[Retracted] Common -Fuzzy Fixed Point Results for Kannan Type Contractions with Application

Abstract

The purpose of this paper is to utilize complex valued extended -metric space and obtain common -fuzzy fixed point theorems under Kannan type contractions. As application, we derive common -fixed point results for multivalued mappings in the context of complex valued extended -metric space.

1. Introduction

In the theory of fixed points, there is vital role of metric spaces which have useful applications in mathematics as well as in computer science, medicine, physics, and biology (see [1–3]). Many mathematicians generalized, improved, and extended the notion of metric spaces to vector-valued metric spaces of Perov ([4]), -metric space of Czerwik [5], cone metric spaces of Huang and Zhang [6], and others.

In 1969, Kannan [7] gave an analogue sort of contractive condition that demonstrated the existence of fixed point. The basic distinction between Banach fixed point theorem (BFT) and that of Kannan contraction is that continuity of contraction is not required in the latter.

In 2011, Azam et al. [8] introduced the concept of complex valued metric space and obtained some common fixed points results for rational contraction which consist of a pair of single valued mappings. Later, on many researchers, [9–17] worked on this generalized metric space. Ahmad et al. [18] and Azam et al. [19] defined the generalized Hausdorff metric function in the setting of complex valued metric space and obtained common fixed point results for multivalued mappings. In [20], Mukheimer generalized the concept of complex valued metric space to complex valued -metric space. Recently, Naimatullah et al. [21] introduced the notion of complex valued extended -metric space as extension of complex valued -metric space and established some results for rational contractions in this generalized space.

On the other hand, Heilpern [22] introduced the concept of fuzzy mappings in the setting of metric linear spaces and extended Banach contraction principle [23]. In 2014, Kutbi et al. [24] established fuzzy fixed point results in complex valued metric spaces and generalized the results in metric spaces. Owing to the notion of a complex valued metric space, Humaira et al. [25] proved some common fixed point results under contractive condition for rational expressions.

In this paper, we define the generalized fuzzy contraction in the setting of complex valued extended -metric space and obtain some fuzzy fixed point results. As application, we derive the main results of Azam et al. [8], Rouzkard et al. [10], Ahmad et al. [18], and Kutbi et al. [24] for fuzzy and multivalued mappings in complex valued metric spaces.

2. Preliminaries

In 2011, Azam et al. [8] introduced the complex valued metric space as follows:

Definition 1. (see [8]). Let be the set of complex number and . A partial order ≾ on is defined in this way: It follows that if one of these assertions is satisfied:

Definition 2. (see [8]). Let . A mapping is said to be a complex valued metric if the following assertions hold:
(CV1) for all and if and only if
(CV2) for all
(CV3) for all
Then, is called a complex valued metric space.

Example 3. (see [8]). Let and let Define by Then, is a complex valued metric space (CVMS).

In 2014, Mukheimer [20] introduced the notion of complex valued -metric space as follows:

Definition 4. (see [20]). Let and be a real number. A mapping is said to be a complex valued -metric space if the following assertions hold:
(CVB1) for all and if and only if
(CVB2) for all
(CVB3) for all
Then, is called a complex valued -metric space (CVbMS).

Example 5. (see [20]). Let Define by for all Then, is a complex valued -metric space with .

Recently, Naimatullah et al. [21] defined the notion of complex valued extended b-metric space in this way.

Definition 6. (see [21]). Let and . A mapping is called a complex valued extended -metric if following conditions hold:
(ECVB1) for all and if and only if
(ECVB2) for all
(ECVB3) for all
Then, is called a complex valued extended -metric space (CVEbMS).

Example 7. (see [21]). Let and be defined by and by (i) (ii) Then, is a CVEbMS.

Example 8. (see [21]). Let and be a function defined by and by Then, is a CVEbMS.

Lemma 9. (see [21]). Let be a CVEbMS, and let . Then, converges to as .

Lemma 10. (see [21]). Let be a CVEbMS, and let . Then, is a Cauchy sequence where .
Let be a CVEbMS, and then denotes the family of all nonempty, closed, and bounded subsets of .
From now on, we denote for and for and
For we denote

Lemma 11. (see [21]). Let be a CVEbMS: (i)Let . If then (ii)Let and If then (iii)Let and let and If then for all or for all Let be a complex valued extended -metric space and be a collection of nonempty closed subsets of . Let be a multivalued mapping. For and , define Thus, for ,

Definition 12. (see [21]). Let be a complex valued metric space. A subset of is called bounded below if there exists such that for all .

Definition 13. (see [21]). Let be a complex valued metric space. A multivalued mapping is called bounded below if for each there exists such that for all
In 1981, Heilpern [22] utilized the concept of fuzzy set and introduced the notion of fuzzy mappings in metric spaces (MS). A fuzzy set in is a function with domain and values in , and is the collection of all fuzzy sets in If is a fuzzy set and , then the function values are called the grade of membership of in . The -level set of is denoted by and is defined as follows: Here, denotes the closure of the set . Let be the collection of all fuzzy sets in a metric space

Definition 14. (see[22]). Let be a nonempty set and be a MS. A mapping is called fuzzy mapping if is a mapping from into . A fuzzy mapping is a fuzzy subset on with membership function . The function is the grade of membership of in .

Definition 15. (see[22]). Let be a MS and : A point is said to be a fuzzy fixed point of if for some The point is said to be a common fuzzy fixed point of and if for some .
In 2014, Kutbi et al. [24] used the above notion of fuzzy mappings in complex valued metric space (CVMS) and established the following result.

Theorem 16. (see [24]). Let be a complete CVMS and let : satisfying g.l.b property. Assume that there exists such that for each , and there exists nonnegative real numbers with such that for all , and then such that
In this paper, we establish fuzzy fixed point results in the setting of complex valued extended -metric spaces (CVEbMS) and derive the above result of Kutbi et al. [24] for fuzzy mappings and some fixed point result for multivalued mappings in CVMS.

3. Main Result

Definition 17. Let be a CVEbMS. The fuzzy mapping is said to have g.l.b. property on , if for any and any the greatest lower bound of exists in for all . We denote by the g.l.b of That is,

Now, we state our main result in this way.

Theorem 18. Let be a complete CVEbMS, , and let : satisfying g.l.b property. Assume that there exists such that for each , , and there exists nonnegative real numbers with and where such that for all . If for each then there exists such that

Proof. Let be an arbitrary point in By assumption, we can find So, we have That is, Since so, we have By definition, This implies that such that That is, By the meaning of and for we get This implies which implies that Similarly, for we have That is, Since so, we have By definition of “” function, we have By definition of “” function, there exists some such that That is, By the meaning of and for we get which implies that which implies Inductively, we can construct a sequence in such that for all Now, by triangular inequality, for , we have Since so the series converges by ratio test for each Let Thus, for , the above inequality can be written as Now, by taking , we get By Lemma 10, we conclude that is a Cauchy sequence. Since is complete, then there exists an element such that as Now, to show and from (16), we have That is, Since , we have This implies that there exists such that That is, The g.l.b property of yields We know that Hence, It follows that Letting , we get By using Lemma 9, we have . Since is closed, so . Following the similar steps, we can prove that . Hence, there exists such that
By setting in Theorem 18, we get the following Corollary:

Corollary 19. Let be a complete CVEbMS, , and let : satisfying g.l.b property. Assume that there exists such that for each , and there exists nonnegative real numbers with and where such that for all . If for each then there exists such that

By setting in Theorem 18, we get the following Corollary:

Corollary 20. Let be a complete CVEbMS, , and let : satisfying g.l.b property. Assume that there exists such that for each , and there exists nonnegative real numbers with and where such that for all . If for each then there exists such that

Remark 21. If we take in the above Theorem 18, then we get the main result of Kutbi et al. [24].

4. Application

In this section, we derive some fixed point results for multivalued mappings as application of our main result.

Theorem 22. Let be a complete CVEbMS, , and let : satisfying g.l.b property. If there exists nonnegative real numbers with and where such that for all . If for each then there exists such that

Proof. Consider defined by where Then, Thus, Theorem 18 can be applied to obtain such that If we consider just one multivalued mapping, then we get the following result.

Corollary 23. Let be a complete CVEbMS, , and let : satisfying g.l.b property. If there exists nonnegative real numbers with and where such that for all . If for each then there exists such that

Corollary 24. Let be a complete CVEbMS, , and let : satisfying g.l.b property. If there exists nonnegative real numbers with and where such that for all . If for each then there exists such that

Proof. Taking in Theorem 22.

Corollary 25. Let be a complete CVEbMS, and let : satisfying g.l.b property. If there exists nonnegative real number such that for all . If for each then there exists such that

Proof. Taking in Theorem 22, if we take in the above Theorem 22, then we get main result of Ahmad et al. [18] in this way.

Corollary 26. (see [18]). Let be a complete CVMS, and let : satisfying g.l.b property. If there exists nonnegative real numbers with such that for all , then there exists such that

The following result is a direct consequence of Theorem 22 if we replace multivalued mappings with single valued mappings.

Theorem 27. Let be a complete CVEbMS, , and let :. If there exists nonnegative real numbers with and where such that for all . If for each then there exists such that

If we take in the above Theorem 27, then we get main result of Rouzkard et al. [10] in this way.

Corollary 28. (see [10]). Let be a complete CVMS, and let :. If there exists nonnegative real numbers with such that for all then there exists such that
If we take and in the above Theorem 27, then we get the main result of Azam et al. [8] in this way.

Corollary 29. (see [8]). Let be a complete CVMS, and let :. If there exists nonnegative real numbers with such that for all then there exists such that

5. Conclusion

In this paper, we have defined fuzzy contraction in the setting of complex valued extended -metric space and proved fuzzy fixed point results. As application of our main theorems, we have derived fixed point results for multivalued mappings. We hope that the results proved in this paper will form new connections for those who are working in complex valued extended -metric space.

In this direction, the future work will focus on studying the fixed points of -fuzzy mappings in complex valued extended -metric spaces. Also, coincidence point results for single valued mappings and left total relations can be established as future work. As applications, one can investigate the fractional differential inclusion problems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.