Volume 2010, Article ID 367274,11pages doi:10.1155/2010/367274
Research Article
Approximating Fixed Points of
Nonexpansive Nonself Mappings in CAT(0) Spaces
W. Laowang and B. Panyanak
Department of Mathematics, Faculty of Science, Chaing Mai University, Chiang Mai 50200, Thailand
Correspondence should be addressed to B. Panyanak,banchap@chiangmai.ac.th
Received 23 July 2009; Accepted 30 November 2009
Academic Editor: Tomonari Suzuki
Copyrightq2010 W. Laowang 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.
Suppose thatK is a nonempty closed convex subset of a complete CAT0 spaceX with the nearest point projectionP fromXontoK. LetT : K → Xbe a nonexpansive nonself mapping withFT : {x ∈ K : Tx x}/∅. Suppose that {xn}is generated iteratively by x1 ∈ K,
xn1P1−αnxn⊕αnTP1−βnxn⊕βnTxn,n≥1, where{αn}and{βn}are real sequences inε,1−εfor someε∈0,1. Then{xn}Δ-converges to some pointx∗inFT. This is an analog of a result in Banach spaces of Shahzad2005and extends a result of Dhompongsa and Panyanak
2008to the case of nonself mappings.
1. Introduction
A metric space X is a CAT0 space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT0 space. Other examples include Pre-Hilbert spaces, R-trees
see1, Euclidean buildingssee2, the complex Hilbert ball with a hyperbolic metricsee
3, and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry see Bridson and Haefliger1. The work by Burago et al.4contains a somewhat more elementary treatment, and by Gromov5a deeper study.
Fixed point theory in a CAT0space was first studied by Kirksee6,7. He showed that every nonexpansive single-valued mapping defined on a bounded closed convex subset of a complete CAT0space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT0spaces has been rapidly developed and much papers have appearedsee, e.g.,8–19.
weak convergence, and Dhompongsa and Panyanak22obtainedΔ-convergence theorems for the Picard, Mann and Ishikawa iterations in the CAT0space setting.
The purpose of this paper is to study the iterative scheme defined as follows. LetK is a nonempty closed convex subset of a complete CAT0spaceX with the nearest point projectionP fromXontoK. IfT:K → Xis a nonexpansive mapping with nonempty fixed point set, and if{xn}is generated iteratively by
x1∈K, xn1P
1−αnxn⊕αnTP
1−βn
xn⊕βnTxn
, 1.1
where{αn} and{βn} are real sequences inε,1−ε for someε ∈ 0,1,we show that the sequence{xn}defined by1.1 Δ-converges to a fixed point ofT.This is an analog of a result in Banach spaces of Shahzad23and also extends a result of Dhompongsa and Panyanak
22 to the case of nonself mappings. It is worth mentioning that our result immediately applies to any CATκspace withκ≤0 since any CATκspace is a CATκspace for every κ ≥κsee1, page 165.
2. Preliminaries and Lemmas
LetX, dbe a metric space. Ageodesic pathjoiningx∈Xtoy∈Xor, more briefly, ageodesic fromxtoyis a mapcfrom a closed interval0, l ⊂ RtoX such thatc0 x, cl y, anddct, ct |t−t|for allt, t ∈ 0, l.In particular,c is an isometry anddx, y l. The imageαofcis called a geodesicormetricsegmentjoiningxand y. When it is unique this geodesic segment is denoted byx, y. The spaceX, dis said to be ageodesic spaceif every two points ofXare joined by a geodesic, andXis said to beuniquely geodesicif there is exactly one geodesic joiningxandyfor eachx, y∈X.A subsetY ⊆Xis said to beconvexif Y includes every geodesic segment joining any two of its points.
Ageodesic trianglex1, x2, x3in a geodesic metric spaceX, dconsists of three points
x1, x2, x3inX theverticesofand a geodesic segment between each pair of verticesthe
edgesof. Acomparison trianglefor the geodesic trianglex1, x2, x3inX, dis a triangle
x1, x2, x3 : x1, x2, x3 in the Euclidean plane E2 such that dE2xi, xj dxi, xj for i, j∈ {1,2,3}.
A geodesic space is said to be a CAT0space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
CAT0: Letbe a geodesic triangle inXand letbe a comparison triangle for. Then is said to satisfy the CAT0inequalityif for allx, y ∈and all comparison points x, y∈,
dx, y≤dE2
x, y. 2.1
Ifx, y1, y2are points in a CAT0space and ify0is the midpoint of the segmenty1, y2,
then the CAT0inequality implies
dx, y0
2≤ 1
2d
x, y1
21
2d
x, y2
2−1
4d
y1, y2
2
This is theCNinequality of Bruhat and Tits24. In factcf.1, page 163, a geodesic space is a CAT0space if and only if it satisfies theCNinequality.
We now collect some elementary facts about CAT0 spaces which will be used frequently in the proofs of our main results.
Lemma 2.1. LetX, dbe a CAT(0) space.
i[1, Proposition 2.4] LetKbe a convex subset ofXwhich is complete in the induced metric. Then, for everyx ∈X,there exists a unique pointPx ∈K such thatdx, Px inf{dx, y : y∈K}.Moreover, the mapx→Pxis a nonexpansive retract fromXontoK.
ii[22, Lemma 2.1(iv)] Forx, y ∈X andt ∈ 0,1,there exists a unique pointz ∈x, y such that
dx, z tdx, y, dy, z 1−tdx, y. 2.2
one uses the notation1−tx⊕tyfor the unique pointzsatisfying (2.2).
iii[22, Lemma 2.4] Forx, y, z∈Xandt∈0,1,one has
d1−tx⊕ty, z≤1−tdx, z tdy, z. 2.3
iv[22, Lemma 2.5] Forx, y, z∈Xandt∈0,1,one has
d1−tx⊕ty, z2≤1−tdx, z2tdy, z2−t1−tdx, y2. 2.4
LetKbe a nonempty subset of a CAT0spaceX and letT:K → Xbe a mapping.T is callednonexpansiveif for eachx, y∈K,
dTx, Ty≤dx, y. 2.5
A pointx∈Kis called a fixed point ofTifxTx. We shall denote byFTthe set of fixed points ofT. The existence of fixed points for nonexpansive nonself mappings in a CAT0 space was proved by Kirk6as follows.
Theorem 2.2. LetKbe a bounded closed convex subset of a complete CAT(0) spaceX. Suppose that T:K → Xis a nonexpansive mapping for which
inf{dx, Tx:x∈K}0. 2.6
ThenT has a fixed point inK.
Let{xn}be a bounded sequence in a CAT0spaceX. Forx∈X,we set
rx,{xn} lim sup
n→ ∞ dx, xn. 2.7
Theasymptotic radiusr{xn}of{xn}is given by
and theasymptotic centerA{xn}of{xn}is the set
A{xn} {x∈X:rx,{xn} r{xn}}. 2.9
It is knownsee, e.g.,12, Proposition 7that in a CAT0space,A{xn}consists of exactly one point.
We now give the definition ofΔ-convergence.
Definition 2.3see20,21. A sequence{xn}in a CAT0spaceX is said toΔ-converge to x∈Xifxis the unique asymptotic center of{un}for every subsequence{un}of{xn}. In this case one writesΔ-limnxnxand callxtheΔ-limit of{xn}.
The following lemma was proved by Dhompongsa and Panyanak see22, Lemma 2.10.
Lemma 2.4. LetKbe a closed convex subset of a complete CAT(0) spaceX,and letT:K → Xbe a nonexpansive mapping. Suppose{xn}is a bounded sequence inKsuch thatlimndxn, Txn 0and {dxn, v}converges for allv∈FT, thenωwxn⊂FT.Hereωwxn:A{un}where the
union is taken over all subsequences{un}of{xn}.Moreover,ωwxnconsists of exactly one point.
We now turn to a wider class of spaces, namely, the class of hyperbolic spaces, which contains the class of CAT0spacesseeLemma 2.8.
Definition 2.5see16. A hyperbolic space is a tripleX, d, WwhereX, dis a metric space andW:X×X×0,1 → Xis such that
W1dz, Wx, y, α≤1−αdz, x αdz, y;
W2dWx, y, α, Wx, y, β |α−β|dx, y;
W3Wx, y, α Wy, x,1−α;
W4dWx, z, α, Wy, w, α≤1−αdx, y αdz, w
for allx, y, z, w∈X, α, β∈0,1.
It follows fromW1that for eachx, y∈Xandα∈0,1,
dx, Wx, y, α≤αdx, y, dy, Wx, y, α≤1−αdx, y. 2.10
In fact, we have
dx, Wx, y, ααdx, y, dy, Wx, y, α 1−αdx, y, 2.11
since if
we get
dx, y≤dx, Wx, y, αdWx, y, α, y
< αdx, y 1−αdx, y
dx, y,
2.13
which is a contradiction. By comparing between2.2and2.11, we can also use the notation
1−αx⊕αyforWx, y, αin a hyperbolic spaceX, d, W.
Definition 2.6see16. The hyperbolic spaceX, d, Wis called uniformly convex if for any r >0,andε∈0,2there exists aδ∈0,1such that for alla, x, y∈X,
dx, a≤r dy, a≤r dx, y≥εr
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭⇒d
1 2x⊕
1 2y, a
≤1−δr. 2.14
A mappingη:0,∞×0,2 → 0,1providing such aδ:ηr, εfor givenr >0 and ε∈0,2is called a modulus of uniform convexity.
Lemma 2.7see16, Lemma 7. LetX, d, Wbe a uniformly convex hyperbolic with modulus of uniform convexityη.For anyr >0, ε∈0,2,λ∈0,1anda, x, y∈X,
dx, a≤r dy, a≤r dx, y≥εr
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭⇒d
1−λx⊕λy, a≤1−2λ1−ληr, εr. 2.15
Lemma 2.8 see16, Proposition 8. Assume that X is a CAT(0) space. Then X is uniformly convex, and
ηr, ε ε
2
8 2.16
is a modulus of uniform convexity.
The following result is a characterization of uniformly convex hyperbolic spaces which is an analog of Lemma 1.3 of Schu25. It can be applied to a CAT0space as well.
Lemma 2.9. Let X, d, W be a uniformly convex hyperbolic space with modulus of convexityη, and letx ∈ X. Suppose thatη increases withr (for a fixedε) and suppose that{tn}is a sequence in b, c for someb, c ∈ 0,1and {xn},{yn} are sequences in X such that lim supndxn, x ≤ r,lim supndyn, x≤r,andlimnd1−tnxn⊕tnyn, x rfor somer≥0.Then
lim n→ ∞d
xn, yn
Proof. The caser 0 is trivial. Now supposer >0. If it is not the case thatdxn, yn → 0 as n → ∞,then there are subsequences, denoted by{xn}and{yn}, such that
inf n d
xn, yn
>0. 2.18
Chooseε∈0,1such that
dxn, yn
≥εr1>0 ∀n∈N. 2.19
Since 0 < b1−c ≤ 1/2 and 0 < ηr, ε ≤ 1, 0 < 2b1−cηr, ε ≤ 1.This implies 0 ≤ 1−2b1−cηr, ε<1.ChooseR∈r, r1such that
1−2b1−cηr, εR < r. 2.20
Since
lim sup
n dxn, x
≤r, lim sup n d
yn, x
≤r, r < R, 2.21
there are further subsequences again denoted by{xn}and{yn}, such that
dxn, x≤R, d
yn, x
≤R, dxn, yn
≥εR ∀n∈N. 2.22
Then byLemma 2.7and2.20,
d1−tnxn⊕tnyn, x
≤1−2tn1−tnηR, ε
R
≤1−2b1−cηr, εR < r 2.23
for alln∈N.Takingn → ∞,we obtain
lim n→ ∞d
1−tnxn⊕tnyn, x
< r, 2.24
which contradicts to the hypothesis.
3. Main Results
In this section, we prove our main theorems.
Theorem 3.1. Let K be a nonempty closed convex subset of a complete CAT(0) space X and let T:K → Xbe a nonexpansive mapping withx∗∈FT:{x∈K:Txx}.Let{αn}and{βn}be sequences inε,1−εfor someε∈0,1.Starting from arbitraryx1 ∈K,define the sequence{xn}
Proof. ByLemma 2.1ithe nearest point projectionP:X → Kis nonexpansive. Then
dxn1, x∗ d
P1−αnxn⊕αnTP
1−βn
xn⊕βnTxn
, P x∗
≤d1−αnxn⊕αnTP
1−βn
xn⊕βnTxn
, x∗
≤1−αndxn, x∗ αnd
TP1−βn
xn⊕βnTxn
, Tx∗
≤1−αndxn, x∗ αnd
P1−βn
xn⊕βnTxn
, x∗
≤1−αndxn, x∗ αn
1−βn
dxn, x∗ βndTxn, Tx∗
≤1−αndxn, x∗ αn
1−βn
dxn, x∗ βndxn, x∗
dxn, x∗.
3.1
Consequently, we have
dxn, x∗≤dx1, x∗ ∀n≥1. 3.2
This implies that{dxn, x∗}∞n1is bounded and decreasing. Hence limndxn, x∗exists.
Theorem 3.2. Let K be a nonempty closed convex subset of a complete CAT(0) space X and let T:K → Xbe a nonexpansive mapping withFT/∅.Let{αn}and{βn}be sequences inε,1−ε for someε∈0,1.From arbitraryx1∈K,define the sequence{xn}by the recursion1.1. Then
lim
n→ ∞dxn, Txn 0. 3.3
Proof. Letx∗∈FT.Then, byTheorem 3.1, limndxn, x∗exists. Let
lim n→ ∞dxn, x
∗ r.
3.4
If r 0,then by the nonexpansiveness of T the conclusion follows. If r > 0, we letyn P1−βnxn⊕βnTxn.ByLemma 2.1ivwe have
dyn, x∗ 2
dP1−βnxn⊕βnTxn
, P x∗2
≤d1−βnxn⊕βnTxn, x∗ 2
≤1−βn
dxn, x∗2βndTxn, x∗2−βn
1−βn
dxn, Txn2
≤1−βn
dxn, x∗2βndxn, x∗2
dxn, x∗2.
3.5
Therefore
dyn, x∗
≤d1−βn
xn⊕βnTxn, x∗
It follows from3.6andLemma 2.1ivthat
dxn1, x∗2d
P1−αnxn⊕αnTyn
, P x∗2
≤d1−αnxn⊕αnTyn, x∗ 2
≤1−αndxn, x∗2αnd
Tyn, x∗ 2−
αn1−αnd
xn, Tyn 2
≤1−αndxn, x∗2αndxn, x∗2−αn1−αnd
xn, Tyn 2
dxn, x∗2−αn1−αnd
xn, Tyn 2
.
3.7
Therefore
dxn1, x∗2≤dxn, x∗2−Wαnd
xn, Tyn 2
, 3.8
whereWα α1−α.Sinceαn∈ε,1−ε, Wαn≥ε2. By3.8, we have
ε2
∞
n1
dxn, Tyn 2≤
dx1, x∗2<∞. 3.9
This implies limn→ ∞dxn, Tyn 0.
SinceT is nonexpansive, we get thatdxn, x∗≤dxn, Tyn dyn, x∗,and hence
r≤lim inf n→ ∞ d
yn, x∗
. 3.10
On the other hand, we can get from3.6that
lim sup n→ ∞
dyn, x∗
≤r. 3.11
Thus limndyn, x∗ r. This fact and3.6imply
lim n→ ∞d
1−βn
xn⊕βnTxn, x∗
r. 3.12
SinceT is nonexpansive,
lim sup
n→ ∞ dTxn, x ∗≤r.
It follows from3.4,3.12,3.13, andLemma 2.9that
lim
n→ ∞dxn, Txn 0. 3.14
This completes the proof.
The following theorem is an analog of23, Theorem 3.5and extends22, Theorem 3.3to nonself mappings.
Theorem 3.3. Let K be a nonempty closed convex subset of a complete CAT(0) space X and let T:K → Xbe a nonexpansive mapping withFT/∅.Let{αn}and{βn}be sequences inε,1−ε for someε ∈ 0,1.From arbitraryx1 ∈ K,define the sequence{xn}by the recursion1.1. Then
{xn}Δ-converges to a fixed point ofT.
Proof. ByTheorem 3.2, limndxn, Txn 0.It follows from3.2that{dxn, v}is bounded and decreasing for eachv∈FT,and so it is convergent. ByLemma 2.4,ωwxnconsists of exactly one point and is contained inFT. This shows that the sequence{xn}Δ-converges to an element ofFT.
We now state two strong convergence theorems. Recall that a mappingT:K → Xis said to satisfyCondition I26if there exists a nondecreasing functionf :0,∞ → 0,∞ withf0 0 andfr>0 for allr >0 such that
dx, Tx≥fdx, FT ∀x∈K. 3.15
Theorem 3.4. Let K be a nonempty closed convex subset of a complete CAT(0) space X and let T:K → Xbe a nonexpansive mapping withFT/∅.Let{αn}and{βn}be sequences inε,1−ε for someε∈0,1.From arbitraryx1∈K,define the sequence{xn}by the recursion1.1. Suppose
thatTsatisfies condition I. Then{xn}converges strongly to a fixed point ofT.
Theorem 3.5. LetK be a nonempty compact convex subset of a complete CAT(0) spaceX and let T:K → Xbe a nonexpansive mapping withFT/∅.Let{αn}and{βn}be sequences inε,1−ε for someε ∈ 0,1.From arbitraryx1 ∈ K,define the sequence{xn}by the recursion1.1. Then
{xn}converges strongly to a fixed point ofT.
Another result in23is that the author obtains a common fixed point theorem of two nonexpansive self-mappings. The proof is metric in nature and carries over to the present setting. Therefore, we can state the following result.
Theorem 3.6. Let K be a nonempty closed convex subset of a complete CAT(0) space X and let S, T:K → Kbe two nonexpansive mappings withFS∩FT/∅.Let{αn}and{βn}be sequences inε,1−εfor someε∈0,1.From arbitraryx1∈K,define the sequence{xn}by the recursion
xn1 1−αnxn⊕αnS
1−βn
xn⊕βnTxn
. 3.16
Acknowledgments
The authors are grateful to Professor Sompong Dhompongsa for his suggestions and advices during the preparation of the article. The second author was supported by the Commission on Higher Education and Thailand Research Fund under Grant MRG5280025. This work is dedicated to Professor Wataru Takahashi.
References
1 M. Bridson and A. Haefliger,Metric Spaces of Non-Positive Curvature, vol. 319 ofFundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1999.
2 K. S. Brown,Buildings, Springer, New York, NY, USA, 1989.
3 K. Goebel and S. Reich,Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
4 D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2001.
5 M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, vol. 152 of Progress in Mathematics, Birkh¨auser, Boston, Mass, USA, 1999.
6 W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), vol. 64 of Colecc. Abierta, pp. 195–225, Seville University Publications, Seville, Spain, 2003.
7 W. A. Kirk, “Geodesic geometry and fixed point theory II,” inInternational Conference on Fixed Point Theory and Applications, pp. 113–142, Yokohama Publications, Yokohama, Japan, 2004.
8 P. Chaoha and A. Phon-on, “A note on fixed point sets in CAT0spaces,”Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 983–987, 2006.
9 S. Dhompongsa, W. Fupinwong, and A. Kaewkhao, “Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4268–4273, 2009.
10 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.
11 S. Dhompongsa, W. A. Kirk, and B. Panyanak, “Nonexpansive set-valued mappings in metric and Banach spaces,”Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 35–45, 2007.
12 S. Dhompongsa, W. A. Kirk, and B. Sims, “Fixed points of uniformly Lipschitzian mappings,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 4, pp. 762–772, 2006.
13 N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in metric spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4423–4429, 2009.
14 A. Kaewcharoen and W. A. Kirk, “Proximinality in geodesic spaces,”Abstract and Applied Analysis, vol. 2006, Article ID 43591, 10 pages, 2006.
15 W. A. Kirk, “Fixed point theorems in CAT0spaces andR-trees,”Fixed Point Theory and Applications, vol. 2004, no. 4, pp. 309–316, 2004.
16 L. Leustean, “A quadratic rate of asymptotic regularity for CAT0-spaces,”Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 386–399, 2007.
17 N. Shahzad, “Fixed point results for multimaps in CAT0spaces,”Topology and Its Applications, vol. 156, no. 5, pp. 997–1001, 2009.
18 N. Shahzad, “Invariant approximations in CAT0 spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4338–4340, 2009.
19 N. Shahzad and J. Markin, “Invariant approximations for commuting mappings in CAT0 and hyperconvex spaces,”Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1457–1464, 2008.
20 W. A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 12, pp. 3689–3696, 2008.
21 T. C. Lim, “Remarks on some fixed point theorems,”Proceedings of the American Mathematical Society, vol. 60, pp. 179–182, 1976.
23 N. Shahzad, “Approximating fixed points of non-self nonexpansive mappings in Banach spaces,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 6, pp. 1031–1039, 2005.
24 F. Bruhat and J. Tits, “Groupes r´eductifs sur un corps local,”Publications Math´ematiques de l’Institut des Hautes ´Etudes Scientifiques, no. 41, pp. 5–251, 1972.
25 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”
Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.
26 H. F. Senter and W. G. Dotson, Jr., “Approximating fixed points of nonexpansive mappings,”
