Strong Convergence Theorem of Implicit Iteration Process for Generalized Asymptotically Nonexpansive Mappings in Hilbert Space

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International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 649510,9pages

doi:10.1155/2008/649510

Research Article

Strong Convergence Theorem of Implicit

Iteration Process for Generalized Asymptotically

Nonexpansive Mappings in Hilbert Space

Lili He, Lei Deng, and Jianjun Liu

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Correspondence should be addressed to Lei Deng,denglei [email protected]

Received 21 May 2008; Accepted 14 August 2008

Recommended by Petru Jebelean

LetCbe a nonempty closed and convex subset of a Hilbert spaceH, letT andS : C → Cbe two commutative generalized asymptotically nonexpansive mappings. We introduce an implicit iteration process ofSandTdefined byxnαnx0 1−αn2/n1n2

_n

k0

ijkSiTjxn,

and then prove that{xn}converges strongly to a common fixed point ofSand T. The results

generalize and unify the corresponding results.

Copyrightq2008 Lili He et al. 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

LetHbe a Hilbert space,Ca nonempty closed and convex subset ofH. A mappingSis said to be generalized asymptotically nonexpansive if

_Sn_x₋_Sn_y_≤_k

nx−yrn 1.1

withkn ≥ 0,rn ≥0,kn →1,rn → 0n→ ∞, for eachx, y ∈C,n0,1,2, . . . .Ifkn 1 and

rn 0, 1.1reduces to nonexpansive mapping; if rn 0, 1.1 reduces to asymptotically nonexpansive mapping; if k_n 1, 1.1 reduces to asymptotically nonexpansive-type mapping. So, a generalized asymptotically nonexpansive mapping is much more general than many other mappings.

Browder1introduced the following implicit iteration process of a nonexpansive self-mapping for arbitraryx0∈C:

xnαnx0

1−αnTxn, 1.2

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After that, Baillon 2 proved the first nonlinear ergodic theorem: let T be a nonexpansive self-mapping defined on a nonempty bounded closed and convex set. Then for arbitraryx∈C,{1/n1n_i₀Tix}converges weakly to a fixed point ofT. Those results have been extended by several authorssee, e.g.,3–7.

Recently, Shimizu and Takahashi8studied the following implicit iteration process of asymptotically nonexpansive mappings for arbitraryx0∈C:

xnαnx0

1−αn_n1₁ n

i0

Ti_x

n, 1.3

whereαn bn−1/bn−1a,bn 1/n1ni01|1−ki|e−i,{kn}is the asymptotical coeﬃcient ofT. And then proved that{x_n}converges strongly to the fixed point ofT which is nearest tox0.

They also studied the following explicit iteration process of two commutative nonexpansive self-mappings in9:

xn1αnx0

1−αn_n₁2_n₂ n

k0

ijk

Si_Tj_x

n, 1.4

where {αn} ⊆ 0,1, limn→∞αn 0, ∞n0αn ∞. And then proved that {xn} converges strongly to the common fixed point ofSandTwhich is nearest tox0.

In this paper, we prove the strong convergence theorem of implicit iteration process for generalized asymptotically nonexpansive mappings which is defined by

xnαnx0

1−α_n 2

n1n2

n

k0

ijk

Si_Tj_x

n, 1.5

where{α_n}⊆0,1, lim_n_→∞α_n 0 and lim sup_n_→∞1−α_n2/n1n2n_k₀_i_j_kk_it_j<1. It is well known that a Hilbert space H satisfies Opial’s condition 5, that is, if a sequence{xn}converges weakly to an elementy∈Handy /z, then

lim inf

n→∞ xn−y<lim infn→∞ xn−z. 1.6

Throughout this paper, S and T are two commutative generalized asymptotically nonexpansive mappings. We denote byFSand FTthe set of fixed points of Sand T, respectively; and suppose thatFSFT/∅.SandT satisfy the following conditions:

_Sn_x₋_Sn_y_≤_k

nx−yrn,

_Tn_x₋_Tn_y_≤_t

nx−ybn,

1.7

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2. Auxiliary lemmas

This section collects some lemmas which will be used to prove the main results in the next section.

Lemma 2.1see9. LettingL_n n1n2/2, there holds the identity in a Hilbert spaceH:

_y_n₋_v2 1

Ln n

k0

ijk

_x_i,j₋_v2₋ 1

Ln n

k0

ijk

_x_i,j₋_y_n2

2.1

for{x_i,j}∞_i,j₀⊆H,y_n 1/L_nn_k₀_i_j_kx_i,j ∈H, andv∈H.

Lemma 2.2. LetCbe a nonempty bounded closed convex subset of a Hilbert spaceH,SandT two generalized asymptotically nonexpansive mappings ofCinto itself such thatSTTS. For anyx∈C, putFnx 2/n1n2nk0

ijkSiTjx. Then

lim l→∞nlim→∞supx∈C

_F_n_x₋_Sl_F

nx0, lim

l→∞ lim n→∞

sup x∈C

_F_n_x₋_Tl_F_n_x₀_. 2.2

Proof. Putxi,j SiTjx,v SlFnxandLn n1n2/2. It follows fromLemma 2.1 that

_F_n_x₋_Sl_F nx2

_L1

n n

k0

ijk

_Si_Tj_x₋_Sl_F

nx2−_L1 n

n

k0

ijk

_Si_Tj_x₋_F nx2

_L1

n l−1

k0

ijk

nx2_L1 n

n

kl

ijk, i≤l−1

_Si_Tj_x₋_Sl_F nx2

1

Ln n

kl

ijk, i≥l

nx2− _L1 n

n

k0

ijk

_Si_Tj_x₋_F nx2

≤ _L1 n

l−1

k0

ijk

nx2_L1 n

n

kl

ijk, i≤l−1

_L1

n n

kl

ijk ,i≥l

klSi−lTjx−Fnxrl2−_L1 n

n

k0

ijk

_Si_Tj_x₋_F nx2

1

Ln l−1

k0

ijk

nx2_L1 n

n

kl

ijk, i≤l−1

_L1

n n−l

k0

ijk

k2

lSiTjx−Fnx22klrlSiTjx−Fnxrl2

−_L1 n

n

k0

ijk

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≤ 1

Ln l−1

k0

ijk

nx2_L1 n

n

kl

ijk, i≤l−1

_L1

n n−l

k0

ijk

k2

l −1SiTjx−Fnx2 2klrl

Ln n−l

k0

ijk

_Si_Tj_x₋_F nx

n_n2−ln1−l

2n1 r 2 l.

2.3

Choosep∈FSFT, then there exists a constantM >0 such that

_Si_Tj_x₋_p_≤_k_i_Tj_x₋_p_r_i_≤_k_i_t_j_x₋_p_k_i_b_j_r_i_≤ M 2 ,

_F_n_x₋_p_≤ 1

Ln n

k0

ijk

_Si_Tj_x₋_p_≤ M 2 ,

_Sl_F

nx−p≤klFnx−prl≤ M₂

2.4

for all nonnegative integeri,j,l, andn. Hence,SiTjx−SlFnx ≤M,SiTjx−Fnx ≤M for all nonnegative integeri, j, l,andn. So

sup x∈C

_F_n_x₋_Sl_F nx2

≤ _nl1l 2n1M

2 2n1−ll

n2n1M

2

k2 l −1

n2−ln1−l

n2n1 M

2

2klrln2−ln1−l

n2n1 M

n2−ln1−l

n2n1 r

2 l

−→0n−→ ∞, l−→ ∞.

2.5

Similarly, we can prove that

lim l→∞

lim n→∞

sup x∈C

_F_n_x₋_Tl_F

nx0. 2.6

Remark 2.3. Lemma 2.2extends8, Lemma 3and9, Lemma 1.

Lemma 2.4. Let S and T be two continuous generalized asymptotically nonexpansive mappings defined on a nonempty bounded closed convex subsetC of a Hilbert spaceH withST TS. Let

Ln n1n2/2. If{xn}is a sequence inCsuch that{xn}converges weakly to somex∈C

and{xn−1/Lnnk0

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Proof. We claim that {Slx} converges strongly toxas l → ∞. If not, there exist a positive numberε0 and a subsequence{lm}of{l}such thatSlmx−x ≥ ε0 for allm. However, we have

_x_n_i₋_Slm_x_≤_x

ni−

1

Lni

ni

k0

ijk

Si_Tj_x ni _L1 ni ni

k0

ijk

Si_Tj_x ni−Slm

1

Lni

ni

k0

ijk

Si_Tj_x ni

Slm

1

Lni

ni

k0

ijk

Si_Tj_x

ni −Slmx

≤xni−

1

Lni

ni

k0

ijk

k0

ijk

1

Lni

ni

k0

ijk

Si_Tj_x ni

klm _L1_n

i

ni

k0

ijk

Si_Tj_x ni−x

rlm

≤xni−

1

Lni

ni

k0

ijk

k0

ijk

1

Lni

ni

k0

ijk

Si_Tj_x ni

klm _L1_n

i

ni

k0

ijk

Si_Tj_x ni−xni

klmxni −xrlm.

2.7

By Opial’s condition, for anyy∈Cwithy /x, we have

lim inf

n→∞ xn−x<lim infn→∞ xn−y. 2.8 Letrlim infn→∞xn−xand choose a positive numberρsuch that

ρ <

r2 ε 2 0

4 −r. 2.9

Then, there exists a subsequence{x_n_i}of{x_n}such that lim_i_→∞x_n_i −x randx_n_i −x<

rρ/5 for alli. By definition of{klm}and{rlm}, there exists a positive integerm0such that

klmxni−x< r

ρ

5, rlm <

ρ

5 2.10

for allm > m0. Since

lim n→∞

xn−_L1

n n

k0

ijk

Si_Tj_x

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and{klm}is bounded , there exists a positive integeri0such that

xni−

1

Lni

ni

k0

ijk

Si_Tj_x ni

< ρ

5,

klm _L1_n

i

ni

k0

ijk

Si_Tj_x ni−xni

< ρ

5

2.12

for allmandi > i0. As{xni} ⊂Cis bounded and byLemma 2.2, there existm1> m0andi1>0 such that

_L1_n

i

ni

k0

ijk

Si_Tj_x ni−Slm1

1

Lni

ni

k0

ijk

Si_Tj_x ni <

ρ

5 2.13

for alli > i1. By2.7,2.10,2.12, and2.13, we have

_x_n_i₋_Slm1x< ρ 5

ρ

5

ρ

5 r

ρ

5

ρ

5 rρ 2.14

for alli >max{i0, i1}. However,

xni−

Slm1xx 2

2 1

2xni−Slm1x

21

2xni−x

2₋1 4S

lm1x−x2

< rρ2

2

rρ/52

2 −

ε2 0 4

<rρ2₋ε20 4 < r

2

2.15

for alli >max{i0, i1}. This contradicts with2.8. So{Slx}converges strongly toxand then

x∈FS. Similarly, we can getx∈FT. Hence,xis a common fixed point ofSandT.

3. Main results

In this section, we prove our main theorem.

Theorem 3.1. LetCbe a nonempty closed convex subset of a Hilbert spaceHand letS,T be two continuous generalized asymptotically nonexpansive mappings ofCinto itself such that

iSnx−Sny ≤knx−yrn,kn≥0,rn≥0,kn→1,rn→0 (n→ ∞),

iiTnx−Tny ≤tnx−ybn,tn≥0,bn≥0,tn→1,bn→0 (n→ ∞),

iiiSTTSandFSFT/∅.

For arbitraryx0∈C, the sequence{xn}∞n0is defined by

xnαnx0

1−αn_n₁2_n₂ n

k0

ijk

Si_Tj_x

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withαn∈0,1and limn→∞αn0. If lim sup_n_→∞1−αn2/n1n2nk0

ijkkitj<1,

then{x_n}∞_n₀ converges strongly to the common fixed pointPx0 ofSand T, whereP is the metric

projection ofHontoFSFT.

Proof. Firstly, we show that {xn}∞_n₀ is bounded. By lim sup_n_→∞1 − αn2/n 1n 2n_k₀_i_j_kk_it_j < 1, there existN > 0 and 0 < a < 1 such that1−α_n2/n1n 2n_k₀_i_j_kkitj≤afor alln > N. Choosep∈FSFT, then we have

_x_n₋_p_α_n

x0−p

1−αn_n₁2_n₂ n

k0

ijk

Si_Tj_x

n−SiTjp

≤αnx0−p

1−αn_n₁2_n₂ n

k0

ijk

kitjxn−pkibjri

αnx0−p

1−αn_n₁2_n₂ n

k0

ijk

kitjxn−p

1−α_n 2

n1n2

n

k0

ijk

kibj1−αn_n₁2_n₂ n

k0

ijk

ri.

3.2

Hence

1−axn−p≤αnx0−p

1−αn_n₁2_n₂ n

k0

ijk

kibj

1−αn_n₁2_n₂ n

k0

ijk

ri

3.3

for alln > N. So{x_n}is bounded.

Letting {xni} be any subsequence of{xn}, there exists a subsequence {xmt}of {xni}

such that {xmt}converges weakly to some u ∈ C. For{xmt} is bounded, so is{2/mt

1mt2m_kt0

ijkSiTjxmt}. Thus

xmt−

2

mt1mt2 mt

k0

ijk

Si_Tj_x mtαmt

x0− 2

mt1mt2 mt

k0

ijk

Si_Tj_x mt

−→0t−→ ∞.

3.4

It follows fromLemma 2.4thatu∈FSFT. For

αmtPx0Px0−

1−αmt

2

mt1mt2 mt

k0

ijk

Si_Tj_Px

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we have

αmt

x0−Px0, xmt−Px0

xmt−Px0−

1−αmt

2

mt1mt2 mt

k0

ijk

Si_Tj_x

mt−SiTjPx0

, xmt−Px0

xmt−Px02−

1−αmt

2

mt1mt2 mt

k0

ijk

Si_Tj_x

mt−SiTjPx0

, xmt−Px0

≥ xmt−Px02−

1−α_m_t 2

mt1mt2

×mt k0

ijk

kitjxmt−Px0kibjri

xmt−Px0

1−1−αmt

2

mt1mt2 mt

k0

ijk

kitj xmt−Px02

−1−αmt

2

mt1mt2 mt

k0

ijk

kibjrixmt−Px0.

3.6

By hypothesis, there exist M > 0 and 0 < a < 1 such that 1 −α_m_t2/m_t 1m_t 2mt

k0

ijkkitj≤afor allt > M. So

1−ax_m_t−Px02

≤αmt

x0−Px0, xmt−Px0

1−αmt

2

mt1mt2 mt

k0

ijk

kibjrixmt−Px0

3.7

for allt > M. We can easily prove that limn→∞1−αmt2/mt1mt2 _m_t

k0

ijkkibj

rixmt−Px00. SincePis metric projection,x0−Px0, x−Px0 ≤0 for allx∈FS

FT. Hence

x0−Px0, xmt−Px0

x0−Px0, xmt−u

x0−Px0, u−Px0

≤x0−Px0, xmt−u

. 3.8

Since{x_m_t}converges weakly tou, lim sup_t_→∞x0−Px0, xmt−Px0 ≤0.It follows from3.7

that{xmt}converges strongly toPx0. Hence,{xn}converges strongly toPx0. This completes the proof.

Acknowledgment

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