Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces

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Volume 2010, Article ID 579725,21pages doi:10.1155/2010/579725

Research Article

Strong Convergence Theorems of Viscosity

Iterative Methods for a Countable Family of Strict

Pseudo-contractions in Banach Spaces

Rabian Wangkeeree and Uthai Kamraksa

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Correspondence should be addressed to Rabian Wangkeeree,[email protected]

Received 23 June 2010; Accepted 13 August 2010

Academic Editor: A. T. M. Lau

Copyrightq2010 R. Wangkeeree and U. Kamraksa. 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.

For a countable family{Tn}∞n1of strictly pseudo-contractions, a strong convergence of viscosity iteration is shown in order to find a common fixed point of{Tn}∞n1in either ap-uniformly convex Banach space which admits a weakly continuous duality mapping or ap-uniformly convex Banach space with uniformly Gˆateaux diﬀerentiable norm. As applications, at the end of the paper we apply our results to the problem of finding a zero of accretive operators. The main result extends various results existing in the current literature.

1. Introduction

LetEbe a real Banach space andCa nonempty closed convex subset ofE. A mappingf:C → Cis calledk-contraction if there exists a constant 0< k <1 such thatfx−fy ≤kx−y for allx,y∈C. We use_Cto denote the collection of all contractions onC. That is,_C{f: f is a contraction on C}. A mappingT :C → Cis said to beλ-strictly pseudo-contractive mappingsee, e.g.,1if there exists a constant 0≤λ <1, such that

Tx−Ty2≤x−y2λI−Tx−I−Ty2, 1.1

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T :C → Cis said to beλ-strictly pseudo-contractive mapping with respect topif, for allx, y∈C, there exists a constant 0≤λ <1 such that

Tx−Typ≤x−ypλI−Tx−I−Typ. 1.2

A countable family of mapping{Tn:C → C}∞i1is called afamily of uniformlyλ-strict pseudo-contractions with respect top, if there exists a constantλ∈0,1such that

Tnx−Tnyp≤x−ypλI−Tnx−I−Tnyp, ∀x, y∈C, ∀n≥1. 1.3

We denote byFTthe set of fixed points ofT, that is,FT {x∈C:Txx}.

In order to find a fixed point of nonexpansive mappingT, Halpern2was the first to introduce the following iteration scheme which was referred to as Halpern iteration in a Hilbert space:u,x1∈C,{αn} ⊂0,1,

xn1 αnx 1−αnTxn, n≥1. 1.4

He pointed out that the control conditions C1 limn→ ∞αn 0 and C2 _∞

n1 ∞ are necessary for the convergence of the iteration scheme1.4to a fixed point ofT. Furthermore, the modified version of Halpern iteration was investigated widely by many mathematicians. Recently, for the sequence of nonexpansive mappings{Tn}∞n1with some special conditions, Aoyama et al.3introduced a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings {Tn : C → C} satisfying some conditions. Letx1 x∈Cand

xn1αnx 1−αnTnxn 1.5

for all n ∈ N,where Cis a nonempty closed convex subset of a uniformly convex Banach spaceEwhose norm is uniformly Gâteaux differentiable, and{αn}is a sequence in 0,1. They proved that{xn}defined by1.5converges strongly to a common fixed point of{Tn}. Very recently, Song and Zheng4also studied the strong convergence theorem of Halpern iteration1.5for a countable family of nonexpansive mappings{Tn : C → C} satisfying some conditions in either a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm or a reflexive Banach spaceEwith a weakly continuous duality mapping. Other investigations of approximating common fixed points for a countable family of nonexpansive mappings can be found in3,5–10and many results not cited here.

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In 2007, Marino and Xu 12 proved that the Mann iterative sequence converges weakly to a fixed point ofλ-strict pseudo-contractions in Hilbert spaces, which extend Reich’s theorem 14, Theorem 2 from nonexpansive mappings to λ-strict pseudo-contractions in Hilbert spaces.

Recently, Zhou 13 obtained some weak and strong convergence theorems for λ -strict pseudo-contractions in Hilbert spaces by using Mann iteration and modified Ishikawa iteration which extend Marino and Xu’s convergence theorems12.

More recently, Hu and Wang11obtained that the Mann iterative sequence converges weakly to a fixed point ofλ-strict pseudo-contractions with respect topinp-uniformly convex Banach spaces. To be more precise, they obtained the following theorem.

Theorem HW

LetEbe a realp-uniformly convex Banach space which satisfies one of the following:

iEhas a Fr´echet diﬀerentiable norm; iiEsatisfies Opial’s property.

LetCa nonempty closed convex subset ofE. LetT :C → Cbe aλ-strict pseudo-contractions with respect top,λ∈0,min{1,2−p−2_c

p}andFT/∅. Assume that a real sequence{αn}in 0,1satisfy the following conditions:

0< ε≤αn≤1−ε <1− 2p−2_λ

cp

, ∀n≥1. 1.6

Then Mann iterative sequence{xn}defined by

x1x∈C,

xn1αnxn 1−αnTxn, n≥1,

1.7

converges weakly to a fixed point ofT.

Very recently, Hu15 obtained strong convergence theorems on a mixed iteration scheme by the viscosity approximation methods for λ-strict pseudo-contractions in p -uniformly convex Banach spaces with -uniformly Gˆateaux diﬀerentiable norm. To be more precise, Hu15obtained the following theorem.

Theorem H. LetEbe a realp-uniformly convex Banach space with uniformly Gˆateaux diﬀerentiable

norm, and C a nonempty closed convex subset of E which has the fixed point property for nonexpansive mappings. Let T : C → C be a λ-strict pseudo-contractions with respect to p, λ∈0,min{1,2−p−2_c

p}andFT/∅. Letf:C → Cbe ak-contraction withk∈0,1. Assume

that real sequences{αn},{βn}and{γn}in0,1satisfy the following conditions:

iαnβnγn1for alln∈N, iilimn→ ∞αn0and

_∞

n0αn ∞,

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Let{xn}be the sequence generated by the following:

x1x∈C,

xn1αnfxn βnxnγnTxn, n≥1.

1.8

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

In this paper, motivated by Hu and Wang 11, Hu 15, Aoyama et al. 3 and Song and Zheng 4, we introduce a viscosity iterative approximation method for finding

a common fixed point of a countable family of strictly pseudo-contractions which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in eitherp-uniformly convex Banach space which admits a weakly continuous duality mapping orp-uniformly convex Banach space with uniformly Gˆateaux diﬀerentiable norm. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The results presented in this paper improve and extend the corresponding results announced by Hu and Wang11, Hu15, Aoyama et

al.3Song and Zheng4, and many others.

2. Preliminaries

Throughout this paper, letEbe a real Banach space andE∗its dual space. We writexn x resp.,xn∗xto indicate that the sequence{xn}weaklyresp., weak∗converges tox; as usualxn → xwill symbolize strong convergence. LetSE {x ∈E :x 1}denote the unit sphere of a Banach spaceE. A Banach spaceEis said to have

ia Gˆateaux diﬀerentiable normwe also say thatEis smooth, if the limit

lim t→0

_x_ty₋_x

t 2.1

exists for eachx,y∈SE,

iia uniformly Gˆateaux diﬀerentiable norm, if for each y in SE, the limit 2.1 is uniformly attained forx∈SE,

iiia Fr´echet diﬀerentiable norm, if for eachx∈SE, the limit2.1is attained uniformly fory∈SE,

iva uniformly Fr´echet diﬀerentiable normwe also say thatEis uniformly smooth, if the limit2.1is attained uniformly forx, y∈SE×SE.

The modulus of convexity ofEis the functionδE:0,2 → 0,1defined by

δE inf

1−xy 2

: x1, y1, x−y≥

, 0≤≤2. 2.2

Eis uniformly convex if and only if, for all 0 < ≤ 2 such thatδE > 0. Eis said to be

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The following facts are well known which can be found in16,17:

ithe normalized duality mappingJ in a Banach spaceEwith a uniformly Gˆateaux diﬀerentiable norm is single-valued and strong-weak∗ _{uniformly continuous on}

any bounded subset ofE;

iieach uniformly convex Banach spaceEis reflexive and strictly convex and has fixed point property for nonexpansive self-mappings;

iiievery uniformly smooth Banach space E is a reflexive Banach space with a uniformly Gˆateaux diﬀerentiable norm and has fixed point property for nonexpansive self-mappings.

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1see11. LetEbe a realp-uniformly convex Banach space andCa nonempty closed

convex subset ofE. letT :C → Cbe aλ-strict pseudo-contraction with respect top, and{ξn}a real

sequence in0,1. IfTn :C → Cis defined byTnx: 1−ξnxξnTx, for allx∈C, then for allx,

y∈C, the inequality holds

Tnx−Tnyp≤x−yp−

wpξncp−ξnλI−Tx−I−Typ, 2.3

wherecpis a constant in [18, Theorem1]. In addition, if0≤λ <min{1,2−p−2cp},ξ1−2p−2λcp−1,

andξn∈0, ξ, thenTnx−Tny ≤ x−y, for allx,y∈C.

Lemma 2.2 see19,20. Let Cbe a nonempty closed convex subset of a Banach spaceEwhich

has uniformly Gˆateaux diﬀerentiable norm,T :C → Ca nonexpansive mapping withFT/∅and f : C → Cak-contraction. Assume that every nonempty closed convex bounded subset ofChas the fixed points property for nonexpansive mappings. Then there exists a continuous path:t → xt,

t∈0,1satisfyingxttfxt 1−tTxt, which converges to a fixed point ofT ast → 0.

Lemma 2.3see21. Let{xn}and{yn}be bounded sequences in Banach spaceEsuch that

xn1αnxn 1−αnyn, n≥0, 2.4

where{αn}is a sequence in0,1such that0<lim infn→ ∞αn≤lim sup_n_{→ ∞}αn<1. Assume

lim sup n→ ∞

yn1−yn− xn1−xn

≤0. _2.5

Thenlimn→ ∞yn−xn0.

Definition 2.4see3. Let{Tn}be a family of mappings from a subsetCof a Banach spaceE intoEwith ∞_n₁FTn/∅. We say that{Tn}satisfies the AKTT-condition if for each bounded subsetBofC,

∞

n1 sup

z∈B

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Remark 2.5. The example of the sequence of mappings {Tn} satisfying AKTT-condition is supported byLemma 4.1.

Lemma 2.6see3, Lemma 3.2. Suppose that{Tn}satisfies AKTT-condition. Then, for eachy∈

C,{Tny}converses strongly to a point inC. Moreover, let the mappingTbe defined by

Ty lim

n→ ∞Tny, ∀y∈C. 2.7

Then for each bounded subsetBofC,limn→ ∞supz∈BTz−Tnz0.

Lemma 2.7see22. Assume that{αn}is a sequence of nonnegative real numbers such that

αn1≤

1−γn

αnδn, 2.8

where{γn}is a sequence in0,1and{δn}is a sequence such that a∞

n1γn∞;

blim sup_n_{→ ∞}δn/γn≤0or _∞

n1|δn|<∞.

Thenlimn→ ∞αn0.

By a gauge functionϕwe mean a continuous strictly increasing functionϕ:0,∞ → 0,∞such thatϕ0 0 andϕt → ∞ast → ∞. LetE∗be the dual space ofE. The duality mappingJϕ:E → 2E

∗

associated to a gauge functionϕis defined by

Jϕx

f∗∈E∗:x, f∗xϕx,f∗ϕx, ∀x∈E. 2.9

In particular, the duality mapping with the gauge function ϕt t, denoted by J, is referred to as the normalized duality mapping. Clearly, there holds the relation Jϕx ϕx/xJxfor allx /0see23. Browder23initiated the study of certain classes of nonlinear operators by means of the duality mappingJϕ. Following Browder23, we say that a Banach spaceEhas aweakly continuous duality mapping if there exists a gaugeϕfor which the duality mappingJϕxis single-valued and continuous from the weak topology to the weak∗ topology, that is, for any{xn}withxn x, the sequence{Jϕxn}converges weakly∗toJϕx. It is known thatlphas a weakly continuous duality mapping with a gauge functionϕt tp−1_{for all 1}_{< p <}_∞_{. Set}

Φt t

0

ϕτdτ, ∀t≥0, 2.10

then

Jϕx ∂Φx, ∀x∈E, 2.11

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The following lemma is an immediate consequence of the subdiﬀerential inequality. The first part of the next lemma is an immediate consequence of the subdiﬀerential inequality and the proof of the second part can be found in24.

Lemma 2.8see24. Assume that a Banach spaceEhas a weakly continuous duality mappingJϕ

with gaugeϕ.

iFor allx,y∈E, the following inequality holds:

Φxy≤Φx y, Jϕ

xy. 2.12

In particular, in a smooth Banach spaceE, for allx,y∈E,

xy2≤ x22y, Jxy. 2.13

iiAssume that a sequence{xn}inEconverges weakly to a pointx∈E.

Then the following identity holds:

lim sup n→ ∞ Φ

_x_n₋_y_{lim sup}

n→ ∞ Φxn−x Φ

_y₋_x_, _∀_{x, y}_∈_E. _2.14

3. Main Results

ForT :C → Ca nonexpansive mapping,t∈0,1andf∈_C,tf 1−tT :C → Cdefines a contraction mapping. Thus, by the Banach contraction mapping principle, there exists a unique fixed pointxf_t satisfying

xf_t tfxt 1−tTxft. 3.1

For simplicity we will writextforxft provided no confusion occurs. Next, we will prove the following lemma.

Lemma 3.1. LetEbe a reflexive Banach space which admits a weakly continuous duality mappingJϕ

with gaugeϕ. LetCbe a nonempty closed convex subset ofE,T :C → Ca nonexpansive mapping withFT/∅andf∈_C. Then the net{xt}defined by3.1converges strongly ast → 0to a fixed

pointxofT which solves the variational inequality:

I−fx, J ϕx−z

≤0, z∈FT. 3.2

Proof. We first show that the uniqueness of a solution of the variational inequality3.2.

Suppose bothx∈FTandx∗∈FTare solutions to3.2, then

I−fx, J ϕx−x∗

≤0,

I−fx∗, Jϕx∗−x

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Adding3.3, we obtain

I−fx−I−fx∗, Jϕx−x∗

≤0. 3.4

Noticing that for anyx,y∈E,

I−fx−I−fy, Jϕ

x−yx−y, Jϕ

x−y−fx−fy, Jϕ

x−y

≥x−yϕx−y−fx−fyϕx−y

≥Φx−y−αΦx−y

1−αΦx−y≥0.

3.5

From3.4, we conclude thatΦx−x∗ 0. This implies thatxx∗and the uniqueness is proved. Below we usexto denote the unique solution of3.2. Next, we will prove that{xt} is bounded. Take ap∈FT; then we have

xt−ptfxt 1−tTxt−p 1−tTxt−1−tpt

fxt−p

≤1−txt−pt

αxt−pf

p−p.

3.6

It follows that

xt−p≤ 1 1−αf

p−p. 3.7

Hence{xt}is bounded, so are{fxt}and{Txt}. The definition of{xt}implies that

xt−Txttfxt−Txt−→0, ast−→0. 3.8

If follows from reflexivity ofEand the boundedness of sequence{xt}that there exists{xtn}

which is a subsequence of{xt}converging weakly tow ∈Casn → ∞. SinceJϕis weakly sequentially continuous, we have byLemma 2.8that

lim sup

n→ ∞ Φxtn −x lim supn→ ∞ Φxtn −w Φx−w, ∀x∈E. 3.9

Let

Hx lim sup n→ ∞ Φ

xtn −x, ∀x∈E. 3.10

It follows that

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Since

xtn −Txtntnfxtn−Txtn−→0, asn−→ ∞, 3.12

we obtain

HTw lim sup n→ ∞ Φ

xtn −Tw lim sup

n→ ∞ Φ

Txtn −Tw

≤lim sup

n→ ∞ Φxtn −w Hw.

3.13

On the other hand, however,

HTw Hw ΦTw−w. 3.14

It follows from3.13and3.14that

ΦTw−w HTw−Hw≤0. 3.15

This implies thatTw w. Next we show that xtn → w asn → ∞. In fact, since Φt

t

0ϕτdτ, for all t ≥ 0, andϕ : 0,∞ → 0,∞ is a gauge function, then for 1 ≥ k ≥ 0, ϕkx≤ϕxand

Φkt kt

0

ϕτdτ k t

0

ϕkxdx≤k t

0

ϕxdxkΦt. 3.16

FollowingLemma 2.8, we have

Φxtn −w Φ1−tnTxtn−1−tnwtn

fxtn−w

Φ1−tnTxtn−1−tnw tn

fxtn−w, Jxtn−w

≤Φ1−tnxtn −w tn

fxtn−fw, Jxtn−w

tn

fw−w, Jxtn−w

≤1−tnΦxtn −w tnfxtn−fwJxtn−w

tn

fw−w, Jxtn−w

≤1−tnΦxtn −w tnαxtn−wJϕxtn−w

tn

fw−w, Jxtn−w

1−tnΦxtn −w tnαΦxtn −w

tn

fw−w, Jxtn−w

1−tn1−αΦxtn −w tn

fw−w, Jxtn−w

.

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This implies that

Φxtn−w≤

1 1−α

fw−w, Jxtn−w

. 3.18

Now observing thatxtn wimpliesJϕxtn−w 0, we conclude from the last inequality

that

Φxtn−w−→0, asn−→ ∞. 3.19

Hencextn → wasn → ∞. Next we prove thatwsolves the variational inequality3.2. For

anyz∈FT, we observe that

I−Txt−I−Tz, Jϕxt−z

xt−z, Jϕxt−z

Txt−Tz, Jϕxt−z

Φxt−z−

Tz−Txt, Jϕxt−z

≥Φxt−z− Tz−TxtJϕxt−z

≥Φxt−z− z−xtJϕxt−z Φxt−z−Φxt−z 0.

3.20

Since

xttfxt 1−tTxt, 3.21

we can derive that

I−fxt −1

tI−Txt I−Txt. 3.22

Thus

I−fxt, Jϕxt−z

−1

t

I−Txt−I−Tz, Jϕxt−z

I−Txt, Jϕxt−z

≤I−Txt, Jϕxt−z

.

3.23

Noticing that

xtn−Txtn −→w−Tw w−w0. 3.24

Now replacingtin3.23withtnand lettingn → ∞, we have

I−fw, Jϕw−z

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So, w ∈ FT is a solution of the variational inequality 3.2, and hence w x by the uniqueness. In a summary, we have shown that each cluster point of{xt}att → 0equals

x. Therefore,xt → xast → 0. This completes the proof.

Theorem 3.2. LetEbe a realp-uniformly convex Banach space with a weakly continuous duality

mappingJϕ, andCa nonempty closed convex subset ofE. Let{Tn:C → C}be a family of uniformly

λ-strict pseudo-contractions with respect to p,λ ∈ 0,min{1,2−p−2_c

p}and ∞n1FTn/∅. Let

f :C → Cbe ak-contraction withk ∈ 0,1. Assume that real sequences{αn},{βn}and{γn}in 0,1satisfy the following conditions:

iαnβnγn1for alln∈N;

iilimn→ ∞αn0and _∞

n0αn ∞;

iii0<lim infn→ ∞γn≤lim supn→ ∞γn< ξ, whereξ1−2p−2λcp−1.

x1x∈C,

xn1 αnfxn βnxnγnTnxn, n≥1.

3.26

Suppose that{Tn}satisfies the AKTT-condition. LetT be a mapping ofCinto itself defined byTz limn→ ∞Tnzfor allz ∈Cand suppose thatFT ∞n1FTn. Then the sequence{xn}converges

strongly toxwhich solves the variational inequality:

I−fx, J ϕx−z

≤0, z∈FT. 3.27

Proof. Rewrite the iterative sequence3.26as follows:

xn1αnfxn βnxnγnSnxn, n≥1, 3.28

whereβ_n βn−γn/ξ1−ξ,γn γn/ξandSn : 1−ξIξTn,Iis the identity mapping. By Lemma 2.1,Sn is nonexpansive such thatFSn FTnfor all n ∈ N. Taking anyq ∈

∞

n1FTn, from3.28, it implies that

xn1−q≤αnfxn−qβnxn−qγnSnxn−q

≤αnkxn−qαnf

q−q 1−αnxn−q

αn1−k 1 1−kf

q−q 1−αn1−kxn−q

≤maxx1−q, 1 1−kf

q−q

.

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Therefore, the sequence {xn}is bounded, and so are the sequences{fxn},{Snxn}. Since

Snxn 1−ξnxnξnTnxnand lim infξn>0, we know that{Tnxn}is bounded. We note that for any bounded subsetBofC,

sup z∈B

Sn1z−Snzsup z∈B

1−ξn1zξn1Tn1z−1−ξnzξnTnz

≤ |ξn1−ξn|sup z∈B

zξn1sup

z∈B

Tn1z−Tnz|ξn1−ξn|sup z∈B

Tnz

|ξn1−ξn|sup z∈B

zTz ξn1sup

z∈B

Tn1z−Tnz.

3.30

From∞_n₁|ξn1−ξn|<∞and{Tn}satisfing AKTT-condition, we obtain that

∞

n1 sup

z∈B

Sn1z−Snz<∞, 3.31

that is, the sequence {Sn}satisfies AKTT-condition. ApplyingLemma 2.6, we can take the mappingS:C → Cdefined by

Sz lim

n→ ∞Snz, ∀z∈C. 3.32

Moreover, we haveSis nonexpansive and

Sz lim

n→ ∞Snznlim→ ∞1−ξnzξnTnz 1−ξzξTz. 3.33

It is easy to see that FS FT. Hence FS ∞_n₁FTn ∞n1FSn.The iterative sequence3.28can be expressed as follows:

xn1βnxn

1−β_nyn, 3.34

where

yn αn 1−βn

fxn

γ_n 1−βn

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We estimate from3.35

yn1−yn

αn1

1−β_n₁fxn1

γ_n₁

1−β_n₁Sn1xn1− αn 1−βn

fxn

γ_n 1−βn

Snxn

≤ αn1

1−β_n₁kxn1−xn γ_n₁

1−β_n₁Sn1xn1−Snxn

αn1 1−β_n₁ −

αn 1−βn

fxn−Snxn

≤ αn1

1−β_n₁kxn1−xn γ_n₁

1−β_n₁Sn1xn1−Sn1xnSn1xn−Snxn

αn1 1−β_n₁ −

αn 1−βn

fxn−Snxn

≤ αn1

1−β_n₁kxn1−xn γ_n₁ 1−β_n₁

xn1−xn sup z∈{xn}

Sn1z−Snz

αn1 1−β_n₁ −

αn 1−βn

fxn−Snxn.

3.36

Hence

yn1−yn− xn1−xn ≤

αn1

1−β_n₁kxn1−xn γ_n₁

1−β_n₁_zsup_∈{_x_n_}Sn1z−Snz

αn1 1−β_n₁ −

αn 1−βn

fxn−Snxn.

3.37

Since limn→ ∞αn0, and limn→ ∞supz∈{xn}Sn1z−Snz0, we have from3.37that

lim sup n→ ∞

yn1−yn− xn1−xn

≤0. _3.38

Hence, byLemma 2.3, we obtain

lim

n→ ∞yn−xn0. 3.39

From3.35, we get

lim

n→ ∞yn−Snxnnlim→ ∞ αn 1−βn

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and so it follows from3.39and3.40that

lim

n→ ∞xn−Snxn0. 3.41

It follows fromLemma 2.6and3.41, we have

xn−Sxn ≤ xn−SnxnSnxn−Sxn

≤ xn−Snxnsup{Snz−Sz:z∈ {xn}} −→0, asn−→ ∞.

3.42

SinceSis a nonexpansive mapping, we have fromLemma 3.1that the net{xt}generated by

xttfxt 1−tSx 3.43

converges strongly tox∈FS, ast → 0. Next, we prove that

lim sup n→ ∞

fx−x, J ϕxn−x

≤0. _3.44

Let{xnk}be a subsequence of{xn}such that

lim k→ ∞

fx−x, J ϕxnk −x

lim sup n→ ∞

. _3.45

If follows from reflexivity ofEand the boundedness of sequence{xnk}that there exists{xnki}

which is a subsequence of{xnk}converging weakly tow ∈Casi → ∞. SinceJϕ is weakly

continuous, we have byLemma 2.8that

lim sup i→ ∞ Φ

_x nki −x

lim sup i→ ∞ Φ

_x nki −w

Φx−w, ∀x∈E. _3.46

Let

Hx lim sup i→ ∞ Φ

_x nki −x

, ∀x∈E. _3.47

It follows that

Hx Hw Φx−w, ∀x∈E. 3.48

From3.42, we obtain

HSw lim sup i→ ∞

Φxn_ki −Sw

lim sup i→ ∞

ΦSxn_ki −Sw

≤lim sup i→ ∞ Φ

_x nki −w

Hw.

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On the other hand, however,

HSw Hw ΦSw−w. 3.50

It follows from3.49and3.50that

ΦSw−w HSw−Hw≤0. 3.51

This implies thatSww, that is,w∈FS FT. Since the duality mapJϕis single-valued and weakly continuous, we get that

lim sup n→ ∞

lim k→ ∞

fx−x, J ϕxnk−x

lim i→ ∞

fx−x, J ϕ

xn_ki −x

I−fx, J ϕx−w

≤0

3.52

as required. Finally, we show thatxn → xasn → ∞.

Φxn1−x Φαn

fxn−fx

β_nxn−x γnSnxn−x αn

fx−x

≤Φαn

fxn−fx

β_nxn−x γnSnxn−x αn

fx−x, J ϕxn1−x

≤Φαnkxn−xβnxn−xγnxn−x

αn

Φ1−αn1−kxn−x αn

≤1−αn1−kΦxn−x αn

.

3.53

It follows that from conditioniand3.44that

lim

n→ ∞αn0, ∞

n1

αn∞, lim sup n→ ∞

≤0. 3.54

ApplyLemma 2.7to3.53to concludeΦxn1−x → 0 asn → ∞; that is,xn → xas

n → ∞. This completes the proof.

If{Tn :C → C}is a family of nonexpansive mappings, then we obtain the following results.

Corollary 3.3. LetEbe a realp-uniformly convex Banach space with a weakly continuous duality

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nonexpansive mappings such that ∞_n₁FTn/∅. Letf :C → Cbe ak-contraction withk∈0,1.

Assume that real sequences{αn},{βn}and{γn}in0,1satisfy the following conditions:

iαnβnγn1for alln∈N; iilimn→ ∞αn0and∞n0αn ∞; iii0<lim infn→ ∞γn≤lim supn→ ∞γn<1.

x1x∈C,

3.55

stronglyxwhich solves the variational inequality:

I−fx, J ϕx−z

≤0, z∈FT. 3.56

Corollary 3.4. LetEbe a realp-uniformly convex Banach space with a weakly continuous duality

mappingJϕ, andC a nonempty closed convex subset ofE. Let T : C → C be aλ-strict

pseudo-contraction with respect top, λ ∈ 0,min{1,2−p−2_c_p_}_and _F_T_/_∅_{. Let} _f _: _C _→ _C _{be a}

k-contraction with k ∈ 0,1. Assume that real sequences {αn},{βn} and{γn} in0,1satisfy the

following conditions:

iαnβnγn1for alln∈N; iilimn→ ∞αn0and

_∞

n0αn ∞;

Let{xn}be the sequence generated by the following

x1x∈C,

3.57

Then the sequence{xn}converges strongly toxwhich solves the following variational inequality:

I−fx, J ϕx−z

≤0, z∈FT. 3.58

Theorem 3.5. LetEbe a realp-uniformly convex Banach space with uniformly Gˆateaux diﬀerentiable

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p,λ∈0,min{1,2−p−2_c

p}and ∞n1FTn/∅. Letf:C → Cbe ak-contraction withk∈0,1.

Assume that real sequences{αn},{βn}and{γn}in0,1satisfy the following conditions: iαnβnγn1for alln∈N;

iilimn→ ∞αn0and∞n0αn ∞;

x1x∈C,

3.59

strongly to a common fixed pointxof{Tn}.

Proof. It follows from the same argumentation as Theorem 3.2 that {xn} is bounded and limn→ ∞xn − Sxn 0, where S is a nonexpansive mapping defined by 3.32. From

Lemma 2.2 that the net {xt} generated by xt tfxt 1− tSxt converges strongly to

x∈FS FT, ast → 0. Obviously,

xt−xn 1−tSxt−xn t

fxt−xn

. 3.60

In view ofLemma 2.8, we calculate

xt−xn2 ≤1−t2Sxt−xn22t

fxt−xn, Jxt−xn

≤1−2tt2xt−xnSxn−xn2

2tfxt−xt, Jxt−xn

2txt−xn2

3.61

and therefore

fxt−xt, Jxn−xt

≤ t

2xt−xn

21t2xn−Sxn

2t 2xt−xnxn−Sxn. 3.62

Since{xn},{xt}and{Sxn}are bounded and limn→ ∞xn−Sxn/2t 0, we obtain

lim sup n→ ∞

≤ t

2M, 3.63

whereMsup_n_≥₁_{, t}_∈₀_,₁{xt−xn2}. We also know that

fx−x, J xn−x

fx−fxt xt−x, J xn−xt

fx−x, J xn−x−Jxn−xt

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From the fact thatxt → x∈FT, ast → 0,{xn}is bounded and the duality mappingJis norm-to-weak∗uniformly continuous on bounded subset ofE, it follows that ast → 0,

fx−x, J xn−x−Jxn−xt

−→0, ∀n∈N,

fx−fxt xt−x, J xn−xt

−→0, ∀n∈N. 3.65

Combining3.63,3.64and two results mentioned above, we get

lim sup n→ ∞

fx−x, J xn−x

≤0. _3.66

From3.28andLemma 2.8, we get

xn1−x2≤αn

fxn−fx

β_nxn−x γnSnxn−x2 2αn

fx−x, J xn1−x

≤1−αn1−kxn−x22αn

fx−x, J xn1−x

.

3.67

Hence applying inLemma 2.7to3.67, we conclude that limn→ ∞xn−x0.

Corollary 3.6. LetEbe a realp-uniformly convex Banach space with uniformly Gˆateaux diﬀerentiable norm, andCa nonempty closed convex subset ofEwhich has the fixed point property for nonexpansive mappings. Let{Tn :C → C}be a family of nonexpansive mappings such that n∞1FTn/∅. Let

f :C → Cbe ak-contraction withk ∈ 0,1. Assume that real sequences{αn},{βn}and{γn}in 0,1satisfy the following conditions:

iαnβnγn1for alln∈N; iilimn→ ∞αn0and

_∞

n0αn ∞; iii0<lim infn→ ∞γn≤lim supn→ ∞γn<1.

x1x∈C,

3.68

strongly to a common fixed pointxof{Tn}.

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p,λ ∈ 0,min{1,2−p−2_c

p}and FT/∅. Let f : C → C be ak-contraction with k ∈ 0,1.

Assume that real sequences{αn},{βn}and{γn}in0,1satisfy the following conditions:

iαnβnγn1for alln∈N;

iilimn→ ∞αn0and∞n0αn ∞;

x1x∈C,

3.69

Then the sequence{xn}converges strongly to a common fixed pointxof{Tn}.

4. Some Applications for Accretive Operators

We consider the problem of finding a zero of an accretive operator. An operatorΨ⊂E×Eis said to be accretive if for eachx1, y1andx2, y2∈Ψ, there existsj∈Jx1−x2such thaty1− y2, j ≥0.An accretive operatorΨis said to satisfy the range condition ifDΨ⊂RIλΨ for allλ >0,whereDΨis the domain ofΨ, I is the identity mapping onE,RIλΨis the range ofIλΨ,andDΨis the closure ofDΨ. IfΨis an accretive operator which satisfies the range condition, then we can define, for eachλ >0,a mappingJλ :RIλΨ → DΨ byJλ IλΨ−1,which is called the resolvent of Ψ. We know that Jλ is nonexpansive andFJλ Ψ−10for all λ > 0.We also know the following25: For eachλ,μ > 0 and

x∈RIλΨ∩RIμΨ, it holds that

Jλx−Jμx≤

λ−μ

λ x−Jλx. 4.1

By the proof of Theorem 4.3 in3, we have the following lemma.

Lemma 4.1. LetEbe a Banach space andCa nonempty closed convex subset ofE.LetΨ⊆E×Ebe

an accretive operator such thatΨ−1₀_/_∅_and_D_Ψ_⊂ _C_⊂

λ>0RIλΨ.Suppose that{λn}is a

sequence of0,∞such thatinf{λn :n∈N}>0and∞n1|λn1−λn|<∞.Then

iThe sequence{Jλn}satisfies the AKTT-condition.

iilimn→ ∞JλnzJλzfor allz∈CandFJλ ∞

n1FJλnwhereλn → λasn → ∞.

ByCorollary 3.3, we obtain the following result.

Theorem 4.2. LetEbe a realp-uniformly convex Banach space with a weakly continuous duality

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thatΨ−1₀_/_∅_{. Let}_f _:_C _→ _C_{be a}_{k-contraction with}_k _∈₀_,₁_{. Assume that real sequences}_{_α

n},

{βn}and{γn}in0,1satisfy the following conditions:

iαnβnγn1for alln∈N; iilimn→ ∞αn0and∞n0αn ∞; iii0<lim infn→ ∞γn≤lim sup_n_{→ ∞}γn<1;

iv{λn}is a sequence of0,∞such thatinf{λn:n∈N}>0and∞n1|λn1−λn|<∞.

x1x∈C,

xn1αnfxn βnxnγnJλnxn, n≥1.

4.2

Then the sequence{xn}converges stronglyxwhich solves the following variational inequality:

I−fx, J ϕx−z

≤0, z∈FJλ. 4.3

ByCorollary 3.6, we obtain the following result.

Theorem 4.3. LetEbe a realp-uniformly convex Banach space with uniformly Gˆateaux diﬀerentiable

norm, andCa nonempty closed convex subset ofE. LetΨis anm-accretive operator inEsuch that Ψ−1₀_/_∅_{. Let}_f _:_C _→ _C_{be a}_{k-contraction with}_k_∈₀_,₁_{. Assume that real sequences}_{_α

n},{βn}

and{γn}in0,1satisfy the following conditions:

iαnβnγn1for alln∈N; iilimn→ ∞αn0and∞n0αn ∞; iii0<lim infn→ ∞γn≤lim sup_n_{→ ∞}γn<1;

iv{λn}is a sequence of0,∞such thatinf{λn:n∈N}>0and∞n1|λn1−λn|<∞.

x1x∈C,

xn1αnfxn βnxnγnJλnxn, n≥1.

4.4

Then the sequence{xn}converges stronglyxinΨ−10.

Acknowledgments

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