[Retracted] On Fuzzy Fixed-Point Results in Complex Valued Extended b-Metric Spaces with Application

Abstract

The aim of this paper is to define fuzzy contraction in the context of complex valued extended -metric space and prove fuzzy fixed-point results. Our results improve and extend certain recent results in literature. Moreover, we discuss an illustrative example to highlight the realized improvements. As application, we derive fixed-point results for multivalued mappings in the setting 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 2011, Azam et al. [7] introduced the concept of complex valued metric space and obtained some common fixed-point results for rational contraction which consist of a pair of single valued mappings. Later on, many researchers [8–15] worked on this generalized metric space. Ahmad et al. [16] and Azam et al. [17] defined the generalized Housdorff metric function in the setting of complex valued metric space and obtained common fixed-point results for multivalued mappings. In [18], Mukheimer generalized the concept of complex valued metric space to complex valued -metric space. Recently, Naimatullah et al. [19] 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 contrary, Heilpern [20] introduced the concept of fuzzy mappings in the setting of metric linear spaces and extended Banach Contraction Principle [21]. In 2014, Kutbi et al. [22] 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. [23] 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. [7], Rouzkard and Imdad [9], Ahmad et al. [16], and Kutbi et al. [22] for fuzzy and multivalued mappings in complex valued metric spaces.

2. Preliminaries

In 2011, Azam et al. [7] introduced the complex valued metric space as follows.

Definition 1. (see [7]). Let be the set of complex numbers and . A partial order on is defined in this way:

It follows thatif one of these assertions is satisfied: (CV1) , for all and if and only if  (CV2) , for all  (CV3) , for all .

Definition 2. (see [7]). Let . A mapping is said to be a complex valued metric if the following assertions hold.

Then, is called a complex valued metric space (CVMS).

In 2014, Mukheimer [18] introduced the notion of complex valued -metric space as follows. (CVB1) , for all and if and only if  (CVB2) , for all  (CVB3) , for all

Definition 3 (see [18]). Let and be a real number. A mapping is said to be a complex valued -metric space if the following assertions hold.

Then, is called a complex valued - metric space (CVbMS).

Recently, Naimatullah et al. [19] defined the notion of complex valued extended b-metric space in the following way. (ECVB1) , for all and if and only if  (ECVB2) , for all  (ECVB3) , for all

Definition 4. (see [19]). Let and . A mapping is called a complex valued extended -metric if following conditions hold:

Then, is called a complex valued extended - metric space (CVEbMS).

Lemma 1 (see [19]). Let be a CVEbMS and let . Then, converges to .

Lemma 2 (see [19]). Let be a CVEbMS and let . Then, is a Cauchy sequence , where .

Let be a CVEbMS; then, denotes the family of all nonempty, closed, and bounded subsets of .

From now on, we denote for , and for and .

For , we denote(i)Let . If , then .(ii)Let and . If , then .(iii)Let and let and . If , then , for all , or , for all .

Lemma 3 (see [19]). Let be a CVEbMS.

Let be a complex valued extended -metric space and be a collection of nonempty closed subsets of . Let be a multivalued mapping. For and , we define

Thus, for ,

Definition 5. (see [19]). Let be a complex valued metric space. A subset of is called bounded below if , such that , for all .

Definition 6. (see [19]). Let be a complex valued metric space. A multivalued mapping is called bounded below if, for each ,for all .

In 1981, Heilpern [20] 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 is 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 7. (see [20]). 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 8. (see [20]). 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. [22] used the above notion of fuzzy mappings in complex valued metric space (CVMS) and established the result for these mappings.

In this paper, we establish fuzzy fixed-point results in the setting of complex valued extended b-metric spaces (CVEbMS) and derive the above result of Kutbi et al. [22] for fuzzy mappings and some fixed-point result for multivalued mappings in CVMS.

3. Main Result

Definition 9. Let be a CVEbMS. The fuzzy mapping is said to have g.l.b. property on if, for any and any , greatest lower bound of exists in , . We denote by the g.l.b of . That is,

Now, we state our main result in this way.

Theorem 1. Let be a complete CVEbMS, , and let : satisfy g.l.b property. Assume that , such that, for each , and there exist nonnegative real numbers with and , where such thatfor all . If, for each , , then such that .

Proof. Let be an arbitrary point in . By assumption, we can find . So, we haveSince , so, we haveBy definition,This implies that such thatThat is,By the meaning of and for , we obtainThis impliessuch thatSimilarly, for , we haveSince , so, we haveBy definition of “” function, we haveBy definition of “” function, there exists some , such thatThat is,By the meaning of and , for , we obtainwhich implies thatwhich impliesInductively, we can construct a sequence in such thatfor all . Now, by triangular inequality, for , we haveSince , so the series converges by ratio test for each . LetThus, for , the above inequality can be written asNow, by taking , we obtainBy Lemma 2, we conclude that is a Cauchy sequence. Since is complete, then there exists an element such that as . Now, to show and , from (1), we haveSince , we haveThis implies that such thatThat is,The g.l.b property of yieldsWe know thatHence,It follows thatLetting , we get . By using Lemma 1, we have . Since is closed, so . Following the similar steps, we can prove that . Hence, there exists such that .
By setting in Theorem 1, we get the following Corollary.

Corollary 1. Let be a complete CVEbMS, , and let : satisfy g.l.b property. Assume that , such that, for each , and there exist nonnegative real numbers with and , where such thatfor all . If, for each , , then such that .

By setting in Theorem 1, we get the following corollary.

Corollary 2. Let be a complete CVEbMS, , and let : satisfy g.l.b property. Assume that , such that, for each , and there exist nonnegative real numbers with and , where such thatfor all . If, for each , , then such that .

Remark 1. If we take in Theorem 1, then we get main result of Kutbi et al. [22].

4. Application

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

Theorem 2. Let be a complete CVEbMS, , and let : satisfy g.l.b property. If there exist nonnegative real numbers with and , where such thatfor all . If, for each , , then such that .

Proof. Consider defined bywhere . Then,Thus, Theorem 1 can be applied to obtain such thatIf we consider just one multivalued mapping, then we get the following result.

Corollary 3. Let be a complete CVEbMS, , and let : satisfy g.l.b property. If there exist nonnegative real numbers with and , where , thenfor all . If for each , , then such that .

Corollary 4. Let be a complete CVEbMS, , and let : satisfy g.l.b property. If there exist nonnegative real numbers with and , where such thatfor all . If, for each , , then such that .

Proof. Take in Theorem 2.

Corollary 5. Let be a complete CVEbMS, , and let : satisfy g.l.b property. If there exists nonnegative real number , thenfor all . If, for each , , then such that .

Proof. Take in Theorem 2.
If we take in Theorem 2, then we get main result of Ahmad et al. [16] as follows.

Corollary 6 (see [16]). Let be a complete CVMS, and let : satisfy g.l.b property. If there exist nonnegative real numbers with such thatfor all , then such that .

The following result is a direct consequence of Theorem 2 if we replace multivalued mappings with single valued mappings.

Theorem 3. Let be a complete CVEbMS, , and let : . If there exist nonnegative real numbers with and , where such thatfor all . If, for each , , then such that .

If we take in Theorem 3, then we get main result of Rouzkard and Imdad [9] as follows.

Corollary 7 (see [9]). Let be a complete CVMS and let : . If there exist nonnegative real numbers with such thatfor all , then such that .

If we take and in Theorem 3, then we get the main result of Azam et al. [7] as follows.

Corollary 8 (see [7]). Let be a complete CVMS, and let : . If there exist nonnegative real numbers with such thatfor all , then 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.