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.