New Iterative Approximation Methods for a Countable Family of Nonexpansive Mappings in Banach Spaces

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Volume 2011, Article ID 671754,24pages doi:10.1155/2011/671754

Research Article

New Iterative Approximation Methods for

a Countable Family of Nonexpansive Mappings in

Banach Spaces

Kamonrat Nammanee

1, 2

_{and Rabian Wangkeeree}

2, 3

1_{Department of Mathematics, School of Science and Technology, Phayao University,} Phayao 56000, Thailand

2_{Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand}

3_{Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand}

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

Received 5 October 2010; Accepted 13 November 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 K. Nammanee and R. Wangkeeree. 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.

We introduce new general iterative approximation methods for finding a common fixed point of a countable family of nonexpansive mappings. Strong convergence theorems are established in the framework of reflexive Banach spaces which admit a weakly continuous duality mapping. Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator. The results presented in this paper mainly improve on the corresponding results reported by many others.

1. Introduction

In recent years, the existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authorssee1–4and the references therein. The

well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappingssee5,6. The problem of finding

an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance see 2, 7. A simple algorithmic solution to the problem of minimizing a

quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimationsee7,8.

LetEbe a normed linear space. Recall that a mappingT :E → Eis callednonexpansive

if

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We useFTto denote the set of fixed points ofT, that is,FT {x∈ E : Tx x}. A self mappingf:E → Eis a contraction onEif there exists a constantα∈0,1such that

_f_x₋_f

y≤αx−y, ∀x, y∈E. 1.2

One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping9–11. More precisely, take t ∈ 0,1 and define a contractionTt:E → Eby

Ttxtu 1−tTx, ∀x∈E, 1.3

whereu∈Eis a fixed point. Banach’s contraction mapping principle guarantees thatTthas a unique fixed pointxtinE. It is unclear, in general, what is the behavior ofxtast → 0, even if

T has a fixed point. However, in the case ofT having a fixed point, Browder9proved that ifEis a Hilbert space, then{xt}converges strongly to a fixed point ofT. Reich10extended Browder’s result to the setting of Banach spaces and proved that ifEis a uniformly smooth Banach space, then{xt} converges strongly to a fixed point of T and the limit defines the uniquesunny nonexpansive retraction fromE ontoFT. Xu 11 proved Reich’s results hold in reflexive Banach spaces which have a weakly continuous duality mapping.

The iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example,12–14and the references therein. LetHbe a real Hilbert space, whose inner product and norm are denoted by·,·and·, respectively. LetAbe a strongly positive bounded linear operator onH; that is, there is a constantγ >0 with property

Ax, x ≥γx2, ∀x∈H. 1.4

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH

min x∈FT

1

2Ax, x − x, b, 1.5

wherebis a given point inH. In 2003, Xu13proved that the sequence{xn}defined by the iterative method below, with the initial guessx0∈Hchosen arbitrarily

xn1 I−αnATxnαnu, n≥0 1.6

converges strongly to the unique solution of the minimization problem 1.5 provided the sequence {αn}satisfies certain conditions. Using the viscosity approximation method, Moudafi15introduced the following iterative process for nonexpansive mappingssee16

for further developments in both Hilbert and Banach spaces. Letf be a contraction onH. Starting with an arbitrary initialx0∈H, define a sequence{xn}recursively by

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where {σn} is a sequence in 0,1. It is proved 15, 16 that under certain appropriate conditions imposed on{σn}, the sequence{xn}generated by1.7strongly converges to the unique solutionx∗inCof the variational inequality

I−fx∗, x−x∗≥0, x∈H. 1.8

Recently, Marino and Xu 17 mixed the iterative method 1.6 and the viscosity appro-ximation method1.7and considered the following general iterative method:

xn1 I−αnATxnαnγfxn, n≥0, 1.9

where A is a strongly positive bounded linear operator on H. They proved that if the sequence{αn}of parameters satisfies the following conditions:

C1limn→ ∞αn0, C2∞n1αn∞, C3∞

n1|αn1−αn|<∞,

then the sequence{xn}generated by1.9converges strongly to the unique solutionx∗inH of the variational inequality

A−γfx∗, x−x∗≥0, x∈H, 1.10

which is the optimality condition for the minimization problem: minx∈C1/2Ax, x −hx, wherehis a potential function forγf i.e., hx γfxforx∈H.

On the other hand, in order to find a fixed point of nonexpansive mappingT, Halpern 18was the first who introduced the following iteration scheme which was referred to as Halpern iteration in a Hilbert space:x, x0∈C,{αn} ⊂0,1,

xn1 αnx 1−αnTxn, n≥0. 1.11

He pointed out that the control conditionsC1limn→ ∞αn 0 andC2∞n1αn ∞are necessary for the convergence of the iteration scheme1.11to 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. 1 studied the strong convergence of the following modified version of Halpern iteration forx0, x∈C:

xn1αnx 1−αnTnxn, n≥0, 1.12

where C is a nonempty closed convex subset of a uniformly convex Banach space E

whose norm is uniformly G´ateaux diﬀerentiable,{αn}is a sequence in0,1satisfyingC1 limn→ ∞ αn0,C2

_∞

n1αn∞, and eitherC3 _∞

n1|αn−αn1|<∞orC3

αn∈0,1for alln∈Nand limn→ ∞αn/αn1 1. Very recently, Song and Zheng19also introduced the

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strong convergence theorems of the modified Halpern iteration1.12and the sequence{yn} defined by

y0, y∈C, yn1Tn

αny 1−αnyn

, n≥0, 1.13

in a reflexive Banach spaceEwith a weakly continuous duality mapping and in a reflexive strictly convex Banach space with a uniformly G´ateaux diﬀerentiable norm.

Other investigations of approximating common fixed points for a countable family of nonexpansive mappings can be found in1,20–24and many results not cited here.

In a Banach spaceEhaving a weakly continuous duality mapping Jϕ with a gauge functionϕ, an operatorAis said to be strongly positive 25if there exists a constantγ >0 with the property

Ax, Jϕx

≥γxϕx, 1.14

_αI₋_βA sup x≤1

_αI₋_βA

x, Jϕx, α∈0,1, β∈−1,1, 1.15

whereIis the identity mapping. IfE: His a real Hilbert space, then the inequality1.14

reduces to1.4.

In this paper, motivated by Aoyama et al. 1, Song and Zheng 19, and Marino

and Xu17, we will combine the iterative method1.12with the viscosity approximation method1.9and consider the following three new general iterative methods in a reflexive Banach spaceEwhich admits a weakly continuous duality mappingJϕwith gaugeϕ:

x0x∈E,

xn1αnγfTnxn I−αnATnxn, n≥0,

1.16

z0z∈E,

zn1 αnγfzn I−αnATnzn, n≥0,

y0y∈E,

yn1Tn

αnγf

yn

I−αnAyn

, n≥0,

1.17

where A is strongly positive defined by 1.15, {Tn : E → E} is a countable family of nonexpansive mappings, and f is an α-contraction. We will prove in Section 3 that if the sequence {αn} of parameters satisfies the appropriate conditions, then the sequences

{xn},{zn}, and{yn}converge strongly to the unique solutionxof the variational inequality

A−γfx, Jϕ

x−p≤0, ∀p∈ ∞

n1

FTn. 1.18

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2. Preliminaries

Throughout this paper, letEbe a real Banach space, andE∗be its dual space. We writexn x resp.,xn∗xto indicate that the sequence{xn}weaklyresp., weak∗converges tox; as usualxn → xwill symbolize strong convergence. LetU{x∈E:x1}. A Banach space

Eis said touniformly convexif, for any∈0,2, there existsδ >0 such that, for anyx, y∈U,

x−y ≥impliesxy/2 ≤1−δ. It is known that a uniformly convex Banach space is reflexive and strictly convexsee also26. A Banach spaceEis said to besmoothif the limit limt→0xty − x/texists for allx, y ∈U. It is also said to beuniformly smoothif the

limit is attained uniformly forx, y∈U.

By a gauge functionϕ, we 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. 2.1

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 all_{x /}0 see27. Browder27initiated the study of certain classes of nonlinear operators by means of the duality mappingJϕ. Following Browder27, we say that a Banach space Ehas a weakly continuous duality mappingif there exists a gauge ϕ for 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 thatlp_{has a weakly continuous duality mapping with a gauge} functionϕt tp−1for all 1< p <∞. Set

Φt

_t

0

ϕτdτ, ∀t≥0, 2.2

then

Jϕx ∂Φx, ∀x∈E, 2.3

where∂denotes the subdiﬀerential in the sense of convex analysis.

Now, we collect some useful lemmas for proving the convergence result of this paper. 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 in28.

Lemma 2.1see28. Assume that a Banach spaceEhas a weakly continuous duality mappingJϕ

with gaugeϕ.

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

Φxy≤Φx y, Jϕ

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In particular, for allx, y∈E,

_x_y2_≤

x22y, Jxy. 2.5

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

then the following identity holds:

lim sup n→ ∞ Φ

_x_n₋_ylim sup

n→ ∞ Φxn−x Φ

_y₋_x_, _∀_{x, y}_∈_E. _2.6

Lemma 2.2see1, Lemma 2.3. Let{an}be a sequence of nonnegative real numbers such that

satisfying the property

an1≤1−αnanαncnbn, ∀n≥0, 2.7

where{αn},{bn},{cn}satisfying the restrictions i∞n1αn∞;ii

_∞

n1bn<∞;iiilim supn→ ∞cn≤0.

Then,limn→ ∞an0.

Definition 2.3see1. 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

Tn1z−Tnz<∞. 2.8

Remark 2.4. The example of the sequence of mappings {Tn} satisfying AKTT-condition is supported byLemma 4.6.

Lemma 2.5see1, 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. 2.9

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

The next valuable lemma was proved by Wangkeeree et al.25. Here, we present the

proof for the sake of completeness.

Lemma 2.6. Assume that a Banach spaceEhas a weakly continuous duality mappingJϕwith gauge

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Proof. From 1.15, we obtain thatA sup_x_≤₁|Ax, Jϕx|. Now, for any x ∈ E with

x1, we see that

I−ρAx, Jϕxϕ1−ρAx, Jϕx≥ϕ1−ρA ≥0. 2.10

That is,I−ρAis positive. It follows that

_I₋_ρA_sup

I−ρAx, Jϕx

:x∈E,x1

supϕ1−ρAx, Jϕx

:x∈E,x1

≤ϕ1−ργϕ1 ϕ11−ργ.

2.11

LetEbe a Banach space which admits a weakly continuous dualityJϕ with gaugeϕ such thatϕis invariant on0,1that is,ϕ0,1⊂ 0,1. LetT :E → Ebe a nonexpansive mapping,t ∈ 0,1,f anα-contraction, andAa strongly positive bounded linear operator with coeﬃcientγ >0 and 0< γ < γϕ1/α. Define the mappingSt:E → Eby

Stx tγfx I−tATx, ∀x∈E. 2.12

Then,Stis a contraction mapping. Indeed, for anyx, y∈E,

_S_t_x₋_S_t

ytγfx−fy I−tATx−Ty

≤tγfx−fyI−tATx−Ty

≤tγαx−yϕ11−tγx−y

≤1−tϕ1γ−γαx−y.

2.13

Thus, by Banach contraction mapping principle, there exists a unique fixed pointxtinE, that is

xttγfxt I−tATxt. 2.14

Remark 2.7. We note thatlp_{space has a weakly continuous duality mapping with a gauge} functionϕt tp−1_{for all 1}_{< p <}_∞_{. This shows that}_ϕ_{is invariant on}₀_,_1.

Lemma 2.8 see 25, Lemma 3.3. Let E be a reflexive Banach space which admits a weakly continuous duality mapping Jϕ with gauge ϕ such thatϕ is invariant on0,1. Let T : E → E

be a nonexpansive mapping withFT/∅,f anα-contraction, andAa strongly positive bounded linear operator with coeﬃcient γ > 0 and0 < γ < γϕ1/α. Then, the net{xt}defined by 2.14

converges strongly ast → 0to a fixed pointxofT which solves the variational inequality

A−γfx, Jϕ

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3. Main Results

We now state and prove the main theorems of this section.

Theorem 3.1. LetEbe a reflexive Banach space which admits a weakly continuous duality mapping

Jϕ with gaugeϕsuch thatϕis invariant on0,1. Let{Tn : E → E}∞n0 be a countable family of

nonexpansive mappings satisfyingF :∞_n₀FTn/∅. Letf be anα-contraction andAa strongly positive bounded linear operator with coeﬃcientγ >0and0 < γ < γϕ1/α. Let the sequence{xn}

be generated by1.16, where{αn}is a sequence in0,1satisfying the following conditions:

C1 lim n→ ∞αn0, C2 ∞

n0αn∞

,

C3 ∞ n0

|αn−αn1|<∞.

Suppose that{Tn}satisfies the AKTT-condition. LetT be a mapping ofEinto itself defined byTz limn→ ∞Tnzfor allz∈E, and suppose thatFT

_∞

n0FTn. Then,{xn}converges strongly tox

which solves the variational inequality

A−γfx, Jϕ

x−p≤0, ∀p∈F. 3.1

Proof. Applying Lemma 2.8, there exists a point x ∈ FT which solves the variational inequality3.1. Next, we observe that{xn}is bounded. Indeed, pick anyp∈Fto obtain

_x_n₁₋_p_α_n_γf_T_n_x_n_I₋_α_n_A_T_n_x_n₋_p αn

γfTnxn−A

p I−αnATnxn−I−αnAp

I−αnATnxn−TnpαnγfTnxn−A

p

≤ϕ11−αnγxn−pαnγαxn−pαnγf

p−Ap

≤ϕ1−αn

ϕ1γ−γαxn−pαnγf

p−Ap

≤1−αn

ϕ1γ−γαxn−pαn

ϕ1γ−γαγf

p−Ap ϕ1γ−γα .

3.2

It follows from induction that

_x_n₁₋_p_≤max

_x₀₋_p_,γfp−Ap ϕ1γ−γα

, n≥0. 3.3

Thus,{xn}is bounded, and hence so are{ATnxn}and{fTnxn}. Now, we show that

lim

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We observe that

xn1−xnαnγfTnxn I−αnATnxn−αn−1γfTn−1xn−1−I−αn−1ATn−1xn−1

αnγfTnxn−αnγfTn−1xn−1 αnγfTn−1xn−1

−αn−1γfTn−1xn−1 I−αnATnxn−I−αnATn−1xn−1

I−αnATn−1xn−1−I−αn−1ATn−1xn−1

≤αnγαTnxn−Tn−1xn|αn−αn−1|γfTn−1xn−1−ATn−1xn−1

I−αnATnxn−Tn−1xn−1

≤αnγαTnxn−Tnxn−1αnγαTnxn−1−Tn−1xn|αn−αn−1|M

ϕ11−αγTnxn−Tnxn−1ϕ1

1−αγTnxn−1−Tn−1xn−1

≤1−αn

ϕ1γ−γαxn−xn−1

1−αn

ϕ1γ−γαTnxn−1−Tn−1xn−1|αn−αn−1|M

≤1−αn

ϕ1γ−γαxn−xn−1Tnxn−1−Tn−1xn−1|αn−αn−1|M,

3.5

for alln≥1, whereMis a constant satisfyingM≥sup_n_≥₁γfTn−1xn−1−ATn−1xn−1. Putting

μnTnxn−1−Tn−1xn−1|αn−αn−1|M. From AKTT-condition andC3, we have

∞

n1

μn≤ ∞

n1

sup x∈{xn}

Tnx−Tn−1x

∞

n1

|αn−αn−1|M <∞. 3.6

Therefore, it follows fromLemma 2.2that limn→ ∞xn1−xn 0. Since limn→ ∞αn 0, we obtain

Tnxn−xn ≤ xn−xn1xn1−Tnxn

≤ xn−xn1αnγfTnxn−ATnxn−→0.

3.7

UsingLemma 2.5, we obtain

Txn−xn ≤ Txn−TnxnTnxn−xn

≤sup{Tz−Tnz:z∈ {xn}}Tnxn−xn −→0.

3.8

Next, we prove that

lim sup n→ ∞

γfx−Ax, Jϕxn−x

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

lim k→ ∞

γfx−Ax, Jϕxnk −x

lim sup n→ ∞

. 3.10

If follows from reflexivity ofE and the boundedness of a sequence{xnk} that there exists

{xn_ki}which is a subsequence of{xnk}converging weakly to w ∈ Easi → ∞. SinceJϕ is

weakly continuous, we have byLemma 2.1that

lim sup n→ ∞ Φ

_x n_ki −x

lim sup n→ ∞ Φ

_x

n_ki −w

Φx−w, ∀x∈E. 3.11

Let

Hx lim sup n→ ∞ Φ

_x n_ki −x

, ∀x∈E. 3.12

It follows that

Hx Hw Φx−w, ∀x∈E. 3.13

Then, from limn→ ∞xn−Txn0, we have

HTw lim sup i→ ∞ Φ

_x

n_ki −Tw

lim sup i→ ∞ Φ

_Tx

n_ki −Tw

≤lim sup i→ ∞ Φ

_x

n_ki −w

Hw.

3.14

On the other hand, however,

HTw Hw ΦTw−w. 3.15

It follows from3.14and3.15that

ΦTw−w HTw−Hw≤0. 3.16

Therefore,Tww, and hencew∈FT. Since the duality mapJϕis single valued and weakly continuous, we obtain, by3.1, that

lim sup n→ ∞

lim k→ ∞

γfx−Ax, Jϕxnk−x

lim i→ ∞

γfx−Ax, Jϕ

xnki −x

A−γfx, Jϕx−w≤0.

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Next, we show thatxn → xasn → ∞. In fact, since Φt _t

0ϕτdτ,for all t ≥ 0, and

ϕ: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. 3.18

Finally, we show thatxn → xasn → ∞. FollowingLemma 2.1, we have

Φxn1−x Φαn

γfTnxn−Ax

I−αnATnxn−I−αnAx

≤ΦI−αnATnxn−I−αnAx αn

γfTnxn−Ax, Jϕxn1−x

≤ϕ11−αnγ

ΦTnxn−x αn

≤1−αnγ

Φxn−x αn

1−αnγ

Φxn−x αn

γfTnxn−γfTnxn1, Jϕxn1−x

αn

γfTnxn1−γfx, Jϕxn1−x

αn

γfx−Ax, Jϕxn1−x

≤1−αnγ

Φxn−x αnγαxn−xn1Jϕxn1−x

αnγαxn1−xJϕxn1−xαn

1−αnγ

Φxn−x αnγαxn−xn1Jϕxn1−x

αnγαΦxn1−x αn

.

3.19

It then follows that

Φxn1−x

≤ 1−αnγ

1−αnγαΦxn−x

αn

γα

1−αnγαxn−xn1M

1

1−αnγα

1−αn

γγα

1−αnγα

Φxn−x αn

γγα

1−αnγα

× ₁₋

αnγα

γγα

γα

1

1−αnγα

,

3.20

whereMsup_n_≥₀Jϕxn1−x. Put

γn

αn

γγα

1−αnγα ,

δn

1−αnγα

γγα

γα

1

1−αnγα

.

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It follows that from conditionC1, limn→ ∞xn1−xn0 and3.9that

lim n→ ∞γn0,

∞

n1

γn∞, lim sup

n→ ∞ δn≤0. 3.22

ApplyingLemma 2.2 to3.20, we conclude that Φxn1−x → 0 as n → ∞; that is,

xn → xasn → ∞. This completes the proof.

Settingγ 1, A ≡ I, whereI is the identity mapping andfx xfor allx ∈ Ein

Theorem 3.1, we have the following result.

Corollary 3.2. LetEbe a reflexive Banach space which admits a weakly continuous duality mapping

Jϕ with gauge ϕ. Suppose that {Tn : E → E} is a countable family of nonexpansive mappings

satisfyingF: ∞

n0FTn/∅

. Assume that{xn}is defined by, forx0, x∈E,

xn1αnx 1−αnTnxn, n≥0, 3.23

where{αn}is a sequence in0,1satisfying the following conditions: C1 lim

n→ ∞αn0, C2 ∞

n0αn

∞,

C3 ∞

n0|αn−αn1|<∞

.

Suppose that{Tn}satisfies the AKTT-condition. LetT be a mapping ofEinto itself defined byTz limn→ ∞Tnzfor allz∈E, and suppose thatFT ∩_n∞₀FTn, then{xn}converges strongly toxof

Fwhich solves the variational inequality

I−fx, Jϕ

x−p≤0, ∀p∈F. 3.24

ApplyingTheorem 3.1, we can obtain the following two strong convergence theorems for the iterative sequences{zn}and{yn}defined by1.17.

nonexpansive mappings satisfyingF :∞n0FTn/∅. Letf be anα-contraction andAa strongly

positive bounded linear operator with coeﬃcientγ >0and0< γ < γϕ1/α. Let the sequence{zn}

be generated by1.17, where{αn}is a sequence in0,1satisfying the following conditions: C1 lim

n→ ∞αn0, C2 ∞

n0αn∞

,

C3 ∞

n0|αn−αn1|<∞

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_∞

n0FTn, then{zn}converges strongly tox

which solves the variational inequality3.1.

Proof. Let{xn}be the sequence given byx0z0and

xn1αnγfTnxn I−αnATnxn, n≥0. 3.25

FormTheorem 3.1,xn → x. We claim thatzn → x. ApplyingLemma 2.6, we estimate

xn1−zn1 ≤αnγfzn−fTnxnI−αnATnxn−Tnzn

≤αnγαzn−Tnxnϕ1

1−αnγ

xn−zn

≤αnγαzn−TnxαnγαTnx−Tnxnϕ1

1−αnγ

xn−zn

≤αnγαzn−xαnγαTnx−Tnxnϕ1

1−αnγ

xn−zn

≤αnγαzn−xnαnγαxn−xαnγαx−xnϕ1

1−αnγ

xn−zn

1−αn

ϕ1γ−γαxn−znαn

ϕ1γ−γα 2αγ

ϕ1γ−γαx−xn.

3.26

It follows from ∞_n₁αn ∞, limn→ ∞xn −x 0, and Lemma 2.2that xn −zn → 0. Consequently,zn → xas required.

nonexpansive mappings satisfyingF :∞n0FTn/∅. Letf be anα-contraction andAa strongly

positive bounded linear operator with coeﬃcientγ >0and0 < γ < γϕ1/α. Let the sequence{yn}

be generated by1.17, where{αn}is sequence in0,1satisfying the following conditions:

n0αn

∞,

C3 ∞ n0

|αn−αn1|<∞.

Suppose that{Tn}satisfies the AKTT-condition. LetT be a mapping ofEinto itself defined byTz limn→ ∞Tnzfor allz∈E, and suppose thatFT ∞n0FTn, then{yn}converges strongly tox

which solves the variational inequality3.1.

Proof. Let the sequences{zn}and{βn}be given by

znαnγf

yn

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Takingp∈∞n0FTn, we have

_y_n₁₋_p_T_n_z_n₋_T_n_p_≤_z_n₋_p αnγf

yn

I−αnAyn−p

≤1−αn

ϕ1γ−γαyn−pαnγf

p−Ap

1−αn

ϕ1γ−γαyn−pαn

ϕ1γ−γαγf

p−Ap ϕ1γ−γα .

3.28

It follows from induction that

_y_n₁₋_p_≤max

_y₀₋_p_,γfp−Ap ϕ1γ−γα

, n≥0. 3.29

Thus, both{yn}and{zn}are bounded. We observe that

zn1αn1γf

yn1

I−αn1Ayn1βnγfTnzn

I−βnA

Tnzn. 3.30

Thus,Theorem 3.1implies that{zn}converges strongly to some pointx. In this case, we also have

_y_n₋_x_≤_y_n₋_z_n_z_n₋_x_α_n_γf

yn

−Aynzn−x −→0. 3.31

Hence, the sequence{yn}converges strongly tox. This competes the proof.

Settingγ 1, A ≡ I, whereI is the identity mapping andfx xfor allx ∈ Ein

Theorem 3.4, we have the following result.

Corollary 3.5. LetEbe a reflexive Banach space which admits a weakly continuous duality mapping

Jϕ with gauge ϕ. Suppose that {Tn : E → E} is a countable family of nonexpansive mappings

satisfyingF:∞n0FTn/∅. Assume that{xn}is defined by forx0, x∈E,

xn1Tnαnx 1−αnxn, n≥0, 3.32

where{αn}is a sequence in0,1satisfying the following conditions:

n0

αn∞,

C3 ∞

n0|αn−αn1|<∞

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_∞

n0FTn, then{xn}converges strongly toxof

Fwhich solves the variational inequality

I−fx, Jϕ

x−p≤0, ∀p∈F. 3.33

4. Applications

4.1. W

-Mappings

LetT1, T2, . . .be infinite mappings ofCinto itself, and let{ξi}be a nonnegative real sequence with 0≤ξi<1, for all i≥1. For anyn∈N, define a mappingWnofCinto itself as follows:

Un,n1I,

Un,nξnTnUn,n1 1−ξnI,

Un,n−1ξn−1Tn−1Un,n 1−ξn−1I,

.. .

Un,kξkTkUn,k1 1−ξkI,

Un,k−1ξk−1Tk−1Un,k 1−ξk−1I,

.. .

Un,2μ2T2Un,3 1−ξ2I,

WnUn,1ξ1T1Un,2 1−ξ1I.

4.1

Nonexpansivity of each Ti ensures the nonexpansivity ofWn. The mappingWn is called a

W-mapping generated byT1, T2, . . . , Tnandξ1, ξ2, . . . , ξn.

Throughout this section, we will assume that 0< ξn ≤ξ <1,for alln≥1. Concerning

Wndefined by4.1, we have the following useful lemmas.

Lemma 4.1see 4. LetC be a nonempty closed convex subset of a a strictly convex, reflexive Banach spaceE,{Ti:C → C}a family of infinitely nonexpansive mapping with∞i1FTi/∅, and

{ξi}a real sequence such that0< ξi≤ξ <1, for alli≥1, then: 1Wnis nonexpansive andFWn

_∞

i1FTifor eachn≥1;

2for eachx∈Cand for each positive integerk, the limitlimn→ ∞Un,kxexists; 3the mappingW :C → Cdefine by

Wx: lim

n→ ∞Wnxnlim→ ∞Un,1x, x∈C 4.2

is a nonexpansive mapping satisfyingFW ∞i1FTi, and it is called theW-mapping generated

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From Remark 3.1 of Peng and Yao29, we obtain the following lemma.

Lemma 4.2. LetEbe a strictly convex, reflexive Banach space,{Ti : E → E}a family of infinitely

nonexpansive mappings with ∞_i₁FTi/∅, and {ξi} a real sequence such that 0 < ξi ≤ ξ < 1,

for alli≥1. Then sequence{Wn}satisfies theAKTT-condition.

ApplyingLemma 4.2andTheorem 3.1, we obtain the following result.

nonexpansive mappings withF :∞_n₁FTn/∅andf anα-contraction andAa strongly positive bounded linear operator with coeﬃcient γ > 0 and 0 < γ < γϕ1/α. Let the sequence {xn} be

generated by the following:

x1x∈E, xn1αnγfWnxn I−αnAWnxn, 4.3

where{Wn}is defined by4.1and{αn}is a sequence in0,1satisfying the conditions (C1), (C2),

and (C3). Then{xn}converges strongly toxinF.

Theorem 4.4. LetE,{Tn},{Wn},f,A, and{αn}be as inTheorem 4.3. Let the sequence{zn}be

z1z∈E, zn1αnγfzn I−αnAWnzn, 4.4

then{zn}converges strongly toxinF.

Theorem 4.5. LetE,{Tn},{Wn},f,A, and{αn}be as inTheorem 4.3. Let the sequence{yn}be

y1y∈E, yn1Wn

αnγf

yn

I−αnAyn

, 4.5

then{yn}converges strongly toxinF.

4.2. 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

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and FJλ Ψ−1₀ _{for all}_{λ >} _{0. We also know the following} ₃₀_{: for each} _{λ, μ >} _{0 and}

x∈RIλΨRIμΨ, it holds that

_J_λ_x₋_J_μ_x_≤ λ−μ

λ x−Jλx. 4.6

From the Resolvent identity, we have the following lemma.

Lemma 4.6. LetEbe a Banach space andCa nonempty closed convex subset ofE. LetΨ⊆E×Ebe an accretive operator such thatΨ−1₀

/

∅andDΨ⊂ C⊂ _λ>₀RIλΨ. Suppose that{λn}is a

sequence of0,∞such thatinf{λn :n∈N}>0and _∞

n1|λn1−λn|<∞, then ithe sequence{Jλn}satisfies AKTT-condition,

iilimn→ ∞JλnzJλzfor allz∈CandFJλ

_∞

n1FJλn, whereλn → λasn → ∞.

Proof. By the proof of Theorem 4.3 in1and applyingLemma 4.6andTheorem 3.1, we obtain the following result.

Jϕ with gauge ϕ such that ϕ is invariant on 0,1. LetΨ : DΨ ⊂ E → 2E be an accretive

operator such thatΨ−1₀

/

∅. Assume thatKis a nonempty closed convex subset ofEsuch thatDΨ⊂

K ⊂

λ>0

RIλΨandf is anα-contraction. LetAbe a strongly positive bounded linear operator with coeﬃcientγ > 0and 0 < γ < γϕ1/α. Suppose that{λn}is a sequence of0,∞such that limn→ ∞λn ∞. Let the sequence{xn}be generated by the following:

x0x∈E, xn1αnγfJλnxn I−αnAJλnxn, 4.7

where{αn}is a sequence in0,1satisfying the following conditions (C1), (C2), and (C3), then{xn}

converges strongly toxinΨ−1₀_.

Theorem 4.8. LetE,Ψ,K,f,A,{αn}, and{λn}be as inTheorem 4.7. Let{zn}be generated by the

following:

z0z∈E, zn1αnγfzn I−αnAJλnzn, 4.8

then{zn}converges strongly toxinΨ−10.

Theorem 4.9. LetE,Ψ,K,f,A,{αn}, and{λn}be as inTheorem 4.7. Let{yn}be generated by the

following:

y0z∈E, yn1Jλn

αnγf

yn

I−αnAyn

, 4.9

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4.3. The Equilibrium Problems

LetHbe a real Hilbert space, and letFbe a bifunction ofH×H → R, whereRis the set of real numbers. The equilibrium problem forF :H×H → Ris to findx∈Hsuch that

Fx, y≥0, ∀y∈H. 4.10

The set of solutions of4.10is denoted by EPF. Given a mappingT :H → H, letFx, y

Tx, y−xfor allx, y∈H. Then,z∈EPFif and only ifTx, y−z ≥0 for ally∈H, that is,

zis a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of4.10. Some methods have been proposed to solve the

equilibrium problem; see, for instance, Blum and Oettli31and Combettes and Hirstoaga 32. For the purpose of solving the equilibrium problem for a bifunctionF, let us assume thatFsatisfies the following conditions:

A1Fx, x0 for allx∈H,

A2Fis monotone, that is,Fx, y Fy, x≤0 for allx, y∈H, A3for eachx, y, z∈H,limt→0Ftz 1−tx, y≤Fx, y,

A4for eachx∈H, y→Fx, yis convex and lower semicontinuous.

The following lemmas were also given in31,32, respectively.

Lemma 4.10see31, Corollary 1. LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction ofC×C → RsatisfyingA1–A4. Letr >0andx∈H, then there existsz∈Csuch thatFz, y 1/ry−z, z−x ≥0for ally∈C.

Lemma 4.11see32, Lemma 2.12. Assume thatF :C×C∈RsatisfiesA1–A4. Forr >0

andx∈H, define a mappingTr :H → Cas follows:

Trx

z∈C;Fz, y1 r

y−z, z−x≥0,∀y∈C ∀x∈H, 4.11

then, the following hold:

1Tr is single valued,

2Tr is firmly nonexpansive, that is, for anyx, y∈H,Trx−Try2≤ Trx−Try, x−y, 3FTr EPF,

4EPFis closed and convex.

Theorem 4.12. LetHbe a real Hilbert space. LetF be a bifunction fromH×H → Rsatisfying (A1)–(A4) andEPF/∅. Letf be anα-contraction,Aa strongly positive bounded linear operator with coeﬃcientγ >0and0< γ < γ/α. Let the sequences{xn},{un}be generated byx0∈Hand

Fun, y

1

rn

y−un, un−xn

≥0, ∀y∈H,

xn1αnγfun I−αnAun,

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for alln≥0, where{αn}is a sequence in0,1andrn∈0,∞satisfying the following conditions:

C1limn→ ∞αn0,

C2∞n1αn∞,

C3∞n1|αn1−αn|<∞,

C4lim infn→ ∞rn>0and∞n1|rn1−rn|<∞.

then{xn}and{un}converge strongly tox∈EPF.

Proof. Following the proof technique ofTheorem 3.1, we only need, show that limn→ ∞xn−

Trxn0, for allr >0. From4.12, it follows that

xn2−xn1αn1γfun1 I−αn1Aun1−αnγfun−I−αnAun αn1γfun1 I−αn1Aun1−I−αn1Aun I−αn1Aun

−αnγfun−I−αnAun−αn1γfun αn1γfun

≤ I−αn1Aun1−unαn1γfun1−fun

|αn1−αn|γfunαnA−αn1Aun I−αn1Aun1−unαn1γfun1−fun

|αn1−αn|γfunαn−αn1Aun

≤1−αn1γ

Trn1xn1−Trnxnαn1γfun1−fun

|αn1−αn|γfun|αn−αn1|Aun 1−αn1γ

Trn1xn1−Trn1xnTrn1xn−Trnxn

αn1γfun1−fun|αn1−αn|γfun|αn−αn1|Aun

≤1−αn1γ

Trn1xn1−Trn1xn

1−αn1γ

Trn1xn−Trnxn

αn1γfun1−fun|αn1−αn|γfun|αn−αn1|Aun

≤1−αn1γ

xn1−xn

1−αn1γ

Trn1xn−Trnxn

αn1γαun1−un|αn1−αn|γfun|αn−αn1|Aun

≤1−αn1γ

xn1−xn

1−αn1γ

Trn1xn−Trnxnαn1γαxn1−xn

αn1γαTrn1xn−Trnxn|αn1−αn|γfun|αn−αn1|Aun

1−αn1

γ−γαxn1−xn

1−αn1

γ−γαTrn1xn−Trnxn

|αn1−αn|γfun|αn−αn1|Aun.

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On the other hand, from the definition ofTr we have

FTrnxn, y

1

rn

y−Trnxn, Trnxn−xn

≥0, ∀y∈H,

FTrn1xn, y

1

rny−Trn1xn, Trn1xn−xn ≥

0, ∀y∈H.

4.14

PuttingyTrn1xnandyTrnxnin4.14, we have

FTrnxn, Trn1xn

1

rn

Trn1xn−Trnxn, y, Trnxn, Trn1xn−xn

≥0,

FTrn1xn, Trnxn

1

rnTrnxn−Trn1xn, Trnxn, Trn1xn, Trnxn−xn ≥ 0.

4.15

So, fromA2, we have !

Trnxn−Trn1xn,

Trn1xn−xn rn1 −

Trnxn−xn rn

"

≥0, 4.16

and hence, !

Trnxn−Trn1xn,

Trn1xn−Trnxn

rn1

1

rn1 −

1

rn

Trnxn−xn

"

≥0, 4.17

then we have

Trn1xn−Trnxn

2

rn1 ≤

!

Trnxn−Trn1xn,

1

rn1 −

1

rn

Trnxn−xn

"

≤ Trnxn−Trn1xn

_r_n1₁ − 1

rn

Trnxn−xn

≤ Trnxn−Trn1xn

_r_n1₁ − 1

rn 2M,

4.18

and hence,

Trn1xn−Trnxn ≤

1−rn1

rn

2M, 4.19

whereMis a constant satisfyingM≥sup_n_≥₀Trnxn−xn. Substituting4.19in4.13yields

xn2−xn1 ≤

1−αn1

γ−γαxn1−xn

1−αn1

γ−γα2M

b |rn1−rn|

|αn1−αn|γfun|αn−αn1|Aun

≤1−αn1

γ−γαxn1−xn2M|αn1−αn| 2M

b |rn1−rn|,

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for somebwithrn> b >0the definition lim infn→ ∞rn>0. By the assumptions on{rn}and

{αn}and usingLemma 2.2, we conclude that

lim

n→ ∞xn1−xn0. 4.21

From the definition ofxnand limn→ ∞αn0, it follows that

xn1−unαnγfun I−αnAun−un αnγfun−αnAun

αnγfun−Aun−→0.

4.22

Combining4.21and4.22, we have

lim

n→ ∞xn−unnlim→ ∞xn−Trnxn0. 4.23

From the definition ofTr, it follows that

FTrTrnxn, y

1

r

y−TrTrnxn, TrTrnxn−Trnxn

≥0, ∀y∈H. 4.24

PuttingyTrTrnxnin4.14andyTrnxnin4.24, we have

FTrnxn, TrTrnxn

1

rnTrTrnxn−Trnxn, Trnxn−xn ≥

0, ∀y∈H,

FTrTrnxn, Trnxn

1

rTrnxn−TrTrnxn, TrTrnxn−Trnxn ≥0, ∀y∈H.

4.25

So, fromA2, we have !

Trnxn−TrTrnxn,

TrTrnxn−xn

r −

Trnxn−xn rn

"

≥0, 4.26

and hence, for eachr >0,

TrTrnxn−Trnxn

2

r ≤

!

Trnxn−TrTrnxn,

1

rnxn−Trnxn "

≤ TrTrnxn−Trnxn

1

rnTrnxn−xn,

4.27

then

TrTrnxn−Trnxn ≤

rTrnxn−xn

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Since

xn−Trxn ≤ xn−TrnxnTrnxn−TrTrnxnTrTrnxn−Trxn

≤2xn−Trnxn

rTrnxn−xn

b

2 r

b

xn−Trnxn,

4.29

then for eachr >0, we have from4.23

lim

n→ ∞xn−Trxn0. 4.30

This completes the proof.

ApplyingTheorem 4.12, we can obtain the following result.

Corollary 4.13. LetHbe a real Hilbert space. LetF be a bifunction fromH×H → Rsatisfying (A1)–(A4) andEPF/∅. Letf be anα-contraction,Aa strongly positive bounded linear operator with coeﬃcientγ >0and0< γ < γ/α. Let the sequences{zn},{un}be generated byz0∈Hand

Fun, y

1

rn

y−un, un−zn

≥0, ∀y∈H,

zn1αnγfzn I−αnAun,

4.31

for alln∈N, where{αn}is a sequence in0,1andrn∈0,∞satisfying the following conditions:

C1limn→ ∞αn0,

C2∞n1αn∞,

C3∞n1|αn1−αn|<∞,

C4lim infn→ ∞rn>0and _∞

n1|rn1−rn|<∞,

then{zn}and{un}converge strongly tox∈EPF.

Proof. We observe thatun Trnznfor alln≥0. Then we rewrite the iterative sequence4.31

by the following:

z0 ∈H, zn1αnγfzn I−αnATrnzn. 4.32

Let{xn}be the sequence given byx0z0and

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FormTheorem 4.12, xn → x in EPF. We claim thatzn → x. ApplyingLemma 2.6, we estimate

xn1−zn1 ≤αnγfzn−fTrnxnI−αnATrnxn−Trnzn

≤αnγαzn−Trnxn

1−αnγ

xn−zn

≤αnγαzn−TrnxαnγαTrnx−Trnxn

1−αnγ

xn−zn

≤αnγαzn−xαnγαx−xn

1−αnγ

xn−zn

≤αnγαzn−xnαnγαxn−xαnγαx−xn

1−αnγ

xn−zn

1−αn

γ−γαxn−znαn

γ−γα 2αγ

γ−γαx−xn.

4.34

It follows from∞_n₁αn ∞, limn→ ∞xn−x 0, andLemma 2.2thatxn−zn → 0 as

n → ∞. Consequently,zn → xas required.

Acknowledgments

The authors would like to thank the Centre of Excellence in Mathematics, Thailand for financial support. Finally, They would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

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