Fixed Point Theorems for Set-Valued <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3240995pt" id="M1" height="12.223pt" version="1.1" viewBox="-0.0657574 -11.8989 16.4258 12.223" width="16.4258pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-13" d="M608 643C551 682 486 687 443 687C299 687 188 608 188 504C188 361 341 312 465 312C483 312 501 313 518 315C454 195 366 93 314 59C255 82 201 109 141 109C73 109 33 83 33 52C33 13 76 -9 159 -9C218 -9 267 -1 314 15C354 0 402 -15 478 -15C575 -15 671 36 759 141L740 155C657 63 571 16 475 16C448 16 411 26 372 39C458 74 565 199 627 333C806 381 931 499 931 587C931 634 908 669 848 669C767 669 671 582 620 506C591 462 569 411 534 347C503 343 490 342 471 342C330 342 229 399 229 500C229 603 346 657 451 657C486 657 540 657 596 624L608 643ZM890 588C890 519 822 457 747 414C712 395 678 378 643 369C683 484 760 632 841 632C873 632 890 615 890 588ZM266 35C235 24 200 21 161 21C103 21 74 32 74 54C74 75 122 79 141 79C172 79 209 69 266 36V35Z"/></g></svg>-</span>Contractions in Branciari Distance Spaces

Cho, Seong-Hoon

doi:https://doi.org/10.1155/2021/6384818

Abstract and Applied Analysis

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

Research Article | Open Access

Volume 2021 | Article ID 6384818 | https://doi.org/10.1155/2021/6384818

Fixed Point Theorems for Set-Valued -Contractions in Branciari Distance Spaces

Seong-Hoon Cho¹

Academic Editor: Douglas R. Anderson

Received09 Aug 2021

Revised01 Nov 2021

Accepted10 Nov 2021

Published29 Nov 2021

Abstract

In this paper, the notion of set-valued -contractions is introduced, and a new fixed point theorem for such contractions is established. An example to illustrate main theorem is given.

1. Introduction and Preliminaries

Branciari [1] introduced a Branciari distance by replacing the triangle inequality in a metric with the rectangular inequality as follows.

A map , where is a nonempty set, is said to be Branciari distance on if and only if it satisfies the following conditions:

For all and for all distinct points , each of them different from and is as follows:

(d1) if and only if

(d2)

(d3)

The pair is called a Branciari distance space, whenever is a Branciari distance on .

In many papers, for example, [2–8], it is called generalized metric space, Branciari metric space, or rectangular metric space. However, these names do not reflect and indicate the meaning well of the notion of Branciari distance spaces because Branciari distance can not reduce to the standard metric. Further it is well known that a Branciari distance space does not have a topology which is compatible with (see [8]). For these reasons, we rename and use it as Branciari distance space.

Branciari [1] extended the Banach contraction principle to Branciari distance space.

After that, a lot of authors, for example, [2–15] and references therein, obtained fixed point results in such spaces. Jain et al. [16] obtained fixed point results in extended Branciari -distance spaces [17] by defining the notion of certain contractive conditions, and they gave an application to nonlinear fractional differential equations.

Branciari [1] investigated the existence of fixed points with the following two conditions: (i)The topology of a Branciari distance space is a Hausdorff topological space(ii)Any Branciari distance is continuous in each coordinates

However, it is known that the above two conditions are not correct (see [14, 15]).

Sarma et al. [15] and Samet [14] (see also [3, 4, 8, 18]) show that Branciari distances have the following topological disadvantages.

(B1) A Branciari distance does not need to be continuous in each coordinates

(B2) A convergent sequence in Branciari distance spaces does not need to be Cauchy

(B3) The topology of a Branciari distance space does not need to be a Hausdorff topological space

(B4) An open ball does not need to be an open set

Note that it follows from (B3) that the uniqueness of limits can not be guaranteed.

In despite of the above toplogical feature, the existence of fixed points can be investigated without additional conditions such as continuity of Branciari distances or/and Hausdorffness of the topology of Branciari distance spaces. This is why researchers are interested in Branciari distance spaces.

Let be a function.

Consider the following conditions:

(1) is nondecreasing, i.e., , whenever

(2) For any sequence of points in

(3) There exist and such that

(4) is continuous

(5) is strictly increasing

(6) for all with

(7) For any finite sequence

Jleli and Samet [19] introduced the notion of -contractions and generalized the Banach contraction principle in the setting of Branciari distance spaces, where is a function such that (1), (2), and (3) are satisfied.

Since then, Jleli et al. [7] obtained a generalization of result of [19] with conditions (1), (2), (3), and (4). Arshad et al. [20] extended the result of [19] by using the notion of -orbital admissible mapping with conditions (1), (2), and (3). Also, Ahmad et al. [21] extended the result of Jleli and Samet [19] to metric spaces by using conditions (1), (2), and (4). Durmaz and Altun [22] obtained a generalization of the result of Klim and Wardowski [23] by defining the concept of set-valued -contractions with control function satisfying (1), (2), (3), and (6). Abdeljawad et al. [17] introduced the concept of extended Branciari -distance, and they extended the result of Jleli and Samet [19] to extended Branciari -distance spaces with control function satisfying (2) and (3). Cho [24] introduced the notion of generalized set-valued weak -contractions in metric spaces and obtained fixed point results for such contractions with control function satisfying (1), (2), (4), and (7).

Recently, Cho [25] introduced the concept of -contractions in Branciari distance spaces and established a fixed point theorem for such contractions. He unified concepts of some contractions which exist in literature including -contractions.

Very recently, Saleh et al. [26] extended the result of Cho [25] by introducing the concepts of generalized -contractions in Branciari distance spaces. Aydi et al. [27] extended the result of Cho [25] to partial metric spaces.

In the paper, we introduce notions of set-valued -contractions and set-valued -contractions in Branciari distance spaces and prove the existence of fixed points for both type of contractions.

Khojasteh et al. [28] introduced the notion of -contractions by using the concept of simulation functions and unified the some existing metric fixed point results. The authors of [29–32] gave generalizations of simulation functions and obtained generalizations of results of [28]. Moreover, Demma et al. [33] and Yamaod and Sintunavarat [34] extended the results of [28] to -metric spaces by using the notion of -simulation functions and -simulation functions, respectively.

Let be a function. Consider the following conditions:

(1)

(2)

() , where is continuous and strictly increasing function with

(3) For any sequence

(4) For any sequence with

(5) If for any sequence where , then we have

(6) If for any sequence where , then we have

Let be a function.

Then, we say that (1) is called a simulation function in the sense of Khojasteh et al. [28] if and only if (), (), and () hold(2) is called a simulation function in the sense of Argoubi et al. [30] if and only if () and () hold(3) is called a simulation function in the sense of Roldán-López-de-Hierro et al. [32] if and only if (), (), and () hold(4) is called a simulation function in the sense of Isik et al. [31] if and only if () and () hold(5) is called a -simulation function [29] if and only if () and () hold(6) is called a -simulation function [33] if and only if () and () hold(7) is calld a -simulation function [34] if and only if () and () hold

Denote (resp., , ) by the family of all simulation functions in the sense of Khojasteh et al. (resp., all simulation functions in the sense of Argoubi et al., all -simulation functions).

Note that every simulation function in the sense of Argoubi et al. is a -simulation function. In fact, let . If we take , then .

Proposition 1. The following are satisfied. (1) and ([29])(2) and ([30])

Denote (resp., ) by the class of all simulations functions in the sense of Roldán-López-de-Hierro et al. (resp., in the sense of Isik et al.). Also, we denote by and the set of all -simulation functions and -simulation functions, respectively.

Example 1 (see [28, 32, 35, 36]). Let be a function defined as follows: (1), where (2), where is a function such that and (3), where is a function such that (4), where is a function such that or is continuous such that if and only if (5) where are continuous functions such that if and only if , and is increasing(6), where is upper semicontinuous with and if and only if (7), where is a function such that for each exists and

Then, .

Example 2. Let be a function defined by Then, () is satisfied.
Let be sequnces of points in such that Then, we have Thus, , but , because Hence,

Example 3 (see [31]). Let be a function defined as where .

Then, , but . Hence, Also, we know that and .

Note that . The following examples show that .

Example 4. Let be a function defined as where . Then, () and () are satisfied.

We show that () is not satisfied.

To show this, let be two sequences such that

We may assume that .

Then

Hence, () is not satisfied. Thus,

We now show that () is satisfied.

Let be two sequences such that

Then

Hence, () holds. Thus, Therefore,

Proposition 2. The following inclusion relations are satisfied. (1)(2)(3)

Proposition 3. If is decreasing in the first coordinate, then

Proof. Let . Then, (2) and (6) hold.
Assume that is decreasing in the first coordinate.
We show that (5) holds.
Let be two sequences such that where .
From (19), we infer that which implies It follows from (19) that Thus, we obtain By applying (21) and (23) to (6), we have Since is decreasing in the first coordinate, Hence, it follows from (19) and (25) that (5) is satisfied. Thus, .

Note that simulation functions given in Example 1 are -simulation and decreasing functions in the first coordinate. Hence, (see also [33]).

For more details and examples of simulation functions, we refer to [28, 33, 34, 36–40], and for -simulation functions, we refer to [29, 41, 42].

Now, we recall the concept of -simulation function and give the definition of -simulation function.

Let be a function. Consider the following conditions:

(1) is a decreasing function on the first coordinate

(2)

(3)

(4)

(5) For any sequence with

Let be a function.

Then, we say that (1) is -simulation function [25] if and only if it satisfies conditions (), (), and ()(2) is-simulation function if and only if it satisfies conditions (), (), (), (), and ()

Denote by the family of all -simulation functions, and by the class of all -simulation functions.

Note that , and .

Example 5 (see [25]). Let be functions defined as follows, respectively: (1) where (2) where is nondecreasing and lower semicontinuous such that , where .

Then, .

Example 6. Let be a function defined as follows: (1) where are continuous functions such that if and only if , and is an increasing function(2), where is upper semicontinuous with and if and only if (3), where is a function such that for each exists and , and Then, , and .
Note that if is satisfied condition then .

We recall the following definitions which are in [1].

Let be a Branciari distance space, be a sequence, and .

Then, we say that (1) is convergent to (denoted by ) if and only if (2) is Cauchy if and only if (3) is complete if and only if every Cauchy sequence in is convergent to some point in

Let be a Branciari distance space.

We denote by the class of nonempty closed subsets of . Let be the Hausdorff distance on , i.e., for all , where is the distance from the point to the subset .

For , let

Then, we have for all

Lemma 4 (see [43]). Let be a Branciari distance space, be a Cauchy sequence, and . If there exists a positive integer such that (1)(2)(3)(4)then .

Lemma 5. If is a Branciari distance space, then , where is the class of nonempty compact subsets of .

Proof. Let , and let be a sequence such that It follows from compactness of that there exists a convergent subsequence of .
Let Since from Lemma 4, Hence, .

Lemma 6. Let be a Branciari distance space, and let .
If and , then there exists such that .

Proof. Let .
It follows from definition of infimum that there exists such that . Hence, .

2. Fixed Point Theorems

We denote by the class of all functions such that conditions (2), (4), and (5) hold.

Let be a Branciari distance space.

A set-valued mapping , where is the family of all nonempty subsets of , is called set-valued -contraction with respect to if and only if for all with , and for all , there exists with such that where .

Now, we prove our main result.

Theorem 7. Let be a complete Branciari distance space, and let be a set-valued -contraction with respect to .

Then, has a fixed point.

Proof. Let be a point, and let be such that .
From (33), there exists with such that which implies and so Again, from (33), there exists with such that which implies and so Inductively, we can find a sequence such that, Since is a decreasing sequence, there exists such that We now show that .
Assume that .
Then, it follows from that Let and
Then, and It follows from (5) that which is a contradiction.
Thus, we have and so We now show that is a Cauchy sequence.
On the contrary, assume that is not a Cauchy sequence.
Then, there exists an for which we can find subsequences and of such that is the smallest index for which It follows from (47) and condition (d3) that Letting in above inequality, we have From (33), there exists with such that which implies So Taking limit supremum in above inequality and using (49), we have We deduce that Taking limit infimum in above inequality and using (53), we have It follows from (53) and (55) that Let Then,
It follows from (49), (56), (2), and (4) that It follows from (5) that which is a contradiction.
Thus, is a Cauchy sequence.
Since is complete, there exists a point such that It follows from (33) that there exists with such that which implies and hence Thus, we have Since Because and , .

We give an example to illustrate Theorem 7.

Example 1. Let and define as follows:

Then, is a complete Branciari distance space but not a metric space (see [9]).

Define a map by and a function by

We now show that is a set valued -contraction with respect to , where .

We consider the following cases.

Case 1. and .
For , there exists with such that and for , there exists with such that

Case 2. and .
For , there exists with such that and for , there exists with such that

Case 3. and .
For , there exists with such that and for , there exists with such that

Case 4. and .
For , there exists with such that and for , there exists with such that

Case 5. and .
For , there exists with such that and for , there exists with such that

Case 6. and .
For , there exists with such that and for , there exists with such that

Hence, is a set valued -contraction with respect to

Thus, all hypotheses of Theorem 7 are satisfied, and has fixed points 2, 3, and .

Note that the Nadler’s fixed point theorem does not hold. If and , then so which is a contradiction.

From Theorem 7, we have the following corollary.

Corollary 8. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where and

Then, has a fixed point.

Proof. Since for each . Hence, we have It follows from () that there exists the inverse function of , and so from (85), we have From Lemma 6, there exists such that Hence Thus, we have

Thus, condition (33) holds and from Theorem 7, has a fixed point.

Corollary 9. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where and

Then, has a fixed point.

Proof. It follows from () and (90) that which implies

Thus, from Corollary 8, has a fixed point.

3. Consequence

We have fixed point results by applying simulation functions given in Example 5 and Example 6 to Theorem 7.

By taking in Theorem 7, we obtain Corollary 10.

Corollary 10. Let be a complete Branciari distance space, and let be a set valued map.
Suppose that for all with and , there exists with such that where and .

Then, has a fixed point.

Remark 11. Corollary 10 is a generalization of Theorem 7 of [19] and Theorem 2.2 of [21] without condition () to set-valued maps.

Corollary 12. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where .

Then, has a fixed point.

Proof. Since , we have that, for each , and so Hence, there exists such that By Corollary 10, has a fixed point.

Corollary 13. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where .

Then, has a fixed point.

Remark 14. Corollary 13 is a generalization of Theorem 2 of [44] to Branciari distance spaces with conditions:

By taking in Theorem 7, we obtain Corollary 15.

Corollary 15. Let be a complete Branciari distance space, and let be a set valued map.
Suppose that for all and , there exists with such that where and is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

Remark 16. Corollary 15 is a generalization of Corollary 2.7 [24] to Branciari distance spaces without condition ().

Corollary 17. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where and is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

Proof. Suppose that condition (101) satisfied.
Since , we have that, for each , Thus and so there exists such that By Corollary 15, has a fixed point.

Corollary 18. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where and is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

Corollary 19. Let be a complete Branciari distance space, and let be a set valued map.
Suppose that for all with and for all , there exists with such that where is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

By the same way as proof of Corollary 15 in [24], we can prove Corollary 19.

Corollary 20. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

Corollary 21. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

By taking , where in Corollary 15, we obtain the following result.

Corollary 22. Let be a complete Branciari distance space, and let be a set valued map.
Suppose that for all and , there exists with such that where and is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

Corollary 23. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where and is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

Corollary 24. Let be a complete Branciari distance space, and let be a set valued map such that for all with and where and is nondecreasing and lower semicontinuous such that .

Then, has a fixed point.

4. Conclusion

One can unify and merge some existing fixed point theorems by using -simulation functions and -simulation functions in Branciari distance spaces. One can obtain some concequence of the main theorem by applying -simulation functions and -simulation functions given in Example 1 and Example 2. Further, one can derive all the results of the paper in the setting of metric spaces.

5. Suggestion

We suggest that the -simulation function can be extended in a similar way to the one in which the simulation function is extended to the -simulation function. The main theorem can be extended and generalized to -metric space, Branciari -distance space, and extended Branciari -distance space using certain extended simulation functions, and the existing fixed point theorem can be interpreted.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the Hanseo University.

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Copyright © 2021 Seong-Hoon Cho. 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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