. Since is complete and and are Cauchy sequences in , such that and as , or this can be written as and . Therefore,
Similarly,
Regarding its uniqueness, let and be another couple fixed point pairs in such that and . Now, from (39), for , we have
Hence, we get that . Similarly, again from (39), for , we have
Hence, we get that for .
Corollary 14. Let be a mapping on complete FCM-spaces in which is triangular and satisfies for all , , and with . Then, has a unique couple FP.
Example 2. , is a -norm, and is defined as for all and . Then, it is easy to verify that is triangular and is a complete FCM-space. We define
Now from (39), for , we have
It is easy to verify that all the conditions of Theorem 13 are satisfied with and . Then, has unique coupled FP.
4. Application
In this section, we present an application on Lebesgue integral (LI) mapping to support our main work. In 2002, Branciari proved the following result on complete metric space for unique FP (see [33]):
Theorem 15. Let be a complete metric space, , and a mapping such that for each ,
where is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of ) and for each ,
Then, has a unique FP such that for any , .
Now, we are in the position to use the above concept and to prove a unique coupled FP-theorem in FCM-spaces.
Theorem 16. Let be a mapping on complete FCM-spaces in which is triangular and satisfies where for all , , , and with and is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of ) and for each , Then, has a unique couple FP in .
Proof. Any ; we define sequences and in such that ☐
Now from (64) and from the proof of Theorem 10, for , we have that where . Similarly, again by using the same arguments, we have
Now, from (68) and (69) and by induction, for , we have which shows that the sequence is a fuzzy cone contractive, therefore
Hence, we get that
Now for and for , we have
We get that
Hence proved that the sequence is Cauchy. Now for sequence from (64) and from the proof of Theorem 10, for , we have where . Similarly, again by using the same arguments, we have
Now, from (75) and (76) and by induction, for , we have which shows that the sequence is fuzzy cone contractive; therefore,
Hence, we get that
Now for and for , we have
We get that
Hence, it was proved that the sequence is Cauchy. Since is complete and are Cauchy sequences in , so such that and as or this can be written as and . Therefore,
Hence,
Similarly,
Regarding its uniqueness, suppose and are another couple fixed point pairs in such that and . Now, from (64) and from the proof of Theorem 10, for , we have that
Hence, we get that for ; this implies .
Similarly, again from (64)and from the proof of Theorem 10, for , we have that
Hence, we get that for ; this implies .
5. Conclusion
We presented the concept of coupled FP-results in FCM-spaces and prove some unique coupled FP-theorems under the modified contractive type conditions by using “the triangular property of fuzzy cone metric.” We presented examples in support of our result. Further, we presented an application of Lebesgue integral mapping to uplift our main work. With the help of this new concept, one can prove more modified and general contractive type coupled FP-results with different types of integral contractive type of conditions and applications in complete FCM-spaces.
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.
Acknowledgments
The authors are grateful to the Deanship of Scientific Research, King Saud University for funding through Vice Deanship of Scientific Research Chairs.