and B. Panyanak

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Volume 2011, Article ID 869458,11pages doi:10.1155/2011/869458

Research Article

Strong Convergence of Modified Halpern Iterations in CAT(0) Spaces

A. Cuntavepanit

¹

and B. Panyanak

^{1, 2}

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

2Materials Science Research Center, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to B. Panyanak,[email protected] Received 28 November 2010; Accepted 10 January 2011

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 A. Cuntavepanit 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.

Strong convergence theorems are established for the modified Halpern iterations of nonexpansive mappings in CAT0spaces. Our results extend and improve the recent ones announced by Kim and Xu2005, Hu2008, Song and Chen2008, Saejung2010, and many others.

1. Introduction

LetCbe a nonempty subset of a metric spaceX, d. A mappingT : C → Cis said to be nonexpansive if

d

Tx, Ty

≤d x, y

, ∀x, y∈C. 1.1

A pointx∈Cis called a fixed point ofT ifx Tx. We will denote byFTthe set of fixed points ofT. In 1967, Halpern1introduced an explicit iterative scheme for a nonexpansive mappingT on a subsetCof a Hilbert space by taking any pointsu, x1 ∈Cand defined the iterative sequence{xn}by

xn1 αnu 1−αnTxn, forn≥1, 1.2

where αn ∈ 0,1. He pointed out that the control conditions:C1 lim_nαn 0 and C2 _∞

n1αn ∞are necessary for the convergence of{xn}to a fixed point ofT. Subsequently, many mathematicians worked on the Halpern iterations both in Hilbert and Banach spaces

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see, e.g., 2–11 and the references therein. Among other things, Wittmann 7 proved strong convergence of the Halpern iteration under the control conditionsC1,C2, andC4 _∞

n1|αn1−αn|<∞in a Hilbert space. In 2005, Kim and Xu12generalized Wittmann’s result by introducing a modified Halpern iteration in a Banach space as follows. LetCbe a closed convex subset of a uniformly smooth Banach spaceX, and letT :C → Cbe a nonexpansive mapping. For any pointsu, x1∈C, the sequence{xn}is defined by

x_n1βnu 1−βn

Tαnxn 1−αnTxn, forn≥1, 1.3

where {αn} and {βn} are sequences in 0,1. They proved under the following control conditions:

D1 lim

n αn 0, lim

n βn0, D2 ^∞

n1

αn∞, ^∞

n1

βn∞,

D3 ^∞

n1

|α_n1−αn|<∞, ^∞

n1

β_n1−βn<∞,

1.4

that the sequence{xn}converges strongly to a fixed point ofT.

The purpose of this paper is to extend Kim-Xu’s result to a special kind of metric spaces, namely, CAT0spaces. We also prove a strong convergence theorem for another kind of modified Halpern iteration defined by Hu13in this setting.

2. CAT(0) Spaces

A metric space X is a CAT0 space if it is geodesically connected and if every geodesic triangle inXis 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 spacessee14,R-treessee15, Euclidean buildingssee16, the complex Hilbert ball with a hyperbolic metricsee17, 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 Haefliger14.

Fixed point theory in CAT0 spaces was first studied by Kirk see 18, 19. 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 appearedsee, e.g.,20–31and the references therein. It is worth mentioning that fixed point theorems in CAT0spacesspecially inR-treescan be applied to graph theory, biology, and computer sciencesee, e.g.,15,32–35.

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

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two points of X are joined by a geodesic, andX is 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 triangleΔ x1, x2, x3in a geodesic metric spaceX, dconsists of three points x1, x2, andx3inXthe vertices ofΔand a geodesic segment between each pair of vertices the edges of Δ. A comparison triangle for the geodesic triangle Δx1, x2, x3 in X, d is a triangleΔx1, x2, x3: Δx1, x2, x3in the Euclidean planeE²such thatdE²xi, 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: letΔbe a geodesic triangle inX, and letΔbe a comparison triangle forΔ.

Then,Δis said to satisfy the CAT0inequality if for allx, y ∈ Δand all comparison points x, y∈Δ,

d x, y

≤d_E² x, y

. 2.1

Letx, y ∈X, and by Lemma 2.1ivof23for eacht∈ 0,1, there exists a unique pointz∈x, ysuch that

dx, z td x, y

, d

y, z

1−td x, y

. 2.2

From now on, we will use the notation1−tx⊕tyfor the unique pointzsatisfying2.2. We now collect some elementary facts about CAT0spaces which will be used in the proofs of our main results.

Lemma 2.1. LetXbe a CAT0space. Then,

i(see [23, Lemma 2.4]) for eachx, y, z∈Xandt∈0,1, one has d

1−tx⊕ty, z

≤1−tdx, z td y, z

, 2.3

ii(see [21]) for eachx, y∈Xandt, s∈0,1, one has d

1−tx⊕ty,1−sx⊕sy

|t−s|d x, y

, 2.4

iii(see [19, Lemma 3]) for eachx, y, z∈Xandt∈0,1, one has d

1−tz⊕tx,1−tz⊕ty

≤td x, y

, 2.5

iv(see [23, 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

. 2.6

Recall that a continuous linear functionalμon∞, the Banach space of bounded real sequences, is called a Banach limit ifμμ1,1, . . . 1 andμnan μnan1for all{an} ∈ _∞.

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Lemma 2.2see8, Proposition 2. Let{a1, a2, . . .} ∈∞be such thatμnan≤0 for all Banach limitsμand lim sup_na_n1−an≤0. Then, lim sup_nan≤0.

Lemma 2.3see28, Lemma 2.1. LetCbe a closed convex subset of a complete CAT0spaceX, and letT :C → Cbe a nonexpansive mapping. Letu∈Cbe fixed. For eacht∈0,1, the mapping St:C → Cdefined by

Stztu⊕1−tTz, forz∈C 2.7

has a unique fixed pointzt∈C, that is,

ztStzt tu⊕1−tTzt. 2.8

Lemma 2.4see28, Lemma 2.2. LetCandTbe as the preceding lemma. Then,FT/∅if and only if{zt}given by2.8remains bounded ast → 0. In this case, the following statements hold:

1{zt}converges to the unique fixed pointzofT which is nearestu,

2d²u, z ≤ μnd²u, xn for all Banach limits μ and all bounded sequences {xn} with lim_ndxn, Txn 0.

Lemma 2.5 see 10, Lemma 2.1. Let {αn}^∞_n1 be a sequence of nonnegative real numbers satisfying the condition

αn1≤ 1−γn

αnγnσn, n≥1, 2.9

where{γn}and{σn}are sequences of real numbers such that 1{γn} ⊂0,1and_∞

n1γn∞, 2either lim sup_n_{→ ∞}σn≤0 or_∞

n1|γnσn|<∞.

Then, limn→ ∞αn0.

Lemma 2.6see27,36. Let{xn}and{yn}be bounded sequences in a CAT0spaceX, and let {αn}be a sequence in0,1with 0<lim infnαn≤lim sup_nαn <1. Suppose thatxn1αnyn⊕1− αnxnfor alln∈Nand

lim sup

n→ ∞

d yn1, yn

−dxn1, xn

≤0. 2.10

Then, lim_ndxn, yn 0.

3. Main Results

The following result is an analog of Theorem 1 of Kim and Xu12. They prove the theorem by using the concept of duality mapping, while we use the concept of Banach limit. We also observe that the condition_∞

n1αn∞in12, Theorem 1is superfluous.

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Theorem 3.1. Let Cbe a nonempty closed convex subset of a complete CAT0 spaceX, and let T :C → Cbe a nonexpansive mapping such thatFT/∅. Given a pointu∈Cand sequences{αn} and{βn}in0,1, the following conditions are satisfied:

(A1) lim_nαn0 and_∞

n1|αn1−αn|<∞, (A2) limnβn0,_∞

n1βn∞and_∞

n1|βn1−βn|<∞.

Define a sequence{xn}inCbyx1x∈Carbitrarily, and

xn1βnu⊕ 1−βn

αnxn⊕1−αnTxn, ∀n≥1. 3.1

Then,{xn}converges to a fixed pointz∈FTwhich is nearestu.

Proof. For eachn ≥ 1, we letyn : αnxn ⊕1−αnTxn. We divide the proof into 3 steps.

i We will show that {xn}, {yn}, and {Txn} are bounded sequences. ii We show that limndxn, Txn 0. Finally, we show thatiii{xn}converges to a fixed pointz∈FTwhich is nearestu.

i As in the first part of the proof of 12, Theorem 1, we can show that {xn} is bounded and so is{yn}and{Txn}. Notice also that

d yn, p

≤d xn, p

, ∀p∈FT. 3.2

iiIt suﬃces to show that

nlim→ ∞dxn, xn1 0. 3.3

Indeed, if3.3holds, we obtain

dxn, Txn≤dxn, xn1 d xn1, yn

d yn, Txn

dxn, xn1 d

βnu⊕ 1−βn

yn, yn

dαnxn⊕1−αnTxn, Txn

≤dxn, x_n1 βnd u, yn

αndxn, Txn−→0, asn−→ ∞.

3.4

By usingLemma 2.1, we get

dx_n1, xn d βnu⊕

1−βn

yn, β_n−1u⊕

1−β_n−1 y_n−1

≤d βnu⊕

1−βn

yn, βnu⊕ 1−βn

yn−1 d

βnu⊕ 1−βn

y_n−1, β_n−1u⊕

1−β_n−1 y_n−1

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≤ 1−βn

d

yn, yn−1

βn−βn−1d u, yn−1

1−βn

dαnxn⊕1−αnTxn, α_n−1x_n−1⊕1−α_n−1Tx_n−1 βn−βn−1du, αn−1xn−1⊕1−αn−1Txn−1

≤ 1−βn

dαnxn⊕1−αnTxn, αnxn−1⊕1−αnTxn

dαnx_n−1⊕1−αnTxn, αnx_n−1⊕1−αnTx_n−1 dαnxn−1⊕1−αnTxn−1, αn−1xn−1⊕1−αn−1Txn−1 βn−βn−1αn−1du, xn−1 1−αn−1du, Txn−1

≤ 1−βn

αndxn, xn−1 1−αndTxn, Txn−1 |αn−αn−1|dx_n−1, Txn−1 βn−β_n−1α_n−1du, x_n−1 1−α_n−1du, Tx_n−1

1−βn

dxn, xn−1 1−βn

|αn−αn−1|dxn−1, Txn−1 βn−βn−1αn−1du, xn−1 βn−βn−11−αn−1du, Txn−1

≤ 1−βn

dxn, xn−1 1−βn

|αn−αn−1|dxn−1, Txn−1

βn−βn−1αn−1du, Tx_n−1 dTx_n−1, xn−1 βn−βn−1du, Txn−1−βn−βn−1αn−1du, Txn−1

1−βn

dxn, xn−1 1−βn

|αn−αn−1|dxn−1, Txn−1

βn−βn−1αn−1dxn−1, Txn−1 βn−βn−1du, Txn−1.

3.5

Hence,

dxn1, xn≤ 1−βn

dxn, xn−1 γ

|αn−αn−1|2βn−βn−1, 3.6 whereγ > 0 is a constant such thatγ ≥ max{du, Txn−1, dxn−1, Txn−1} for alln ∈ N. By assumptions, we have

nlim→ ∞βn0,

∞ n1

βn∞, ^∞

n1

|αn−αn−1|2βn−βn−1<∞. 3.7 Hence,Lemma 2.5is applicable to3.6, and we obtain limndx_n1, xn 0.

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iiiFromLemma 2.3, letzlimt→0zt, whereztis given by2.8. Then,zis the point ofFTwhich is nearestu. We observe that

d²xn1, z d² βnu⊕

1−βn

yn, z

≤βnd²u, z 1−βn

d² yn, z

−βn

1−βn

d² u, yn

≤βnd²u, z 1−βn

d²xn, z−βn

1−βn

d² u, yn

1−βn

d²xn, z βn

d²u, z− 1−βn

d² u, yn

.

3.8

ByLemma 2.4, we haveμnd²u, z−d²u, xn ≤ 0 for all Banach limitμ. Moreover, since lim_ndx_n1, xn 0,

lim sup

n→ ∞ d²u, z−d²u, xn1

− d²u, z−d²u, xn

0. 3.9

It follows from limndyn, xn 0 andLemma 2.2that

lim sup

n→ ∞ d²u, z− 1−βn

d² u, yn

lim sup

n→ ∞ d²u, z−d²u, xn

≤0. 3.10

Hence, the conclusion follows fromLemma 2.5.

By using the similar technique as in the proof of Theorem 3.1, we can obtain a strong convergence theorem which is an analog of 13, Theorem 3.1 see also37,38for subsequence comments.

Theorem 3.2. Let C be a nonempty closed and convex subset of a complete CAT0spaceX, and let T :C → Cbe a nonexpansive mapping such thatFT/∅. Given a pointu∈Cand an initial value x1∈C. The sequence{xn}is defined iteratively by

xn1βnxn⊕ 1−βn

αnu⊕1−αnTxn, n≥1. 3.11

Suppose that both{αn}and{βn}are sequences in0,1satisfying (B1) lim_n_{→ ∞}αn0,

(B2)_∞

n1αn∞,

(B3) 0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1.

Then,{xn}converges to a fixed pointz∈FTwhich is nearestu.

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Proof. Letyn:αnu⊕1−αnTxn. We divide the proof into 3 steps.

Step 1. We show that{xn},{yn}, and{Txn}are bounded sequences. Letp ∈ FT, then we have

d xn1, p

d

βnxn⊕ 1−βn

αnu⊕1−αnTxn, p

≤βnd xn, p

1−βn

d

αnu⊕1−αnTxn, p

≤βnd xn, p

1−βn

αnd u, p

1−βn

1−αnd Txn, p

≤ βn

1−βn

1−αn d

xn, p

1−βn

αnd u, p

1− 1−βn

αn

d xn, p

1−βn

αnd u, p

≤max d

xn, p , d

u, p .

3.12

Now, an induction yields

d x_n1, p

≤max d

x1, p , d

u, p

, n≥1. 3.13

Hence,{xn}is bounded and so are{yn}and{Txn}.

Step 2. We show that lim_ndxn, Txn 0. By usingLemma 2.1, we get

d yn1, yn

dαn1u⊕1−αn1Txn1, αnu⊕1−αnTxn

≤αndα_n1u⊕1−α_n1Tx_n1, u

1−αndαn1u⊕1−αn1Txn1, Txn

≤αn1−αn1dTxn1, u 1−αnαn1du, Txn 1−αn1−α_n1dTx_n1, Txn

≤αn1−αn1dTxn1, u 1−αnαn1du, Txn 1−αn1−αn1dx_n1, xn.

3.14

This implies that

d yn1, yn

−dxn1, xn≤αn1−αn1dTxn1, u 1−αnαn1du, Txn

αnα_n1−αn−α_n1dx_n1, xn. 3.15

Since{xn}and{Txn}are bounded and limn→ ∞αn0, it follows that

lim sup

n→ ∞

d yn1, yn

−dxn1, xn

≤0. 3.16

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Hence, byLemma 2.6, we get

nlim→ ∞d xn, yn

0. 3.17

On the other hand, d

yn, Txn

dαnu⊕1−αnTxn, Txn≤αndu, Txn−→0, asn−→ ∞. 3.18

Using3.17and3.18, we get dxn, Txn≤d

xn, yn

d yn, Txn

−→0, asn−→ ∞. 3.19

Step 3. We show that{xn}converges to a fixed point ofT. Letzlim_t→0zt, whereztis given by2.8, thenz∈FT. Finally, we show that limnxn z

d²xn1, z d²

βnxn⊕ 1−βn

yn, z

≤βnd²xn, z 1−βn

d² yn, z

−βn

1−βn

d² xn, yn

≤βnd²xn, z 1−βn

d²αnu⊕1−αnTxn, z−βn

1−βn

d² xn, yn

≤ 1−βn

αnd²u, z 1−αnd²Txn, z−αn1−αnd²u, Txn

−βn

1−βn

d² xn, yn

βnd²xn, z

≤ βn

1−βn

1−αn

d²xn, z 1−βn

αn

d²u, z−1−αnd²u, Txn

1− 1−βn

αn

d²xn, z 1−βn

αn

d²u, z−1−αnd²u, Txn .

3.20

ByLemma 2.4, we haveμnd²u, z−d²u, xn≤0 for all Banach limitμ. Moreover, since dx_n1, xn d

βnxn⊕ 1−βn

yn, xn

≤ 1−βn

d yn, xn

−→0, asn−→ ∞,

lim sup

n→ ∞ d²u, z d²u, x_n1−d²u, z−d²u, xn 0,

3.21

it follows from conditionB1, limndxn, Txn 0 andLemma 2.2that

lim sup

n→ ∞ d²u, z−1−αnd²u, Txn

lim sup

n→ ∞ d²u, z−d²u, xn

≤0. 3.22

Hence, the conclusion follows byLemma 2.5.

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Acknowledgments

The authors are grateful to Professor Sompong Dhompongsa for his suggestions and advices during the preparation of the paper. This research was supported by the National Research University Project under Thailand’s Oﬃce of the Higher Education Commission.

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