Fuzzy Fixed Point Results in Convex -Algebra-Valued Metric Spaces

Abstract

The purpose of this note is to come up with some new directions in fuzzy fixed point theory. To this effect, notions of a -algebra-valued fuzzy -contraction and related concepts in a convex -algebra-valued metric space (-AVMS) are set-up. In line with the view of a Hausdorff distance function, an idea of a distance between two approximate quantities is proposed. Consequently, two fixed point results of a -algebra-valued fuzzy mapping (-AVFM) for the new type of contractions are established using Mann and Ishikawa iterative schemes. For some future investigations of our results, two open problems are noted concerning sufficient criteria guaranteeing the existence of fixed points of a -algebra-valued fuzzy -contraction and whether or not the Picard iteration for a -algebra-valued fuzzy -contraction converges.

1. Introduction and Preliminaries

We begin this section with specific notions of -algebras as follows.

Definition 1 (see [1]). Let be a unital algebra with the unit . An involution on is a conjugate linear map such that and , for all . The pair is called a -algebra. A Banach -algebra is a -algebra together with a submultiplicative norm such that , for all ; where a norm on an algebra is said to be submultiplicative if , for all . A -algebra is a Banach -algebra such that , for all .

Throughout this paper, represents a unital -algebra with a unit . Also, we take and denote the zero element in by . An element is called positive, written , if and , where is the spectrum of . Availing positive elements, we set up a partial ordering ° on as follows: if and only if . Hereafter, by , we mean the set and (cf. [2]).

Remark 2. When is a unital -algebra, then for any , we have if and only if (cf. [1]).

With the aid of positive elements in , Ma et al. [2] launched the concept of -AVMS in the following manner.

Definition 3 (see [2]). Let be a nonempty set. Suppose that the mapping satisfies the following conditions:
(c1) and if and only if
(c2) , for all
(c3) , for all
Then, is called a -algebra-valued metric, and is known as a -AVMS.

It is clear that a -AVMS generalizes the idea of a metric space, by replacing the set of real numbers with . For some recent fixed point results in -AVMS, one can consult [3, 4] and the references therein.

Definition 4 (see [2]). Given a -AVMS . Suppose that the sequence and . If for any , there exists such that for all , , then is said to be convergent to with respect to . In this case, we write .
If for any , there is such that , , then the sequence is said to be Cauchy with respect to . We say that is a complete -AVMS, if every Cauchy sequence in is convergent with respect to .

Definition 5 (see [2]). Given a -AVMS , a mapping is called a -algebra-valued contractive mapping on , if there exists with such that for all ,

The following lemma is useful in discussing our main results.

Lemma 6 [1]. Let be a -algebra. Then: (i)For any , if with , then (ii)For any , if , then

In 1970, Takahashi [5] initiated the concept of convexity in metric spaces in the following fashion.

Definition 7 (see [5]). Let be a metric space and . A mapping is called a convex structure on , if for all and ,

Using the notion of convexity, Takahashi [5] complemented some FP results originally obtained in Banach spaces. Following [5], several investigators have come up with FP notions in convex metric spaces; for such results, we can refer [6–9] and the references therein. Using the idea of a convex metric space and with the aid of positive elements in a -algebra, Ghanifard et al. [10] brought up the next definition.

Definition 8 (see [10]). Let be a -AVMS. A mapping is called a convex structure on if for all , it satisfies the following: A -AVMS equipped with a convex structure is said to be a convex -AVMS, denoted by . A subset of is called convex if for all and , .

As an attempt at reducing uncertainties in dealing with practical problems for which conventional mathematics cannot cope effectively, the evolvement of fuzzy mathematics started with the introduction of the concepts of fuzzy sets by Zadeh [11] in 1965. Fuzzy set theory is now well-known as one of the mathematical tools for handling information with nonstatistical uncertainty. As a result, the theory of fuzzy sets has gained greater applications in diverse domains such as management sciences, engineering, environmental sciences, medical sciences, and in other emerging fields. In the meantime, the basic notions of fuzzy sets have been modified and improved in various settings; for example, see [12–15]. Along the lane, Heilpern [16] employed the concept of fuzzy sets to come up with the notion of fuzzy mappings and established a FP result for fuzzy contraction mappings which is a fuzzy version of FP theorems established by Nadler [17] and Banach [18].

Let be a universal set. A fuzzy set in is a map with domain and range set . Let be the collection of all fuzzy sets in . If is a fuzzy set in , then the function value is called the grade of membership of in . The -level set of a fuzzy set is denoted by and is defined as follows: where by , we mean the closure of the crisp set .

Definition 9 (see [16]). Let be an arbitrary set and be a metric space. A mapping is called a fuzzy mapping. A fuzzy mapping is a fuzzy subset of with membership function . The function value is called the grade of membership of in .

Definition 10 (see [16]). Let be a nonempty set and be a fuzzy mapping. A point is said to be a fuzzy FP of if there exists an such that .

Hereafter, .

Motivated by the ideas of fuzzy mappings and -AVMSs due to Heilpern [16] and Ma et al. [2], respectively, the aim of this research is to initiate the study of fuzzy FP results in convex -AVMSs. To this effect, some new concepts of -algebra-valued fuzzy contractions in convex -AVMSs are proposed, and related fuzzy FP theorems are established. The notions put forward herein are not only novel, but complement and unify a few corresponding results in the existing literature.

2. Main Results

In this section, we introduce notions of -algebra-valued fuzzy contractions and some corresponding fixed point results. First, a few requisite auxiliary concepts are initiated as follows.

Definition 11. A fuzzy set in a -AVMS is said to be convex if for all and , . A fuzzy set in is called an approximate quantity if its -level set is a compact convex subset of for each and .

Throughout, the collection of all approximate quantities in is denoted by . We define a distance function between two approximate quantities in as follows.

Definition 12. Let and . Then, we define: where the Hausdorff distance function is set-up as follows:

Consistent with Heilpern [16], we call the function an -distance and a distance between two approximate quantities in .

We say that a subset of a -AVMS is bounded if . The collection of all closed and bounded subsets of is represented by .

Note that is a -algebra-valued metric on (induced by the Hausdorff metric ), and the completeness of implies the completeness of the corresponding -AVMS . Moreover, are isometric embedding via the relation (crisp set) and , respectively, where is the characteristic function of , and

Similarly,

We now define the idea of a -AVFM in the following manner.

Definition 13. Let be an arbitrary set and be a -AVMS. A mapping is called a -AVFM.

In line with the idea of fuzzy -contraction due to Heilpern [16], we introduce the next concept.

Definition 14. Let be a -AVMS. A -AVFM is called a -algebra-valued fuzzy -contraction, if there exists with such that for all ,

Example 15. Let and (the collection all matrices with real entries) with the norm , where are the entries of the matrix and the involution given by . Define by . Obviously, is a -AVMS. We define a partial ordering on as follows: for and for some . Let and for each , define a fuzzy mapping as follows: If we take the mapping as for all , then Obviously, , for each . We see that for all , It follows that is a -algebra-valued fuzzy -contraction with . Clearly,

Definition 16. Let be a nonempty set. A point is called a stationary point of a fuzzy mapping , if there exists an such that . We say that is a common stationary point of any two fuzzy mappings if , for some .

Our main result is provided hereunder.

Theorem 17. Let be a complete convex -AVMS. Suppose that the mapping is a -algebra-valued fuzzy -contraction such that and every is a stationary point of . Let be the Mann iteration scheme given by where and . Then, converges to a fuzzy FP of , provided .

Proof. Let and for each . Then, we have From (15), We observe that the strict inequality in (16) is valid whenever , for each . Indeed, if we take for a finite , then for each , from which it yields that converges to for finite number of iterations, and hence, we obtain the conclusion of our result.
Now, we prove that the sequence is Cauchy with respect to . Since , for each , there exists such that for all , By (17), there exists such that for all , Using the triangle inequality in , Therefore, taking (18) into consideration, we get for . This proves that is a Cauchy sequence with respect to . The completeness of implies that there exists such that . Next, we establish that is a fuzzy FP of . For this, take . Since as , there exists such that for all , Moreover, yields that there exists such that for all , Hence, there exists such that for all , By triangle inequality in , there exists write Consequently, whenever . It follows that , and thus, , for some .

In what follows, we present a fuzzy coincidence theorem for two -algebra fuzzy mappings using Ishikawa iterative scheme.

Theorem 18. Let be a complete convex -AVMS and be any two -AVFMs satisfying: for all with such that . Assume further that , and every is a common stationary point of and . Then, the sequence of Ishikawa iterative scheme set-up by where , and , converges to an element of , provided .

Proof. Let . Assume that for all . Then from which it follows that Similarly, which gives Hence, in line with the proof of Theorem 17, we can prove that is a Cauchy sequence with respect to , and the completeness of implies that converges to some . Thus, consistent with Theorem 17, we obtain that .

As some consequences of Theorem 17 and Theorem 18, the following two results, using -algebra-valued Hausdorff distance function, can be deduced easily.

Corollary 19. Let be a complete convex -AVMS and be a -AVFM. Suppose that and every is a stationary point of . Let be the Mann iterative scheme set-up by (14). If there exist and with such that for all , then converges to a fuzzy FP of , provided .

Proof. Since for all and , Theorem 17 can be followed to complete the proof.

On the same steps in deriving Corollary 19, we can also deduce the following result.

Corollary 20. Let be a complete convex -AVMS and be any two -AVFMs. Assume further that and every is a common stationary point of and . Let be the Ishikawa iterative scheme set-up by (23). If there exist and with such that for all , then converges to a common fuzzy FP of and , provided

3. Open Problems

For some future examinations of our main results, the following two problems are highlighted:

(P1) It is well-known that the importance of contractive mapping is to guarantee the existence and uniqueness of a fixed point of certain self-mappings in complete spaces. On this note, following Theorem 17 and Theorem 18, sufficient criteria guaranteeing the existence of fixed points of -algebra-valued fuzzy -contractions is still a gap that needed to be filled.

(P2) In this article, Mann and Ishikawa iterations are used to develop the ideas of -algebra-valued fuzzy -contractions and associated fixed point theorems. Hence, it is natural to ask whether Picard iteration for -algebra-valued fuzzy -contraction mapping converges or not.

4. Conclusions

Based on the ideas of fuzzy mappings and -AVMSs in the sense of Heilpern [16] and Ma et al. [2], respectively, analogue notions of -algebra-valued fuzzy contractions in convex -AVMSs and associated FP theorems are established. The obtained fuzzy FP results are analysed using Mann and Ishikawa iterative schemes. It is pertinent to note that the ideas of this paper being discussed in fuzzy setting are very fundamental. Hence, it can be improved upon when presented in the framework of some generalized fuzzy mappings such as -fuzzy, intuitionistic fuzzy, and soft set-valued mappings. The underlying space can also be fine-tuned in some other pseudo or quasi metric spaces. For some future considerations of our results, two open problems are posed regarding sufficient conditions under which -algebra-valued fuzzy -contraction has a fixed point and whether or not the Picard iteration for -algebra-valued fuzzy -contraction converges.

Data Availability

No data were used to support this study.

Disclosure

The statements made and views expressed are solely the responsibility of the author.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors contributed equally to this work.

Acknowledgments

The fourth author would like to acknowledge that this publication was made possible by a grant from Carnegie Corporation of New York (provided through the African Institute for Mathematical Sciences).