# On Best Approximations in Hyperconvex Spaces

Authors: Reny George; Zoran D. Mitrović; Hassen Aydi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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


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## Body

## 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 formfu=u, where f is a self-mapping, see [1–6]. Nevertheless, an equation of the type fu=u does not necessarily have a solution if f is a non-self-mapping. Let X,d be a metric space. Here, we search an optimal solution in the sense that du,fu is minimum. That is, we resolve a problem of searching an element u∈X so that u is in best proximity to f in some sense. A best proximity point result presents the condition under which the optimisation problem, i.e., infu∈Adu,fu, possesses a solution. The element u is called the best proximity point of f:A⟶B if du,fu=dA,B=infda,b,a∈A,b∈B. Observe that the best proximity point is reduced to a fixed point if f 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 spaceΛ,ω is named to be a hyperconvex space if for any set of points ℓα of Λ and for any family of nonnegative real numbers rα with ωℓα,ℓβ≤rα+rβ, we have ∩αBℓα,rα≠∅, where Bℓ,r=j∈Λ:ωℓ,j≤r represents the closed ball with center ℓ∈Λ and radius r.Suppose that a subsetA of Λ is bounded. Consider,
(1)coA=∩B⊆Λ:BisaclosedballsothatA⊆B,Aℓ=A⊆Λ:A=coA,i.e., A∈Aℓ iff A is an intersection of closed balls. Here, A is named to be an admissible subset of Λ. In the linear case, the notation convA describes the convex hull of A. Note that coA is always defined and is in Aℓ. If Λ,ω is a hyperconvex space, then it is complete [17].LetΛ be a nonempty set. We denote by Λ and 2Λ the set of all nonempty finite subsets of Λ and the set of all nonempty subsets of Λ, respectively. Let Λ and Ω be topological spaces with A⊆Λ and B⊆Ω. Given a set-valued map F:Λ⟶2Ω, the image of A under F is the set FA=∪a∈AFa and the inverse image of B under F is F−B=ℓ∈Λ:Fℓ∩B≠∅. The map F is lower (upper) semicontinuous if, for each open (closed) set B⊆Ω, F−B is open (closed) set in Λ. The map F is continuous if F is both upper semicontinuous and lower semicontinuous.LetA be an admissible subset of Λ. The set-valued map F:A⟶2Λ is named to be quasiconvex if for any admissible set A of Λ, F−A is also admissible (see [15]). Observe that if F is a quasiconvex map, then the set F−Bℓ,r is admissible for each closed ball Bℓ,r. Note that, if A∈Aℓ, then A+r∈Aℓ (see [25]), where A+r=∪a∈ABa,r.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 F and G: for a set K, find ℓ0∈K so that
(2)ωGℓ0,Fℓ0≤ωGℓ,Fℓ0forallℓ∈K.LetΛ be a metric space and K⊂Λ. A multivalued map H:K⟶2Λ is said to be a KKM map if
(3)coA⊂HAforanyA∈K.Theorem 1 (see [17], KKM principle).
LetΛ be a hyperconvex space, K be an arbitrary subset of Λ, and H:K⟶2Λ be a KKM map so that Hℓ0 is compact for some ℓ0∈K and Hℓ is closed for any ℓ∈K. Then, ⋂ℓ∈KHℓ≠∅.Theorem 2 (see [17], best approximation).
LetΛ be a hyperconvex space and K∈Aℓ be compact. Given a continuous map f:K⟶Λ, there is ℓ0∈K so that
(4)ωℓ0,fℓ0=infℓ∈Kωℓ,fℓ0.
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]).
LetΛ be a metric space. An arbitrary function θ:2Λ⟶0,∞ is named to be a measure of noncompactness on Λ if
(1)
θA=0 iff A is a totally bounded set(2)
forA,B∈2ΛA⊆B, implies θA≤θB(3)
for allA⊆Λ and ℓ∈Λ, θA∪ℓ=θADefinition 4 (see [19]).
LetΛ be a metric space, θ be a measure of noncompactness on Λ, and K⊂Λ. The map H:K⟶2Λ is condensing if for any ε>0, there is A∈K so that θ⋂a∈AHa<ε. A condensing map H:K⟶2Λ is a condensing KKM map if it is a KKM map.In this paper, we present further extensions of the best approximation result (Theorem2) obtained by Khamsi. Finally, we present a problem related to the Schauder conjecture.


## 2. Results

The following result generalizes Theorem1. The proof is essentially the same as Theorem 3.1 in [19].Theorem 5.
Letθ be a measure of noncompactness on Λ a hyperconvex space, K be an arbitrary subset of Λ, and H:K⟶2Λ be a condensing KKM map such that each Hℓ is closed, then ⋂ℓ∈KHℓ is nonempty and compact set.We introduce the concept of aφ-quasiconvex map in hyperconvex spaces.Definition 6.
LetΛ be a hyperconvex space and K∈Aℓ. A set-valued map G:K⟶2Λ is said to be a φ-quasiconvex if for any ℓ∈K and r>0,
(5)coG−Bℓ,r⊆G−Bℓ,φr,where φ:0,∞⟶0,∞ is a continuous monotone increasing function so that φu≥u for any u≥0 and φ0=0.
Letiω:0,∞⟶0,∞ be the identity map and Iω:K⟶2K be the identity set-valued map so that Iωℓ=ℓ for any ℓ∈K. Note that a quasiconvex map is iω-quasiconvex and Iω is iω-quasiconvex in hyperconvex spaces.IfΛ,ω is a linear metric space, then Iω may not be iω-quasiconvex.Example 1.
Denote byS the linear space of real sequences. The Fréchet metric ωF for S is given as follows (see [27]):
Letℓ=ℓ1,ℓ2,⋯,ℓn,⋯, j=j1,j2,⋯,jn,⋯, and 0=0,0,⋯,0,⋯,(6)dFℓ,j=∑n=1+∞12nℓn−jn1+ℓn−jn,then, we obtain
(7)convB0,13⊆B0,13.
Namely, forℓ=1,0,1,0,⋯ and j=1/2,1/2,⋯, we have that ℓ∈B0,1/3,j∈B0,1/3, and ℓ+j/2∉B0,1/3.
Note that for anyℓ∈S and r>0, one writes
(8)convBℓ,r⊆Bℓ,2r.
So, in the linear metric spaceS,ωF, the map Iω is not iω-quasi-convex and Iω is φ-quasiconvex, where φu=2u,u∈0,∞.Theorem 7.
LetΛ,ω be a hyperconvex space, K∈Aℓ be closed, θ be a measure of noncompactness on Λ, F,G:K⟶Aℓ be continuous maps with compact values and G be φ-quasi-convex. If for any u>0 there is j∈K so that
(9)θℓ∈K:ωGℓ,Fℓ≤φωGj,Fℓ≤u,then, there is ℓ0∈K so that
(10)ωGℓ0,Fℓ0≤infℓ∈KφωGℓ,Fℓ0.Proof.
DefineH:K⟶2Λ by
(11)Hj=ℓ∈K:ωGℓ,Fℓ≤φωGj,Fℓ.
From conditionφt≥t, we obtain
(12)ωGj,Fj≤φωGj,Fjforallj∈K,so Hj is a nonempty set for all j∈K, because j∈Hj.
From condition (9), one asserts that H is a condensing map.
SinceF and G are continuous maps and φ is a continuous function, we get that Hj is a closed set for all j∈K.
The mapj⟼Hj is KKM. Indeed, suppose that for some A∈K,
(13)coA⊈HA.
Then, there isj∈coA so that j∉Ha for any a∈A. Thus,
(14)ωGj,Fj>φωGa,Fjforalla∈A.
The functionφ is increasing, so
(15)φ−1ωGj,Fj>ωGa,Fjforalla∈A.
Letε>0 be so that
(16)φ−1ωGj,Fj−ε≥ωGa,Fjforalla∈A.
Then,(17)A⊆G−Fj+φ−1ωGj,Fj−ε,and hence,
(18)coA⊆coG−Fj+φ−1ωGj,Fj−ε.
SinceG is φ-quasiconvex, one gets from (18),
(19)coA⊆G−Fj+φφ−1ωGj,Fj−ε.
Sincej∈coA, we deduce that
(20)ωGj,Fj≤φφ−1ωGj,Fj−ε.
Consequently,(21)φ−1ωGj,Fj≤φ−1ωGj,Fj−ε,which is not possible. Therefore, H must be a KKM map. Now, form Theorem 5, there is ℓ0∈K so that
(22)ℓ0∈⋂ℓ∈KHℓ.
Therefore,(23)ωGℓ0,Fℓ0≤infℓ∈KφωGℓ,Fℓ0.Taking the setK to be compact, we state from Theorem 7 the next results.Theorem 8.
LetΛ,ω be a hyperconvex space, K∈Aℓ be compact, F,G:K⟶Aℓ be continuous maps with compact values, and G be φ-quasiconvex. Then, there is ℓ0∈K so that
(24)ωGℓ0,Fℓ0≤infℓ∈KφωGℓ,Fℓ0.Theorem 9.
LetΛ,ω be a hyperconvex space, K∈Aℓ be compact, F,G:K⟶AK be continuous maps with compact values, and G be φ-quasiconvex onto a map. Then, there is ℓ0∈K so that Gℓ0∩Fℓ0≠∅.Theorem 10.
LetΛ,ω be a hyperconvex space, K∈Aℓ be compact, and F,G:K⟶Aℓ be continuous maps with compact values. If there is λ≥1 such that for any ℓ∈X and r>0,
(25)coG−Bℓ,r⊆G−Bℓ,λr,then there is ℓ0∈K so that
(26)ωGℓ0,Fℓ0≤infℓ∈KλωGℓ,Fℓ0.Theorem 11.
LetΛ,ω be a hyperconvex space, K∈Aℓ be compact, and F:K⟶Aℓ be a continuous map with compact values so that Fℓ∩K≠∅ for any ℓ∈K. Then, there is ℓ0∈K so that ℓ0∈Fℓ0.Remark 12.
IfF 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).
LetΛ,ω be a hyperconvex space, K∈Aℓ be compact, F,G:K⟶Aℓ be continuous maps with compact values, and G be quasiconvex. Then, there is ℓ0∈K so that
(27)ωGℓ0,Fℓ0≤infℓ∈KωGℓ,Fℓ0.Theorem 14 (see [16], Theorem 2.9).
LetΛ,ω be a hyperconvex space, K∈Aℓ be compact, and F:K⟶Aℓ be a continuous map with compact values. Then, there is ℓ0∈K so that
(28)ωℓ0,Fℓ0=infℓ∈Kωℓ,Fℓ0.
Finally, we give the following problems.Problem 15.
Does for every linear metric spaceΛ,ω and for a compact subset K of Λ, there is a map φ so that the identity map Iω:K⟶2K (i.e., Iωℓ=ℓ,ℓ∈K) is φ-quasiconvex?In other words, does for a linear metric space, there is a continuous monotone increasing functionφ:0,∞⟶0,∞ so that φu≥u for any u≥0, φ0=0 and
(29)convBℓ,r⊆Bℓ,φr,for any ℓ∈K and r>0?In 2001, Cauty [28] obtained the affirmative solution to the Schauder conjecture as follows:Problem 16.
LetK be a compact convex subset of a (metrizable) topological vector space. Does any continuous map f:K⟶K have a fixed point?Remark 17.
Note that if Problem15 is affirmative, then Problem 16 is affirmative.



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