International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 163580,9pages
doi:10.1155/2008/163580
Research Article
Fixed Points for Multivalued Mappings in
Uniformly Convex Metric Spaces
A. Kaewcharoen1 and B. Panyanak2
1Department of Mathematics, Faculty of Science, Naresuan University,
Phitsanulok 65000, Thailand
2Department of Mathematics, Faculty of Science, Chiang Mai University,
Chiang Mai 50200, Thailand
Correspondence should be addressed to B. Panyanak,banchap@chiangmai.ac.th
Received 6 June 2007; Revised 11 October 2007; Accepted 12 December 2007
Recommended by Robert H. Redfield
The purpose of this paper is to ensure the existence of fixed points for multivalued nonexpansive weakly inward nonself-mappings in uniformly convex metric spaces. This extends a result of Lim
1980in Banach spaces. All results of Dhompongsa et al.2005and Chaoha and Phon-on2006 are also extended.
Copyrightq2008 A. Kaewcharoen and B. Panyanak. 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
In 1974, Lim1developed a result concerning the existence of fixed points for multivalued nonexpansive self-mappings in uniformly convex Banach spaces. This result was extended to nonself-mappings satisfying the inwardness condition independently by Downing and Kirk
2and Reich3. This result was extended to weak inward mappings independently by Lim
4 and Xu 5. Recently, Dhompongsa et al.6presented an analog of Lim-Xu’s result in CAT0spaces. In this note, we extend the result to uniformly convex metric spaces which improve results of both Lim-Xu and Dhompongsa et al. In addition, we also give a new proof of a result of Lim7by using Caristi’s theorem8. Finally, we give some basic properties of fixed point sets for quasi-nonexpansive mappings for these spaces.
2. Preliminaries
A concept of convexity in metric spaces was introduced by Takahashi9.
dz, Wx, y, λ≤λdz, x 1−λdz, y. 2.1
A metric spaceX, dtogether with a convex structureWis called a convex metric space which will be denoted byX, d, W.
Definition 2.2. A convex metric spaceX, d, Wis said to be uniformly convex 10if for any
ε >0, there existsδδε>0 such that for allr >0 andx, y, z∈Xwithdz, x≤r, dz, y≤r
anddx, y≥rε,
d
z, W
x, y,1
2
≤r1−δ. 2.2
Obviously, uniformly convex Banach spaces are uniformly convex metric spaces. By using the
CNinequality11, it is easy to see that CAT0spaces are also uniformly convex.
Forx, y∈X,C⊂X, andλ∈I, we denoteWx, y, λ:λx⊕1−λy,x, y:{λx⊕1−
λy:λ∈I},x, y: x, y\ {x}, andλx⊕1−λC:{λx⊕1−λz:z∈C}.So we can define the inward set ICxofxas follows:
ICx:{x} ∪z:x, z∩C / ∅. 2.3
Let C be a nonempty subset of a metric space X. Then C is called convex if for
x, y ∈ C, x, y ⊂ C.We will denote byFCthe family of nonempty closed subsets ofC, byKCthe family of nonempty compact subsets ofC,and byKCCthe family of nonempty compact convex subsets ofC.LetH·,·be the Hausdorffdistance onFX.That is,
HA, B max
sup
a∈A
dista, B,sup
b∈B
distb, A , A, B∈ FX. 2.4
Definition 2.3. A multivalued mappingT : C → FX is said to be inward onCif for some
p∈C,
λp⊕1−λTx⊂ICx ∀x∈C, ∀λ∈0,1, 2.5
and weakly inward onCif for somep∈C,
λp⊕1−λTx⊂ICx ∀x∈C, ∀λ∈0,1, 2.6
whereAdenotes the closure of a subsetAofX. In a Banach space setting, ifCis convex, then so isICx. Therefore, the conditions above can be replaced byTx ⊂ ICxandTx ⊂ ICx,
respectively.
Definition 2.4. A multivalued mappingT :C→ FXsatisfying
HTx, Ty≤kdx, y, x, y∈C, 2.7
is called a contraction ifk ∈0,1and nonexpansive ifk 1.A pointxis a fixed point ofT if
Given a metric space X,one way to describe a metric space ultrapower X ofX is to first embedXas a closed subset of a Banach spaceEsee, e.g.,12, page 129. LetEdenote a Banach space ultrapower ofErelative to some nontrivial ultrafilterUsee, e.g.,13. Then take
X:xxn
∈E:xn∈X∀n
. 2.8
One can then letddenote the metric onXinherited from the ultrapower norm·UinE. IfXis
complete, then so isXsinceXis a closed subset of the Banach spaceE. In particular, the metric
donXis given by
dx, y lim
U xn−ynlimU d
xn, yn
, 2.9
with{un} ∈{xn}if and only if limUxn−un0.
IfX, d, Wis a convex metric space, we consider a metric space ultrapowerX, dof
X, dand define a functionW:X×X×I→X by
Wx, y, λ Wxn, yn, λ
. 2.10
In order to show that the functionWis well defined, we need the following condition. For each
p, x, y∈Xandλ∈0,1,
d1−λp⊕λx,1−λp⊕λy≤λdx, y, 2.11
which is equivalent to
d1−λp⊕λx,1−λq⊕λy≤λdx, y 1−λdp, q, 2.12
for allp, q, x, y∈Xandλ∈0,1.
By using condition2.12, it is easy to see thatWis a convex structure onX. This implies thatX, d,Wis a convex metric space.
Example 2.5. Every Banach space satisfies condition2.11.
Example 2.6. Condition2.11is satisfied for spaces of hyperbolic typefor more details of these spaces see14. This is also true for CAT0spaces andR-trees.
Example 2.7. LetH be a hyperconvex metric space. Then there exists a nonexpansive retract
R:l∞H→Hsee, e.g.,15for more on this. For anyx, y∈Handt∈0,1,we let
tx⊕1−tyRtx 1−ty. 2.13
Sincel∞His a Banach space,Halso satisfies condition2.11.
X, d,Wrelative toU is also uniformly convex. Indeed, let anyε > 0 and letδ be a posi-tive number corresponding to the uniform convexity ofX.Letr > 0 andx, y, z ∈X be such that
dz, x≤r, dz, y≤r, dx, y≥rε. 2.14
Then there are some representativesxnandynofxandyand a setI∈ Usuch that
dzn, xn
≤r, dzn, yn
≤r, dxn, yn
≥rε ∀n∈I. 2.15
For suchn, dzn, Wxn, yn,1/2≤r1−δ.This impliesdz, Wx, y, 1/2≤r1−δ.
Recall that a subsetC of a metric spaceX is said to be uniquely proximinal if each
point x ∈ X has a unique nearest point in C. A convex metric space X is said to have
property (C) if every decreasing sequence of nonempty bounded closed convex subsets ofX
has nonempty intersection. In10, Shimizu and Takahashi proved that propertyCholds in complete uniformly convex metric spaces. This implies that every nonempty closed convex subset of a complete uniformly convex metric space is uniquely proximinal. Indeed, letCbe a nonempty closed convex subset of a complete uniformly convex metric spaceX, andx0∈X.
LetN{x∈C:dx0, x distx0, C}.For eachn,we define
Cn:
y∈C:dx0, y
≤distx0, C
1
n . 2.16
ThenCnis a decreasing sequence of nonempty bounded closed convex subsets ofC. More-over,
N
∞
n1
Cn,
which is nonempty by the above observation. The uniqueness follows from the uniform con-vexity ofX.
3. Main results
We first establish the following lemma.
Lemma 3.1. Let X be a complete uniformly convex metric space satisfying condition 2.11, C a nonempty closed convex subset ofX,x∈X,andpxthe unique nearest point ofxinC.Then
dx, px< dx, y ∀y∈ICpx\px. 3.1
Proof. Lety∈ICpx\ {px}. Then there is a sequenceyninICpxandyn→y.Choose
n0∈Nsuch thatpx, yn∩C / ∅for alln≥n0. For suchn, letzn∈px, yn∩Cand write
zn 1−αnpx⊕αnyn, αn∈0,1.Then
dx, px≤dx, zn
≤1−αn
dx, pxαnd
x, yn
. 3.2
This implies
dx, px≤dx, y. 3.3
Ifdx, px dx, y,we letu 1/2px⊕1/2y.By the uniform convexity ofX,we have
dx, u< dx, px.On the other hand, for eachn≥n0, letun 1/2px⊕1/2yn.We will
Case 1.αn1/2.We are done.
Case 2. 1/2< αn.Letvn 1−1/2αnpx⊕1/2αnzn.This implies
dpx, vn
1
2αn
dpx, zn
1
2d
px, yn
dpx, un
, 3.4
dvn, yn
≤dvn, zn
dzn, yn
1− 1 2αn
dpx, zn
1−αn
dpx, yn
αn−
1 2
dpx, yn
1−αn
dpx, yn
1
2d
px, yn
dun, yn
.
3.5
Therefore
dvn, yn
≤dun, yn
. 3.6
We claim thatunvn.If not, letwn 1/2un⊕1/2vn.
From3.4,3.6, and the uniform convexity ofX, we have
dpx, wn
< dpx, un
,
dwn, yn< dun, yn.
3.7
This implies
dpx, yn
≤dpx, wn
dwn, yn
< dpx, un
dun, yn
dpx, yn
, 3.8
which is a contradiction. Henceunvn∈px, znand soun∈ICpxby the convexity ofC.
Case 3. αn<1/2.letvn 1−2αnpx⊕2αnun.By the same arguments in the proof ofCase 2,
we can show thatzn∈px, un.This meansun∈ICpx.
By condition2.11, limnun u,which impliesu ∈ IEpx\ {px}.By the same
ar-guments in the first part of the proof,dx, px ≤ dx, u which is a contradiction. Hence
dx, px< dx, yas desired.
From 6, Lemma 3.4, we observe that the space is not necessarity assumed to be a CAT0space since the proof is only involved with condition2.11which is weaker than the
CNinequalitysee11, Lemma 3. Therefore, we can obtain the following lemma.
Lemma 3.2. LetXbe a complete convex metric space satisfying condition2.11,Ca nonempty closed subset ofX,andT :C→ FXa contraction mapping satisfying, for allx∈C,
Tx⊂ICx. 3.9
ThenThas a fixed point.
Proof ofLemma 3.2. Let 0 ≤ k < 1 be the contraction constant ofT and letε > 0 be such that
ε k2ε1ε < 1.LetM {x, z : z ∈ Tx, x ∈ C}and define a metric ρ onMby
ρx, z,u, v max{dx, u, dz, v}.It is easy to see thatM, ρis a complete metric space. Now defineψ:M→0,∞byψx, z dx, z/ε. Thenψis continuous onM.Suppose thatT has no fixed points, that is, distx, Tx > 0 for allx ∈ C.Letx, z ∈M.By3.9, we can findz ∈ ICxsatisfyingdz, z < εdistx, Tx.Now chooseu ∈ x, z∩Cand write
u 1−δx⊕δzfor some 0< δ≤1.For suchδ, we have
δε 1−δ k2εδ1ε<1. 3.10
Sincedx, u>0, we can findv∈Tusatisfying
dz, v≤HTx, Tu εdx, u≤kεdx, u. 3.11
Now we define a mappingg :M→Mbygx, z u, vfor allx, z∈M.We claim thatgsatisfies
ρx, z, gx, z< ψx, z−ψgx, z ∀x, z∈M. 3.12 Caristi’s theorem8then implies thatghas a fixed point, which contradicts to the strict in-equality3.12and the proof is complete. So it remains to prove3.12. In fact, it is enough to show that
ρx, z,u, v<1 ε
dx, z−du, v. 3.13
Butdz, v≤dx, u,it only needs to prove thatdx, u<1/εdx, z−du, v.
Now,
dx, u δdx, z≤δdx, z εdistx, Tx≤δdx, z εdx, z≤δ1εdx, z.
3.14 Therefore
dx, u≤δ1εdx, z. 3.15
It follows that
dz, v≤kεdx, u≤kεδ1εdx, z. 3.16 We now lety 1−δx⊕δz,then by the condition2.11,
du, v≤du, y dy, z dz, v≤δdz, z 1−δdx, z kεδ1εdx, z
≤δεdx, z 1−δ kεδ1εdx, z.
3.17 Thus
du, v≤δε 1−δ kεδ1εdx, z. 3.18
Inequalities3.15,3.18, and3.10imply that
εdx, u du, v≤εδ1εdx, z δε 1−δ kεδ1εdx, z
δε 1−δ k2εδ1εdx, z< dx, z. 3.19
By Lemmas3.1and3.2with the same arguments in the proof of Theorem 3.3 of6, we can obtain the following theorem which extends4, Theorem 8by Lim and6, Theorem 3.3 by Dhompongsa et al.
Theorem 3.3. LetX be a complete uniformly convex metric space satisfying condition2.11,Ca nonempty bounded closed convex subset ofX,andT : C → KXa nonexpansive weakly inward mapping. ThenThas a fixed point.
As an immediate consequence ofTheorem 3.3, we obtain the following corollary.
Corollary 3.4. LetX be a complete uniformly convex metric space satisfying condition 2.11,Ca nonempty bounded closed convex subset ofX, andT :C→ KCa nonexpansive mapping. ThenT has a fixed point.
In fact, this corollary is a special case of10, Theorem 2in which condition2.11was not assumed. An interesting question is whether condition2.11inTheorem 3.3can be dropped.
LetCbe a nonempty subset of a metric spaceX.Recall that a single-valued mapping
t : C→ Cand a multivalued mapping T : C →2C\∅are said to be commuting ifty ∈ Ttx
for ally ∈Txandx∈ C.Ift: C→Cis nonexpansive withCbeing bounded closed convex andXcomplete uniformly convex satisfying condition2.11, then Fixtis nonempty by the above corollary. Moreover, by a standard argument, we can show that it is also closed and convex. So we can obtain a common fixed point theorem in uniformly convex metric spaces as
6, Theorem 4.1 see also16, Theorem 4.2for a related result in Banach spaces.
Theorem 3.5. LetXbe a complete uniformly convex metric space satisfying condition2.11, letCbe a nonempty bounded closed convex subset ofX,and lett:C→CandT :C→ KCCbe nonexpansive. Assume that for somep∈Fixt,
αp⊕1−αTxis convex ∀x∈C, ∀α∈0,1. 3.20
IftandTare commuting, then there exists a pointz∈Csuch thattzz∈Tz.
4. Fixed point sets of quasi-nonexpansive mappings
LetXbe a metric space. Recall that a mappingf : X → Xis said to be quasi-nonexpansive if
dfx, p≤dx, pfor allx∈Xandp∈Fixf.In this case, we will assume that Fixf/∅.
In17, Chaoha and Phon-on showed that ifXis a CAT0space, then Fixfis closed convex. Furthermore, they gave an explicit construction of a continuous function defined onXwhose fixed point set is any prescribed closed subset ofX.In this section, we extend these results to uniformly convex metric spaces.
We begin by proving the following lemma.
Lemma 4.1. LetXbe a uniformly convex metric space, and letx, y, z∈Xfor which
dx, z dz, y dx, y. 4.1
Proof. Letu ∈ x, ybe such thatdx, u dx, z.Thendx, y dx, u du, yand also
dz, y du, yby4.1. We will show thatz u.Suppose not, we letv 1/2z⊕1/2u
andr dx, u dx, z.Sincedz, u>0,chooseε >0 so thatdz, u > rε.By the uniform convexity ofX,there existsδ >0 such that
dx, v≤r1−δ< rdx, z. 4.2
By using the same arguments, we can show thatdy, v< dy, z.Therefore
dx, y≤dx, v dy, v< dx, z dy, z dx, y, 4.3 which is a contradiction.
By using the above lemma with the proof of Theorem 1.3 of17, we obtain the following result.
Theorem 4.2. LetXbe a convex subset of a uniformly convex metric space andf :X →X a quasi-nonexpansive mapping whose fixed point set is nonempty. Then Fixfis closed convex.
In17, the authors constructed a continuous function defined on a CAT0 space X
whose fixed point set is any prescribed closed subset ofXby using the following two implica-tions of theCNinequality:
d1−tx⊕ty,1−sx⊕sy|t−s|dx, y ∀x, y∈X, t, s∈0,1, 4.4
d1−tx⊕ty,1−tx⊕tz≤dy, z ∀x, y, z∈X, t∈0,1. 4.5 In fact, condition4.4holds in uniformly convex metric spaces as the following lemma shows.
Lemma 4.3. Condition4.4holds in uniformly convex metric spaces.
Proof. We first note that the conclusion holds ifs 0 ort 0.We now letX be a uniformly convex metric space,x, y ∈ X, andt, s ∈ 0,1.Letu 1−tx⊕tyandz 1−sx⊕sy. Without loss of generality, we can assume thatt < s. Letv 1−t/sx⊕t/sz,then
dx, v t
sdx, z tdx, y,
dv, y≤
1− t
s
dx, y t
sdz, y 1−tdx, y.
4.6
Ifu / v,we letw 1/2u⊕1/2v.Then by the uniform convexity ofX,we can show that
dx, w< dx, uanddy, w< dy, u.This implies
dx, y< dx, u du, y dx, y, 4.7
which is a contradiction, henceuv.Therefore
dz, u dz, v
1−t
s
dx, z |s−t|dx, y. 4.8
Theorem 4.4. LetAbe a nonempty subset of a uniformly convex metric spaceXsatisfying condition
4.5. Then there exists a continuous functionf:X→Xsuch that Fixf A.
Acknowledgment
This research was supported by the Commission on Higher Education and Thailand Research Fund under Grant MRG5080188.
References
1 T. C. Lim, “A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space,” Bulletin of the American Mathematical Society, vol. 80, pp. 1123–1126, 1974.
2 D. Downing and W. A. Kirk, “Fixed point theorems for set-valued mappings in metric and Banach spaces,” Mathematica Japonica, vol. 22, no. 1, pp. 99–112, 1977.
3 S. Reich, “Approximate selections, best approximations, fixed points, and invariant sets,” Journal of
Mathematical Analysis and Applications, vol. 62, no. 1, pp. 104–113, 1978.
4 T. C. Lim, “On asymptotic centers and fixed points of nonexpansive mappings,” Canadian Journal of
Mathematics, vol. 32, no. 2, pp. 421–430, 1980.
5 H. K. Xu, “Multivalued nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory,
Meth-ods & Applications, vol. 43, no. 6, pp. 693–706, 2001.
6 S. Dhompongsa, A. Kaewkhao, and B. Panyanak, “Lim’s theorems for multivalued mappings in CAT0spaces,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 478–487, 2005.
7 T. C. Lim, “A fixed point theorem for weakly inward multivalued contractions,” Journal of Mathematical
Analysis and Applications, vol. 247, no. 1, pp. 323–327, 2000.
8 J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,” Transactions of the
American Mathematical Society, vol. 215, pp. 241–251, 1976.
9 W. Takahashi, “A convexity in metric space and nonexpansive mappings. I,” K¯odai Mathematical
Semi-nar Reports, vol. 22, pp. 142–149, 1970.
10 T. Shimizu and W. Takahashi, “Fixed points of multivalued mappings in certain convex metric spaces,” Topological Methods in Nonlinear Analysis, vol. 8, no. 1, pp. 197–203, 1996.
11 W. A. Kirk, “Geodesic geometry and fixed point theory. II,” in Proceedings of the International Conference
on Fixed Point Theory and Applications, pp. 113–142, Valencia, Spain, July 2003.
12 M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 2001.
13 M. A. Khamsi and B. Sims, “Ultra-methods in metric fixed point theory,” in Handbook of Metric Fixed
Point Theory, pp. 177–199, Kluwer Academic, Dordrecht, The Netherlands, 2001.
14 W. A. Kirk, “Krasnosel’ski˘ı’s iteration process in hyperbolic space,” Numerical Functional Analysis and
Optimization, vol. 4, no. 4, pp. 371–381, 1982.
15 M. A. Khamsi, “On asymptotically nonexpansive mappings in hyperconvex metric spaces,”
Proceed-ings of the American Mathematical Society, vol. 132, no. 2, pp. 365–373, 2004.
16 S. Dhompongsa, A. Kaewcharoen, and A. Kaewkhao, “The Dom´ınguez-Lorenzo condition and multi-valued nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 958–970, 2006.
17 P. Chaoha and A. Phon-on, “A note on fixed point sets in CAT0spaces,” Journal of Mathematical
