Common Fixed Point Theorems for a Pair of Self-Mappings in Fuzzy Cone Metric Spaces

Abstract

In this paper, we present some common fixed point theorems for a pair of self-mappings in fuzzy cone metric spaces under the generalized fuzzy cone contraction conditions. We extend and improve some recent results given in the literature.

1. Introduction

In 1965, Zadeh [1] came up with a fabulous idea. He introduced the theory of fuzzy set, which is the generalization of crisp sets. A mapping is from to ; then is known as a fuzzy set. Later on, the fuzzy metric space concept was given by Kramosil and Michalek [2], which is performing the probabilistic metric space and would approach the fuzzy set. In [3], George and Veeramani were given the stronger form of the fuzz metric. Some more set-valued mapping results for fixed point on fuzzy metric spaces can be seen, for example, in [4–6] and the references therein.

In 2007, Som [7] proved some continuous self-mapping results for common fixed point in fuzzy metric spaces. He generalized the results of Pant [8], Som [9], and Vasuki [10]. Some other common fixed point results in the fuzzy metric space can be found in [11–16] and the references therein.

Huang and Zhang [17] introduced the concept of cone metric space. They proved the convergent sequences, Cauchy sequences, and some fixed point theorems for contractive-type mappings in cone metric spaces. Later on, Abbas and Jungck [18] proved some noncommuting mapping results in cone metric spaces. After that, a series of authors proved some fixed point and common fixed point results for different contractive-type mappings in cone metric spaces (see, e.g., [19–25]).

Recently, the concept of fuzzy cone metric space was introduced by Oner et al. [26]. They proved some basic properties and a Banach contraction theorem for fixed point with the assumption of Cauchy sequences. Rehman and Li [27] generalized the result of Oner et. al. [26] and proved some fixed point theorems in fuzzy cone metric spaces without the assumption of Cauchy sequences. Some more fixed point and common fixed point results in fuzzy cone metric spaces can be found in [27–31].

In the demonstration of this research work, we generalize the results of Oner [26] and Rehman [27] for a pair of self-mappings in fuzzy cone metric spaces and prove some unique common fixed theorems with illustrative examples.

2. Preliminaries

Definition 1 ([32]). An operation is known as a continuous -norm if it holds the following: (1) is commutative, associative, and continuous.(2), .(3), whenever and , for every .

Meanwhile, the basic -norm continuous conditions are as follows.

The minimum, product, and Lukasiewicz -norms are defined, respectively, as (see [32])

Definition 2 ([17]). A subset of a real Banach space is called a cone if (1), closed and , where represents the zero element of ,(2), if and ,(3), if both .

All the cones have nonempty interior and the natural numbers set is denoted by .

Definition 3 ([26]). A 3-tuple is known as a fuzzy cone metric space, if is a continuous -norm, is an arbitrary set, is a cone of , and is a fuzzy set on if the following hold: (i) and if ,(ii),(iii),(iv) is continuous,

for all and .

Remark 4 ([27]). If we suppose that , , and , then every fuzzy metric space becomes a fuzzy cone metric space.

Definition 5 ([26]). Let be a fuzzy cone metric space, , and a sequence in is (i)converging to if and such that , . We can write this as or as .(ii)Cauchy sequence if and such that , .(iii) is complete if every Cauchy sequence is convergent in .(iv)fuzzy cone contractive if , satisfying for all , .

Definition 6 ([27]). Let be a fuzzy cone metric space. A fuzzy cone metric is triangular if and .

Lemma 7 ([26]). Let and let be a sequence in . Then in a fuzzy cone metric space if as , for each .

For more properties of fuzzy cone metric spaces, see [26].

Definition 8 ([26]). A mapping is known as fuzzy cone contractive in a fuzzy cone metric space , if such thatfor all , , and is known as a contraction constant of .

Theorem 9 ([26]). A self-mapping in a complete fuzzy cone metric space, in which the fuzzy cone contractive sequences are Cauchy, has a unique fixed point.

Further, in this paper, we shall study some common fixed point results in . Let be two self-mappings satisfying the following more generalized fuzzy cone contraction condition:where and the constants . It is noted that (5) is the same as (4) if , , and . On the other hand, the mappings and may not hold the fuzzy cone contraction condition if (5) is satisfied, which is shown in Example 14. Thus, in this research work, we generalize some recent results given in the literature (see Remark 13 and Example 14).

3. Main Result

Theorem 10. Let , be two self mappings and is triangular in a complete fuzzy cone metric space which satisfies (5) with . Then and have a unique common fixed point in .

Proof. Fix and we define the iterative sequences in as By view of (5), for , Thenwhere , since .
Let us denote by ; then, from (8), we haveSimilarly, Thenwhere , since . Then (11) can be written asNow, from (9) and (12), we can get the following inequalities: Thus, we have Hence, from the above, we conclude that a sequence is fuzzy cone contractive in ; that is,Let and let be the above sequence; we assume that . Then, two cases arise.
Case (i). If is an even number,Case (ii). If is an odd number, Thus, the right-hand sides of (16) and (17) converge to zero as , which yields that is a Cauchy sequence. Since is complete, such thatSince is triangular, By using (5), (15), and (18), for , ThenThe above (21) together with (18) and (19) implies that, since ; then ; that is, . Similarly, by triangular,Again, by using (5), (15), and (18), similar to the above, after simplification, we can getThe above (24) together with (18) and (23) implies that, since ; then ; that is, . Hence, the fact that is the common fixed point of and in is proven.
Uniqueness: let be the other common fixed point of and in . Then, again by view of (5), for , We note that , where . Therefore , implying that . Hence the fact that the common fixed point of and is unique is proven.

Corollary 11. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfiesfor all , , and such that . Then and have a unique common fixed point in .

Corollary 12. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfiesfor all , , and such that . Then and have a unique common fixed point in .

Remark 13. (i) In special case, Theorem 10, Corollaries 11 and 12, and [26, Theorem 3.3] (i.e., Theorem 9) all have the same results. In fact, if , and in (5).
(ii) Theorem 10 and [27, Theorem 3.1] both have similar proof. If , and in (5).

Example 14. Let ; is a continuous -norm and is defined as and . Then, one can easily prove that is triangular and is a complete fuzzy cone metric space. Now we define as And Then and are not fuzzy cone contractive, since In special case, if , then Theorem 9 does not hold. But it can be easily proven that all the conditions of Theorem 10 hold with , , . Thus, and have a unique common fixed point in , that is, .

Theorem 15. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfiesfor all , , and . Then and have a unique common fixed point in .

Proof. Fix and a point such that and such that . If , then we have that which implies that if and only if . Then the proof is complete. Otherwise, we assume that and let us take . Now we define the iterative sequence in such asBy view of (33), Now there are three possibilities.
(i) If is minimum, then will be the maximum in the above (36). Then, we have(ii) If is minimum, then will be the maximum in the above (36). Then, we havewhich is not possible.
(iii) If is minimum, then will be the maximum in the above (36). Then, we have which implies thatwhere , since . Thus, . Now, from (i), (ii), and (iii), for all and , which shows that a sequence is fuzzy cone contractive. Thus,Since is triangular, for all , we have which shows that is a Cauchy sequence. Since is complete and , we haveNow we shall show that . By the triangular property of , we haveNow, by using (33), (42), and (44), for , we have Thus,The above (47) together with (44) and (45) implies that and , since . This implies that ; that is, . Similarly, we can prove that . Thus, .
Uniqueness: let such that . Then, by using (33), for every , we have This implies that , since . This implies that ; that is, . Hence the fact that and have a unique common fixed point is proven. That is, .

Corollary 16. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfiesfor all , , and . Then and have a unique common fixed point in .

Example 17. From Example 14, we define as Then, and are fuzzy cone contractive, since for all . Then, all the conditions of Theorem 15 easily hold with , as well as Theorem 9, if . Thus, and have a unique common fixed point in , that is, .

4. Conclusion

We gave the concept of common fixed point for a pair of self-mappings in fuzzy cone metric spaces and proved some unique common fixed point results in fuzzy cone metric spaces. We also proved that a pair of self-mappings may not be a fuzzy cone contraction if it satisfies (5), which is shown in Example 14. According to this concept, one can study some more common fixed point results for two or more self-mappings in fuzzy cone metric spaces for different contractive-type mappings.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors share equal contributions to the final manuscript.