Volume 2008, Article ID 648985,5pages doi:10.1155/2008/648985
Research Article
Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces
A. Amini-Harandi,1 A. P. Farajzadeh,2 D. O’Regan,3 and R. P. Agarwal4
1Department of Mathematics, Faculty of Basic Sciences, University of Shahrekord, Shahrekord 88186-34141, Iran
2Department of Mathematics, School of Science, Razi University, Kermanshah 67149, Iran
3Department of Mathematics, College of Arts, Social Sciences and Celtic Studies, National University of Ireland, Galway, Ireland
4Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901, USA
Correspondence should be addressed to A. Amini-Harandi,aminih [email protected] Received 14 July 2008; Accepted 27 October 2008
Recommended by Nan-jing Huang
A best proximity pair for a set-valued mapF:ABwith respect to a mapg:A→ Ais defined, and new existence theorems of best proximity pairs for upper semicontinuous set-valued maps with respect to a homeomorphism are proved in hyperconvex metric spaces.
Copyrightq2008 A. Amini-Harandi 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.
1. Introduction and preliminaries
LetM, dbe a metric space and let AandB be nonempty subsets ofM. Letg : A → A and letF :A Bbe a set-valued map. Now,ga, Fais called a best proximity pair for F with respect togifdga, Fa dA, B,wheredA, B inf{da, b :a∈ A, b ∈ B}.
Best proximity pair theorems establish conditions under which the problem of minimizing the real-valued functionx → dgx, Fxhas a solution. In the setting of normed linear spaces, the best proximity pair problem has been studied by many authors forg I, see 1–5. Very recently, Al-Thagafi and Shahzad1proved some existence theorems for a finite family of Kakutani set-valued maps in a normed space setting. In the present paper, our aim is to prove new results in hyperconvex metric spaces. In the rest of this section, we recall some definitions and theorems which are used inSection 2.
LetX andY be topological spaces withA ⊆ X andB ⊆ Y. LetF :X Y be a set- valued map with nonempty values. The image ofAunderFis the setFA
x∈AFxand the inverse image ofBunderFisF−B {x∈X:Fx∩B /∅}. Now,Fis said to be upper semicontinuous, if for each closed setB⊆Y,F−B {x∈X :Fx∩B /∅}is closed inX.
A topological spaceX is said to be contractible if the identity mapIX ofX is homotopic to a constant map and acyclic if all of its reduced ˇCech homology groups over the rationals vanish.
Note that a contractible space is acyclic. For topological spacesXandY,we define
F∈VX, Y⇐⇒F:X Y is an acyclic map; that is,
F is upper semicontinuous with compact acyclic values. 1.1
We denote byVcX, Ythe set of all finite composites of maps inVX, Y. LetM, d be a metric space and letBx, r {y∈M:dx, y≤r}denote the closed ball with centerxand radiusr. Let
coA
{B⊆M:Bbe a closed ball inMsuch thatA⊆B}. 1.2
IfAcoA, we say thatAis admissible subset ofM. Note that coAis admissible and the intersection of any family of admissible subsets ofMis admissible. The following definition of a hyperconvex metric space is due to Aronszajn and Pantichpakdi6.
Definition 1.1. A metric space M, d is said to be a hyperconvex metric space if for any collection of points xα of M and any collection rα of nonnegative real numbers with dxα, xβ≤rαrβ, one has
α
Bxα, rα/∅. 1.3
The simplest examples of hyperconvex spaces are finite dimensional real Banach spaces endowed with the maximum norm. For other examples of hyperconvex metric spaces which are not linear spaces, see7. Note that an admissible subset of a hyperconvex metric space is hyperconvex and contractible 8. LetA be a subset of M. The r-parallel ofA is defined as
Ar
a∈A
Ba, r. 1.4
The following result is due to Sine9.
Lemma 1.2. The r-parallel sets of an admissible subset of a hyperconvex metric space are also admissible sets.
For A ⊆ M, the set PAx {a ∈ A : da, x dx, A}is called the set of best approximations inAtox ∈M. The mapPA :M Ais called the metric projection onA.
The following lemma is well known. We give its proof for completeness.
Lemma 1.3. Let Abe a nonempty, admissible, and compact subset of a hyperconvex metric space M, d. ThenPA∈VM, A.
Proof. SinceAis compact, thenPA is nonempty. We now show thatPA is contractible and so is acyclic. To see this, notice that
PAx A∩ {a∈M:da, x≤dx, A}A∩Bx, dx, A. 1.5
ThenPAxis admissiblenote thatAis admissibleand therefore is contractible. Now, we show thatPAis upper semicontinuous. LetCbe a closed subset ofA,xn∈ {x:PAx∩C /∅}, andxn → x0. Then there exists a sequencean ∈PAxn∩Csuch thatdan, xn dxn, A.
SinceAis compact andan∈A, without loss of generality, we may assume thatan → a. Thus,
da, x0 lim
n→ ∞dan, xn lim
n→ ∞dxn, A dx0, A. 1.6 Therefore,x0∈ {x:PAx∩C /∅}and the set{x:PAx∩C /∅}is closed.
To prove our main result, we need the following fixed point theorem, which is particular form of Theorem 4 in10.
Theorem 1.4. LetXbe a nonempty compact admissible subset of a hyperconvex metric spaceM, d andF∈VcX, X. ThenFhas a fixed point.
Corollary 1.5. LetXbe a nonempty compact admissible subset of a hyperconvex metric spaceM, d, g :X → X a homeomorphism, andF ∈VcX, X. Then there exists anx0 ∈Xsuch thatgx0∈ Fx0.
Proof. Sincegis a homeomorphism, theng−1◦F∈VcX, X. Hence, byTheorem 1.4,g−1◦F has a fixed point, sayx0. Therefore,gx0∈Fx0.
2. Best proximity theorems
LetAandBbe nonempty subsets ofM. Define
A0 :{a∈A:da, b dA, Bfor someb∈B},
B0 :{b∈B:da, b dA, Bfor somea∈A}. 2.1
Notice thatA0is nonempty if and only ifB0is nonempty.
Theorem 2.1. LetAandBbe nonempty subsets ofM. Then the following statement holds.
iIfA0andBare admissible, thenB0is admissible.
iiIfA0andBare compact, thenB0is compact.
Proof. To provea, notice that
B0:{b∈B:da, b dA, Bfor somea∈A}
{b∈B:da, b dA, Bfor somea∈A0} B∩ {b∈M:da, b≤dA, Bfor somea∈A0}
B∩
a∈A0
Ba, dA, B
B∩A0dA, B.
2.2
Since A0 is admissible, then by Lemma 1.2, A0 dA, B is also admissible. Thus, B0 is admissiblenote, thatBis admissible.
bLet{bn}be a sequence inB0such thatbn → b∈B. Then there exists sequence{an} inAsuch thatdan, bn dA, B. SinceA0 is compact, we may assume thatan → a∈A0. Then
da, b lim
n→ ∞dan, bn dA, B. 2.3
Thus,b∈B0. Therefore,B0is closed and so compact.
Theorem 2.2. LetM, dbe a hyperconvex metric space,A ⊆ MandB ⊆ Mare admissible. Let g :A0 → A0be a homeomorphism, and letF :ABbe an upper semicontinuous set-valued map with admissible values. Assume thatA0is compact and admissible andFx∩B0 is nonempty, for eachx∈A0. Then there existsa∈A0such thatdga, Fa dA, B.
Proof. We use some ideas from1, Theorem 3.2. From11, Proposition 2.8,A0is nonempty.
SinceA0 and Bare admissible, it follows fromTheorem 2.1athatB0 is admissible. From Lemma 1.3,PA0 ∈ VB0, A0 note, thatA0is nonempty, admissible, and compact andB0 is hyperconvex sinceB0is an admissible subset ofM. DefineG:A0B0byGx Fx∩B0. SinceFis upper semicontinuous with nonempty admissible values andB0is admissible, then Gis upper semicontinuous with admissiblein particular acyclicvalues. FromLemma 1.3 see proof,PA0 :B0 A0 is upper semicontinuous with admissiblein particular acyclic values. Since PA0 ◦G ∈ VcA0, A0, it follows from Corollary 1.5 that there existsa ∈ A0 such thatga ∈ PA0◦Ga. Thus, there exists b ∈ Gasuch that ga ∈ PA0b. Hence, ga∈ PA0b ⊆ A0 andb∈ Ga Fa∩B0. Sinceb ∈ B0, there existsa0 ∈ A0such that da0, b dA, Band hence
dA, B≤dga, Fa≤dga, b db, A0≤da0, b dA, B. 2.4 Then
dga, Fa dA, B. 2.5
If we takeg IA0, we get the following corollary.
Corollary 2.3. LetM, dbe a hyperconvex metric space,A ⊆ MandB ⊆ Mare admissible. Let F:ABbe an upper semicontinuous set-valued map with admissible values. Assume thatFx∩B0 is nonempty, for eachx∈A0andA0is compact and admissible. Then there existsa∈A0such that da, Fa dA, B.
Corollary 2.4 see8, Corollary 5.6. Let M, d be a hyperconvex metric space and A ⊆ M nonempty, compact, and admissible. LetF : A Mbe an upper semicontinuous set-valued map with admissible values. Assume thatFx∩Ais nonempty, for eachx∈A. ThenFhas a fixed point.
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