#### Abstract

In this manuscript, we present further extensions of the best approximation theorem in hyperconvex spaces obtained by Khamsi.

#### 1. Introduction

The importance of fixed point theory emerges from the fact that it gives a unified approach and constitutes an essential tool in resolving problems which are not necessarily linear. A variant number of problems can be expressed as nonlinear equations of the form , where is a self-mapping, see [1–6]. Nevertheless, an equation of the type does not necessarily have a solution if is a non-self-mapping. Let be a metric space. Here, we search an optimal solution in the sense that is minimum. That is, we resolve a problem of searching an element so that is in best proximity to in some sense. A best proximity point result presents the condition under which the optimisation problem, i.e., , possesses a solution. The element is called the best proximity point of if . Observe that the best proximity point is reduced to a fixed point if is a self-mapping. For more related works, see [7–11].

The concept of a hyperconvex space was initiated in [12] by Aronszajn and Panitchpakdi. In hyperconvex spaces, many results on coincidence points, fixed points, best approximations, and coupled best approximations are obtained. See, for example, [13–23]. For more details on the best approximation and KKM principle, we refer readers to the classic book [24]. Due to Aronszajn and Panitchpakdi [12], the definition of a hyperconvex metric space is as follows.

A metric space is named to be a hyperconvex space if for any set of points of and for any family of nonnegative real numbers with , we have , where represents the closed ball with center and radius .

Suppose that a subset of is bounded. Consider, i.e., iff is an intersection of closed balls. Here, is named to be an admissible subset of . In the linear case, the notation describes the convex hull of . Note that is always defined and is in . If is a hyperconvex space, then it is complete [17].

Let be a nonempty set. We denote by and the set of all nonempty finite subsets of and the set of all nonempty subsets of , respectively. Let and be topological spaces with and . Given a set-valued map , the image of under is the set and the inverse image of under is . The map is lower (upper) semicontinuous if, for each open (closed) set , is open (closed) set in . The map is continuous if is both upper semicontinuous and lower semicontinuous.

Let be an admissible subset of . The set-valued map is named to be quasiconvex if for any admissible set of , is also admissible (see [15]). Observe that if is a quasiconvex map, then the set is admissible for each closed ball Note that, if , then (see [25]), where

Khamsi [17] presented a hyperconvex version of the KKM principle in hyperconvex spaces. As an application, he gave a hyperconvex version of the best approximation result of Fan for continuous single-valued maps. In this manuscript, we ensure the existence of a solution of a best approximation problem for set-valued maps and : for a set , find so that

Let be a metric space and . A multivalued map is said to be a KKM map if

Theorem 1 (see [17], KKM principle). *Let be a hyperconvex space, be an arbitrary subset of , and be a KKM map so that is compact for some and is closed for any . Then, *

Theorem 2 (see [17], best approximation). *Let be a hyperconvex space and be compact. Given a continuous map , there is so that
**This result has been generalized to other forms of maps. For more details, see [13–16, 18, 22].**Now, we give the definition of a measure of noncompactness of Pasicki [26].*

*Definition 3 (see [26]). *Let be a metric space. An arbitrary function is named to be a measure of noncompactness on if
(1) iff is a totally bounded set(2)for , implies (3)for all and ,

*Definition 4 (see [19]). *Let be a metric space, be a measure of noncompactness on , and . The map is condensing if for any , there is so that A condensing map is a condensing KKM map if it is a KKM map.

In this paper, we present further extensions of the best approximation result (Theorem 2) obtained by Khamsi. Finally, we present a problem related to the Schauder conjecture.

#### 2. Results

The following result generalizes Theorem 1. The proof is essentially the same as Theorem 3.1 in [19].

Theorem 5. *Let be a measure of noncompactness on a hyperconvex space, be an arbitrary subset of , and be a condensing KKM map such that each is closed, then is nonempty and compact set.*

We introduce the concept of a -quasiconvex map in hyperconvex spaces.

*Definition 6. *Let be a hyperconvex space and . A set-valued map is said to be a -quasiconvex if for any and ,
where is a continuous monotone increasing function so that for any and .

Let be the identity map and be the identity set-valued map so that for any . Note that a quasiconvex map is -quasiconvex and is -quasiconvex in hyperconvex spaces.

If is a linear metric space, then may not be -quasiconvex.

*Example 1. *Denote by the linear space of real sequences. The Fréchet metric for is given as follows (see [27]):

Let , , and then, we obtain
Namely, for and , we have that , and

Note that for any and , one writes
So, in the linear metric space , the map is not -quasi-convex and is -quasiconvex, where

Theorem 7. *Let be a hyperconvex space, be closed, be a measure of noncompactness on , be continuous maps with compact values and be -quasi-convex. If for any there is so that
then, there is so that
*

*Proof. *Define by
From condition , we obtain
so is a nonempty set for all , because

From condition (9), one asserts that is a condensing map.

Since and are continuous maps and is a continuous function, we get that is a closed set for all .

The map is KKM. Indeed, suppose that for some ,
Then, there is so that for any . Thus,
The function is increasing, so
Let be so that
Then,
and hence,
Since is -quasiconvex, one gets from (18),
Since , we deduce that
Consequently,
which is not possible. Therefore, must be a KKM map. Now, form Theorem 5, there is so that
Therefore,

Taking the set to be compact, we state from Theorem 7 the next results.

Theorem 8. *Let be a hyperconvex space, be compact, be continuous maps with compact values, and be -quasiconvex. Then, there is so that
*

Theorem 9. *Let be a hyperconvex space, be compact, be continuous maps with compact values, and be -quasiconvex onto a map. Then, there is so that *

Theorem 10. *Let be a hyperconvex space, be compact, and be continuous maps with compact values. If there is such that for any and ,
then there is so that
*

Theorem 11. *Let be a hyperconvex space, be compact, and be a continuous map with compact values so that for any . Then, there is so that *

*Remark 12. *If is a single-valued map, we deduce from Theorem 11 the main theorem of Park [22] (Theorem 5. (iv)).

Theorem 13 (see [18], Theorem 3.2). *Let be a hyperconvex space, be compact, be continuous maps with compact values, and be quasiconvex. Then, there is so that
*

Theorem 14 (see [16], Theorem 2.9). *Let be a hyperconvex space, be compact, and be a continuous map with compact values. Then, there is so that
**Finally, we give the following problems.*

*Problem 15. *Does for every linear metric space and for a compact subset of , there is a map so that the identity map (i.e., ) is -quasiconvex?

In other words, does for a linear metric space, there is a continuous monotone increasing function so that for any , and for any and ?

In 2001, Cauty [28] obtained the affirmative solution to the Schauder conjecture as follows:

*Problem 16. *Let be a compact convex subset of a (metrizable) topological vector space. Does any continuous map have a fixed point?

*Remark 17. *Note that if Problem 15 is affirmative, then Problem 16 is affirmative.

#### Data Availability

Data sharing is not applicable to this article as no data set was generated or analyzed during the current study.

#### Conflicts of Interest

The authors declare no conflict of interest.

#### Authors’ Contributions

All authors have read and agreed to the published version of the manuscript.

#### Acknowledgments

The authors are thankful to the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research.