Coincidence Best Proximity Point Results in Branciari Metric Spaces with Applications

Saleem, Naeem; De la Sen, Manuel; Farooq, Sadia

doi:https://doi.org/10.1155/2020/4126025

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 2020 | Article ID 4126025 | https://doi.org/10.1155/2020/4126025

Coincidence Best Proximity Point Results in Branciari Metric Spaces with Applications

Naeem Saleem,¹Manuel De la Sen,²and Sadia Farooq¹

Academic Editor: Nawab Hussain

Received16 Jul 2020

Revised30 Aug 2020

Accepted27 Nov 2020

Published14 Dec 2020

Abstract

This paper is aimed at studying the uniqueness of coincidence best proximity point for -proximal contractions in complete Branciari metric space. Throughout this article, discontinuity of the Branciari metric space is used and we obtained the desired results without assuming it as a continuous. Some examples are provided to validate the results proved herein. As an application, we derive the best proximity point results in the setup of complete Branciari metric space endowed with graph. Further, our results extend and generalize the existing ones in literature.

1. Introduction and Preliminaries

Let be a mapping, where be any nonempty set. “An element is a fixed point of if satisfies the equation (known as a fixed point equation) or A collection of all “fixed points” of will be represented as that is,

In this direction, Banach [1] gives the existence and uniqueness of the “fixed point” of the self mapping if mapping is a contraction and is a complete, but it becomes more interesting, if is a nonself mapping then it is not necessary that the operator equation has a solution. In this situation, we can find a point which is closest to and we have the following minimization/optimization problem

Now, consider be a metric space, and are nonempty subsets of , and consider a mapping , we can find a point in such that is minimum. In other words, we have to minimize for all in and in It is important to see that the where which cannot be further reduced. If such point in exists then is called an “approximate fixed point” of [2].

Later, several authors studied the results dealing with “approximate fixed points” in different spaces (for detail, see [3–14]).

The best proximity point of the mapping is actually “a point such that .” Note that if then ; in this case, every “approximate fixed point” becomes “fixed point” of the mapping . From this perspective, we can say that “the best proximity point results” are natural generalization of “fixed point results.”

The concept of “coincidence best proximity point” was introduced in [5] for a pair of mappings in metric space. “A point is called the coincidence best proximity point of a pair of mappings and if .” We denote the set of all “coincidence best proximity points” of a pair of mappings and by , that is,

This is an extension of a “best proximity point problem.” If is an identity mapping on then a “coincidence best proximity point” will reduce to a “best proximity point” of mapping .

Recently, Branciari [15] defined a Branciari (by some author generalized/rectangular) metric space. Branciari metric space generalizes the deterministic metric space in a natural way. Few examples are also provided to show that a Branciari metric space is not a metric space.

Definition 1 (see [15]). Let be a nonempty set. Any mapping is a Branciari metric on if for all and the following conditions are satisfied: (1) if and only if (2)(3)Then, the pair is called a Branciari metric space.

Definition 2 (see [15]). Let be a Branciari metric space. Then, a sequence in is as follows: (a)Convergent sequence which converges to if and only if as In this case, we can write(b)Cauchy sequence if and only if as (c)A Branciari metric space is complete if every Cauchy sequence in converges to some element in

Lemma 3 (see [16]). Let be a Branciari metric space and be a Cauchy sequence in such that whenever Then, the sequence can converge to utmost one point.

In a Branciari metric space, if a sequence is both Cauchy and convergent, then pathologies provided in an example [17] cannot happen, as shown in the following Lemma.

Lemma 4 (see [18]). Suppose that is a Cauchy sequence in a Branciari metric space with where Then, for all In particular, the sequence cannot converge to if

Recently, Jleli and Samet [4] introduced the concept of -contraction and proved a fixed point result for such mappings in the setup of Branciari metric spaces.

Definition 5 (see [4]). Let be the set of all functions satisfying the following conditions:
is increasing
for any sequence in if and only if
there exist and such that

Definition 6 (see [4]). Let be a complete Branciari metric space and A mapping is called -contraction if for any where and

Theorem 7 (see [4]). Let be a Branciari metric space and be a -contraction. Then, has a unique fixed point in .

Following definitions are also needed in the sequel.

Definition 8. Let be a Branciari metric space and and be two nonempty subsets of . Define

Definition 9 (see [19]). Let be a nonempty subset of and be a nonempty subset of ; then, the pair satisfies the “weak -property,” if for all and

Definition 10. A mapping is called the following: (1)[20] An “isometry,” if(2)[21] An “expansive mapping,” iffor any

Proposition 11 (see [22]). A self mapping is said to satisfy property if there exist a mapping ) such that for all

Definition 12 (see [3]). Let be a metric space, is a nonempty subset of ), and is a mapping. Then, is said to be -regular, if is a sequence in such that and as then for all

Now we have the following definitions:

Definition 13. If ) then mapping is called “triangular proximal -admissible,” if
(T1) (T2) for all

Example 1. Consider with the usual metric defined by . Let and be subsets of defined by It is easy to see that Let be the mapping defined as Also, define ) by

Case 1. If we take and in , we get which implies that If we take , we get

Case 2. If we take and in , we get which implies that If we take , we get

Definition 14. If and then pair of mappings satisfies (1)-proximal contraction, if(2)-generalized proximal contraction, ifwhere , and with

Definition 15. If and then pair of mappings satisfies (1)-proximal contraction, if(2)“-generalized proximal contraction,” ifwhere , and with

Definition 16. Let be a mapping satisfying (1)-proximal contraction, if(2)-generalized proximal contraction, ifwhere , and with.

Definition 17. Let be a mapping satisfying (1)-proximal contraction, if(2)-generalized proximal contraction, ifwhere , and with .

Remark 18. By taking (identity, mapping over ) then every -proximal contraction will reduce to -proximal contraction and -generalized proximal contraction will reduce to -generalized proximal contraction.

Remark 19. By taking every -proximal contraction will reduce to -proximal contraction and -generalized proximal contraction will reduce to -generalized proximal contraction.

Remark 20. By taking (identity, mapping over ) and then every -proximal contraction will reduce to -proximal contraction and -generalized proximal contraction will reduce to -generalized proximal contraction.

Note that now and onwards in this article, we assumed that and are nonempty, distinct, and disjoint subsets of a complete Branciari metric space and will represent a “complete Branciari metric space” until otherwise it is stated. Also, note that if is nonempty, weakly compact, and convex pair in Banach space , then and are nonempty ([23]).

2. Main Results

Now we state and prove our main result which runs as follows.

Theorem 21. Let ), be a pair of mappings satisfying -proximal contraction, where be a triangular proximal -admissible and be a one to one expansive mapping satisfying property. Moreover, if is a nonempty -regular closed set in and satisfy the weak -property. Then, there exists a coincidence best proximity point of the pair of mappings provided that there exists such that Moreover, if for every such that then is the unique coincidence best proximity point of the pair of mappings

Proof. Let such that and As and , there exists such that . Since is a triangular proximal -admissible, we have . Since satisfies property, hence . Similarly, by and there exists a point such that Since is a triangular proximal -admissible, this further implies that Following the same arguments, we have Continuing this way, we can obtain a sequence in such that If for all , the pair of mappings is -proximal contraction, and we have Since, is a triangular proximal -admissible, we obtain that Since pair satisfies the weak -property and is one to one on then we have which further implies that since is an expansive and is an increasing mapping; hence, After simplification, we have Further, we can write as which implies that where . Taking limit in the above inequality, we have which implies that Since then there exist and , such that the following limit holds true: Assume that and . Thus, there exist such that Hence, we have Further, we can write as where . If then there exists such that which implies that where . Hence, in all cases, there exist and such that From inequalities (39) and (48), we have Then, by taking limit as on both sides of the above inequality, we obtain Thus, there exists such that Now suppose that for all and . If for all then Since is a triangular proximal -admissible and , by (T2) of Definition 13, we have Again, since the pair of mappings is -proximal contraction, we have The pair satisfies the weak -property; is one to one on , and is increasing; then, Then, from inequalities (54) and (55), we have which further implies that Since is an expansive mapping and is an increasing mapping, then we have After further simplifications, we have which implies that where . Taking the limit in above inequality, we have Similarly, from condition , we have Similarly, from condition , there exists such that

Now we have the following cases:

Case 1. If and is odd. Consider using (51), and we obtain for all where

Case 2. If is even. Consider using (51) and (63), and we obtain for all where Thus, by combining all the cases, we have where Then, by -series test, converges as . We deduce that is a Cauchy sequence in . By the completeness of space and is closed, there exists such that . Since , we have Since is -regular then . Since then ; thus, there exists a point such that and Since the pair of mappings is -proximal contraction and by using weak -property, we obtain

Further, the above inequality can be written as follows:

Since is an expansive mapping, we have

This implies that

As is increasing, we have

Then, by rectangular property, (51) and (72), we have

Taking limit in the above inequality, we conclude that Hence,

Thus, is a coincidence best proximity point of pair of mappings .

Uniqueness. Now we have to show that is a unique coincidence best proximity point of pair of mappings . Suppose that and be two coincidence best proximity points of a pair of mappings that is,

Since , for every , and by using properties of and and reasoning as above, we obtain that

Further, we have which is a contradiction. Therefore, . Hence, pair of mappings has a unique coincidence best proximity point.

If is an isometry in Theorem 21 then it yields the following theorem.

Theorem 22. Let ), be a pair of mappings satisfying -proximal contraction, where be a triangular proximal -admissible and be a one to one isometry mapping satisfying property. Moreover, if is a nonempty -regular closed set in , and satisfy the weak -property. Then, there exists a coincidence best proximity point of the pair of mappings provided that there exists such that Moreover, if for every such that then is the unique coincidence best proximity point of the pair of mappings

Proof. The result follows from Theorem 21 by choosing as an isometry mapping instead of expansive mapping, and the remaining proof follows under the same lines.

Corollary 23. Let be a -proximal contraction and be a one to one expansive mapping satisfying property. Moreover, if is a nonempty closed set in , , , and satisfy the weak -property. Then, there exists a unique coincidence best proximity point of pair of mappings provided that there exists such that

Proof. The result follows from Theorem 21 by choosing and the remaining proof follows under the same lines.

Corollary 24. Let ) and be a triangular proximal -admissible and -proximal contraction. Moreover, if is a nonempty -regular closed set, and satisfy the weak -property. Then, there exists a best proximity point of provided that there exists such that Moreover, if for every such that then is a unique best proximity point of

Proof. The result follows from Theorem 21 by choosing and the remaining proof follows under the same lines.

Corollary 25. Let be a -proximal contraction. Moreover, if is a nonempty closed set, and satisfy the weak -property. Then, there exists a unique best proximity point of provided that there exists such that

Proof. The result follows from Theorem 21 by choosing and and the remaining proof follows under the same lines.

To support the Corollary 25, we provide the following example.

Example 2. Let and ) which is defined as Then, is a Branciari metric space. Since then is not a metric space.
Suppose that and are subsets of Branciari metric space Note that and . It is easy to see that satisfy the weak -property. Define a mapping as follows: Clearly, has no fixed point; also, note that Define a function as follows: Hence, satisfies the conditions of -proximal contractive mapping. Further, by taking , and in such a way that , we have This implies that satisfies Hence, is -proximal contraction for all . Thus, all conditions of Corollary 25 hold true; after simple calculation, we can find is a unique best proximity point of .

Remark 26. If we take and in the Example 2, we obtained the example of the main Theorem (1.7) in [4].

Theorem 27. Let ), be a pair of mappings satisfying -generalized proximal contraction, where be a triangular proximal -admissible and be a one to one expansive mapping satisfying property. Moreover, if is a nonempty -regular closed set in , and satisfy the weak -property. Then, there exists a coincidence best proximity point of the pair of mappings provided that there exists such that Moreover, if for every such that then is the unique coincidence best proximity point of the pair of mappings

Proof. Following the arguments similar to those given in the proof of Theorem 21, we obtain a sequence in such that If for all also the pair of mappings is -generalized proximal contraction and is an expansive mapping, we have Further, the above inequality becomes Again by using the arguments similar to those given in the proof of Theorem 21, we obtain Now suppose that for every such that . If for all since the pair of mappings is -generalized proximal contraction and is an expansive mapping, we have Further, the above inequality becomes By using the arguments similar to those given in the proof of Theorem 21, we deduce that is a Cauchy sequence in and Since the pair of mappings is -generalized proximal contraction and is an expansive mapping, we have

By using the arguments similar to those given in the proof of Theorem 21, we conclude that is a coincidence best proximity point of pair of mappings .

Uniqueness. Now we have to show that is a unique coincidence best proximity point of pair of mappings . Suppose that and be two coincidence best proximity points of a pair of mappings that is,

Since , for every and by using properties of and and reasoning as above, we obtain that

which is a contradiction. Therefore, . Hence, pair of mappings has a unique coincidence best proximity point.

If is an isometry then the preceding theorem yields the following Theorem.

Theorem 28. Let ), be a pair of mappings satisfying -generalized proximal contraction, where be a triangular proximal -admissible and be a one to one isometry mapping satisfying property. Moreover, if a is nonempty -regular closed set in , and satisfy the weak -property. Then, there exists a coincidence best proximity point of the pair of mappings provided that there exists such that

Moreover, if for every such that then is the unique coincidence best proximity point of the pair of mappings

Proof. The result follows from Theorem 27 by choosing as an isometry mapping instead of an expansive mapping, and the remaining proof follows under the same lines.

Corollary 29. Let be a pair of mappings satisfying -generalized proximal contraction and be an expansive mapping satisfying property. Moreover, if is nonempty closed set in and satisfy the weak -property. Then, there exists a unique coincidence best proximity point of the pair of mappings provided that there exists such that

Proof. The result follows from Theorem 27 by choosing and the remaining proof follows under the same lines.

Corollary 30. Let ), be a -generalized proximal contraction. Moreover, if is a nonempty -regular closed set in and satisfy the weak -property. Then, there exists a best proximity point of mapping provided that there exists such that Moreover, if for every such that then is the unique best proximity point of the mapping

Proof. The result follows from Theorem 27 by choosing and the remaining proof follows under the same lines.

Corollary 31. Let be a -generalized proximal contraction. Moreover, if is a nonempty closed set, and satisfy the weak -property. Then, there exists a unique best proximity point of provided that there exists such that

Proof. The result follows from Theorem 27 by choosing and and the remaining follows under the same lines.

To support the Corollary 31, we provide the following example.

Example 3. Let and ) which is defined as where Then, is a Branciari metric space. Since then is not a metric space. Suppose that and Note that and . Define a mapping as Clearly, has no fixed point; also, note that Define a function as Hence, satisfies the conditions of -generalized proximal contractive mapping. Further, by taking and in such a way that we have Further, satisfies

Hence, is -generalized proximal contraction for all Thus, all conditions of Corollary 31 hold true; after simple calculation, we can find is a unique best proximity point of .

3. Application to Coincidence Point and Fixed Point Theory

If we take then from Definition 13, triangular proximal -admissible implies which becomes

(P1)

(P2)

Remark 32. Note that, for self mapping, every triangular proximal -admissible self mapping is triangular -admissible mapping.

Remark 33. If then -proximal contraction becomes and -generalized proximal contraction becomes where and for all

Definition 34. A self mapping satisfying inequality (113) is called -contraction, and the mapping which satisfies inequality (114) is called -generalized contraction.

Remark 35. If and a self mapping on is -proximal contraction and -generalized proximal contraction then implies that where and for all

Definition 36. A self mapping on and satisfying inequality (115) is called -contraction.

Corollary 37. Let be a complete Branciari metric space. Let be a -proximal contraction. Then, the pair of mappings has a coincidence point.

Proof. Let We show that satisfies -contraction. Then, we have for all , since implies that and and since satisfies condition (113). So, which implies that which further implies that is -contraction. As then is a coincidence point of and .

Corollary 38. Let be a complete Branciari metric space and pair of mapping be a -generalized proximal contraction. Then, pair of mappings has a coincidence point.

Proof. Let We show that satisfies -generalized contraction. Then, we have for all , since implies that and and since satisfies condition (114). So, which implies that which further implies that is -generalized contraction. As then is a coincidence point of and .

If is an identity mapping then we will have the following corollary.

Corollary 39. Let be a complete Branciari metric space, , and be a -contraction. If is a sequence in such that and as then for all Then, there exists a fixed point of provided that there exists such that Moreover, if for all then is the unique fixed point of the mapping .

Proof. Let We show that satisfies -proximal contraction and -generalized proximal contraction. Then, we have for all since implies that and and since satisfies condition (115). So, which implies that which further implies that is -proximal contraction. Also, we have which implies that which further implies that is -generalized proximal contraction. Since is triangular proximal -admissible, by Remark 32, is a triangular -admissible mapping. If there exists such that and , it gives that is a fixed point of

Uniqueness. Let for all Now we will show that is a unique fixed point of . On contrary, suppose that be another fixed point of mapping with . Hence,

Then, by using the properties of , we obtain that

which is a contradiction. Therefore, . Hence, the mapping has a unique fixed point.

Remark 40. By taking and for all in the above Corollary 39, we obtain the main Theorem (1.7) in [4] as Corollary.

4. Application to Graph Theory

Let be a nonempty set and define a set A graph is a pair where is a set of vertices coinciding with and is a set of its edges such that . Moreover, we suppose that the graph is without parallel edges. By reversing the direction of edges in , we obtain a graph whose edge set and vertex set are defined as follows:

Consider the graph comprising of all vertices and edges of and that is,

We denote the undirected graph by obtained by ignoring the direction of edges of

Definition 41 (see [24]). (1)A subgraph of a graph is consisting upon a subset of edges of graph and associated vertices.(2)Let and be two vertices in a graph A path of length (where ) in from to is a sequence of distinct vertices such that , and for (3)A graph is called a connected graph if there exists a path between any two vertices of graph , and it is said to be a weakly connected graph if is connected.(4)A path is called elementary if no vertices appear more than one time in it.Throughout this section, we suppose that be a Branciari metric space and a graph may be transformed to a weighed graph by appointing to each edge the distance given by the Branciari metric between its vertices. In order to apply the rectangular inequality to the vertices of the graph , we will consider the graph of length greater than which signifies that between two vertices, we will obtain a path between two vertices.

Definition 42. Let be a pair of nonempty subsets of a Branciari metric space and be a directed graph without parallel edges such that . A mapping is said to be -proximal contraction if for all with such that where , and with

Definition 43. Let be a pair of nonempty subsets of a Branciari metric space and be a directed graph without parallel edges such that . A mapping is said to be -generalized proximal contraction if for all with such that where , and with

Corollary 44. Let a-proximal contraction. If and is nonempty closed subset in , and satisfy the weak -property. Also, there exists such that there exists an elementary path between them and Moreover, if there exists a path in between any two elements and then has a unique best proximity point.

Proof. Let such that A path in is a sequence containing points of since and from definition of there exists such that Similarly, for each there exists such that As is a path in then From the above argument, we have and Since, is -proximal contraction, it follows that In similar manner, we have the following: Let then is a path from to For each as and then by definition of there exists such that In addition, we have . As above mentioned, we have Similarly, by there exists a point where Then, is a path from and Continuing this way, we can obtain a sequence where and by producing a path from and in such a way that for all Thus, we have Next, we claim that where is a constant. To prove the claim, we need to consider the following two cases where is a path from to . Note that for all are different owing to the fact that the considered path is elementary. Then, we can apply the rectangular inequality.

Case 1. ( is odd). For any positive integer , we get Since is -proximal contraction, we have Since pair of subsets satisfies the weak -property then we have Then, from above inequalities, we have After simplification, we have By induction, it follows that for all , which implies that where . Taking limit in the above inequality, we have which implies that Since then there exist and , such that the following limit holds true: Assume that and . Thus, there exists such that Hence, we have and so we obtain where . If then there exists such that which implies where . Hence, in all cases, there exist and such that From inequalities (144) and (153), we have and taking limit as on both sides of the above inequality, we obtain Thus, there exists such that By putting inequality (156) in inequality (138), we get where

Case 2. ( is even). By the same arguments used in Case 1, we deduce that Indeed, from (136), we have Since is -proximal contraction, we have The pair satisfies weak -property; then, we can write Since is increasing then the above inequality becomes Then, from inequalities (161) and (163), we have After further simplifications, By induction, it follows that for all , where . Taking the limit in above inequality, we have Similarly, from condition , we have Similarly, from condition , there exists such that Then, from inequalities (158), (159), and (169), we have where Let us prove that is a Cauchy sequence. Let such that . We suppose that is odd () since the case is similar. Note that , and for all since the path is elementary. Then, by using the rectangular metric, we obtain for all where Hence, we obtain We deduce that is a Cauchy sequence in . By using the completeness of Branciari metric space and closeness of , there exists such that . Since then , thus there exists a point such that and As is -proximal contraction, we obtain Then, by using the weak -property of subsets of Branciari metric space,we have Hence, we have Further, This implies that As is increasing, we have Then, by rectangular property and using above inequality and inequality (170), we have Now, by taking limit as , we have

Thus, is a coincidence best proximity point of pair of mappings .

Uniqueness. We now show that is a unique best proximity point of . Let be another best proximity point of with . Hence,

Then, by using properties of and and reasoning as above, we obtain that which is a contradiction. Therefore, . Hence, has a unique best proximity point.

Corollary 45. Let be a -generalized proximal contraction. Moreover, is a nonempty closed subset in Branciari metric space and . If there exists a path in between two elements and then there exists a unique best proximity of provided that there exists such that there is an elementary path in between them and

Proof. Let such that A path in is a sequence containing points of Following arguments similar to those given in the proof of Corollary 44, we obtain Since is -generalized proximal contraction, we have Again by using the arguments similar to those given in the proof of Corollary 44, we have where Again by using the arguments similar to those given in the proof of Corollary 44 and since is -proximal contraction, we have By using the arguments similar to those given in the proof of Corollary 44, we deduce that is a Cauchy sequence in . By using the completeness of space and is closed, there exists such that . Since then ; thus, there exists a point such that and As is -generalized proximal contraction, we have

Following arguments similar to those given in the proof of Corollary 44, we obtain that is the best proximity point of the mapping .

Uniqueness. Now we have to show that is a unique best proximity point of . Let be another best proximity point of with . Hence,

Then, by using properties of and and reasoning as above, we obtain that which is a contradiction. Therefore, . Hence, has a unique best proximity point.

5. Conclusion

In this paper, we define -proximal contraction and provide the existence results for coincidence best proximity point in Branciari metric space. The important aspect of Branciari metric space is that it is not continuous; we dealt with discontinuity of Branciari metric space and obtained the desired results. As an application, we derive the coincidence point and fixed point results for some self mappings. We also introduce the notion of -proximal contraction and provide an application to graph theory in the setting of Branciari metric space. Some examples are also provided to illustrate the novelty of the result proved herein.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

We appreciate the reviewer’s careful reading and remarks which helped us to improve the paper. The authors are grateful to the Basque Government for Grant IT1207-19.

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Copyright © 2020 Naeem Saleem 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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