Volume 2008, Article ID 607926,9pages doi:10.1155/2008/607926
Research Article
Best Proximity Pairs Theorems for Continuous Set-Valued Maps
A. Amini-Harandi,1 A. P. Farajzadeh,2 D. O’Regan,3 and R. P. Agarwal4
1Department of Mathematics, University of Shahrekord, Shahrekord 88186-34141, Iran
2Department of Mathematics, Razi University, Kermanshah 67149, Iran
3Department of Mathematics, National University of Ireland, Galway, Ireland
4Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
Correspondence should be addressed to A. Amini-Harandi,aminih [email protected] Received 15 July 2008; Accepted 16 September 2008
Recommended by Nan-jing Huang
A best proximity pair for a set-valued mapF:ABwith respect to a set-valued mapG:AA is defined, and a new existence theorem of best proximity pairs for continuous set-valued maps is proved in nonexpansive retract metric spaces. As an application, we derive a coincidence point theorem.
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
Let M, dbe a metric space and let A and B be nonempty subsets ofM. Let dA, B inf{da, b:a∈ A, b∈B}, and ProxA, B {a, b∈A×B :da, b dA, B}.Ais said to be approximately compact if for eachy ∈ Mand each sequencexninAsatisfying the conditiondxn, y → dy, Athere is a subsequence ofxnconverging to an element ofA.
Let
B0:
b∈B:da, b dA, Bfor somea∈A , A0:
a∈A:da, b dA, Bfor someb∈B
. 1.1
Let G : A A and F : A B be set-valued maps. Gx0, Fx0 is called a best proximity pair for F with respect to G if dGx0, Fx0 dA, B. Best proximity pair theorems analyze the 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; see1–5. In 2000, Sadiq Basha and Veeramani4proved the following theorem.
Theorem 1.1. Let Ebe a normed linear space. Let Abe a nonempty, approximately compact and convex subset ofEand letBbe a nonempty, closed and convex subset ofEsuch that ProxA, Bis nonempty andA0is compact. Suppose that
aF :ABis a set-valued map such that for everyx∈A0,Fx∩B0/∅, and for every y∈B0, the fiberF−1yis open;
bfor every open setUinA, the set∩{Fu:u∈U}is convex;
cg : A → Ais a continuous, proper, quasi-affine, and surjective single-valued map such thatg−1A0⊆A0.
Then there exists an elementx0∈A0such that d
g x0
, F x0
dA, B. 1.2
In the rest of this section we recall some definitions and theorems which are used in the next section. LetXandY be topological spaces withA⊆XandB⊆Y. LetF:X Y be a set-valued map with nonempty values. The image ofAunderFis the setFA
x∈AFx
and the inverse image ofBunderFisF−B {x∈X:Fx∩B /∅}. NowFis said to be aclosed if its graph, GrF {x, y∈X×Y :y ∈Fx}, is a closed set in product
spaceX×Y;
bupper semicontinuous, if for each closed setB⊆Y,F−Bis closed inX;
clower semicontinuous, if for each open setB⊆Y, the setF−Bis open;
dcontinuous ifFis both lower semicontinuous and upper semicontinuous.
We say thatF:XY is onto ifFX Y. IfF :X Y is onto thenF−:Y X, the lower inverse ofF, is defined byF−y {x∈X :y∈Fx}.f:X → Yis called a homeomorphism iffis a bijective, continuous, and open map. We say that the set-valued mappingF :XY has a continuous selection if there exists a continuous functionf :X → Y such thatfx∈ Fxfor eachx∈X. We let
SX, Y {F :XY :F has a continuous selection}. 1.3 For a nonempty finite subsetDofX, letDdenote the set of all nonempty finite subsets of D.
Definition 1.2. LetXbe a nonempty subset of a topological vector spaceY. A set-valued map F:XY is said to be a generalized KKM mappingGKKMif for each nonempty finite set {x1, . . . , xn} ⊆X, there exist a set{y1, . . . , yn}of points ofY, not necessarily all different, such that for each subset{yi1, . . . , yik}of{y1, . . . , yn}, we have
conv
yi1, . . . , yik
⊆k
j1
F xij
. 1.4
The following extension of the classical KKM principle in topological vector spaces is due to Chang and Zhang6.
Theorem 1.3. LetX be a nonempty subset of a topological vector spaceY and letF : X Y be a GKKM mapping with closed values. Then, the family{Fx : x ∈ X}has the finite intersection property, that is,
x∈A
Fx/∅ for eachA∈ X. 1.5
Furthermore, if there exists anx0∈Xsuch thatFx0is a compact set inY, then
x∈X
Fx/∅. 1.6
Let X be a nonempty subset of a topological vector spaceY. Let F : X Y and G :Y Y be set-valued mappings such that for each nonempty finite set{x1, . . . , xn} ⊆ X, there exists a set{y1, . . . , yn}of points ofY, not necessarily all different, such that for each subset{yi1, . . . , yik}of{y1, . . . , yn}, we have
G conv
yi1, . . . , yik
⊆k
j1
F xij
. 1.7
ThenF is called a generalized KKM mapping with respect toG. If the set-valued mapping G:Y Ysatisfies the requirement that for any generalized KKM mappingF:X Ywith respect toGthe family{Fx:x∈X}has the finite intersection property, thenGis said to be have the KKM property. We denote
KKMY {G:Y Y :Ghas the KKM property}. 1.8
By Theorem 1.3, the identity mapIY has the KKM property. It is well known, and easy to see, that the continuous functions have the KKM property. Thus if a set-valued mappingGhas a continuous selection, thenGhas trivially the KKM property.
LetM, dbe a metric space and letBx, r {y∈M:dx, y≤r}denote the closed ball with centerxand radiusr. Let
coA
{B⊆M:B is a closed ball inMsuch thatA⊆B}. 1.9 If A coA, we say thatA is an admissible subset of M. 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 Panitchpakdi7.
Definition 1.4. 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, we have
α
B xα, rα
/∅. 1.10
The simplest examples of hyperconvex spaces are finite dimensional real Banach spaces andl∞endowed with the maximum norm.
Now we introduce an important class of metric spaces.
Definition 1.5 see 8. A nonexpansive retract metric space i.e., an NR-metric space M, E, rconsists of a metric spaceM, d, a convex subsetE, ρof a metrizable topological vector spaceV, ρin which every closed ball is convex such thatM, dcan be isometrically embedded intoE, ρandr:E → Mis a nonexpansive retraction.
LetA⊆M. We say thatAisr-convex if, for eachD ∈ A,rconvD⊆Anote we identifyMwith the isometric embedding image set inE.
Remark 1.6. Every closed ball inE, ρis convex if and only if ρ
αx1βx2, αy1βy2
≤max ρ
x1, y1
, ρ x2, y2
, 1.11
for eachx1, x2, y1, y2∈E, αβ1, α, β≥0.
Examples 1.7. aLetX,·be a normed linear space. LetEX,ρx, y x−y, andrI the identity mapping. ThenX,·is a nonexpansive retract metric space. In this caseA⊆X isr-convex if and only ifAis convex.
b Let M, d be a hyperconvex metric space. It is well known that there exists an index set I and a natural isometric embedding from M into l∞I. Also there exists a nonexpansive retractionr : l∞I → M. Thus every hyperconvex metric space is an NR- metric space. In hyperconvex metric spaces, every admissible set isr-convex . To see this, let A⊆Mbe admissible andD∈ A. ThenrconvD⊆coD 9. SinceAis admissible, then coD⊆coA A. ThusrconvD⊆A, which implies thatAisr-convex.
cLetX, dbe a metrizable Hausdorfftopological vector space in which every closed ball is convex. LetEX,ρx, y dx, y, andr Ibe the identity mapping. ThenX, dis anNR-metric space. In this case,A⊆Xisr-convex if and only ifAis convex.
2. Main theorems
This section is devoted to main results on best proximity pairs.
Theorem 2.1. LetM, E, rbe anNR-metric space. LetA ⊆Mbe nonempty, compact,r-convex, and letBbe a nonempty subset ofM. LetG : A Abe a continuous, onto set-valued map with compact values such thatG− ∈ SA, A. LetF : A Bbe a continuous set-valued map withr- convex, compact values. Assume thatFx∩B0/∅, for eachx∈A. Then there existsx0 ∈Asuch that
d G
x0
, F x0
dA, B. 2.1
Proof. Define a set-valued mapH:AAby
Hy
x∈A:d
Gx, Fx
≤d
Gy, Fx
. 2.2
Since y ∈ Hy, then Hy/∅for each y ∈ A. We show that for eachy ∈ A,Hy is closed and therefore is a compact subset ofA. Letxn∈Hyandxn → x. SinceFandGare compact-valued, then there exists∈Gy, t∈Fx, un∈Gxn, andvn ∈Fxnsuch that
d G
xn , F
xn d
un, vn , d
Gy, Fx
ds, t. 2.3
NowF is lower semicontinuous so for eachn∈N, there existstn ∈Fxnsuch thattn → t.
SinceFAandGAare compact andFandGare closed, without loss of generality, we may assume thatun → u,vn → v,u∈Gxandv∈Fx. Therefore sincexn∈Hy, we have
d
Gx, Fx
≤du, v lim
n d un, vn lim
n d G
xn , F
xn
≤lim sup
n
d
Gy, F xn
≤lim
n d s, tn
ds, t d
Gy, Fx ,
2.4
which shows thatx∈Hy. Now, we prove that
H:A⊆EE 2.5
is a generalized KKM mapping with respect toG−◦r. To show this, suppose thatx1, . . . , xn are inAand take anyy0withy0/∈n
i1Hxi. Then we have d
G y0
, F y0
> d G
xk , F
y0
, ∀k1, . . . , n. 2.6
Let
S y0
:
x∈A:∃y∈Gxsuch thatd G
y0 , F
y0
> d y, F
y0
. 2.7
Clearly xk ∈ Sy0 for k 1, . . . , n. Let g : A → A be a selection of G not necessary continuous. We takezk∈Fy0such that
d G
y0
, F y0
> d g
xk
, zk
, for 1≤k≤n. 2.8
Letλi ≥0 and ni1λi 1. Nowr is nonexpansive andRemark 1.6yieldsnote we identify Mwith the isometric embedding image set inE
d
r n
i1
λig xi
, r
n
i1
λizi
≤ρ n
i1
λig xi
, n i1
λizi
≤max
1≤i≤nρ g
xi , zi max
1≤i≤nd g
xi , zi
< d G
y0
, F y0
.
2.9
SinceFy0andAarer-convex, then
r n
i1
λizi
∈F y0
, r
n
i1
λig xi
∈A. 2.10
Thus
d
r n
i1
λig xi
, F
y0
< d G
y0
, F y0
. 2.11
Hence, we deduce that note that G is onto and see the definition of Sy0 with y r ni1λigxi
G− r
conv g
x1
, . . . , g xn
⊆S y0
. 2.12
Asy0/∈Sy0, we havey0/∈G−rconv{gx1, . . . , gxn}. Consequently,
G−◦r conv
g x1
, . . . , g xn
⊆n
i1
H xi
. 2.13
Sincex1, . . . , xnare arbitrary elements ofA, then we deduce that for each subset{i1, . . . , ik} ⊆ {1, . . . , n}we have
G−◦r conv
g xi1
, . . . , g xik
⊆k
j1
H xij
. 2.14
Now sinceG− ∈ SA, Aandr is continuous, then G−◦r ∈ SE, Aand soG−◦r has the KKM property. Hence the family{Hx : x ∈ A}has the finite intersection property. Now sinceHxis compact for anyx ∈ A, we have immediately that
x∈AHx/∅. Therefore, there exists anx0∈Asuch that
x0∈
x∈A
Hx. 2.15
Then, it is clear that d
G x0
, F x0
≤d
Gx, F x0
∀x∈A. 2.16
Sincex0∈A, then
d G
x0 , F
x0 inf
x∈Ad
Gx, F x0
. 2.17
SinceG:AAis onto, then for eachy∈Athere existsx∈Asuch thaty∈Gx. Thus d
A, F x0
≤d
Gx, F x0
≤d y, F
x0
. 2.18
Hence
x∈Ainfd
Gx, F x0
d A, F
x0
. 2.19
Pickb∈Fx0∩B0/∅. Then there existsa∈Asuch thatda, b dA, B. Thus d
A, F x0
≤dA, b≤da, b dA, B. 2.20
By2.17,2.19, and2.20, we get d
G x0
, F x0
≤dA, B. 2.21
On the other hand, trivially d
G x0
, F x0
≥dA, B. 2.22
Thus by2.21and2.22, we get d
G x0
, F x0
dA, B. 2.23
Remark 2.2. a Let G : A → A be a single-valued homeomorphism. Then G obviously satisfies all conditions ofTheorem 2.1.
bThere are many conditions under whichG−has a continuous selection10–13.
The following corollary is immediate.
Corollary 2.3. LetXbe a normed linear space. LetA⊆Xbe a nonempty compact convex and letB be a nonempty subset ofX. LetG:AAbe a continuous, onto set-valued map with compact values such thatG− ∈ SA, A. LetF : A Bbe a continuous set-valued map with convex, compact values. Assume thatFx∩B0/∅, for eachx∈A. Then there existsx0∈Asuch that
d G
x0
, F x0
dA, B. 2.24
Remark 2.4. A similar result to that ofCorollary 2.3holds in every topological vector space in which every closed ball is convex.
Since hyperconvex metric spaces areNR-metric spaces, then we have the following corollary.
Corollary 2.5. Let M, d be a hyperconvex metric space. Let A ⊆ M be a nonempty compact admissible and letBbe a nonempty subset ofM. LetG : A Abe a continuous, onto set-valued map with compact values such thatG− ∈ SA, A. LetF :ABbe a continuous set-valued map with admissible, compact values. Assume thatFx∩B0/∅, for eachx∈A. Then there existsx0∈A such that
d G
x0 , F
x0
dA, B. 2.25
Corollary 2.6. LetM, dbe a hyperconvex metric space. LetAbe a nonempty compact admissible subset ofM. LetG : A Abe a continuous, onto set-valued map with compact values such that G− ∈ SA, A. LetF : A Mbe a continuous set-valued map with admissible, compact values.
Assume thatFx∩A /∅, for eachx∈A. Then there existsx0∈Asuch that
Gx0∩Fx0/∅. 2.26
Proof. LetBMand applyCorollary 2.5noteB0A.
Remark 2.7. If we takeGIA,Corollary 2.6reduces to Corollary 3.5 of Kirk and Shin14.
References
1 M. A. Al-Thagafi and N. Shahzad, “Best proximity pairs and equilibrium pairs for Kakutani multimaps,” Nonlinear Analysis: Theory, Methods & Applications. In press.
2 W. K. Kim and K. H. Lee, “Corrigendum to “Existence of best proximity pairs and equilibrium pairs”
J. Math. Anal. Appl. 3162006433–446,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 1482–1483, 2007.
3 W. K. Kim and K. H. Lee, “Existence of best proximity pairs and equilibrium pairs,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 433–446, 2006.
4 S. Sadiq Basha and P. Veeramani, “Best proximity pair theorems for multifunctions with open fibres,”
Journal of Approximation Theory, vol. 103, no. 1, pp. 119–129, 2000.
5 P. S. Srinivasan and P. Veeramani, “On best proximity pair theorems and fixed-point theorems,”
Abstract and Applied Analysis, vol. 2003, no. 1, pp. 33–47, 2003.
6 S.-S. Chang and Y. Zhang, “Generalized KKM theorem and variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 159, no. 1, pp. 208–223, 1991.
7 N. Aronszajn and P. Panitchpakdi, “Extension of uniformly continuous transformations and hyperconvex metric spaces,” Pacific Journal of Mathematics, vol. 6, no. 3, pp. 405–439, 1956.
8 A. Amini, M. Fakhar, and J. Zafarani, “KKM mappings in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 6, pp. 1045–1052, 2005.
9 W. A. Kirk, B. Sims, and G. X.-Z. Yuan, “The Knaster-Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 39, no. 5, pp. 611–627, 2000.
10 H. Ben-El-Mechaiekh and M. Oudadess, “Some selection theorems without convexity,” Journal of Mathematical Analysis and Applications, vol. 195, no. 2, pp. 614–618, 1995.
11 J.-C. Hou, “Michael’s selection theorem under an assumption weaker than lower semicontinuous in H-spaces,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 501–508, 2001.
12 J. T. Markin, “A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces,”
Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 862–866, 2006.
13 D. Repovˇs and P. V. Semenov, Continuous Selections of Multivalued Mappings, vol. 455 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
14 W. A. Kirk and S. S. Shin, “Fixed point theorems in hyperconvex spaces,” Houston Journal of Mathematics, vol. 23, no. 1, pp. 175–188, 1997.
