Strong Convergence of a Modified Halpern’s Iteration for Nonexpansive Mappings

Volume 2008, Article ID 649162,9pages doi:10.1155/2008/649162

Research Article

Strong Convergence of a Modified Halpern’s Iteration for Nonexpansive Mappings

Liang-Gen Hu

Department of Mathematics, Ningbo University, Ningbo 315211, China

Correspondence should be addressed to Liang-Gen Hu,[email protected] Received 12 September 2008; Accepted 9 December 2008

Recommended by Jerzy Jezierski

The purpose of this paper is to consider that a modified Halpern’s iterative sequence {xn} converges strongly to a fixed point of nonexpansive mappings in Banach spaces which have a uniformly Gˆateaux differentiable norm. Our result is an extension of the corresponding results.

Copyrightq2008 Liang-Gen Hu. 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

LetEbe a real Banach space andCa nonempty closed convex subset ofE. We denote byJ the normalized duality map from E to 2^E∗E^∗is the dual spaces ofEdefined by

Jx

f ∈E^∗:x, fx²f²,∀x∈E

. 1.1

A mappingT : C → C is said to be nonexpansive ifTx−Ty ≤ x−y, for all x, y ∈ C. We denote by FixT {x ∈ C : Tx x}the set of fixed points ofT. In the last ten years, many papers have been written on the approximation of fixed point for nonlinear mappings by using some iterative processessee, e.g.,1–18.

An explicit iterative process was initially introduced, in 1967, by Halpern3in the framework of Hilbert spaces, that is, Halpern’s iteration. For anyu, x0 ∈C, the sequence{xn}is defined by

xn1αnu 1−αn

Txn, ∀n≥0, 1.2

where{αn} ⊂0,1. He proved that the sequence{xn}converges weakly to a fixed point ofT, whereα_nn^−a,a∈0,1. In 1977, Lions8further proved that the sequence{x_n}converges

strongly to a fixed point ofTin a Hilbert space, where{αn}satisfies the following conditions:

C1 lim

n→ ∞αn0, C2 ^∞

n0

α_n ∞, C3 lim

n→ ∞

αn1−αn α²_n1 0.

1.3

But, in3,8, the real sequence{αn}excluded the canonical choiceαn 1/n1. In 1992, Wittmann16proved, still in Hilbert spaces, the strong convergence of the sequence1.2to a fixed point ofT, where{α_n}satisfies the following conditions:

C1 lim

n→ ∞α_n0, C2 ^∞

n0

α_n ∞,

C4 ^∞

n1

αn1−αn<∞.

1.4

The strong convergence of Halpern’s iteration to a fixed point of T has also been proved in Banach spacessee, e.g.,2,6,10–12,14,15,17,18. Reich10,11has showed the strong convergence of the sequence1.2, where{αn}satisfies the conditionsC1,C2, and C5.{αn}is decreasingnoting that the conditionC5is a special case of conditionC4.

In 1997, Shioji and Takahashi12extended Wittmann’s result to Banach spaces. In 2002, Xu 17obtained a strong convergence theorem, where{αn}satisfies the following conditions:

C1,C2, and C6. limn→ ∞|αn1−αn|/αn1 0. In particular, the canonical choice of α_n1/n1satisfies the conditionsC1,C2, andC6.

However, is a real sequence {αn} satisfying the conditions (C1) and (C2) sufficient to guarantee the strong convergence of the Halpern’s iteration 1.2 for nonexpansive mappings? It remains an open question.

Some mathematician considered the open question. A partial answer to this question was given independently by C. E. Chidume and C. O. Chidume2and Suzuki14. They defined the sequence{x_n}by

x_n1α_nu

1−α_n

1−δx_nδTx_n

, 1.5

whereδ∈0,1, Iis the identity, and obtained the strong convergence of the iteration1.5, where {α_n} satisfies the conditions C1and C2. Recently, Xu18 gave another partial answer to this question. He obtained the strong convergence of the iterative sequence

xn1 αn

1−δuδxn

1−αn

Txn, 1.6

whereδ∈0,1and{α_n}satisfies the conditionsC1andC2.

Inspired and motivated by the above researches, we introduce a modified Halpern’s iteration. For anyu, x0∈C, the sequence{x_n}is defined by

xn1αnuβnxnγnTxn, n≥0, I where{αn}, {βn},and{γn}are three real sequences in0,1, satisfyingαnβnγn 1. Clearly, the iterative sequenceIis a natural generalization of the well-known iterations.

iIf, for alln, we takeβ_n≡0 inI, then the sequenceIreduces to Halpern’s iteration 1.2.

iiIf, for alln, we takeα_n≡0 inI, then the sequenceIreduces to Mann iteration.

The purpose of this paper is to present a significant answer to the above open question. we will show that the sequence {α_n} satisfying the conditionsC1 and C2 is sufficient to guarantee the strong convergence of the modified Halpern’s iterative sequence for nonexpansive mappings. As we will see, our theorem extends the corresponding results in three aspects.1The real sequence{α_n}satisfies only the conditionsC1andC2.2 In contrast with the results2,14, we replace the sequence{1−αn1−δ}in1.5by an arbitrary sequence{β_n}in0,1.3In contrast with the result18, we replace the sequence {α_nδ}in1.6by an arbitrary sequence{β_n}in0,1.

2. Preliminaries

A Banach spaceEis said to have a Gˆateaux differentiable norm if the limit limt→0

xty − x

t 2.1

exists for eachx, y ∈U, whereU {x ∈E :x 1}.Eis called a uniformly Gˆateaux dif- ferentiable norm if for eachy∈U, the limit is attained uniformly forx∈U. It is well known that if the norm ofEis uniformly Gˆateaux differentiable norm, then the duality mapping is single-valued and norm-to-weak^∗uniformly continuous on each bounded subset ofE.

Lemma 2.1see9,10. LetCbe a nonempty closed convex subset of a Banach spaceEwhich has uniformly Gˆateaux differentiable norm andT :C → Ca nonexpansive mapping with FixT/∅.

Assume that every nonempty closed convex bounded subset of C has the fixed points property for nonexpansive mappings. Then there exists a continuous path:t → zt, t ∈ 0,1 satisfyingzt tu 1−tTz_t, for anyu∈C, which converges to a fixed point ofT.

Lemma 2.2see13. Let{xn}and{yn}be two bounded sequences in a Banach spaceEsuch that x_n1β_nx_n

1−β_n

y_n, n≥0, 2.2

where{β_n}is a sequence in0,1such that 0<lim inf_{n→ ∞}β_n≤lim sup_{n→ ∞}β_n<1. Assume lim sup

n→ ∞

yn1−yn−xn1−xn≤0. 2.3 Then lim_{n→ ∞}y_n−x_n0.

Lemma 2.3. LetEbe a real Banach space. Then the following inequality holds:

xy²≤ x²2 y, jxy

, ∀x, y∈E, ∀jxy∈Jxy. 2.4 Lemma 2.4 see 17. Let {a_n} be a sequence of nonnegative real numbers such that an1 ≤ 1 −δnan δnξn, ∀n ≥ 0, where {δn} is a sequence in 0,1 and {ξn} is a sequence inRsatisfying the following conditions:

i_∞

n1δ_n ∞;

iilim sup_{n→ ∞}ξ_n≤0 or_∞

n1δn|ξn|<∞;

Then lim_{n→ ∞}an 0.

3. Main results

Theorem 3.1. Let C be a nonempty closed convex subset of a real Banach space E which has a uniformly Gˆateaux differentiable norm. LetT :C → Cbe a nonexpansive mapping with FixT/∅.

Assume that{zt}converges strongly to a fixed pointzofTast → 0, whereztis the unique element of Cwhich satisfiesz_ttu 1−tTz_tfor anyu∈C. Let{α_n}, {β_n},and{γ_n}be three real sequences in0,1which satisfy the following conditions:C1lim_{n→ ∞}αn 0 andC2_∞

n0αn ∞. For anyx0 ∈C, the sequence{xn}is defined by the iteration inI. Then the sequence{xn}converges strongly to a fixed point ofT.

Proof. Take anyp∈FixT. FromI, it implies that x_n1−pα_nu−p β_n

x_n−p γ_n

Tx_n−p

≤αnu−p

1−αnxn−p. 3.1

Adopting mathematical induction, we obtain, for alln≥0, x_n−p≤maxx0−p,u−p

. 3.2

Therefore, we conclude that the sequence{x_n}is bounded. Next, we separate the proof into two cases.

Case 10<lim inf_{n→ ∞}βn≤lim sup_{n→ ∞}βn<1. Rewrite the iterative processIas follows:

xn1αnuβnxnγnTxn

βnxn

1−βnαnuγnTxn

1−βn

β_nx_n 1−β_n

y_n,

3.3

where

yn αn

1−β_nu γn

1−β_nTxn, ∀n≥0. 3.4

SinceT is a nonexpansive mapping and{xn}is bounded, we get that{yn} and{Txn}are bounded. Manipulating3.4yields

y_n1−y_n α_n1

1−βn1 − α_n 1−βn

u

1− α_n1 1−βn1

Tx_n1−Tx_n

αn

1−βn − αn1

1−βn1

Txn.

3.5

Consequently, we have

yn1−yn−xn1−xn≤ αn1

1−β_n1 − αn

1−β_n

u

1− αn1

1−β_n1

xn1−xn

αn

1−β_n − αn1

1−β_n1

Txn−xn1−xn.

3.6

From the fact that{x_n}and{Tx_n}are bounded and lim_{n→ ∞}α_n 0, it follows that lim sup

n→ ∞

y_n1−y_n−x_n1−x_n≤0. 3.7

Hence, byLemma 2.2, we get

nlim→ ∞y_n−x_n0. 3.8

Using lim_{n→ ∞}α_n0 and3.8, we obtain x_n1−β_nx_nγ_nTx_n

1−αn

≤α_n

u−β_nx_nγ_nTx_n 1−αn

−→0, n−→ ∞, xn1−xn≤

1−βnyn−xn−→0, n−→ ∞.

3.9

Clearly,

xn−β_nx_nγ_nTx_n 1−α_n γ_n

1−α_n

xn−Txn

. 3.10

Since lim_{n→ ∞}α_n 0 and lim sup_{n→ ∞}β_n < 1, we get that lim inf_{n→ ∞}γ_n >0. Therefore, from 3.9, and3.10, we find

xn−Txn 1−αn

γ_n

xn− βnxnγnTxn

1−α_n

−→0, n−→ ∞. 3.11

From lim_{n→ ∞}xn−Txn 0, it follows that there exists a positive numberNsuch thattn max{

x_n−Tx_n,1/n}, n > N. Obviously, we find

n→ ∞lim

xn−Txn

t_n 0. 3.12

Sincez_tis a unique solution of the equationz_ttu 1−tTz_t, we can write z_tn−x_n

1−t_n

Tz_tn−x_n t_n

u−x_n

. 3.13

UsingLemma 2.3, we get ztn−xn²≤

1−tn₂Tztn−xn²2tn u−xn, j

ztn−xn

≤

1−tn₂Tztn−TxnTxn−xn²2tn u−xn, j

ztn−xn

≤

1t²_nztn−xn²

1−2tnt²_nxn−Txn2ztn −xnxn−Txn 2tn u−ztn, j

ztn−xn .

3.14

Thus,

u−z_tn, j

x_n−z_tn

≤ t_n

2z_tn−x_n²

1t²_nxn−Txn 2t_n

2z_tn−x_nx_n−Tx_n. 3.15

From the fact that{x_n}, {z_tn},and{Tx_n}are bounded and3.12, it implies that lim sup

n→ ∞ u−ztn, j xn−ztn

≤0. 3.16

We know that u−z, j

x_n−z

u−z, j x_n−z

−j

x_n−z_tn

u−z_tn, j

x_n−z_tn z_tn−z, j

x_n−z_tn

. 3.17

Noting the hypothesis thatz_tn → z∈FixT,n → ∞, and the fact that{x_n}is bounded and the duality mappingjis norm-to-weak^∗uniformly continuous on bounded subset ofE, we have

z−z_tn, j

x_n−z_tn

−→0, n−→ ∞, u−z, j

xn−ztn

−j

xn−z

−→0, n−→ ∞. 3.18

Consequently, from3.16,3.17, and the two results mentioned above, we obtain lim sup

n→ ∞ u−z, j

xn−z

≤0. 3.19

EstimatingIyields

xn1−z²≤βn xn−z

γn

Txn−z²2αn u−z, j

xn1−z

≤

1−α_nx_n−z²2α_n u−z, j

x_n1−z

. 3.20

Therefore, combiningLemma 2.4with3.19and3.20, we get that limn→ ∞xn−z0.

Case 2lim_{n→ ∞}β_n1. Assume thatT_nxβ_n/1−α_nxγ_n/1−α_nTx, for alln≥0, then it is easy to see that for eachn∈N,Tnxxif and only ifTxx, that is,Tnhas the same set of fixed points ofT. Rewrite the iterative processIas follows:

xn1αnu 1−αn

Tnxn. 3.21

Since lim_{n→ ∞}αn0 and lim_{n→ ∞}βn1, and{Tnxn}is bounded, we have

xn1−xn≤αnuγnTnxn−→0, n−→ ∞. 3.22 Consequently, we get that

xn1−βnxnγnTxn

1−αn

≤αn

u−βnxnγnTxn

1−αn

−→0, n−→ ∞. 3.23

Combining3.22and3.23yields

xn−Tnxn≤xn−xn1xn1−Tnxn−→0, n−→ ∞. 3.24

Adopting the same proof as Case1, we can easily conclude that lim_{n→ ∞}x_n−z0.

If lim sup_{n→ ∞}β_n1, then we take arbitrarily the two subsequences{β_ni}and{β_nj}in {βn}such that lim_{i→ ∞}β_ni 1 and lim sup_{j→ ∞}βnj <1, and we obtain the strong convergence of the sequence{x_n}by employing the above-mentioned proof method.

Remark 3.2. If lim_{n→ ∞}β_n0 and lim inf_{n→ ∞}β_n0, then, from Xu’s results18and proof of Theorem 3.1, we obtain the strong convergence theorem.

As direct consequences ofTheorem 3.1, we obtain the two following corollaries.

Corollary 3.3see2, Theorem 3.1. LetE,C,andT be asTheorem 3.1. For a fixedδ ∈ 0,1, defineS:C → CbySx: 1−δxδTx, ∀x∈C. Assume that{zt}converges strongly to a fixed pointzofTast → 0, wherez_tis the unique element ofCwhich satisfiesz_ttu 1−tTz_t, for any

u∈C. Let{αn}be a real sequence in0,1satisfying the conditionsC1andC2. For anyx0∈C, the sequence{x_n}is defined by

x_n1α_nu 1−α_n

Sx_n. 3.25

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

Proof. If, in proof ofTheorem 3.1, we takeβ_n 1−α_n1−δ, lim_{n→ ∞}β_n 1−δ, then we get the desired conclusion.

Corollary 3.4see18, Theorem 3.1. LetEbe a uniformly smooth Banach space,Ca closed convex subset ofE, andT :C → Ca nonexpansive mapping such that FixT/∅. For any sequence{α_n} in0,1satisfying the conditionsC1andC2, numberδ ∈0,1,and vectorsu, x0 ∈C, define a sequence{xn}by the iterative algorithm

x_n1α_n

δu 1−δx_n

1−α_n

Tx_n, n≥0. 3.26

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

Remark 3.5. For any real sequence{β_n} ⊂ 0,1, the real sequence {α_n}satisfying the two conditionsC1andC2is sufficient for the strong convergence of the iterative sequenceII for nonexpansive mappings. Therefore, our results give a significant partial answer to the open question.

Acknowledgments

The authors are grateful to anonymous referees for careful reading of the manuscript and helpful suggestions. This work was supported partly by National Science Foundation of China 60872095, the Natural Science Foundation of Zhejiang Province Y606093, K.

C. Wong Magna Fund in Ningbo University, and Ningbo Natural Science Foundation 2008A610018.

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