Fixed Point Results in Fuzzy Strong Controlled Metric Spaces with an Application to the Domain Words

Hussain, Aftab; Ishtiaq, Umar; Al Sulami, Hamed

doi:https://doi.org/10.1155/2023/4350504

Advances in Mathematical Physics

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4350504 | https://doi.org/10.1155/2023/4350504

Fixed Point Results in Fuzzy Strong Controlled Metric Spaces with an Application to the Domain Words

Aftab Hussain,¹Umar Ishtiaq,²and Hamed Al Sulami¹

Academic Editor: John Clayton

Received20 Sept 2022

Revised01 Jun 2023

Accepted19 Jul 2023

Published14 Aug 2023

Abstract

In this manuscript, we introduce the notions of fuzzy strong controlled metric spaces, fuzzy strong controlled quasi-metric spaces, and non-Archimedean fuzzy strong controlled quasi-metric spaces and generalize the famous Banach contraction principle. We prove several fixed point results in the context of non-Archimedean fuzzy strong controlled quasi-metric space. Furthermore, we use our main result to obtain the existence of a solution for a recurrence problem linked with the study of Quicksort algorithms.

1. Introduction and Preliminaries

In 1965, Zadeh [1] introduced the notion of fuzzy sets. The term “fuzzy” appears to be highly common and prevalent in modern research linked to the logical and set-theoretical aspects of mathematics. We believe that the primary cause of this rapid change is simple to comprehend. The surrounding world is full of uncertainty, the information we obtain from the environment, the notions we use and the data resulting from our observation or measurement are, in general, vague, and incorrect. Because of this, each explicit representation of the world’s reality or a portion of it is, in each instance, merely an estimate and an idealization of the real situation. Fuzzy concepts, such as fuzzy sets, fuzzy orderings, fuzzy languages, etc., make it possible to deal with and explore the aforementioned level of uncertainty in a mathematical and formal manner.

In 1988, Grabiec [2] proved a famous fuzzy version of the Banach contraction principle by employing the notion of a fuzzy metric space in the sense of Ivan Kramosil [3]. Although Grabiec’s fixed point theorem has the drawback of not being applicable to the fuzzy metric induced by the Euclidean metric on , it is nevertheless useful (for more detail, see [4, 5]). Rakić et al. [6] proved several fixed-point theorems in the context of fuzzy b-metric spaces. As an important result, they gave a sufficient condition for a sequence to be Cauchy in a fuzzy b-metric space and they simplified the proofs of many fixed-point theorems in fuzzy b-metric spaces with the well-known contraction conditions. Mecheraoui et al. [7] proved several interesting fixed-point results in the context of E-fuzzy metric spaces. Moussaoui et al. [8] established several fixed-point results for contraction mappings via admissible functions and FZ-simulation functions in the context of fuzzy metric spaces. Zhou et al. [9] proved several fixed-point results for contraction mappings in the sense of non-Archimedean fuzzy metric spaces. Recently, Kanwal et al. [10] have established the notion of fuzzy strong b-metric spaces and generalized a fuzzy version of the Banach contraction principle. Sezen [11] presented a generalized version of Banach contraction principle in the context of controlled fuzzy metric spaces. Ishtiaq et al. [12] and Farhan et al. [13] used controlled function in generalization of metric spaces and proved several fixed point results with applications. Al-Omeri et al. [14] introduced (Φ, Ψ)-weak contractions in neutrosophic cone metric spaces and established several fixed point theorems. Al-Omeri et al. [15] and Al-Omeri [16] introduced several contraction mappings and topological structures in generalized spaces and derived some interesting results to find the fixed point for contraction mappings. Ghareeb and Al-Omeri [17] introduced new degrees for functions in (L, M)-fuzzy topological spaces based on (L, M)-fuzzy semiopen and (L, M)-fuzzy preopen operators. Batul et al. [18] examined several fuzzy fixed point results of fuzzy mappings on b-metric spaces. Mohammadi et al. [19] proved some fixed point results for generalized fuzzy contractive mappings in fuzzy metric spaces with application to integral equations. Rezaee et al. [20] worked on JS-Presic contractive mappings in extended modular S-metric spaces and extended fuzzy S-metric spaces.

We aim to extend the fuzzy version of the Banach contraction principle in the context of fuzzy strong controlled (fsc) metric spaces, fuzzy strong controlled quasi-metric spaces and non-Archimedean fuzzy strong controlled quasi-metric spaces. In fact, we prove results in the broader setting of non-Archimedean fuzzy strong controlled quasi-metric spaces, because in this case, measuring the distance between two words and automatically shows whether is a prefix of or not. Finally, we will use our approaches to show that some recurrence equations related to the complexity analysis of Quicksort algorithms have a solution (and that it is unique) (see [21–23]).

Kanwal et al. [10] established the following definition: Consider as an arbitrary set, is a continuous t-norm (Ct-norm), , and is a fuzzy set (F-set) on It is said to be a fsc-metric if it verifies for all (i)(ii)(iii)(iv)(v) is left continuous.

Then, is known as a fuzzy strong b-metric space.

2. Main Results

In this section, several new concepts and fixed-point results are demonstrated.

Definition 1. Consider is an arbitrary set, is a Ct-norm, and is a F-set on It is said to be a fsc-metric if it verifies for all

(i)(ii)(iii)(iv)(v) is left continuous and

Then is known as fsc-metric space.

Remark 1. If we take , then any fsc-metric space is a fuzzy strong b-metric space.

Proposition 1. Assumeandis defined by

Let be defined by the following:for all Then is a fsc-metric space with product and minimum Ct-norms.

Proposition 2. Let and defined by

Let defined by the following:for all Then is a fsc-metric space with product and minimum Ct-norms.

Example 2.1. Let and be a one-to-one function. Assume a continuous and increasing function , fix and define by the following:

Then, is a fsc-metric space with product Ct-norm and is defined by the following:

Proof. We examine only triangular inequality. Let we have three cases:
(1)(2)(3)Now, if we put the following:Then, it is easy to examine the above three cases of inequalitysatisfied, since is increasing.
The proof in case is similar.

Definition 2. Suppose is a fsc-metric space.

(i) Suppose is a sequence in The sequence is said to be convergent to if (ii) We say that a sequence is Cauchy if for each and any , there exists a natural number such that for all (iii) A fsc-metric space is known as a complete space if every Cauchy sequence is convergent in

We will utilize continuous fsc-metric space in the next study.

Theorem 1. Supposeis a complete fsc-metric space, and letbe a mapping verifying

Also, suppose that for each we deduceexists and are finite. Then has a unique fixed point in

Proof. Assume is an arbitrary point and be a sequence in so that
Now,That is,for each and Thus, for any integer by utilizing triangular inequality, we deduce the following:By utilizing Equations (12) and (13), we deduce the following:As implies that , so by utilizing the definition of fsc-metric space, we get the following:Thus, and this implies that is a Cauchy sequence. Given is complete, so there exists in such that
Using triangular inequalityAs , we get the following:That is, So,
Uniqueness: Let be two fixed points of the operator; then hence,Then,for all By taking limit as in the preceding inequality we get for all hence

Example 2.2. Let and be defined by

Let be defined by the following:for all Then is a complete fsc-metric space with product t-norm. Let then

That is,

So, has a unique fixed point 0.

Theorem 2. Supposeis a complete fsc-metric space, and letbe a mapping verifying

Also, suppose that for each exists and are finite. Then has a unique fixed point in

Proof. Assume is an arbitrary point and be a sequence in so that
Now,Since, is strictly increasing and we cannot write the following:Therefore,For every and . Thus, for any integer and by utilizing triangular inequality, we deduce the following:By utilizing Equations (28) and (29), we deduce the following:Taking as this implies that , so by utilizing definition of a fsc-metric space, we get the following:Thus, and this implies that is a Cauchy sequence. Given is complete and so, there exists in such that
Now, utilizing the contractive condition,As we have the following:which is a contradiction. Hence, So, is a fixed point of
Uniqueness: Let . So, thenThat is, Hence,

Corollary 1. Supposeis a complete fsc-metric space, and letbe a mapping verifying.

Also, suppose that for each we obtain the following:exists and are finite. Then has a unique fixed point in

Proof. Immediate from Theorem 2.

Definition 3. Suppose and for some is an orbit of . A function is known as -orbitally lower semi continuous at if for such that then we obtain

Example 2.3. Let and defined by Pick an element in then we obtain the following:

Observe that for any sequence we examine Let be defined by the following:

Now, and implies thatwhich implies that is orbital lower semicontinuous.

Theorem 3. Supposeis a complete fsc-metric space, and letbe a mapping verifying

for everywhereThenFurthermore, is a fixed point of if and only ifisis orbital lower semi continuous at

Proof. Assume is an arbitrary point and be a sequence in so that
Now,That is,Same manners of Theorem 1, we get is a Cauchy sequence. From the completeness of , we have Suppose that is orbitally lower semicontinuous at then we obtain the following:Conversely, suppose and with then we have the following:

Corollary 2. Supposebe a complete fsb-metric space, and letbe a mapping verifying

for everywhereThenFurthermore, is a fixed point ofif and only ifisis orbital lower semi continuous at

Proof. Immediate from Theorem 3.

Definition 4. Consider be an arbitrary set, be a Ct-norm, and be a F-set on It is said to be a fsc-quasi-metric if it verifies for all

(i)(ii)(iii)(iv) is left continuous and

Then is known as a fsc-quasi-metric space.

Remark 2. Every fsc-quasi-metric space is nondecreasing.

Remark 3. If is a fsc-quasi-metric space, then is also a fsc-quasi-metric space, where is a fuzzy set defined by Moreover, we denote fuzzy set by then is a fsc-metric space.

Theorem 4. Supposebe a complete fsc-quasi-metric space, and letbe a mapping verifying

Also, suppose that for each we deduceexists and are finite. Then has a unique fixed point in

Theorem 5. Supposeis a complete fsc-quasi-metric space, and letbe a mapping verifying

Also, suppose that for each we deduce the following:exists and are finite. Then has a unique fixed point in

Theorem 6. Supposeis a complete fsc-quasi-metric space, and letbe a mapping verifying

Also, suppose that for each we deduce the following:exists and are finite. Then has a unique fixed point in

Theorem 7. Supposeis a complete fsc-quasi-metric space, and letbe a mapping verifying

for every where Then Furthermore, is a fixed point of if and only if is orbital lower semi continuous at

Definition 5. If is a fsc-quasi-metric space, then it is known as bicomplete fsc-quasi-metric space if is complete.

Theorem 8. Supposeis a bicomplete fsc-quasi-metric space, and letbe a mapping verifies

Also, suppose that for each we deduce the following:exists and are finite. Then has a unique fixed point in

Proof. Immediate if we take and proceeding on the lines of Theorem 1.

Definition 6. A fsc-quasi-metric space such that

for all is known as a non-Archimedean fsc-quasi-metric space.

Theorem 9. Supposeis a bicomplete non-Archimedean fsc-quasi-metric space, and letis a mapping verifying

Also, suppose that for each we deduce the following:exists and are finite. Then has a unique fixed point in

3. Quicksort Algorithm

Let be the size of the input and be the average (anticipated value) of the number of times the algorithm performs the fundamental operation for an input size of for a given algorithm. Now we look at the quicksort algorithm, which was established by Hoare [22] (for more details, see [21]). Quicksort performs the sort by dividing the array into partitions and then recursively sorting each partition. Average-case time complexity. The basic operation compares to pivot items in a partition. The number of items in the array determines the size of the input.

We suppose that there is no reason to believe the numbers in the array are in any particular order and that the value of the pivot point provided by partition might be any integer from 1 to . This study would be invalid if there were cause to believe the different distributions. When every conceivable ordering is sorted the same number of times, the average achieved is the average sorting time. The following recurrence gives the average-case time complexity in this case: probability pivot point is

average time to sort sub arrays when pivot point is ,

times of partition.

Therefore,

Combination of Equations (59) and (60) yieldsmultiplying on both sides, we get the following:

Utilizing Equation (62) to yields

Subtracting Equation (62) from Equation (63) giveswhich yields

4. Application to Domain Words

Suppose a nonempty alphabet ∑ and assume the set of all finite and infinite sequences (words) over that is . Where we assume that The prefix order on denoted by i.e., is a prefix of For every defined by , the length of That is, whenever and . For every suppose be the common prefix of and Thus the function defined on by the following:is a quasi-metric on . We take the convention

Let

Remark 4. is a bicomplete non-Archimedean fsc-quasi-metric space with minimum Ct-norm and defined by

Let be defined as follows:

Remark 5. is a bicomplete non-Archimedean fsc-quasi-metric space with minimum Ct-norm and defined by

Next, for complexity analysis of the quicksort algorithm, utilize Theorem 9. The below recurrence equationis examined in the average case analysis of the quicksort algorithm. Assume as an alphabet We associate with the functional defined by the following:

If has length we write otherwise we write Now we show that satisfies Theorem 9 on with contraction constant From construction, we obtain for all (in particular, whenever Furthermore, it is obvious that if and only if and Therefore, . From the following:and if is not a prefix of thenfor all That is, That is, has a unique fixed point which is a unique solution for the recurrence equation i.e., and

Remark 6. The above procedure can also be used with product Ct-norm instead of minimum Ct-norm.

5. Conclusion

In this manuscript, we established fuzzy strong controlled metric spaces, fuzzy strong controlled quasi-metric spaces, and non-Archimedean fuzzy strong controlled quasi-metric spaces and generalized the famous Banach contraction principle. In fact, we proved our findings in the broader setting of non-Archimedean fuzzy strong controlled quasi-metric spaces, because in this case, measuring the distance between two words and automatically shows whether is a prefix of or not. Finally, we utilized our approaches to show that some recurrence equations related to the complexity analysis of the quicksort algorithms have a solution (and that it is unique). In future, we will work on generalizations of fuzzy metric spaces and fixed point results for new types of contraction mappings.

Data Availability

On request, the data used to support the findings of this study can be obtained from the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

This article was written equally by all contributors. The final manuscript was read and approved by all of the authors who contributed equally to this work.

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Copyright © 2023 Aftab Hussain 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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