Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces

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,rabianw@nu.ac.th 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 a p-uniformly convex Banach space which admits a weakly continuous duality mapping or a p-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

for allx,y∈C. Note that the class ofλ-strict pseudo-contractions strictly includes the class of nonexpansive mappings which are mappingTonCsuch thatTx−Ty ≤ x−y, for allx, y∈C. That is,T is nonexpansive if and only ifTis a 0-strict pseudo-contraction. A mapping

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−Ty^p≤x−y^pλI−Tx−I−Ty^p. 1.2 A countable family of mapping{Tn:C → C}^∞_i1is called a family of uniformlyλ-strict pseudo-contractions with respect top, if there exists a constantλ∈0,1such that

T_nx−T_ny^p≤x−y^pλI−T_nx−I−T_ny^p, ∀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,

x_n1 α_nx 1−α_nTxn, n≥1. 1.4

He pointed out that the control conditions C1 lim_{n→ ∞}α_n 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

x_n1αnx 1−αnTnxn 1.5

for all n ∈ N,where Cis a nonempty closed convex subset of a uniformly convex Banach spaceEwhose norm is uniformly Gˆateaux 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 Gˆateaux 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.

On the other hand, in the last twenty years or so, there are many papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudo- contractive mappings by using the Mann and Ishikawa iteration process. Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and a more general class of mappings see, e.g.,1,11–13and the references therein.

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−2c_p}andFT/∅. Assume that a real sequence{αn}in 0,1satisfy the following conditions:

0< ε≤α_n≤1−ε <1−2^p−2λ cp

, ∀n≥1. 1.6

Then Mann iterative sequence{xn}defined by

x₁x∈C,

x_n1αnxn 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−2cp}andFT/∅. Letf:C → Cbe ak-contraction withk∈0,1. Assume that real sequences{αn},{βn}and{γn}in0,1satisfy the following conditions:

iαnβnγn1 for alln∈N, iilim_{n→ ∞}α_n0 and_∞

n0α_n ∞,

iii0<lim infn→ ∞γn≤lim sup_{n→ ∞}γn< ξ, whereξ1−2^p−2λc_p⁻¹.

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

x1x∈C,

x_n1α_nfxn β_nx_nγ_nTx_n, 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 writex_n x resp.,x_n^∗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

limt→0

xty− 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

δ_E 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 p-uniformly convex, if there exists a constanta >0 such thatδ_E≥a^p.

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

T_nx−T_ny^p≤x−y^p−

w_pξncp−ξ_nλI−Tx−I−Ty^p, 2.3 wherecpis a constant in [18, Theorem 1]. In addition, if 0≤λ <min{1,2^−p−2cp},ξ1−2^p−2λc_p⁻¹, andξn∈0, ξ, thenTnx−Tny ≤ x−y, for allx,y∈C.

Lemma 2.2 see19,20. LetCbe 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 → x_t, 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

x_n1α_nx_n 1−α_nyn, n≥0, 2.4

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

n→ ∞

y_n1−y_n− x_n1−x_n

≤0. 2.5

Then lim_{n→ ∞}yn−x_n0.

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∈BTn1z−Tnz<∞. 2.6

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→ ∞T_ny, ∀y∈C. 2.7

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

Lemma 2.7see22. Assume that{αn}is a sequence of nonnegative real numbers such that α_n1≤

1−γ_n

α_nδ_n, 2.8

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

n1γ_n∞;

blim sup_{n→ ∞}δ_n/γ_n≤0 or_∞

n1|δn|<∞.

Then limn→ ∞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 → 2^E∗associated to a gauge functionϕis defined by

J_ϕx

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 J_ϕx ϕx/xJxfor allx /0see23. Browder23initiated the study of certain classes of nonlinear operators by means of the duality mappingJ_ϕ. 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}withx_n x, the sequence{Jϕxn}converges weakly^∗toJ_ϕx. It is known thatl^phas a weakly continuous duality mapping with a gauge functionϕt t^p−1for all 1< p <∞. Set

Φt _t

0

ϕτdτ, ∀t≥0, 2.10

then

J_ϕx ∂Φx, ∀x∈E, 2.11

where ∂ denotes the subdifferential in the sense of convex analysis recall that the subdifferential of the convex functionφ:E → Ratx∈Eis the set∂φx {x^∗∈E^∗;φy≥ φx x^∗, y−x, for all y∈E}.

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, J_ϕ

xy

. 2.12

In particular, in a smooth Banach spaceE, for allx,y∈E, xy²≤ x²2

y, J xy

. 2.13

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

Then the following identity holds:

lim sup

n→ ∞ Φxn−ylim sup

n→ ∞ Φxn−x Φy−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 pointx^f_t satisfying

x^f_t tfxt 1−tTx^f_t. 3.1 For simplicity we will writex_tforx^f_t 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 → 0 to a fixed pointxofT which solves the variational inequality:

I−f

x, Jϕx−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−f

x, J_ϕx−x^∗

≤0, I−f

x^∗, Jϕx^∗−x

≤0. 3.3

Adding3.3, we obtain

I−f x−

I−f

x^∗, Jϕx−x^∗

≤0. 3.4

Noticing that for anyx,y∈E, I−f

x− I−f

y, Jϕ

x−y

x−y, Jϕ

x−y

−

fx−f y

, Jϕ

x−y

≥x−yϕx−y−fx−f

yϕx−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−tx_t−pt

αx_t−pf p

−p.

3.6

It follows that

x_t−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 sup

n→ ∞ Φxtn −w Φx−w, ∀x∈E. 3.9 Let

Hx lim sup

n→ ∞ Φxtn −x, ∀x∈E. 3.10

It follows that

Hx Hw Φx−w, ∀x∈E. 3.11

Since

xtn −Tx_tnt_nfxtn−Tx_tn−→0, asn−→ ∞, 3.12 we obtain

HTw lim sup

n→ ∞ Φxtn −Tw lim sup

n→ ∞ ΦTxtn −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−t_nTxtn−1−t_nw t_n

fxtn−w, Jxtn−w

≤Φ1−tnxtn −w tn

fxtn−fw, Jxtn−w t_n

fw−w, Jxtn−w

≤1−t_nΦxtn −w t_nfxtn−fwJxtn−w tn

fw−w, Jxtn−w

≤1−t_nΦxtn −w t_nαxtn−wJ_ϕxtn−w t_n

fw−w, Jxtn−w

1−tnΦxtn −w tnαΦxtn −w t_n

fw−w, Jxtn−w 1−tn1−αΦxtn −w tn

fw−w, Jxtn−w .

3.17

This implies that

Φxtn−w≤ 1 1−α

fw−w, Jxtn−w

. 3.18

Now observing thatx_tn wimpliesJ_ϕxtn−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

x_t−z, J_ϕxt−z

Tx_t−Tz, J_ϕxt−z Φxt−z−

Tz−Txt, Jϕxt−z

≥Φxt−z− Tz−Tx_tJ_ϕxt−z

≥Φxt−z− z−x_tJ_ϕxt−z Φxt−z−Φxt−z 0.

3.20

Since

xttfxt 1−tTxt, 3.21

we can derive that

I−f

xt −1

tI−Txt I−Txt. 3.22 Thus

I−f

xt, 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 replacingtin3.23witht_nand lettingn → ∞, we have

I−f

w, Jϕw−z

≤0. 3.25

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,x_t → xast → 0. This completes the proof.

Theorem 3.2. LetEbe a realp-uniformly convex Banach space with a weakly continuous duality mappingJ_ϕ, 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−2c_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:

iα_nβ_nγ_n1 for alln∈N;

iilim_{n→ ∞}α_n0 and_∞

n0α_n ∞;

iii0<lim inf_{n→ ∞}γ_n≤lim sup_{n→ ∞}γ_n< ξ, whereξ1−2^p−2λc_p⁻¹. Let{xn}be the sequence generated by the following:

x1x∈C,

x_n1 α_nfxn β_nx_nγ_nT_nx_n, n≥1. 3.26

Suppose that{Tn}satisfies the AKTT-condition. LetT be a mapping ofCinto itself defined byTz lim_{n→ ∞}T_nzfor allz ∈Cand suppose thatFT ^∞_n1FTn. Then the sequence{xn}converges strongly toxwhich solves the variational inequality:

I−f

x, J_ϕx−z

≤0, z∈FT. 3.27

Proof. Rewrite the iterative sequence3.26as follows:

x_n1α_nfxn β_nx_nγ_nS_nx_n, n≥1, 3.28

whereβ_n β_n−γn/ξ1−ξ,γ_n γ_n/ξandS_n : 1−ξIξT_n,Iis the identity mapping.

By Lemma 2.1,S_n is nonexpansive such thatFSn FTnfor all n ∈ N. Taking anyq ∈

∞n1FTn, from3.28, it implies that

x_n1−q≤α_nfxn−qβ_nx_n−qγ_nS_nx_n−q

≤αnkxn−qαnf q

−q 1−αnxn−q α_n1−k 1

1−kf q

−q 1−α_n1−kx_n−q

≤maxx₁−q, 1 1−kf

q

−q .

3.29

Therefore, the sequence {xn}is bounded, and so are the sequences{fxn},{Snxn}. Since S_nx_n 1−ξ_nxnξ_nT_nx_nand lim infξ_n>0, we know that{Tnx_n}is bounded. We note that for any bounded subsetBofC,

sup

z∈BS_n1z−S_nzsup

z∈B1−ξ_n1zξ_n1T_n1z−1−ξ_nzξ_nT_nz

≤ |ξn1−ξn|sup

z∈Bzξ_n1sup

z∈BTn1z−Tnz|ξn1−ξn|sup

z∈BTnz |ξn1−ξn|sup

z∈BzTz ξ_n1sup

z∈BTn1z−Tnz.

3.30

From_∞

n1|ξ_n1−ξ_n|<∞and{Tn}satisfing AKTT-condition, we obtain that ∞

n1

sup

z∈BS_n1z−S_nz<∞, 3.31

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

Sz lim

n→ ∞S_nz, ∀z∈C. 3.32

Moreover, we haveSis nonexpansive and

Sz lim

n→ ∞Snz lim

n→ ∞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:

x_n1β_nxn 1−β_n

yn, 3.34

where

yn αn

1−β_nfxn

γ_n

1−β_nSnxn. 3.35

We estimate from3.35 y_n1−y_n

α_n1

1−β_n1fx_n1 γ_n1

1−β_n1 S_n1x_n1− α_n 1−β_nfxn

γ_n

1−β_nS_nx_n

≤ α_n1

1−β_n1kxn1−xn γ_n1

1−β_n1Sn1x_n1−Snxn

α_n1

1−β_n1 − αn

1−β_n

fxn−S_nx_n

≤ α_n1

1−β_n1kx_n1−x_n γ_n1

1−β_n1S_n1x_n1−S_n1x_nS_n1x_n−S_nx_n

α_n1

1−β_n1 − αn

1−β_n

fxn−Snxn

≤ α_n1

1−β_n1kxn1−xn γ_n1 1−β_n1

xn1−xn sup

z∈{xn}Sn1z−Snz

α_n1

1−β_n1 − α_n 1−β_n

fxn−S_nx_n.

3.36

Hence

y_n1−y_n− x_n1−x_n ≤ α_n1

1−β_n1 kx_n1−x_n γ_n1 1−β_n1 sup

z∈{xn}S_n1z−S_nz

α_n1

1−β_n1 − αn

1−β_n

fxn−Snxn.

3.37

Since lim_{n→ ∞}α_n0, and lim_{n→ ∞}sup_z∈{x

n}Sn1z−S_nz0, we have from3.37that lim sup

n→ ∞

y_n1−y_n− x_n1−x_n

≤0. 3.38

Hence, byLemma 2.3, we obtain

nlim→ ∞yn−xn0. 3.39

From3.35, we get

nlim→ ∞yn−Snxn lim

n→ ∞

αn

1−β_nfxn−Snxn0, 3.40

and so it follows from3.39and3.40that

nlim→ ∞xn−Snxn0. 3.41

It follows fromLemma 2.6and3.41, we have xn−Sxn ≤ xn−SnxnSnxn−Sxn

≤ x_n−S_nx_nsup{Snz−Sz:z∈ {x_n}} −→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

klim→ ∞

fx−x, J ϕxnk −x

lim sup

n→ ∞

fx−x, J ϕxn−x

. 3.45

If follows from reflexivity ofEand the boundedness of sequence{xnk}that there exists{xn_ki} which is a subsequence of{xnk}converging weakly tow ∈Casi → ∞. SinceJ_ϕ is weakly continuous, we have byLemma 2.8that

lim sup

i→ ∞ Φxn_ki −x

lim sup

i→ ∞ Φxn_ki −w

Φx−w, ∀x∈E. 3.46

Let

Hx lim sup

i→ ∞ Φxn_ki −x

, ∀x∈E. 3.47

It follows that

Hx Hw Φx−w, ∀x∈E. 3.48

From3.42, we obtain

HSw lim sup

i→ ∞ Φx_nki −Sw

lim sup

i→ ∞ ΦSx_nki −Sw

≤lim sup

i→ ∞ Φxn_ki −w

Hw.

3.49

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→ ∞

fx −x, J ϕxn−x lim

k→ ∞

fx −x, J ϕxnk−x lim

i→ ∞

fx−x, J _ϕ

x_nki −x

I−f

x, J_ϕx−w

≤0

3.52

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

Φxn1−x Φ αn

fxn−fx

β_nxn−x γ_nSnxn−x αn

fx −x

≤Φαn

fxn−fx

β_nxn−x γ_nSnxn−x α_n

fx −x, J _ϕxn1−x

≤Φ

αnkxn−x β_nxn−x γ_nxn−x α_n

fx −x, J _ϕx_n1−x Φ1−α_n1−kxn−x α_n

fx−x, J _ϕxn1−x

≤1−αn1−kΦxn−x αn

fx−x, J ϕxn1−x .

3.53

It follows that from conditioniand3.44that

nlim→ ∞αn0,

∞ n1

αn∞, lim sup

n→ ∞

fx −x, J ϕxn1−x

≤0. 3.54

ApplyLemma 2.7to3.53to concludeΦxn1−x → 0 asn → ∞; that is,x_n → 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 mapping J_ϕ, and C a nonempty closed convex subset of E. Let {Tn : C → C} be a family of

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:

iα_nβ_nγ_n1 for alln∈N;

iilim_{n→ ∞}α_n0 and_∞

n0α_n ∞;

iii0<lim inf_{n→ ∞}γ_n≤lim sup_{n→ ∞}γ_n<1.

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

x1x∈C,

x_n1 αnfxn βnxnγnTnxn, n≥1. 3.55

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 stronglyxwhich solves the variational inequality:

I−f

x, Jϕx−z

≤0, z∈FT. 3.56

Corollary 3.4. LetEbe a realp-uniformly convex Banach space with a weakly continuous duality mappingJ_ϕ, 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−2cp}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:

iα_nβ_nγ_n1 for alln∈N;

iilim_{n→ ∞}α_n0 and_∞

n0α_n ∞;

iii0<lim inf_{n→ ∞}γ_n≤lim sup_{n→ ∞}γ_n< ξ, whereξ1−2^p−2λc_p⁻¹. Let{xn}be the sequence generated by the following

x1x∈C,

x_n1α_nfxn β_nx_nγ_nTx_n, n≥1. 3.57

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

I−f

x, J_ϕx−z

≤0, z∈FT. 3.58

Theorem 3.5. 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 uniformlyλ-strict pseudo-contractions with respect to

p,λ∈0,min{1,2^−p−2cp}and ^∞_n1FTn/∅. Letf:C → Cbe ak-contraction withk∈0,1.

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

iα_nβ_nγ_n1 for alln∈N;

iilimn→ ∞αn0 and_∞

n0αn ∞;

iii0<lim inf_{n→ ∞}γ_n≤lim sup_{n→ ∞}γ_n< ξ, whereξ1−2^p−2λc_p⁻¹. Let{xn}be the sequence generated by the following:

x₁x∈C,

x_n1 αnfxn βnxnγnTnxn, n≥1. 3.59 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 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,

x_t−x_n 1−tSxt−x_n t

fxt−x_n

. 3.60

In view ofLemma 2.8, we calculate

xt−x_n² ≤1−t²Sxt−x_n²2t

fxt−x_n, Jxt−x_n

≤

1−2tt²

xt−xnSxn−xn² 2t

fxt−x_t, Jxt−x_n

2txt−x_n²

3.61

and therefore

fxt−xt, Jxn−xt

≤ t

2xt−xn²1t²xn−Sx_n

2t 2xt−xnxn−Sxn. 3.62

Since{xn},{xt}and{Sxn}are bounded and lim_{n→ ∞}xn−Sx_n/2t 0, we obtain lim sup

n→ ∞

fxt−xt, Jxn−xt

≤ t

2M, 3.63

whereMsup_{n≥1, t∈0,1}{xt−xn²}. We also know that fx −x, Jx n−x

fxt−x_t, Jxn−x_t

fx −fxt x_t−x, Jx n−x_t

fx −x, Jx n−x −Jxn−xt

. 3.64

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, Jx n−x −Jxn−x_t

−→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−x ²≤αn

fxn−fx

β_nxn−x γ_nSnxn−x ² 2α_n

fx −x, Jx _n1−x

≤1−αn1−kxn−x ²2αn

fx −x, Jx n1−x .

3.67

Hence applying inLemma 2.7to3.67, we conclude that limn→ ∞xn−x 0.

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 ^∞_n1FTn/∅. Let f :C → Cbe ak-contraction withk ∈ 0,1. Assume that real sequences{αn},{βn}and{γn}in 0,1satisfy the following conditions:

iαnβnγn1 for alln∈N;

iilim_{n→ ∞}α_n0 and_∞

n0α_n ∞;

iii0<lim inf_{n→ ∞}γ_n≤lim sup_{n→ ∞}γ_n<1.

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

x₁x∈C,

x_n1 αnfxn βnxnγnTnxn, n≥1. 3.68

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 to a common fixed pointxof{Tn}.

Corollary 3.7. 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