On Two Banach-Type Fixed Points in Bipolar Metric Spaces

Gaba, Yaé Ulrich; Aphane, Maggie; Sihag, Vizender

doi:https://doi.org/10.1155/2021/4846877

Abstract and Applied Analysis

On this page

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

Research Article | Open Access

Volume 2021 | Article ID 4846877 | https://doi.org/10.1155/2021/4846877

On Two Banach-Type Fixed Points in Bipolar Metric Spaces

Yaé Ulrich Gaba,^1,2,3Maggie Aphane,⁴and Vizender Sihag⁵

Academic Editor: Roberto Paoletti

Received03 May 2021

Accepted17 Jul 2021

Published13 Aug 2021

Abstract

In this article, we propose two Banach-type fixed point theorems on bipolar metric spaces. More specifically, we look at covariant maps between bipolar metric spaces and consider iterates of the map involved. We also propose a generalization of the Banach fixed point result via Caristi-type arguments.

Y. U. Gaba dedicates this paper to his father Adimi P. Gaba, on the occasion of his 70th birthday

1. Introduction and Preliminaries

Metric fixed point theory deals with the existence of a point such that where is a self-mapping defined on a metric space . The notion of metric space has many generalizations in literature. One of the most recent is that of bipolar metric space which was introduced by Mutlu and Gürdal [1]. They explored the link between metric spaces and bipolar metric spaces and proved some well-known fixed point theorems in that new setting. Mutlu et al. [2] also proved coupled fixed point theorems in the setting of bipolar metric spaces. In [3], they also studied fixed point theorems for multivalued mappings on bipolar metric spaces. Finding fixed points for different type of contractions has become the focus of fruitful research activity, and numerous fixed point results in that theory can be found in the literature. Recently, many investigators have published various papers on the fixed point in bipolar metric spaces, and these can be read in [4–13] and the references therein, just to name a few. Bartwal et al. [14] even initiated the concept of fuzzy bipolar metric spaces. An account of the theory bipolar metric spaces can be read in detail in the paper by Mutlu and Gürdal [1]. Recent work in fixed-point characterization of completeness in bipolar metric spaces can be read in [15].

In the present paper, we extend certain fixed point theorems, which we think are genuine generalizations of the Banach fixed point theorem. A similar work has been done by Gaba [16]. We give examples which present the applicability of our obtained results. In concluding this introductory part, we recall the definitions of some fundamental notions related to bipolar metric spaces.

Definition 1 (see [1], Definition 8). A bipolar metric space is a triple such that are two nonempty sets, and is a function satisfying the following properties: (1) whenever (2) whenever (3), whenever Moreover, is called a bipolar metric on the pair .

Definition 2 (see [1], Definition 2.3). Let and be pairs of sets and given a function : (1)If and , we call a covariant map from to and denote this with (2)If and , we call a contravariant map from to and denote this with (3)Moreover, if and are bipolar metrics on and , respectively, the following notations are self-explanatory: and

Definition 3 (see [1], Definition 4.1). Let be a bipolar metric space. (1)A sequence on the set is called a left sequence and a sequence on is called a right sequence. In a bipolar metric space, a left or a right sequence is called simply a sequence(2)A sequence is said to be convergent to a point , if and only if is a left sequence, and , or is a right sequence, and (3)A bisequence on is a sequence on the set . Furthermore, if the sequences and are convergent, then the bisequence is said to be convergent. In addition, if and converge to a common point , then is called biconvergent(4)A bisequence is a Cauchy bisequence, if

Remark 4 (see [1], Proposition 4.3). In a bipolar metric space, every convergent Cauchy bisequence is biconvergent.

Definition 5 (see [1], Definition 4.4). A bipolar metric space is called complete, if every Cauchy bisequence is convergent, hence biconvergent.

Example 6 (see [1], Example 5.7). Let be the class of all singleton subsets of and be the class of all nonempty compact subsets of . We define as Then, the triple is a complete bipolar metric space.

Definition 7 (see [1], Definition 3.4). A covariant or a contravariant map from the bipolar metric space to the bipolar metric space is continuous, if and only if on implies on.

2. Some Interesting Facts around Banach Fixed Point

In this section, we show that if is a self-map on a bipolar metric space and has a power which is a contraction, i.e., there exists and such that then there is a transformation another bipolar metric such that a contraction on .

We introduce it with the following definitions.

Definition 8. Two bipolar metrics and on a set are said to be equivalent if there exist such that

Theorem 9 (see [1], Theorem 5.1). Let be a complete bipolar metric space and covariant contraction a covariant contraction. Then, the function has a unique fixed point.

Theorem 10 (see [1], Theorem 5.2). Let be a complete bipolar metric space given a contravariant contraction . Then, the function has a unique fixed point.

We give the following natural corollary, a consequence of the above theorems:

Corollary 11. Let be a complete bipolar metric type space.
Assume that and be mappings such that there exists satisfying for some , whenever Then, the functions and have a unique fixed point.

Proof. By Theorem 9, has a unique fixed point, say with . Since it follows that is a fixed point of , and thus, by the uniqueness of , we have , that is, has a fixed point. Since, the fixed point of is necessarily a fixed point of , so it is unique.
By Theorem 10, and using the same reasoning as above, one can easily see that has a unique fixed point.☐

Moving in the same line as the Banach contraction, we introduce the following.

Definition 12. Let be a complete bipolar metric space and a covariant contraction. is called Lipschitzian if there exists a constant such that whenever The smallest constant verifying (6) will be denoted .

Definition 13. Let be a complete bipolar metric space and . The subset is said to be -bounded, if there exists such that whenever

Theorem 14. Let be a complete bipolar metric space and a covariant map such that is Lipschitzian for any and Then, has a unique fixed point if and only if there exists such that is -bounded. Moreover, if has a fixed point , then for any , the bisequence is biconvergent to .

Proof. Clearly, if has a fixed point, say , then the subset is -bounded.
Conversely, let such that is -bounded, that is, there exists such that for any
Letting we have Since , then is a Cauchy bisequence. Hence, converges to some point since is complete. guarantees that has a unique limit. Since the map is continuous, implies that and combining this with gives . Hence, is a fixed point of .☐

We are now in position to state the first main result of this paper.

Theorem 15. Let be a complete bipolar metric space and given a covariant contraction be a self mapping such that for some and , whenever If is a nonnegative real such that then the application defined by satisfies the following: (i) is a bipolar metric on the space (ii) a self-mapping such thatthat is, is a contraction with constant with respect to .

Proof. We first prove that is a bipolar metric:
(b0) For if then for . In particular, for , we have , which implies that since is a bipolar metric.
(b1) For if , then and then , i.e., (b2) For that is, for all
(b3) For all and , since we get So (b3) holds.
Hence, is a bipolar metric on .
We now prove that is a covariant contraction with constant .
It is readily seen, by a simple computation, that Since is a covariant contraction with constant , it follows that because of the choice This completes the proof.☐

Remark 16. The map can be thought of as an approximation of order of a certain bipolar metric , equivalent to . Indeed, under the assumptions of Theorem 15, it is readily seen that

The sum therefore defines a bipolar metric , equivalent to , as long as the series happen to converge for some . Moreover, whenever is finite, the map is a contraction with contraction constant

We introduce the following notion of completeness, namely, the -completeness which seems quite more natural than the one introduced by Definition 4.4 of [1]. However, these two notions are equivalent, and to see this, it is enough to look at Proposition 4.3 of [1]. The main idea is that in a bipolar metric space, every convergent Cauchy bisequence is biconvergent and every biconvergent bisequence is a Cauchy bisequence (see Proposition 4.2 of [1]). So we rather express completeness in terms of biconvergent Cauchy bisequences.

Definition 17 (compare [1], Definition 4.4). A bipolar metric space is called -complete, if every Cauchy bisequence in this space is biconvergent.

Remark 18. As already observed by Mutlu and Gürdal [1], in bipolar metric spaces, the notion of completeness is defined via bisequences, rather than sequences. In the present manuscript, we observe that the newly introduced concept of completeness, namely, the -completeness, is equivalent to the notion of completeness in metric spaces, when a bipolar metric space represents a metric space, that is, .

Proposition 19. A metric space is complete if and only if the corresponding bipolar metric space is -complete.

Proof. Let be a complete metric space and be a Cauchy bisequence on . By the fact that is a Cauchy bisequence on , we know that Following the proof of Proposition 4.5 of [1], it is easy to see that and are Cauchy sequences in , and since is complete, there exist such that Since taking the limit as we get that Therefore, the Cauchy bisequence is biconvergent. Hence, the bipolar metric space is -complete.
Conversely, assume that the bipolar metric space is -complete. Let be the metric space, such that the corresponding bipolar metric space is . Let be a Cauchy sequence in . Then, since can be made arbitrarily small, the bisequence is a Cauchy bisequence on the complete bipolar metric space , so it is biconvergent. So there exist such that Therefore, the Cauchy sequence is convergent. Hence, the metric space is complete.☐

In order to move forward, we introduce the following notion of uniform continuity.

Definition 20. Given two bipolar metric type spaces and , we say that is uniformly continuous if for every real number , there exists such that for every with , we have that .

Next, we establish that whenever the mapping is uniformly continuous and the bipolar metric is complete, then so is the bipolar metric .

Theorem 21. We repeat the assumptions of Theorem 15. If is uniformly continuous and the bipolar metric is -complete, then so is the bipolar metric .

Proof. Since we know that any Cauchy bisequence in is also a Cauchy bisequence in . It is therefore enough to prove that, under uniform continuity of in , any biconvergent bisequence in is also biconvergent in .
So let be a biconvergent bisequence in the bipolar metric space such that right converges to some and left converges to the same Set and observe that Since all the powers of are also uniformly continuous in , we can write that, for any , there exists such that for all and , Since left converges to some there exists such that Then, i.e., Thus, right converges to with respect to the bipolar metric
Similarly, since right converges to one can show that right converges to with respect to the bipolar metric
In conclusion, the bisequence is biconvergent in the bipolar metric space , i.e., the bipolar metric space in -complete.
This completes the proof.☐

We conclude this section by an example, illustrating the importance of the requirement for to be uniformly continuous.

Example 22. Let and let be endowed with the partial metric , defined as Observe that for , every mapping is as .
Define by The mapping is covariant , since . It is discontinuous at , but for all i.e., is a contraction and the unique fixed point is Moreover, any real can be used as contraction constant for . We can then apply Theorem 15 with any such that . From Theorem 15, the bipolar metric is given by The bipolar metric is not -complete. Indeed, the bisequence given by is bi-Cauchy in but is not biconvergent, as the only candidate for a limit is but

3. Generalizations of Banach Fixed Point Theorem

The aim of this section is to propose a generalization of the Banach fixed point result via a Caristi-type argument. We begin by this first technical lemma.

Lemma 23. Let be a bipolar metric space and consider a bisequence . If there exists one constant such that then is a Cauchy bisequence.

Proof. The following proof is inspired by the first part of the proof of Theorem 5.1 of [1].
Since , it is obvious that Since , it is obvious that Now let us set . For we have Hence, Similarly, one can establish that Since , for any there exists such that Then, and is a Cauchy bisequence.☐

We are now ready to give the first Banach contraction principle (BCP) generalization.

Theorem 24. Let be a -complete bipolar metric space with and be a covariant map. Suppose that there exists a function satisfying (i) is bounded from below(ii), whenever Then, has at least one fixed point in .

Proof. Let Define the bisequence by .
We prove that the bisequence is Cauchy. (1)If there exist such that , then andTherefore, And the bisequence is Cauchy. (2)Now, assume that for all Owing to (ii), we derive thatSo we have Thus, the sequence is positive and nonincreasing. Hence, it converges to some Observe that for each , we have The above entails that It follows, that for , there exists such that Similarly, one can prove that for , there exists such that Now using Lemma 23, we obtain that the bisequence is Cauchy.
Since the bisequence is Cauchy, it is biconvergent to some . We claim that is the fixed point of . Assuming that and employing assumption (ii) of the theorem, we find that Consequently, we obtain , that is, ☐

Example 25. Let and let us endow with the following bipolar metric: It is easy to see that is a -complete bipolar metric space because any Cauchy bisequence is eventually constant, i.e., there exists such that for , for some . Hence, it is biconvergent.
Let defined by Define as is a covariant map, and for all , We have Thus, the mapping satisfies our conditions and also has a fixed point.
Note that ; thus, does not satisfy the Banach contraction principle.

Remark 26. In a few words, Example 25 tells us that Theorem 24 is not a consequence of the Banach contraction principle on complete bipolar metric spaces. It is therefore very natural to ask if the reverse holds, i.e., if the Banach contraction principle is a consequence of Theorem 24 on complete bipolar metric spaces. The authors plan to look at that question in another study.

Here goes the second Banach contraction principle (BCP) generalization.

Theorem 27. Let be a -complete bipolar metric space with and be a continuous covariant map. If there exists a function such that and satisfying the additional condition: whenever then has a unique fixed point in .

Proof. We follow similar steps as in the proof of Theorem 24. Let Define the bisequence by . For any Hence, the sequence is nonincreasing of nonnegative numbers. Since is bounded below, there exists such that
We can then write that i.e., Similarly, one can see that So the bisequence is Cauchy and then is biconvergent to some , and guarantees that has a unique limit. Since the map is continuous, implies that and combining this with gives . Hence, is a fixed point of .
If is any fixed point of , then implies that and we have, by (57), and so ☐

Remark 28. Note that Theorem 27 is indeed a generalization of the Banach contraction principle (see [1], Theorem 5.1.) over -complete bipolar metric spaces, as we demonstrate it in the coming lines.

If is a Banach contraction, there exists such that

Hence,

Consequently,

This yields

By choosing the linear function we can easily rewrite the above as which is of the form (57).

Moreover, setting , we deduce the following corollary.

Corollary 29. Let be a -complete bipolar metric space with and be a continuous covariant map. If there exists a function such that and satisfying the additional condition: whenever . Then, has a unique fixed point in .

Example 30. Let and let us endow with the following bipolar metric: Let be defined by

We need only check the following two cases.

Case 1. and , so We have

Case 2. , and , and so One could observe that for and without loss of generality, we may assume that . So we get

So, by Corollary 29, has a unique fixed point. Here, .

We point out that the mapping is not a Banach contraction. Indeed, just observe that

4. Conclusion and Going Further

In this paper, we introduced new generalizations/variants of the Banach contraction principle using Caristi-type arguments in the setting of bipolar metric spaces. We plan to investigate how these generalizations can be powerful tools in finding solutions of integral differential equations, for instance. Another interesting line of research could be to think of the multivalued version of the new contraction considered in the present manuscript, as this could provide a powerful tool for the existence solution of Volterra-integral inclusions. Furthermore, it is our opinion that the conclusions of Theorem 21 and the subsequent results remain valid if we consider contravariant mappings. The authors plan to take up this investigation in another manuscript. The contravariant case appears to be more easy but contains, however, some tricky steps than need to be taken care of wisely.

Data Availability

No data was used in 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 concerning the publication of this article.

Authors’ Contributions

All authors contributed equally and significantly in writing this article.

Acknowledgments

The first author (Y. U. G.) would like to acknowledge that this publication was made possible by a grant from the Carnegie Corporation of New York.

References

A. Mutlu and U. Gürdal, “Bipolar metric spaces and some fixed point theorems,” Journal of Nonlinear Sciences and Applications, vol. 9, no. 9, pp. 5362–5373, 2016.
View at: Publisher Site | Google Scholar
A. Mutlu, K. Özkan, and U. Gürdal, “Coupled fixed point theorems on bipolar metric spaces,” European journal of pure and applied Mathematics, vol. 10, no. 4, pp. 655–667, 2017.
View at: Google Scholar
A. Mutlu, K. Özkan, and U. Gürdal, “Fixed point theorems for multivalued mappings on bipolar metric spaces,” Fixed Point Theory, vol. 21, no. 1, pp. 271–280, 2020.
View at: Publisher Site | Google Scholar
U. Gürdal, A. Mutlu, and K. Özkan, “Fixed point results for α-ψ-contractive mappings in bipolar metric spaces,” Journal of Inequalities & Special Functions, vol. 11, no. 1, 2020.
View at: Google Scholar
G. N. V. Kishore, R. P. Agarwal, B. Srinuvasa Rao, and R. V. N. Srinivasa Rao, “Caristi type cyclic contraction and common fixed point theorems in bipolar metric spaces with applications,” Fixed point theory and Applications, vol. 2018, no. 1, 2018.
View at: Publisher Site | Google Scholar
G. N. V. Kishore, H. Isk, H. Aydi, B. S. Rao, and D. R. Prasad, “On new types of contraction mappings in bipolar metric spaces and applications,” Journal of Linear and Topological Algebra, vol. 9, no. 4, pp. 253–266, 2020.
View at: Google Scholar
G. N. V. Kishore, D. Ram Prasad, B. S. Rao, and V. S. Baghavan, “Some applications via common coupled fixed point theorems in bipolar metric spaces,” Journal of Critical Reviews, vol. 7, no. 2, 2020.
View at: Publisher Site | Google Scholar
G. N. V. Kishore, K. P. R. Rao, H. IsIk, B. S. Rao, and A. Sombabu, “Covarian mappings and coupled fixed point results in bipolar metric spaces,” International Journal of Nonlinear Analysis and Applications, vol. 12, no. 1, pp. 1–15, 2021.
View at: Google Scholar
G. N. V. Kishore, K. P. R. Rao, A. Sombabu, and R. V. N. S. Rao, “Related results to hybrid pair of mappings and applications in bipolar metric spaces,” Journal of Mathematics, vol. 2019, Article ID 8485412, 7 pages, 2019.
View at: Publisher Site | Google Scholar
G. Naveen, V. Kishore, B. S. Rao, S. Radenovic, and H. Huang, “Caristi type cyclic contraction and coupled fixed point results in bipolar metric spaces,” Sahand Communications in Mathematical Analysis, vol. 17, no. 1, pp. 1–22, 2020.
View at: Google Scholar
A. Mutlu, K. Özkan, and U. Gürdal, “Locally and weakly contractive principle in bipolar metric spaces,” TWMS Journal of Applied and Engineering Mathematics, vol. 10, no. 2, pp. 379–388, 2020.
View at: Google Scholar
A. Mutlu, K. Özkan, and U. Gürdal, “Caristi's and downing-kirk's fixed point theorems on bipolar metric spaces,” Advances in Fixed Point Theory, vol. 22, no. 2, 2021.
View at: Google Scholar
B. S. Rao, G. N. V. K. ishore, and G. K. Kumar, “Geraghty type contraction and common coupled fixed point theorems in bipolar metric spaces with applications to homotopy,” International Journal of Mathematics Trends and Technology, vol. 63, no. 1, pp. 25–34, 2018.
View at: Publisher Site | Google Scholar
A. Bartwal, R. C. Dimri, and G. Prasad, “Some fixed point theorems in fuzzy bipolar metric spaces,” Journal of Nonlinear Sciences & Applications, vol. 13, no. 4, pp. 196–204, 2020.
View at: Publisher Site | Google Scholar
Ö. Kübra and U. Gürdal, “The fixed point theorem and characterization of bipolar metric completeness,” Konuralp Journal of Mathematics, vol. 8, no. 1, pp. 137–143, 2020.
View at: Google Scholar
Y. U. Gaba, “RelatedG-metrics and fixed points,” Annals of West University of Timisoara - Mathematics and Computer Science, vol. 56, no. 1, pp. 64–72, 2018.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Yaé Ulrich Gaba 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.

PDF Download Citation

Download other formats

Order printed copies

Views

565

Downloads

1025

Citations