Common Best Proximity Point Theorems for Generalized Proximal Weakly Contractive Mappings in b-Metric Space

Gebru, Yohannes; Negede, Yitages

doi:https://doi.org/10.1155/2022/5982401

Abstract and Applied Analysis

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

Research Article | Open Access

Volume 2022 | Article ID 5982401 | https://doi.org/10.1155/2022/5982401

Common Best Proximity Point Theorems for Generalized Proximal Weakly Contractive Mappings in b-Metric Space

Yohannes Gebru¹and Yitages Negede¹

Academic Editor: Alberto Fiorenza

Received11 Feb 2022

Accepted05 Aug 2022

Published24 Aug 2022

Abstract

In this paper, common best proximity point theorems for weakly contractive mapping in b-metric spaces in the cases of nonself-mappings are proved; we introduced the notion of generalized proximal weakly contractive mappings in b-metric spaces and proved the existence and uniqueness of common best proximity point for these mappings in complete b-metric spaces. We also included some supporting examples that our finding is more generalized with the references we used.

1. Introduction

The metric fixed point theory gained impetus due to its wide range of applicability to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, and so forth. The first fixed point theorem was given by Brouwer [1], but the credit of making concept useful and popular goes to polish mathematician, Banach [2] who proved the famous contraction mapping theorem in 1922 in the setting of metric space. This principle guarantees the existence and uniqueness of fixed point of certain self-maps of metric spaces and provides a constructive method to find those fixed points. This principle includes different directions in different spaces adopted by mathematicians for example metric spaces, G-metric spaces, partial metric spaces, and cone metric spaces.

A classical best approximation theorem was introduced by Fan [3], which states that “if is a non-empty compact convex subset of a Hausdorff locally convex topological vector space and is a continuous mapping, then there exists an element such that .” Afterwards, Prolla [4], Reich [5], and Sehgal and Singh [6] have derived extensions of Fan Theorem in many directions. The common fixed point theorem insists to the authors to investigation on common best proximity point theorem for nonself-mappings. The common best proximity point theorem assures a common optimal solution at which both the real valued multiobjective functions and attain the global minimal value . A number of authors have improved, generalized, and extended this basic result either by defining a new contractive mapping in the context of a complete metric space or extend best proximity results from fixed point theory (see [7–12]).

Definition 1. Let be a nonempty set and a self-map. A point is said to be fixed point of if .

Example 2. Let and defined by , for each .
, and we get , which is a fixed point of .

Definition 3 (see [13]). A function is called an altering distance function if the following properties are satisfied: (i) is monotone increasing and continuous(ii) if and only if

Example 4. Define by .
, which shows is nondecreasing, satisfies that , and is continuous.

Definition 5 (see [14]). Let be a nonempty set, and a mapping is said to be metric if and only if, for all , the following conditions are satisfied: (i) if and only if and if (ii)(iii)

Example 6. Let ; then, that means , for all which is a metric space.

Definition 7 (see [2]). Let be a metric space and be a self-map; then, is said to be a contraction mapping if there exists a constant , such that , .

Example 8. Let , , and a mapping defined by , .
Then, , which implies that and , for all . Therefore, is contraction mapping.

Definition 9 (see [15]). Let be a metric space. The mapping is said to be contractive mapping if

Example 10. Let and , and a mapping defined by , .
Then, , which implies that with .
Therefore, is contractive mapping.

Definition 11 (see[16]). Let be a metric space and , a mapping is said to be weakly contractive if where is altering function.

Remark 12. If with and , a weak contraction reduces to a contraction.

Example 13. Let be endowed by , and let define by for each, .
Define by .
Claim: is weakly contractive.
which shows is nondecreasing and satisfies that and is continuous.
Then, , for all .
So is weakly contractive.

Definition 14 (see [17]). Let be a nonempty set and be a given real number. A mapping is said to be a b-metric if and only if, for all , the following conditions are satisfied: (i) if and only if and if (ii)(iii)

Remark 15 (see [18]). We should note that a b-metric space with is a metric space. We can find several examples of b-metric spaces which are not metric spaces.

Example 16 (see [19]). Let be a metric space, and , where is a real number. Then, is a b-metric space with

Definition 17 (see [20]). Let be a b-metric space with parameter . Then, a sequence {} in is said to be (i)b-convergent if and only if there exists such that as (ii)a b-Cauchy sequence if and only if as , for all In addition, a b-metric space is called complete if and only if each Cauchy sequence in this space is b-convergent.

Example 18. Let and ; then, the space is a complete b-metric space.

Definition 19 (see [21]). Let and be two self-mappings on a nonempty set . If , for some , then is said to be the coincidence point of and , where is called the point of coincidence of and . Let denote the set of all coincidence points of and .

Definition 20 (see [21]). Let and be two self-mappings defined on a nonempty set . Then, and are said to be weakly compatible if they commute at every coincidence point, that is, , for every .

Example 21. (i) defined by and , . In this example, and have coincidence point at , and but and are not weakly compatible(ii) equipped with the usual metric space Define by the following: This example shows, for any , . Therefore, and are weakly compatible maps on .
In this study, motivated and inspired by Yan Hao and Hongyan Guan [22], we introduce the notion of generalized proximal weakly contractive mappings in b-metric spaces and prove a common best proximity point theorem for generalized proximal weakly contractive mapping defined on complete b-metric spaces.

2. Preliminaries

Definition 22 (see [23]). Let and be nonempty subsets of a metric space . We denote by and the following sets: where is the distance between and .

Definition 23 (see [24]). Let be nonempty subset of metric space . Given a nonself-mapping , then an element is called best proximity point of the mapping if

Definition 24 (see [25]). Let be nonself-mappings. An element is said to be a common best proximity point of the pair if this condition is satisfied:

Definition 25 (see [26]). Let be mappings. A pair is said to commute proximally if for each ,

Lemma 26 (see [19]). Let be a b-metric space with parameter . Assume that and are b-convergent to and , respectively. Then, we have the following: In particular, if , then we have .
Moreover, for each , we have the following:

Definition 27 (see [22]). A function , where is a b-metric space, is called lower semicontinuous if for all , and a sequence is b-convergent to , and we have Consider the following:
such that is continuous and nondecreasing function}.
Also, we denote such that is nondecreasing and lower semicontinuous, and Hao and Guan [22] proved the following common fixed point results for generalized weakly contractive mapping in b-metric spaces:

Theorem 28 (see [22]). Let be a complete b-metric space with parameter , and let be given self-mappings satisfying as injective and where is closed. Suppose is a lower semicontinuous function and is a constant. If there are functions such that where then and have a unique coincidence point in . Moreover, and have a unique common fixed point provided that and are weakly compatible.

3. Result and Discussion

Definition 29. Let be a b-metric space and and be two nonempty subset of a b-metric space with parameter and is a constant. A pair of map is said to be a generalized proximal weakly contractive mapping, if for all , then where ,, and is a lower semicontinuous function.

Theorem 30. Let be a pair of nonempty subsets of a complete b-metric space , and assume that and are nonempty such that is closed. Define a pair of mapping satisfying the following conditions: (i) and are generalized proximal weakly contractive mapping(ii), and (iii) and are continuous mapping(iv) and are commute proximityThen, and have a unique common best proximity point.

Proof. We prove the existence of common best proximity point.
Let . Since , there exists such that Also, . Since , there exists such that Continuing this process in a similar fashion, obtain the sequence and in such that for each .
Since and is nonempty set, there exists such that for all
Further, we obtain that for all
Our first goal is to show that , for some .
Suppose that , for some , by (2) and (3), we get that Since and commute proximally, , and so we are done.
Assume that , for all . From (3), note that for all . Since a pair is generalized proximal weakly contractive map with , , we have that where If , for some , in view of (5)–(8), we have which implies .
Hence, , a contradiction.
Thus, we have It follows from (10) that {} is a nonincreasing sequence, and so there exists such that By (5), (11), and (12), we can obtain Now assume that . Taking the upper limit as in (15), we have which implies that , a contradiction. Thus, we have It follows that Now, we claim that {} is a Cauchy sequence.
Suppose contradiction, that is, {}, is not a Cauchy sequence. Then, there exists such that there are subsequences {} and {} of {} so that for all with , we obtain By triangular inequality in b-metric space and (19) and (20), we have Taking the upper limit as in the above inequality, we have Also, we have Then, by taking the upper limit as in (42), we have which implies It is from By taking the upper limit as in (43), we have In similar fashion by taking the lower limit, we can obtain Since and satisfy equations (26) and (27), we obtain that for each . Since and are generalized proximal weakly contractive mapping with and , we have From the definition, we have Taking the upper limit as , we obtain Also, we have By taking the lower limit as , we have By applying generalized proximal weakly contractive mapping with and , we have which implies that a contradiction to (53). Hence, the sequence is Cauchy. Since be a closed subset of the complete b-metric space , there exists such that By the definition of , we have Consider, by (2) and (3), that Since and are commute proximally, for all . By continuity of and , Now, we claim the existence of common best proximity point of and . Since , there exists such that By the assumption that and commute proximally, .
According to the assumption that , there exists such that Next, we claim that . Suppose that , that is, . By applying generalized proximal weakly contractive mapping with and , we observe that where From (63)–(65), we have which implies This contradicts the assumption . Thus, . Hence, That is, the element is a common best proximity point of and .
Finally, we have to show that the point is unique.
Let be another common best proximity point of and . Then, Since and are generalized proximal weakly contractive mapping, we obtain that where Now, from (71) and (72), we have By the properties of and from (73), we have which contradict the supposition that . Thus, .
Therefore, and have a unique common best proximity point.
The proof is completed.

Example 31. Let and be defined by , for all and is a complete b-metric space with parameter .
Suppose Let be the mapping defined by , defined by , and define a mapping with , and . Clearly, is lower semicontinuous function, and is continuous and nondecreasing function. Further, is nondecreasing and lower semicontinuous, and .
, which implies that and implies that .
Notice that and are continuous. Now, we check that and are generalized proximal weakly contractive mapping.
In fact, for all , we have
implies that , which implies that
; this implies , which implies that
implies that , implies that
implies that , implies that
Then, by applying the generalized proximal weakly contractive mapping definition with , we have the following: Now, we have According to above inequalities, we get that Hence, and are generalized proximal weakly contractive mappings.
Next, consider, by the definition of , that and ; thus, and . Additionally,
Now, it remains to show that and commute proximally. Let such that Consequently, , , and , where and . Thus, Therefore, Now, we claim that which implies .
Hence,
Therefore, and are commute proximally.
Finally, by Theorem 30, we can conclude that there is a unique common best proximity point of the pair . In fact, the point is the unique common best proximity point of .
To show this, there exists such that where is common best proximity point of and . Now find and this implies that implying that From this, we get Hence, , and also, from we have implying that From this, we get Hence, .
Therefore, the point is a unique common best proximity point of and .
If in Theorem 30, we can get the following result:

Corollary 32. Let be a pair of nonempty subsets of a complete b-metric space , and assume that and are nonempty such that is closed. Define a pair of mapping satisfying the following conditions: (i)For all ,then where (ii) and (iii) and are continuous mapping(iv) and are commute proximityThen, and have a unique common best proximity point.
If we consider the corresponding problem in the setting of metric space, that is, in Theorem 30, we can obtain the following:

Corollary 33. Let be a pair of nonempty subsets of a complete b-metric space and assume that and are nonempty such that is closed. Define a pair of mapping satisfying the following conditions:
(i) For all , then where is the same as Theorem 30, , , and is a lower semicontinuous function. (ii) and are continuous mapping
(iii) and are commute proximity
Then, and have a unique common best proximity point.

4. Conclusion

This study is concerned with the existence and uniqueness of common best proximity point for generalized proximal weakly contractive mapping in complete b-metric spaces, and in this study, we have defined the notion of generalized proximal weakly contractive mapping in b-metric spaces.

Data Availability

All datas are included.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Yohannes Gebru and Yitages Negede. 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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