Global Optimal Solutions for Proximal Fuzzy Contractions Involving Control Functions

Moussaoui, Abdelhamid; Hussain, Nawab; Melliani, Said

doi:https://doi.org/10.1155/2021/6269304

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Recent Advances in Fixed Point Theory in Abstract Spaces

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Research Article | Open Access

Volume 2021 | Article ID 6269304 | https://doi.org/10.1155/2021/6269304

Global Optimal Solutions for Proximal Fuzzy Contractions Involving Control Functions

Abdelhamid Moussaoui,¹Nawab Hussain,²and Said Melliani¹

Academic Editor: Naeem Saleem

Received07 May 2021

Accepted09 Jun 2021

Published30 Jun 2021

Abstract

In this study, we introduce new concepts of -contraction and -contraction and we discuss existence results of the best proximity points of such types of non-self-mappings involving control functions in the structure of complete fuzzy metric spaces. Our results extend, generalize, enrich, and improve diverse existing results in the current literature.

1. Introduction

Recent advancements in fixed point theory are one of the central and active research areas of nonlinear functional analysis, which provides a variety of mathematical methods, principles, and techniques for solving a variety of problems arising from various branches of mathematics as well as various fields in science and engineering. The Banach fixed point theorem is considered as one of the most fruitful results in this theory. Due to its vast and significant applicability in pure and applied mathematics, this principle has been generalized and developed in various approaches (see, e.g., [1–22]). In particular, Khojasteh et al. [23] presented an impressive technique to the investigation of fixed point theory by developing the notion of simulation functions, which exhibit a significant unifying power. The idea of simulation functions has been generalized, improved, and extended in different metric spaces (see, e.g., [11, 14, 24, 25]).

The best proximity theory is another expanding and prominent aspect of fixed point theory which plays a fundamental role in the investigation of requirements that guarantee the existence of an optimal approximate fixed point when the functional equation has no solution. Indeed, a non-self-mapping does not possess necessarily a fixed point, with and are two nonempty subsets of a classical metric space . Best proximity theory is a remarkable generalization of fixed point theorems. In fact, the best proximity point turned out to be a fixed point in a natural way if the mapping in question is a self-mapping. For more recent developments in best proximity theory and related techniques, refer to [9–11, 19, 26–32].

In the present study, following this line of research interest, we present a simulation function approach to best proximity point problems in fuzzy metric spaces. We initiate new concepts of -contraction, -contraction, and generalized -contraction, and we discuss existence results of best proximity point of such classes of non-self-mappings involving control functions in the structure of complete fuzzy metric spaces. The furnished results enrich, generalize, and extend various existing findings in the literature.

2. Preliminaries

Throughout this study, and will represent natural and real numbers, respectively. First, we start with some notions and main properties of fuzzy metric spaces.

Definition 1. (see [33]). A binary operation is a continuous t-norm if it fulfills the following conditions: (CT1) is continuous (CT2) is commutative and associative (CT3) for all (CT4) whenever and , for all

Example 1. Three standard instances are as follows:(a)(b)(c)

Definition 2. (see George and Veeramani [34]). Let be an arbitrary set, is a continuous t-norm, and is a fuzzy set on . The ordered triple is said to be a fuzzy metric space if if and only if is continuousfor all and .
For , the open ball with centre and radius , where , is defined byA subset of a fuzzy metric space is said to be open if given any point , there exists and such that . Let denote the collection of all open subsets of ; hence, is a topology on . This topology is Hausdorff and first countable. For further topological results, refer to [2, 34].

Example 2. (see [34]). Let be a metric space and be the product t-norm, and define the function byfor all . Then, is a fuzzy metric space on .

Lemma 1. (see [1]). is nondecreasing for all in .

Definition 3. (see [34]). Let be a fuzzy metric space.(1)A sequence is said to be convergent to if an only if for all (2)A sequence is said to be a Cauchy sequence iff for each and , there exists such that for all (3)A fuzzy metric space is called complete if every Cauchy sequence in has a limit in In [2], Gregori and Sapena initiated the notion of a fuzzy contractive mapping as follows.

Definition 4. (see [2]). Let be a fuzzy metric space. A mapping is called a fuzzy contractive mapping if there exists such thatfor each and .

Definition 5. (see [4]). Let be the class of nondecreasing functions fulfilling the following two conditions: is continuous for all A self-mapping on a fuzzy metric space is called a fuzzy -contractive mapping ifAfterwards, Wardowski [5] proposed the idea of a fuzzy -contractive mapping as follows.

Definition 6. (see [5]). Let be the set of functions satisfying the two conditions and given by transforms onto is strictly decreasingA self-mapping on a fuzzy metric space is called a fuzzy -contractive with respect to the function if there exists such that the following inequality holds:The following class of control functions has been introduced in [8], where we used the term class instead of the present -simulation functions.

Definition 7. (see [8]). The function is said to be a -simulation function if the following properties hold: for each if are sequences in such that , then By , we denote the collection of all -simulation functions.

Definition 8. (see [8]). Let be a fuzzy metric space, a mapping, and . Then, is said to be a -contraction with respect to if the following condition is satisfied:

Example 3. (see [8]). The type of fuzzy contractive mappings developed by Gregori and Sapena [2] is a perfect example of -contraction. It can be expressed facilely from the previous definition by taking the -simulation function aswhere .

Example 4. (see [8]). The corresponding -simulation function for the fuzzy -contractive mapping is defined by

Definition 9. (see [6]). Let be a fuzzy metric space. We say that a mapping is -admissible if there exists a function such that for all , ,In line with [15] (see also [16]), we use the notion of triangular weak--admissible function in the form that is as follows.

Definition 10. Let be a mapping and be a function. We say that is a triangular weak--admissible iffor all , .

Definition 11. (see [19]). Let and be nonempty subsets of a fuzzy metric space . Define and by the following sets:whereNote that, a point is said to be a fuzzy best proximity point of the mapping , where , and are nonempty subsets of an abstract nonempty set if for all .

3. Main Results

Firstly, we define the following concepts.

Definition 12. Let and be two nonempty subsets of fuzzy metric space and . We say that is an -proximal admissible iffor all and .

Remark 1. Note that if , then Definition 12 reduces to Definition 9 of -admissibility.

Definition 13. Let and be nonempty subsets of fuzzy metric space and . We say that is an -contraction with respect to if is an -proximal admissible such thatfor all and .

Definition 14. Let and be nonempty subsets of fuzzy metric space , , and . We say that is an -contraction with respect to if is an -proximal admissible such thatfor all and .

Remark 2. Note that Definition 14 cannot be reduced to Definition 13 since does not belong to .

Definition 15. Let and be nonempty subsets of fuzzy metric space and . We say that is a generalized -contraction with respect to if is an -proximal admissible such thatfor all and , whereNext, we give our first main result.

Theorem 1. Let and be nonempty subsets of a complete fuzzy metric space , , , and is nonincreasing in its second argument. Assume that is an -contraction with respect to and(i) is triangular weak--admissible(ii) is closed(iii)(iv)There exists such that and for all (v) is continuous.Then, there exists such that for all ; that is, has a best proximity point .

Proof. Due to condition (iv), there exists such that andRegarding (iii), we deduce that ; hence, there exists such thatSince and is an -proximal admissible, consequently, . Recursively, a sequence can be defined as follows:If there exists such that , we obtainwhich means that is a best proximity point of . Thus, to continue our proof, we suppose that for all . Making use of (20) and (21), we obtainRegarding that is an -contraction with respect to , together with (20), (21), and , we obtainConsequently, we havewhich means that is a nondecreasing sequence of positive real numbers in . Then, there exists such that for all . We shall prove that . Reasoning by contradiction, suppose that for some . Now, if we take the sequences and and considering and and that is nonincreasing with respect to its second argument, we obtainwhich is a contradiction and yieldsNext, we show that the sequence is Cauchy. Reasoning by contradiction, suppose that is not a Cauchy sequence. Thus, there exists , , and two subsequences and of with for all such thatTaking into account Lemma 1, we deriveBy choosing as the smallest index satisfying (29), we haveOn account of (28), (30), and , we haveTaking limit as and employing (27), we deriveOn the other hand, we havewhich imply thatFurthermore, given that is triangular weak--admissible and taking into account (20), we deduce thatSo thatRegarding the fact that is an -contraction with respect to and making use of (35) and (36), we haveFrom (32) and (34), we see that the sequences and have the same limit , taking into consideration that is nonincreasing with respect to its second argument; by the property , we conclude thatwhich is a contradiction. So that is a Cauchy sequence in . As is closed subset of a complete fuzzy metric space , there exists such thatAs is continuous, we conclude that converges to ; thus,Due to the continuity of , we have . From (21), we deducewhich means that is a best proximity point of .
In the next theorem, we substitute the continuity of in Theorem 1 with the following condition.
: if is a sequence in such that for all , , and as , then there exists a subsequence of such that for all and .

Theorem 2. Let and be nonempty subsets of a complete fuzzy metric space and , , and is nonincreasing in its second argument. Assume that is an -contraction with respect to and(i) is triangular weak--admissible(ii) is closed(iii)(iv)There exists such that and for all (v)If is a sequence in such that for all , , and as , then there exists a subsequence of such that for all and Then, there exists such that for all .

Proof. Following the lines of the proof of Theorem 1, we deduce that there exists a Cauchy sequence in which converges to . Since , we have , and thenBy condition (v), there exists a subsequence of such thatRegarding that is an -proximal admissible andwe obtain that . Hence,Applying the property , it follows thatwhich yields . Then, ; from (45), we derive that .

Theorem 3. Let and be nonempty subsets of a complete fuzzy metric space and . Assume that is a generalized -contraction with respect to and(i) is triangular weak--admissible(ii) is closed(iii)(iv)There exists such that and for all (v) is continuousThen, there exists such that for all .

Proof. Using condition (iv), there exists such that and . Regarding (iii), we have which yields that there exists such thatSince and is an -proximal admissible, it therefore follows that . We recursively construct the sequence as follows:Suppose that there exists certain such that . Hence,which means that is a best proximity point of ; thus, the proof is finished. For this reason, to continue our proof, we assume that for all . Making use of (56) and (60), we obtainSince is a generalized -contraction with respect to , we havewhereNow, ifapplying , we get thatThus,which is a contradiction. Consequently,By (55), we obtain thatHence, we deduce that is a nondecreasing sequence in . Thus, there exists such that for all . We shall prove that . Reasoning by contradiction, suppose that for some . Now, if we take the sequences and and consider , we obtainwhich is a contradiction. Therefore,Next, we show that is Cauchy sequence. On the contrary, assume that is not a Cauchy. Hence, there exist , , and two subsequences and of with for all such thatTaking into account Lemma 1, we derive thatBy choosing as the smallest index satisfying (29), we haveMaking use of (61) and (63) and the triangular inequality, we getPassing to the limit and using (60), we derive thatOn the other hand,which imply thatFurthermore, since is triangular weak--admissible, we deduce thatThus,for all . By the fact that is a generalized -contraction with respect to and using (69) and (70), we obtain thatwhereLetting in equality (72) and using (60), we deriveTake the sequences and for all . Applying , we derive thatwhich is a contradiction. Then, is a Cauchy sequence in . Given that is closed subset of a complete fuzzy metric space , there exists such thatAs is continuous, we obtain that converges to ; thus,As the metric function is continuous, we have . In view of (51), we getThus, is a best proximity point of .

Theorem 4. Let and be nonempty subsets of a complete fuzzy metric space and . Assuming that is an -contraction with respect to ,(i) is triangular weak--admissible(ii) is closed(iii)(iv)There exists such that and for all (v) is continuous or holdsThen, there exists such that for all .

Proof. Pursuant to the same arguments as those given in the proof of Theorem 3, we know that there exists a Cauchy sequence in which converges to . Further,If is continuous, thenTaking into account (21), we deducewhich means that is a best proximity point of .
Now, suppose that holds. Since , we have and thenBy condition , there exists a subsequence of such thatRegarding that is an -proximal admissible andwe obtain that . Hence,Applying the property , it follows thatwhich yields . Then, ; from (45), we derive that . This completes the proof.
Note that Theorem 4 cannot be deduced by combining Theorems 1 and 2 since the function does not belong to . Moreover, in Theorems 1 and 2, we have an added condition that is nonincreasing in its second argument.

4. Consequences

Now, we shall clarify that diverse consequences of the existence results can be easily derived and developed from our main results.

Corollary 1. Let and be nonempty subsets of a complete fuzzy metric space , , . Assume that is an -admissible proximal mapping such thatfor all and . Assume also that(i) is triangular weak--admissible(ii) is closed(iii)(iv)There exists such that and for all (v) is continuous or holdsThen, there exists such that for all .

Proof. Define bySince , Theorem 4 leads to the desired results.

Corollary 2. Let and be nonempty subsets of a complete fuzzy metric space , , . Assume that is an -admissible proximal mapping such thatfor all and . Assume also that(i) is triangular weak--admissible(ii) is closed(iii)(iv)There exists such that and for all (v) is continuous or holdsThen, there exists such that for all .

Proof. It follows from Theorem 4 using the -simulation function , for all .

Corollary 3. Let and be nonempty subsets of a complete fuzzy metric space , . Assume that is an -admissible proximal mapping such thatfor all and , where with for all and . Assume also(i) is triangular weak--admissible(ii) is closed(iii)(iv)There exists such that and for all (v) is continuous or holds.Then, there exists such that for all , i.e., has a best proximity point .

Proof. It follows from Theorem 4 by taking for all .

Example 5. Let and be a fuzzy set on given by , , and is a t-norm given by for all . is a complete fuzzy metric space. Consider the mappings by . Define for all , , and byIt is easy to see that and . Therefore, all conditions of Theorem 4 are satisfied, and is a fixed point of .
We must point to the fact that, by defining the control function and the admissible mapping in a proper way, it is possible to particularize and derive a number of varied consequences of our main results. We skip making such a number of corollaries since they seem clear.

5. Conclusion

This paper has dealt with a -simulation function approach to best proximity point problems in fuzzy metric spaces. We have initiated some classes of non-self-mappings and discussed existence results of the best proximity points of such types of non-self-mappings. Our results can be further extended by replacing the fuzzy metric space by various settings (e.g., partially ordered fuzzy metric spaces and complex valued fuzzy metric spaces), and more generalization can be obtained by the study of optimal coincidence points, optimal best proximity coincidence points, and the setting of cyclic mappings.

Data Availability

The data used to support the findings of this study are included in the references within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Abdelhamid Moussaoui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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