Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

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Volume 2012, Article ID 421050,11pages doi:10.1155/2012/421050

Research Article

Strong Convergence of Viscosity

Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Luo Yi Shi and Ru Dong Chen

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Correspondence should be addressed to Ru Dong Chen,[email protected] Received 30 March 2012; Accepted 27 April 2012

Academic Editor: Yonghong Yao

Copyrightq2012 L. Y. Shi and R. D. Chen. 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.

Viscosity approximation methods for nonexpansive mappings in CAT0 spaces are studied.

Consider a nonexpansive self-mappingTof a closed convex subsetCof a CAT0spaceX. Suppose that the set FixTof fixed points ofT is nonempty. For a contractionfonCandt ∈0,1, let xt ∈Cbe the unique fixed point of the contractionx→ tfx⊕1−tTx. We will show that if Xis a CAT0space satisfying some property, then{xt}converge strongly to a fixed point ofT which solves some variational inequality. Consider also the iteration process{xn}, wherex0∈Cis arbitrary andxn1αnfxn⊕1−αnTxnforn≥1, where{αn} ⊂0,1. It is shown that under certain appropriate conditions onαn,{xn}converge strongly to a fixed point ofT which solves some variational inequality.

1. CAT(0) Spaces

A metric space X is a CAT0 space if it is geodesically connected and if every geodesic triangle inX is at least as thin as its comparison triangle in the Euclidean plane. The precise definition is given below. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT0space. Other examples include pre-Hilbert spaces1, R-trees2, Euclidean buildings3, the complex Hilbert ball with a hyperbolic metric4, and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger1.

Fixed-point theory in CAT0spaces was first studied by Kirksee5,6. 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 many papers have appeared2,7–17.

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The purpose of this paper is to study the iterative scheme defined as follows. Consider a nonexpansive self-mappingT of a closed convex subsetCof a CAT0spaceX. Suppose that the set FixTof fixed points ofT is nonempty. For a contractionf onCandt∈ 0,1, letxt ∈ Cbe the unique fixed point of the contractionx → tfx⊕1−tTx. Consider the iteration process{xn}, wherex0∈Cis arbitrary and

x_n1αnfxn⊕1−αnTxn, 1.1

forn ≥ 1, where{αn} ⊂ 0,1. We show that{xn} converge strongly to a fixed point ofT under certain appropriate conditions onαn, and the fixed point ofTsolves some variational inequality.

LetX, dbe a metric space. A geodesic path joiningx∈Xtoy∈Xor, more briefly, a geodesic fromxtoyis a mapcfrom a closed interval0, l⊂RtoXsuch thatc0 x, cl y, anddct, ct |t−t|for allt, t ∈0, l. In particular,cis an isometry anddx, y l.

The imageαofcis called a geodesicor metricsegment joiningxandy. When it is unique, this geodesic segment is denoted byx, y. The spaceX, dis said to be a geodesic space if every two points ofXare joined by a geodesic, andXis said to be uniquely geodesic if there is exactly one geodesic joiningxandyfor eachx, y∈X. A subsetY ⊆Xis said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic trianglex1, x2, x3in a geodesicmetric spaceX, dconsists of three points x1, x2, andx3 inXthe vertices ofand a geodesic segment between each pair of vertices the edges of . A comparison triangle for the geodesic triangle x1, x2, x3 in X, d is a trianglex1, x2, x3:x1, x2, x3in the Euclidean planeE²such thatd_E²xi, xj dxi, xj fori, j∈1,2,3.

A geodesic space is said to be a CAT0 space if all geodesic triangles satisfy the following comparison axiom.

CAT0: letbe a geodesic triangle inX, and letbe a comparison triangle for. Then,is said to satisfy the CAT0inequality if for allx, y ∈ and all comparison points x, y∈ ,

d x, y

≤dE² x, y

. 1.2

Letx, y ∈ X, and by Lemma 2.1ivof18for eacht ∈ 0,1, there exists a unique pointz∈x, ysuch that

dx, z td x, y

, d

y, z

1−td x, y

. 1.3

From now on, we will use the notation1−tx⊕tyfor the unique pointzsatisfying the above equation.

We now collect some elementary facts about CAT0spaces which will be used in the proofs of our main results.

Lemma 1.1. LetXbe a CAT(0) space. Then,

i(see [18, Lemma 2.4]) for eachx, y, z∈Xandt∈0,1, one has

d

1−tx⊕ty, z

≤1−tdx, z td y, z

, 1.4

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ii(see [7]) for eachx, y, z∈Xandt, s∈0,1, one has

d

1−tx⊕ty,1−sx⊕sy

≤ |t−s|d x, y

, 1.5

iii(see [6, Lemma 3]) for eachx, y, z∈Xandt∈0,1, one has

d

1−tz⊕tx,1−tz⊕ty

≤td x, y

, 1.6

iv(see [18, Lemma 2.5]) for eachx, y, z∈Xandt∈0,1, one has

d

1−tx⊕ty, z2≤1−tdx, z²td

y, z2−t1−td x, y2

. 1.7

LetXbe a complete CAT0space, let{xn}be a bounded sequence in a complete X, and forx∈X, set

rx,{xn} lim sup

n→ ∞ dx, xn. 1.8

The asymptotic radiusr{xn}of{xn}is given by

r{xn} inf{rx,{xn}:x∈X}, 1.9

and the asymptotic centerA{xn}of{xn}is the set

A{xn} {x∈X:rx,{xn} r{xn}}. 1.10

It is knownsee, e.g.,11, Proposition 7that in a CAT0space,A{xn}consists of exactly one point.

A sequence{xn}inXis said to-converge tox∈Xifxis the unique asymptotic center of{un}for every subsequence{un}of{xn}. In this case, we write-limnxn xand callxthe -limit of{xn}.

Lemma 1.2. Assume thatXis a CAT(0) space. Then,

i(see [14]) every bounded sequence inXhas a-convergent subsequence,

ii(see [14, Proposition 3.7]) if K is a closed convex subset ofX, and f : K → X is a nonexpansive mapping, then the conditions{xn}-converge toxanddxn, fxn → 0 and implyx∈Kandfx x.

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Lemma 1.3see19, Proposition 3.5. Assume thatX is a CAT(0) space, Cis a closed convex subset ofX. Then the metric (nearest point) projectionPC:X → C, PCx:inf{dx, y;y∈C}is a nonexpansive mapping. one calls a CAT(0) spaceXsatisfying propertyPif forx, u, y1, y2∈X,

d

x, P_x,y₁u d

x, y1

≤d

x, P_x,y₂u d

x, y2

dx, ud y1, y2

. 1.11

Remark 1.4. The propertyPin Hilbert space corresponds to the inequality

u−x, y1−x ≤u−x, y2−xu−x ·y1−y2. 1.12

Recall that a continuous linear functionalμonl^∞, the Banach space of bounded real sequences, is called a Banach limit if μ μ1,1, . . . 1 and μnan μna_n1 for all {an} ∈l^∞.

Lemma 1.5see20, Proposition 2. Leta1, a2, . . .∈l^∞be such thatμnan≤0 for all Banach limitsμand lim sup_na_n1−an≤0. Then lim sup_nan≤0.

Lemma 1.6see21, Lemma 2.3. Let{sn}be a sequence of nonnegative real numbers,{αn}a sequence of real numbers in0,1with_∞

n1αn ∞,{un}a sequence of nonnegative real numbers with_∞

n1un<∞, and{tn}a sequence of real numbers with lim sup_ntn≤0. Suppose that

sn1 1−αnsnαntnun, ∀n∈N. 1.13

Then lim_n_{→ ∞}sn0.

2. Viscosity Iteration for a Single Mapping

In this section, we prove the main results of this paper.

Lemma 2.1. LetCbe a closed convex subset of a complete CAT(0) spaceX, and letT :C → Cbe a nonexpansive mapping. Letf be a contraction onCwith coeﬃcientα <1. For eacht∈0,1, the mappingSt:C → Cdefined by

Stxtfx⊕1−tTx, forx∈C 2.1

has a unique fixed pointxt∈C, that is,

xttfxt⊕1−tTxt. 2.2

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Proof. Forx, y∈C, according toLemma 1.1, we have the following:

d

Stx, St

y d

tfx⊕1−tTx, tf y

⊕1−tTy

≤d

tfx⊕1−tTx, tfx⊕1−tTy d

tf y

⊕1−tTx, tf y

⊕1−tTy

≤td

fx, f y

1−td

Tx, Ty

≤1−t1−αd x, y

.

2.3

This implies thatStis a contraction mapping, and hence, the conclusion follows.

The following result is to prove that the net {xt}converge strongly to a fixed point ofT.

Theorem 2.2. LetCbe a closed convex subset of a complete CAT(0) spaceXsatisfying the property P, and letT :C → Cbe a nonexpansive mapping. Letfbe a contraction onCwith coeﬃcientα <1.

For eacht∈0,1, let{xt}be given by

xttfxt⊕1−tTxt. 2.4

Then one has limt→0xt:xandxPFixTf x.

Proof. We first show that{xt}is bounded. Indeed choose ap∈FixT, and usingLemma 1.1 and the nonexpansive ofT, we derive that

d xt, p

d

tfxt⊕1−tTxt, p

≤td

fxt, p

1−td Txt, p

≤td

fxt, p

1−td xt, p

.

2.5

It follows that

d xt, p

≤d

fxt, p

≤d

fxt, f p

d f

p , p

≤αd xt, p

d f

p , p

. 2.6

Hence,

d xt, p

≤ 1 1−αd

f p

, p

, 2.7

and{xt}is bounded, so are{Txt}and{fxt}. As a result, we can get that

limt→0dxt, Txt lim

t→0d

tfxt⊕1−tTxt, Txt

lim

t→0td

fxt, Txt

0. 2.8

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Assume that{tn} ⊆ 0,1is such thattn → 0 asn → ∞. Putxn :xtn. We will show that{xn}contains a subsequence converging strongly tox, wherex∈FixT.

Since{xn}is bounded, byLemma 1.2i,ii, we may assume that{xn}-converges to a pointx, andx∈FixT.

Next we will prove that{xn}converge strongly tox.

Indeed, according toLemma 1.1and the property ofTandf, we can get that

d²xn,x d²

tnfxn⊕1−tnTxn,x

≤tnd²

fxn,x

1−tnd²Txn,x−tn1−tnd²

fxn, Txn

≤tnd²

fxn,x

1−tnd²xn,x−tn1−tnd²

fxn, Txn

.

2.9

It follows that

d²xn,x≤d²

fxn,x

−1−tnd²

fxn, Txn

d²

fxn,x

−d²

fxn, Txn

tnd²

fxn, Txn

. 2.10

Since lim_t→0dxt, Txt 0, we can get that

limsup

n→ ∞d²xn,x≤limsup

n→ ∞d²

fxn,x

−d²

fxn, xn

. 2.11

LetΔ x, xn, fxnbe a comparison triangle forx, xn, fxninE². Then,

d²

fxn,x

−d²

fxn, xn

d²

fxn,x

−d²

fxn, xn

fxn−x, fxn−x

−

fxn−xn, fxn−xn

2

fxn−x, xn−x

−

xn−x, xn−x 2

fxn−x, xn−x

−d² xn,x 2

fxn−x, xn−x

−d²xn,x.

2.12

Hence,

lim sup

n→ ∞d²xn,x≤lim sup

n→ ∞

fxn−x, xn−x

. 2.13

Letx, xn, fxbe a comparison triangle forx, xn, fxinE². For eachn, letun

be the point of the segment x, fxwhich is nearest to xn, and letun be the point of the segmentx, f xfor whichdun,x dun,x.

By passing to subsequences again, we may suppose that {un} converges to u ∈ x, fx,{un}converges tou∈x, f x.

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Since{xn}-converges to a pointx, we have r{xn} lim

n supdx, xn lim

n supdx, xn

≥lim

n supdun, xn lim

n supdu, xn

≥lim

n supdu, xn.

2.14

Thus,ru,{xn} ≤ r{xn}. This implies thatu xby uniqueness of the asymptotic center.

Hence,ux. That is to say,{un}converges tox, and{un}converges tox.

Moreover, sinceXsatisfies the propertyP, we can get that

fxn−x, xn−xd

x, P_x,fx_nxn

·d

x, fxn d

x, Px,fxnxn

·d

x, fxn

≤d

x, P_x,f_xxn

·d

x, fx d x, xn·d

fxn, fx d

x, P_x,f_xxn

·d

x, fx d x, xn·d

fxn, fx

≤d un,x

d

x, fx

αd²xn,x.

2.15

It follows that

limsup

n→ ∞d²xn,x≤ 1

1−αlimsup

n→ ∞d un,x

d

fx,x

. 2.16

Since{un} converges tox, we obtain that limsup_n_{→ ∞}d²xn,x 0, that is, {xn} converge strongly tox. Since{tn} ⊆0,1is such thattn → 0 asn → ∞is arbitrarily selected, we can get that lim_t→0xtx.

Finally, we will prove thatxsatisfy the equationxPFixTf x.

Indeed, for anyy∈FixT,

d xt, y

d

tfxt⊕1−tTxt, y td

fxt, y

1−td Txt, y

≤td

fxt, y

1−td xt, y

.

2.17

It follows that

d xt, y

≤d

fxt, y

. 2.18

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Since limt→0xtx, we can get that

d x, y

≤d

f x, y

. 2.19

Hence,

d

fx, y

≥d x, y

−d

x, fx, d

x, f x

≥d

f x, y

. 2.20

That is to say,xPFixTfx.

Consider now the iteration process

x0∈C,

x_n1αnfxn⊕1−αnTxn, n≥0, 2.21

where{αn} ⊆0,1satisfies H1αn → 0,

H2_∞

n0αn∞, H3either_∞

n0|αn1−αn|<∞or lim_n_{→ ∞}αn1/αn 1.

Theorem 2.3. LetX be a CAT(0) space satisfying the propertyP,Ca closed convex subset of X, T :C → Ca nonexpansive mapping with FixT/∅, andf :C → Ca contraction with coeﬃcient α < 1. Letx0 ∈ C,{xn}be generated byxn1 αnfxn⊕1−αnTxn, n ≥ 0. Then under the hypotheses (H1 )–(H3 ),xn → x, wherexPFixTfx.

Proof. We first show that the sequence{xn}is bounded. Letp∈FixT. Then,

d xn1, p

d

αnfxn⊕1−αnTxn, p

≤αnd

fxn, p

1−αnd Txn, p

≤αn

d

fxn, f p

d f

p , p

1−αnd xn, p

≤max

d xn, p

, 1 1−αd

f p

, p .

2.22

By induction, we have

d xn, p

≤max

d x0, p

, 1 1−αd

f p

, p

, 2.23

for alln∈N. This implies that{xn}is bounded and so is the sequence{Txn}and{fxn}.

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We claim thatdxn1, xn → 0. Indeed, we have

dx_n1, xn d

αnfxn⊕1−αnTxn, α_n−1fx_n−1⊕1−α_n−1Tx_n−1

≤d

αnfxn⊕1−αnTxn, αnfxn⊕1−αnTxn−1 d

αnfxn⊕1−αnTxn−1, αnfxn−1⊕1−αnTxn−1 d

αnfxn−1⊕1−αnTxn−1, αn−1fxn−1⊕1−αn−1Txn−1

≤1−αndTxn, Txn−1 αnd

fxn, fxn−1

|αn−αn−1|d

fxn−1, Txn−1

≤1−αndxn, x_n−1 αnαdxn, x_n−1 |αn−α_n−1|d

fx_n−1, Tx_n−1 .

2.24

By the conditions H2 and H3, we have

dx_n1, xn−→0. 2.25

Consequently, by the condition H1,

dxn, Txn≤dxn, xn1 dxn1, Txn dxn, x_n1 d

αnfxn⊕1−αnTxn, Txn

dxn, x_n1 αnd

fxn, Txn

−→0.

2.26

Since {xn} is bounded, we may assume that {xn}-converges to a point x. By Lemma 1.2, we havex∈FixT.

Next we will prove that{xn}converge strongly toxandx x. Indeed, according to Lemma 1.1and the property ofT andf, we can get that

d²xn1,x d²

tnfxn⊕1−tnTxn,x

≤αnd²

fxn,x

1−αnd²Txn,x −αn1−αnd²

fxn, Txn

≤1−αnd²xn,x αn

d²

fxn,x

−1−αnd²

fxn, Tn

.

2.27

With a minor modification of the proof of the analogous statement inTheorem 2.2, we can get that

d²

fxn,x

−d²

fxn, xn

2

fxn−x, x n−x

−d²xn,x

≤2d un,x

d

fx, x

2αd²xn,x −d²xn,x, 2.28

anddun,x → 0.

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Thus,

d²x_n1,x ≤1−αnd²xn,x αn

d²

fxn,x

−1−αnd²

fxn, xn

1−αnd²

fxn, xn

−1−αnd²Txn, xn

≤1−21−ααnd²xn,x 21−ααn

1 1−αd

un,x d

fx, x 1

1−αβn

,

2.29

whereβn 1−αnd²fxn, xn−1−αnd²Txn, xn. Sincedun,x → 0 anddxn, Txn → 0, we obtain that

limn sup 1 1−αd

un,x d

fx,x 1

1−αβn≤0. 2.30

According toLemma 1.6, we can getd²xn,x → 0.

Finally, we prove thatxx.

Indeed, for anyz∈FixT,

d²xn1, z≤αnd²

z, fxn

1−αnd²z, Txn−αn1−αnd²

fxn, Txn

≤αnd²

z, fxn

1−αnd²z, xn−αn1−αnd²

fxn, Txn

. 2.31

Letμbe a Banach limit. Then,

μnd²x_n1, z≤μnd²

z, fxn

−μnd²

fxn, Txn

. 2.32

Sincexn → x, we obtain that

d²x, z ≤d²

z, fx

−d²

fx,x

. 2.33

It follows that

d²

fx, x

≤d²

z, fx

, 2.34

that is to say,xPFixTfx. SincePFixTfis a contraction andxPFixTfx, we know that

xx.

Acknowledgment

This paper was supported by NSFC Grant no. 11071279.

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Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Research Article

Strong Convergence of Viscosity

Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Luo Yi Shi and Ru Dong Chen

1. CAT(0) Spaces

2. Viscosity Iteration for a Single Mapping

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

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Complex Analysis

Optimization

Combinatorics

Function Spaces

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