Journal of Mathematics
Volume 2021 (2021), Article ID 5542994, 6 pages
https://doi.org/10.1155/2021/5542994

Research Article

Condensing Mappings and Best Proximity Point Results

Sarah O. Alshehri,1,2 Hamed H. Alsulami,¹ and Naseer Shahzad¹

¹King Abdulaziz University, Department of Mathematics, P.O. Box 80203, Jeddah 2159, Saudi Arabia
²University of Jeddah, College of Science, Department of Mathematics, Jeddah, Saudi Arabia

Correspondence should be addressed to Naseer Shahzad; nshahzad@kau.edu.sa

Received 22 January 2021; Revised 8 February 2021; Accepted 23 February 2021; Published 21 April 2021

Academic Editor: Xiaolong Qin

Copyright © 2021 Sarah O. Alshehri 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.

Abstract.

Best proximity pair results are proved for noncyclic relatively u-continuous condensing mappings. In addition, best proximity points of upper semicontinuous mappings are obtained which are also fixed points of noncyclic relatively u-continuous condensing mappings. It is shown that relative u-continuity of is a necessary condition that cannot be omitted. Some examples are given to support our results.

1. Introduction

The concept of measure of noncompactness was first introduced by Kuratowski [1]. However, the interest in the concept was revived in 1955 when Darbo [2] proved a generalization of Schauder’s fixed point theorem using this concept. Sadovskii [3], in 1967, defined condensing mappings and extended Darbo’s theorem. Since then a lot of work has been done using this concept, and several interesting results have appeared, see, for instance, [4–9].

Let be a nonempty pair in a Banach space (that is, both and are nonempty sets). A mapping is called noncyclic provided and . If there exists which satisfies , , and , then we say that the noncyclic mapping has a best proximity pair. For a multivalued nonself mapping , a point is called a fixed point of if . The necessary condition for the existence of a fixed point for such is . If , then for each . Best proximity point theorems provide sufficient conditions for the existence of at least one solution for the minimization problem, . If , the point is called a best proximity point of . The existence results of best proximity points for multivalued mappings were obtained in [10–14] and [15]. Best proximity point theorems for relatively nonexpansive and relatively u-continuous were established by Elderd et al. in [16, 17] and by Markin and Shahzad in [18]. In recent years, the topics of best proximity points of single-valued and multivalued mappings have attracted the attention of many researchers, see, for example, the work in [6, 7, 19, 20] and the references cited therein. In this paper, we prove best proximity pair theorems for noncyclic relatively u-continuous condensing mappings. In addition, we obtain best proximity points of upper semicontinuous mappings which are fixed points of noncyclic relatively u-continuous condensing mappings. Also, we give examples to support our results and show by giving an example that relative u-continuity of is a necessary condition that cannot be omitted. Our results extend and complement results of [6, 7, 11].

2. Preliminaries

In this section, we present some notions and known results which will be used in the sequel.

Definition 1. Let be a bounded set in a metric space . The Kuratowski noncompactness measure (or simply, measure of noncompactness) is defined as follows:

Theorem 1. Let be a metric space. Then, for any nonempty bounded pair in (that is, both and are nonempty and bounded sets), the following hold:

(1) if and only if is relatively compact(2) implies (3), where denotes the closure of (4)(5)For a normed space :

(i)(ii)(iii), for any number (iv), where represents the convex hull of

Theorem 2. Let be a decreasing sequence of nonempty closed subsets of a complete metric space If as , then .

For more details about the measure of noncompactness, see [4].

Definition 2. Let be a nonempty pair in Banach space and a mapping. Then, is said to be noncyclic relatively u-continuous. If is noncyclic and for each , there is such that

for each and .

Definition 3. Let be a nonempty convex pair in Banach space . A mapping is said to be affine if for each with and (respectively, ),

Definition 4. Let be a nonempty pair in Banach space and a multivalued mapping on , then is said to be upper semicontinuous if for each closed subset in , is closed in .

Lemma 1. (see [21]). Let be a nonempty, convex, and compact subset of a Banach space . If can be written as a finite composition of upper semicontinuous multivalued mappings of nonempty, compact, and convex values, then has a fixed point.

Definition 5. Let be a noncyclic relatively u-continuous mapping and be an upper semicontinuous multivalued mapping (here, denotes the collection of all nonempty, convex, and compact subsets of ), then by the commutativity of and , we mean that holds for each .
Given , a nonempty pair in Banach space, its proximal pair is given by

Moreover, if is a nonempty, convex, and compact pair in , then is also a nonempty, convex, and compact pair.

Definition 6. Let be a normed space. For a nonempty subset of , the metric projection operator is given by

For a nonempty, convex, and compact subset of a strictly convex Banach space, is a single-valued mapping. Furthermore, for a nonempty, convex, and compact subset of a Banach space , is upper semicontinuous with nonempty, convex, and compact values.

Lemma 2. (see [11]). Let be a nonempty, convex, and compact pair in a strictly convex Banach space . Let be a noncyclic relatively u-continuous and be a mapping given by

Then, for each .

Theorem 3. (see [18]). Let be a nonempty, convex, and compact pair in a strictly convex Banach space . If is a noncyclic relatively u-continuous mapping. Then, has best proximity pair.

In [6], Gabeleh and Markin introduced the class of noncyclic condensing operators.

Recall that a nonempty pair in a Banach space is called proximinal if and .

Definition 7. Let be a nonempty convex pair in a strictly convex Banach space . A mapping is called noncyclic condensing operator provided that, for any nonempty, bounded, closed, convex, proximinal, and -invariant pair with , there exists such that

Lemma 3. (see [11]). Let be a nonempty, convex, and compact pair in a strictly convex Banach space . If is a noncyclic relatively u-continuous mapping, then is continuous on and .

3. Main Results

Throughout this paper, we will assume that is a strictly convex Banach space and is the measure of noncompactness on .

Remark 1. Let be condensing in the sense of Definition 7 with . Then, for any bounded subset of , satisfies

To see this, in (7), set and . Since , then

Theorem 4. Let be a nonempty, convex, and closed pair in such that is bounded and is nonempty. Suppose is a noncyclic relatively u-continuous, affine, and condensing mapping. Then, there exists such that , and . Moreover, if is an upper semicontinuous multivalued mapping, and commute, and for each , there exists such that and .

Proof. We follow [6, 11]. Clearly, is a nonempty, closed, convex, proximinal, and -invariant pair. Let be such that . Suppose is a family of nonempty, closed, convex, proximinal, and -invariant pairs such that , then is nonempty. Set , , and . So, and . Furthermore, and , that is, is noncyclic on . Also, for , , where for all with and , . Since is proximinal, there is such that , for each . Set . Then, . Moreover,

So, one can conclude that . Similarly, , and hence, , that is, and . Notice that

But , so . We conclude that is a nonempty, compact, and convex pair with . By Theorem 3, there exists such that , and .
Now, let , , and . By the above part, the pair is nonempty. Also, it is a convex pair. Indeed, for , with and (respectively, ):

and by convexity of (respectively, ), we conclude that (respectively, ). Furthermore, since is condensing,

which implies that the pair is compact.
For and , we have

that is, is invariant under . So, by the invariance of under , is invariant under . So, in view of Remark 3.1, Darbo’s fixed point theorem guarantees the existence of a fixed point for the continuous mapping : . Thus, , for . Define by , for each . Then, is an upper semicontinuous multivalued mapping with nonempty, compact, and convex values. Moreover, : is well-defined. Indeed, for , there is such that . So,

By relative u-continuity of , we conclude that . Thus, and , by (15), . Then, . Consider , by Lemma 1, there is such that , that is, and . So, there is such that . We conclude that . But since , there is such that . Thus,

Hence, .

Example 1. Consider the Hilbert space over with the basis (the canonical basis) and let

Then, be a nonempty, convex, and closed pair of such that is bounded. Furthermore, and

Define the mapping by , for each Then, is a noncyclic relatively u-continuous, affine, and condensing mapping. Now, define by ; then, is an upper semicontinuous multivalued mapping, and commute, and for each , . For , we have and .

Example 2. Consider the Hilbert space over with the basis and let

Then, be a nonempty, convex, and closed pair of such that is bounded with and

Define the mapping by for each . Then, is a noncyclic relatively u-continuous, affine, and condensing mapping. Furthermore, for , we have , , and . Now, let given by , then is an upper semicontinuous multivalued mapping, and commute, and for each , . For , we have and .

Remark 2. The relative u-continuity of is necessary in Theorem 4.
To see this, consider the Hilbert space over with the basis and let , . Then, is a nonempty, convex, and closed pair in such that is bounded. Obviously, and

Define the mapping by

for . Then, is a noncyclic, affine, and condensing mapping. Let given by . Then, is an upper semicontinuous multivalued mapping, and commute, and for each , . Here, is the only fixed point of in , but . Note that for all but .
The following corollary follows immediately from Theorem 4.

Corollary 1. Let be a nonempty, convex, and closed pair in such that is bounded and is nonempty. Suppose is a continuous, affine, and condensing mapping. If is an upper semicontinuous multivalued mapping, and commute, and then there is which satisfies .

Theorem 5. Let be a nonempty, convex, and closed pair in such that is bounded and is nonempty. If are commuting, noncyclic relatively u-continuous, affine, and condensing mappings, then there exists such that , , and .

Proof. Since is nonempty and by relative u-continuity of , for , there exists such that . Consequently, . That is, is invariant under . Thus, Darbo’s fixed point theorem guarantees that there is such that . Notice and so . Thus, , and thus, is compact. Furthermore, . So, is a continuous mapping on a compact convex set. By Schauder's fixed point theorem, there is such that , that is, . Let  in  be the unique closest point to . By relative u-continuity of and , we infer that, since , and . Hence, , .

Lemma 4. Let be a nonempty, convex, and closed pair in such that is bounded and is nonempty. Let be the collection of the commuting, noncyclic relatively u-continuous, affine, and condensing mappings on . Then, the mappings in have common fixed points and .

Proof. For each , consider , and defined previously. Then, is nonempty, compact, and convex. Let be a finite subcollection of . Assume , , and , for . Then, is a decreasing sequence of compact subsets of . Furthermore, for each . Indeed, for and each , then , and this implies that . Thus, is invariant under . By Schauder's fixed point theorem, we get that . Now, for each and , pick :

that is, . So, is continuous on , and then there is such that . Therefore, . By Theorem 2, . Hence, . Similarly, we can show that .

Theorem 6. Let be a nonempty, convex, and closed pair in such that is bounded and is nonempty. Let be the collection of the commuting, noncyclic, relatively u-continuous, affine, and condensing mappings on . Then, there is such that, for each , , and .

Proof. Based on the previous lemma, the mappings in have a fixed point in common , that is, , for each . Let be the unique closest point to . By relative u-continuity of , since ,

Hence, .

Theorem 7. Let be a nonempty, convex, and closed pair in such that is bounded and is nonempty. Let be the collection of the commuting, noncyclic relatively u-continuous, affine, and condensing mappings on . If is an upper semicontinuous multivalued mapping such that, for each : . If commutes with , then there exists such that

Proof. By Lemma 4, is a nonempty compact convex pair. Also, in view to the proof of Theorem 4, for and for each , we have and are invariant under . So, .
Define by , for . Then, is an upper semicontinuous multivalued mapping with nonempty, compact, and convex values. Moreover, : is well-defined. Indeed, for , there exists such that . So,

By relative u-continuity of , one can conclude that . Thus, and , and by (26), . Thus, . Note that , and by Lemma 1, there is such that , that is, for , we have and . So, there is such that . We infer that . But , then there is such that . Then,

Hence, .

Example 3. Let over with the basis and let

Then, be a nonempty, convex, and closed pair in such that is bounded. Furthermore, and

Consider given by

for each . Then, are noncyclic, affine, and condensing mappings. Furthermore, and commute.
Define by , then is an upper semicontinuous multivalued mapping that commutes with and and satisfies that, for each : . For and and . Furthermore, and .

4. Conclusion

We have proved some best proximity pair theorems for noncyclic relatively u-continuous and condensing mappings. We have also obtained best proximity points of upper semicontinuous mappings which are fixed points of noncyclic relatively u-continuous condensing mappings. Moreover, we have given some examples to support our results. It has been shown that relative u-continuity of is a necessary condition that cannot be omitted. We have extended recent results of [6, 11].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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