On Fuzzy b-Metric-Like Spaces

Javed, Khalil; Uddin, Fahim; Aydi, Hassen; Arshad, Muhammad; Ishtiaq, Umar; Alsamir, Habes

doi:https://doi.org/10.1155/2021/6615976

Journal of Function Spaces

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

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Fixed Point Theory and Applications for Function Spaces

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Research Article | Open Access

Volume 2021 | Article ID 6615976 | https://doi.org/10.1155/2021/6615976

On Fuzzy b-Metric-Like Spaces

Khalil Javed,¹Fahim Uddin,¹Hassen Aydi,^2,3Muhammad Arshad,⁴Umar Ishtiaq,¹and Habes Alsamir⁵

Academic Editor: Zoran Mitrovic

Received31 Dec 2020

Revised26 Jan 2021

Accepted01 Feb 2021

Published15 Mar 2021

Abstract

The aim of this manuscript is to introduce the concept of fuzzy b-metric-like spaces and discuss some related fixed point results. Some examples are imparted to illustrate the feasibility of the proposed methods. Finally, to validate the superiority of the obtained results, an application is provided to solve a first kind of Fredholm type integral equations.

1. Introduction and Preliminaries

The concept of fuzzy sets was initiated by Zadeh [1], which gave a new aspect to research activity leading with the improvement of fuzzy systems. Afterwards, several researchers contributed towards some basic significant results in fuzzy sets.

Kramosil and Michlek [2] introduced the concept of fuzzy metric spaces by generalizing the concepts of probabilistic metric spaces to fuzzy metric spaces. George and Veeramani [3] derived a Hausdorff topology which was initiated by a fuzzy metric to modify the concept of fuzzy metric spaces. Later on, the fixed point theory dealing with a fuzzy metric has been enriched with a number of different generalizations. Garbiec displayed the fuzzy version of Banach contraction principle in fuzzy metric spaces. For some necessary definitions, examples, and basic results, we refer to [4–6] and the references herein.

As we know, fixed point theory plays a crucial role in proving the existence of solutions for different mathematical models and has a wide range of applications in different fields related to mathematics. This theory has intrigued many researchers. Recently, Harandi [7] initiated the concept of metric-like spaces, which generalizes the notion of metric spaces in a nice way. Alghamdi et al. [8] used the concept metric-like spaces to introduce the notion of b-metric-like spaces. Since then, a number of authors contributed in same. For more details, we refer to [7, 9–16]. In this sequel, Shukla and Abbas [8] generalized the concept of metric-like spaces and introduced fuzzy metric-like spaces. For more details on this topic, please see [17–29].

In this article, our aim is to generalize the concept of b-metric-like spaces by introducing the concept of fuzzy b-metric-like spaces and prove some related fixed point results. We also support this work by some examples and an application to solve an integral equation.

First, we write some notations used throughout this paper, as C-t-norm for a continuous triangular norm, b-metric-l for b-metric-like, F-metric-l for fuzzy metric-like, F-b-metric for fuzzy b-metric, F-b-metric-l for fuzzy b-metric-like, and s.t. for such that.

Definition 1 [17]. A binary operation: is called a C-t-norm if it satisfies the following assertions: (1)(2)(3)(4)If and with then Some fundamental examples of a t-norm are and

Definition 2 [8]. A b-metric-l on a set is a function such that for all and it satisfies the following conditions: (1)If (2)(3)

The pair is called a b-metric-l space.

Example 1 [8]. Let . Define the function by

Then, is called a b-metric-l space with .

Example 2 [8]. Let . Define the function by.

Then, is named as a b-metric-l space with .

Definition 3 [18]. A 3-tuple is said an F-metric-l space if is a random set, is a C-t-norm, and is a fuzzy set on meeting the conditions below for all :
FL1)
FL2) If then
FL3)
FL4)
FL5) is continuous.

Example 3 [18]. Let , , and Define a t-norm by and the fuzzy set on by

Then, is an F-metric-l space.

Definition 4 [5]. A 3-tuple is said an F-b-metric space if is a random (nonempty) set, is a C-t-norm, and is a fuzzy set on meeting the conditions below for all and a provided real number:
FB1)
FB2) iff
FB3)
FB4)
FB5) is continuous.

Example 4 [9]. Let where is a real number. It is then simple to show that is an F-b-metric with .

2. Main Results

We start this section with the introduction of F-b-metric-l spaces and prove some related fixed point results.

Definition 5. A 4-tuple is named an F-b-metric-l space if is a random set, is a C-t-norm, and is a fuzzy set on meeting the following conditions below for all :
B1)
B2) If then
B3)
B4) for
B5) is continuous.

Remark 6. In the above definition, a set is endowed by an F-b-metric-l with a t-norm (). An F-b-metric-l space does not satisfy the (FB2) condition of F-b-metric spaces, that is, the self-distance may not be equal to 1, i.e., for all for some or may be for all But all other conditions are the same. Hence, the F-b-metric-l may not be a F-b-metric, but the converse is true.

Definition 7. Let be an F-b-metric-l space. For we define the open ball as Then, is a topology on .

The following simplest example shows that an F-b-metric-l need not to be an F-b-metric.

Example 5. Take . Consider the t-norm defined by , then is an F-b-metric-l. But it is not an F-b-metric.

Proof. (B1), (B2), (B3), and (B5) are obvious. Here, we prove (B4). We have Here, is an arbitrary integer. We have Hence, Since is an increasing function for one writes That is, Hence, (B4) satisfied.
Now, we have to prove that is not an F-b-metric space. For this purpose, we investigate the self-distance. Indeed, Hence, is not an F-b-metric space.

The following example shows that an F-b-metric-l space need not be continuous.

Example 6. Let and If we consider the t-norm defined by , then is an F-b-metric-l space with a coefficient To illustrate the discontinuity, we have However, since One can assert that is not continuous.

Proposition 8. Let be a b-metric-l space. Then, is an F-b-metric-l space defined as

Proof. (B1), (B2), (B3), and (B5) are obvious. Here, we prove (B4). We have Therefore, That is, Hence, That is, Hence, (B4) is satisfied and is an F-b-metric-l space.

Definition 9. A sequence in a F-b-metric-l space is said to be convergent to , if

Definition 10. A sequence in an F-b-metric-l space is said to be Cauchy if exists and is finite.

Definition 11. An F-b-metric-l space is said to be complete if every Cauchy sequence in converges to some such that

Definition 12. Let be an F-b-metric-l space. A mapping is said to be fuzzy contractive if there exists such that for all and Here, is called the fuzzy contractive constant of.

Theorem 13. Let be a complete F-b-metric-l space and be a fuzzy contractive mapping with a fuzzy contractive constant, then has a unique fixed point so that for all

Proof. For an arbitrary, define a sequence in by If for some then is a fixed point of. We assume that for all . For and , we get from (24) We have Therefore, Continuing in this way, we get We have Now, for and , we have Continuing in this way, we get By using (30) in the above inequality, we have Here, is an arbitrary positive integer, and as , we deduce from the above expression that Therefore, is a Cauchy sequence in By the completeness of ,
there is such that Now, we prove that is a fixed point for . For this, we obtain from (24) that Using the above inequality, we obtain Taking limit as and using (35) in the above expression, we get that that is, Therefore, is a fixed point of and for all
Now, we investigate the uniqueness of the fixed point of . Let be another fixed point of , s.t. for some . It follows from (24) that a contradiction. Therefore, we must have for all and hence

Corollary 14. Let be a complete F-b-metric-l space and be a mapping satisfying for some , where . Then, has a unique fixed point and .

Proof. is the unique fixed point of by using Theorem 13, and is also a fixed point of as and from Theorem 13, , is the unique fixed point, since the unique fixed point of is also the unique fixed point of

Example 7. Let and the t-norm be defined as. Given as (From Example 2 and Proposition 8, is an F-b-metric-l).

Then, is a complete F-b-metric-l space. Define as

Then, we have 8 cases:

Case 1. If , then .

Case 2. If and , then and

Case 3. If and , then and

Case 4. If and , then and .

Case 5. If and , then and .

Case 6. If and , then and

Case 7. If and then and

Case 8. If and then and

All above cases satisfy the fuzzy contraction: with . Hence is a fuzzy contractive mapping with . All conditions of Theorem 13 are satisfied. Also, 0 is the unique fixed point of and

Theorem 15. Let be a complete F-b-metric-l space such that for all , and be a mapping satisfying the condition for all where Then has a unique fixed point and

Proof. For an arbitrary , define a sequence in by If for some , then is a fixed point of. We assume that for all . For and , we get from (44) that for all and Therefore, by applying the above expression, we can deduce that for all , , and >0. Thus, we have Continuing in this way, we get Using (48) in the above inequality, we deduce Here, is an arbitrary positive integer. We know that and . So, from (51), we deduce that Hence, is a Cauchy sequence. The hypothesis of completeness of the F-b-metric-l space ensures that there exists such that Now, we derive that is a fixed point of . We have Taking limit as , and by (53), we get Therefore, is a fixed point of and
Now, we investigate the uniqueness of fixed point. For this, assume that and are two fixed points of Then, by (44), we have We obtain Taking limit as and using the fact so ; hence the fixed point is unique.

Example 8. Let and the t-norm be defined as Also, is defined as Then, is a complete F-b-metric-l space. Define by

Now, For, we have four cases:

Case 1. If , then Here,

Case 2. If and then and . We have

Case 3. If then and Here,

Case 4. If and then and Then,

From all 4 cases, we obtain that

Hence, all conditions of Theorem 13 are satisfied, and 0 is the unique fixed point of. Also,

3. An Application to an Integral Equation

Consider the following integral equation: where and

Let be the set of all continuous real valued functions defined on. Consider the b-metric-l given as

Then, by Proposition 8,

Clearly, is a complete F-b-metric-l space. Let

for . Observe that the existence of a solution of (66) is equivalent to the existence of a fixed point of.

Theorem 16. Assume that the following hypotheses hold: (1) is continuous(2), there is a continuous function such thatwhere and (3)andThen, the integral Equation (66) has a unique solution.

Proof. For all we have Therefore,. Also, observe that all conditions of Theorem 15 are satisfied. Hence, the operator has a unique fixed point. This means that the integral Equation (66) has a unique solution.

Example 9. Consider the nonlinear integral equation below Then, it has a solution in

Proof. Let be defined by Set in Theorem 16, we get (i) is continuous(ii), there is a continuous function such thatwhere (iii)andHence, all hypotheses of Theorem 16 are fulfilled. Therefore, the problem (72) has a solution on

4. Conclusion

Fixed point techniques are used to solve many mathematical problems, as differential and integral equations, integro-differential equations, game theory, and economics. The intent of this manuscript is to present a new space, so-called a fuzzy b-metric-like space. Topological properties and related examples are addressed, so our fixed point results are new. Ultimately, to illustrate the practical side of the theoretical results, a solution of a nonlinear integral equation is given.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Post Graduate and Scientific Research at Dar Al Uloom University for funding this work.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 3, pp. 338–353, 1965.
View at: Google Scholar
I. Kramosil and J. Michlek, “Fuzzy metric and statistical metric spaces,” Kybernetika, vol. 11, pp. 336–344, 1975.
View at: Google Scholar
A. George and P. Veeramani, “On some results of analysis for fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 90, no. 3, pp. 365–368, 1997.
View at: Publisher Site | Google Scholar
T. Došenovic, A. Javaheri, S. Sedghi, and N. Shobe, “Coupled fixed point theorem in b-fuzzy metric spaces,” Novi Sad Journal of Mathematics, vol. 47, no. 1, pp. 77–88, 2017.
View at: Google Scholar
S. Sedghi and N. Shobe, “Common fixed point theorem in b-fuzzy metric space,” Nonlinear Functional Analysis and Applications, vol. 17, no. 3, pp. 349–359, 2012.
View at: Google Scholar
M. Humaira and T. A. Sarwar, “Existence of solutions for nonlinear impulsive fractional differential equations via common fixed-point techniques in complex valued fuzzy metric spaces,” Mathematical Problems in Engineering, vol. 2020, Article ID 7042715, 14 pages, 2020.
View at: Publisher Site | Google Scholar
A. Harandi, “Metric-like paces, partial metric spaces and fixed point,” Fixed Point Theory and Applications, vol. 2012, 2012.
View at: Google Scholar
M. A. Alghamdi, N. Hussain, and P. Salimi, “Fixed point and coupled fixed point theorems on b-metric-like spaces,” Journal of Inequalities and Applications, vol. 2013, no. 1, Article ID 402, 2013.
View at: Publisher Site | Google Scholar
H. A. Hammad, M. D. la Sen, and H. Aydi, “Generalized dynamic process for an extended multi-valued F-contraction in metric-like spaces with applications,” Alexandria Engineering Journal, vol. 59, no. 5, pp. 3817–3825, 2020.
View at: Publisher Site | Google Scholar
H. A. Hammad and M. De la Sen, “A solution of Fredholm integral equation by using the cyclic η s q -rational contractive mappings technique in b-metric-like spaces,” Symmetry, vol. 11, no. 9, p. 1184, 2019.
View at: Publisher Site | Google Scholar
H. Aydi, H. Lakzian, Z. D. Mitrovic, and S. Radenovic, “Best proximity points of MT-cyclic contractions with property UC,” Numerical Functional Analysis and Optimization, vol. 41, no. 7, pp. 871–882, 2020.
View at: Publisher Site | Google Scholar
C. Chen, L. Wen, J. Dong, and Y. Gu, “Fixed point theorems for generalized F-contractions in b-metric-like spaces,” Journal of Nonlinear Sciences and Applications, vol. 9, no. 5, pp. 2161–2174, 2016.
View at: Publisher Site | Google Scholar
M. U. Ali, H. Aydi, and M. Alansari, “New generalizations of set valued interpolative Hardy-Rogers type contractions in b-metric spaces,” Journal of Function Spaces, vol. 2021, Article ID 6641342, 2021.
View at: Publisher Site | Google Scholar
E. Ameer, H. Aydi, M. Arshad, H. Alsamir, and M. S. Noorani, “Hybrid multivalued type contraction mappings in αK-complete partial b-metric spaces and applications,” Symmetry, vol. 11, no. 1, p. 86, 2019.
View at: Publisher Site | Google Scholar
H. Aydi, A. Felhi, E. Karapinar, and S. Sahmim, “A Nadler-type fixed point theorem in dislocated spaces and applications,” Miskolc Mathematical Notes, vol. 19, no. 1, pp. 111–124, 2018.
View at: Publisher Site | Google Scholar
H. R. Marasi and H. Aydi, “Existence and uniqueness results for two-term nonlinear fractional differential equations via a fixed point technique,” Journal of Mathematics, vol. 2021, Article ID 6670176, 7 pages, 2021.
View at: Publisher Site | Google Scholar
B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 314–334, 1960.
View at: Google Scholar
S. Shukla and M. Abbas, “Fixed point results in fuzzy metric-like spaces,” Iranian Journal of Fuzzy Systems, vol. 11, no. 5, pp. 81–92, 2014.
View at: Google Scholar
H. A. Hammad, H. Aydi, and Y. U. Gaba, “Exciting fixed point results on a novel space with supportive applications,” Journal of Function Spaces, vol. 2021, Article ID 6613774, 12 pages, 2021.
View at: Publisher Site | Google Scholar
D. Rakic, A. Mukheimer, T. Došenovic, Z. D. Mitrovic, and S. Radenovic, “On some new fixed point results in fuzzy b-metric spaces,” Journal of Inequalities and Applications, vol. 2020, no. 1, 2020.
View at: Publisher Site | Google Scholar
S. Nădăban, “Fuzzy b-metric spaces,” International Journal of Computers Communications & Control, vol. 11, no. 2, pp. 273–281, 2016.
View at: Publisher Site | Google Scholar
M. Imdad, J. Ali, and M. Hasan, “Common fixed point theorems in fuzzy metric spaces employing common property (E.A.),” Math. Comput. Modelling, vol. 55, no. 3-4, pp. 770–778, 2012.
View at: Publisher Site | Google Scholar
N. Saleem, J. Vujakovic, W. U. Baloch, and S. Radenovic, “Coincidence point results for multivalued Suzuki type mappings using θ-contraction in b-metric spaces,” Mathematics, vol. 7, no. 11, p. 1017, 2019.
View at: Publisher Site | Google Scholar
N. Saleem, B. Ali, M. Abbas, and M. Raza, “Fixed points of Suzuki type generalized multivalued mappings in fuzzy metric spaces with applications,” Fixed Point Theory and Applications, vol. 1, 18 pages, 2015.
View at: Google Scholar
N. Salem and S. Beloul, “Common fixed point theorems for weakly subsequentially continuous mappings in modified intuitionistic fuzzy metric spaces,” Universal Journal of Applied Mathematics, vol. 5, no. 5, pp. 96–105, 2017.
View at: Publisher Site | Google Scholar
M. Abbas, N. Saleem, and K. Sohail, “Optimal coincidence best approximation solution in b-fuzzy metric spaces,” Commun. Nonlinear Anal, vol. 6, pp. 1–12, 2019.
View at: Google Scholar
M. Abbas, F. Lael, and N. Saleem, “Fuzzy b-metric spaces: fixed point results for ψ-contraction correspondences and their application,” Axioms, vol. 9, no. 2, p. 36, 2020.
View at: Publisher Site | Google Scholar
N. Saleem, M. Abbas, and Z. Raza, “Optimal coincidence best approximation solution in non-Archimedean Fuzzy Metric Spaces,” Iranian Journal of Fuzzy Systems, vol. 13, no. 3, pp. 113–124, 2016.
View at: Google Scholar
M. S. Humairaa and T. Abdeljawad, “Existence of unique solution to nonlinear mixed Volterra Fredholm-Hammerstein integral equations in complex-valued fuzzy metric spaces,” Journal of Intelligent & Fuzzy Systems, pp. 1–10, 2020.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Khalil Javed 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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