Volume 2011, Article ID 643740,21pages doi:10.1155/2011/643740

*Research Article*

**Strong Convergence Theorems of the General**

**Iterative Methods for Nonexpansive Semigroups in** **Banach Spaces**

**Rattanaporn Wangkeeree**

^{1, 2}*1**Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand*

*2**Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand*

Correspondence should be addressed to Rattanaporn Wangkeeree,rattanapornw@nu.ac.th Received 4 February 2011; Accepted 22 March 2011

Academic Editor: Yonghong Yao

Copyrightq2011 Rattanaporn 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.

Let*E*be a real reflexive Banach space which admits a weakly sequentially continuous duality
mapping from*E*to *E*^{∗}. LetS {Ts : 0 ≤ *s <* ∞}be a nonexpansive semigroup on*E*such
that FixS :

*t≥0*FixTt*/*∅, and*f* is a contraction on*E* with coeﬃcient 0 *< α <* 1. Let
*F* be *δ-strongly accretive and* *λ-strictly pseudocontractive with* *δ* *λ >* 1 and *γ* a positive
real number such that*γ <* 1/α1−

1−*δ/λ. When the sequences of real numbers*{α*n*}and
{t*n*} satisfy some appropriate conditions, the three iterative processes given as follows:*x**n1*
*α**n**γfx**n* I −*α**n**FT*t*n*x*n*, *n* ≥ 0, *y**n1* *α**n**γf*Tt*n*y*n* I −*α**n**FTt**n*y*n*, *n* ≥ 0, and
*z**n1**Tt**n*α*n**γfz**n* I−*α**n**Fz**n*,*n*≥0 converge strongly to*x, where* *x*is the unique solution
in FixSof the variational inequalityF−*γfx, jx* −*x ≥* 0,*x*∈FixS. Our results extend and
improve corresponding ones of Li et al.2009Chen and He2007, and many others.

**1. Introduction**

Let*E*be a real Banach space. A mapping*T* of*Einto itself is said to be nonexpansive if*Tx−
*Ty ≤ x*−*y*for each*x, y*∈*E. We denote by FixT*the set of fixed points of*T. A mapping*
*f*:*E* → *E*is called*α-contraction if there exists a constant 0< α <*1 such thatfx−*fy ≤*
*αx*−*y*for all*x, y*∈*E. A family*S{Tt: 0≤*t <*∞}of mappings of*E*into itself is called
*a nonexpansive semigroup onE*if it satisfies the following conditions:

i*T*0x*x*for all*x*∈*E;*

ii*T*s*t T*sTtfor all*s, t*≥0;

iiiTtx−*Tty ≤ x*−*y*for all*x, y*∈*E*and*t*≥0;

ivfor all*x*∈*E, the mappingt*→*Ttx*is continuous.

We denote by FixSthe set of all common fixed points ofS, that is,
FixS:{x∈*E*:*T*tx*x,*0≤*t <*∞}

*t≥0*

FixTt. 1.1

In1, Shioji and Takahashi introduced the following implicit iteration in a Hilbert space

*x**n**α**n**x* 1−*α**n*1
*t**n*

_{t}_{n}

0

*Tsx**n**ds,* ∀n∈^{}*,* 1.2

where {α*n*} is a sequence in 0,1 and {t*n*} is a sequence of positive real numbers which
diverges to ∞. Under certain restrictions on the sequence {α*n*}, Shioji and Takahashi 1
proved strong convergence of the sequence{x*n*}to a member of*FS. In*2, Shimizu and
Takahashi studied the strong convergence of the sequence{x*n*}defined by

*x**n1**α**n**x* 1−*α**n*1
*t**n*

_{t}_{n}

0

*T*sx*n**ds,* ∀n∈^{} 1.3

in a real Hilbert space where{Tt : *t* ≥ 0} is a strongly continuous semigroup of nonex-
pansive mappings on a closed convex subset*C* of a Banach space*E* and lim*n*→ ∞*t**n* ∞.

Using viscosity method, Chen and Song3studied the strong convergence of the following
iterative method for a nonexpansive semigroup{Tt : *t* ≥ 0}with FixS*/*∅ in a Banach
space:

*x**n1* *α**n**f*x 1−*α**n*1
*t**n*

_{t}_{n}

0

*T*sx*n**ds,* ∀n∈^{}*,* 1.4

where*f* is a contraction. Note however that their iterate*x**n*at step*n*is constructed through
the average of the semigroup over the interval0, t. Suzuki 4was the first to introduce
again in a Hilbert space the following implicit iteration process:

*x**n**α**n**u* 1−*α**n*Tt*n*x*n**,* ∀n∈^{}*,* 1.5
for the nonexpansive semigroup case. In 2002, Benavides et al.5, in a uniformly smooth
Banach space, showed that ifSsatisfies an asymptotic regularity condition and{α*n*}fulfills
the control conditions lim_{n→ ∞}*α**n* 0,_{∞}

*n1**α**n* ∞, and lim*n*→ ∞*α**n**/α**n1* 0, then both the
implicit iteration process1.5and the explicit iteration process1.6,

*x**n1**α**n**u* 1−*α**n*Tt*n*x*n**,* ∀n∈^{}*,* 1.6
converge to a same point of *FS. In 2005, Xu* 6 studied the strong convergence of the
implicit iteration process1.2and1.5in a uniformly convex Banach space which admits a

weakly sequentially continuous duality mapping. Recently, Chen and He7introduced the viscosity approximation process:

*x**n1**α**n**fx**n*

1−*β**n* *T*t*n*x*n**,* ∀n∈^{}*,* 1.7

where*f*is a contraction and{α*n*}is a sequence in0,1and a nonexpansive semigroup{Tt:
*t*≥0}. The strong convergence theorem of{x*n*}is proved in a reflexive Banach space which
admits a weakly sequentially continuous duality mapping. In8, Chen et al. introduced and
studied modified Mann iteration for nonexpansive mapping in a uniformly convex Banach
space.

On the other hand, iterative approximation methods for nonexpansive mappings have
recently been applied to solve convex minimization problems; see, for example,9–11and
the references therein. Let *H* be a real Hilbert space, whose inner product and norm are
denoted by·,·and · , respectively. Let*A*be a strongly positive bounded linear operator
on*H; that is, there is a constantγ >*0 with property

Ax, x ≥*γx*^{2} ∀x∈*H.* 1.8

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

min*x∈C*

1

2Ax, x − x, b, 1.9

where*C*is the fixed point set of a nonexpansive mapping*T*on*H*and*b*is a given point in*H.*

In 2003, Xu10proved that the sequence{x*n*}defined by the iterative method below, with
the initial guess*x*0∈*H*chosen arbitrarily,

*x** _{n1}* I−

*α*

*n*

*ATx*

*n*

*α*

*n*

*u,*

*n*≥0, 1.10 converges strongly to the unique solution of the minimization problem 1.9 provided the sequence {α

*n*} satisfies certain conditions. Using the viscosity approximation method, Moudafi12introduced the following iterative process for nonexpansive mappingssee13 for further developments in both Hilbert and Banach spaces. Let

*f*be a contraction on

*H.*

Starting with an arbitrary initial*x*0∈*H, define a sequence*{x*n*}recursively by

*x**n1* 1−*α**n*Tx*n**α**n**fx**n*, *n*≥0, 1.11
where{α*n*}is a sequence in0,1. It is proved12,13that, under certain appropriate con-
ditions imposed on{α*n*}, the sequence{x*n*}generated by1.11 strongly converges to the
unique solution*x*^{∗}in*C*of the variational inequality

*I*−*f* *x*^{∗}*, x*−*x*^{∗}

≥0, *x*∈*H.* 1.12

Recently, Marino and Xu14mixed the iterative method1.10and the viscosity approxima- tion method1.11and considered the following general iterative method:

*x**n1* I−*α**n**ATx**n**α**n**γfx**n*, *n*≥0, 1.13
where *A* is a strongly positive bounded linear operator on *H. They proved that if the*
sequence{α*n*}of parameters satisfies the certain conditions, then the sequence{x*n*}generated
by1.13converges strongly to the unique solution*x*^{∗}in*H*of the variational inequality

*A*−*γf* *x*^{∗}*, x*−*x*^{∗}

≥0, *x*∈*H* 1.14

which is the optimality condition for the minimization problem, min* _{x∈C}*1/2Ax, x −

*hx,*where

*h*is a potential function for

*γf*i.e., h

^{}x

*γfx*for

*x*∈

*H.*

Very recently, Li et al. 15 introduced the following iterative procedures for the
approximation of common fixed points of a one-parameter nonexpansive semigroup on a
Hilbert space*H:*

*x*0*x*∈*H,* *x**n1* I−*α**n**A*1
*t**n*

_{t}_{n}

0

*T*sx*n**dsα**n**γfx**n*, *n*≥0, 1.15

where*A*is a strongly positive bounded linear operator on*H.*

Let*δ* and*λ*be two positive real numbers such that*δ, λ <*1. Recall that a mapping*F*
with domain*DF*and range*RF*in*E*is called*δ-strongly accretive if, for eachx, y*∈*DF,*
there exists*jx*−*y*∈*Jx*−*y*such that

*Fx*−*Fy, j*

*x*−*y* ≥*δx*−*y*^{2}*,* 1.16

where*J* is the normalized duality mapping from*E*into the dual space*E*^{∗}. Recall also that a
mapping*F*is called*λ-strictly pseudocontractive if, for eachx, y*∈*DF, there existsjx*−*y*∈
*Jx*−*y*such that

*Fx*−*Fy, j*

*x*−*y* ≤*x*−*y*^{2}−*λx*−*y* −

*Fx*−*Fy* ^{2}*.* 1.17

It is easy to see that1.17can be rewritten as
I−*Fx*−I−*Fy, j*

*x*−*y* ≥*λ*I−*Fx*−I−*F*y^{2}*,* 1.18
see16.

In this paper, motivated by the above results, we introduce and study the strong con-
vergence theorems of the general iterative scheme{x*n*}defined by1.19in the framework of
a reflexive Banach space*E*which admits a weakly sequentially continuous duality mapping:

*x*0 *x*∈*E,* *x**n1**α**n**γfx**n* I−*α**n**FT*t*n*x*n**,* *n*≥0, 1.19

where*F* is*δ-strongly accretive andλ-strictly pseudocontractive withδλ >* 1,*f* is a con-
traction on*E*with coeﬃcient 0*< α <*1,*γ* is a positive real number such that*γ <*1/α1−
1−*δ/λ, and*S {Tt : 0 ≤ *t <* ∞}is a nonexpansive semigroup on *E. The strong*
convergence theorems are proved under some appropriate control conditions on parameters
{α*n*}and{t*n*}. Furthermore, by using these results, we obtain strong convergence theorems
of the following new general iterative schemes{y*n*}and{z*n*}defined by

*y*0*y*∈*E,* *y**n1* *α**n**γf*

*T*t*n*y*n* I−*α**n**FTt**n*y*n**,* *n*≥0, 1.20
*z*0 *z*∈*E,* *z**n1**T*t*n*

*α**n**γfz**n* I−*α**n**Fz**n* *,* *n*≥0. 1.21
The results presented in this paper extend and improve the main results in Li et al.15, Chen
and He7, and many others.

**2. Preliminaries**

Throughout this paper, it is assumed that*E*is a real Banach space with norm · and let*J*
denote the normalized duality mapping from*E*into*E*^{∗}given by

*Jx *

*f* ∈*E*^{∗}:
*x, f*

x^{2}*f*^{2}

2.1
for each*x*∈ *E, whereE*^{∗}denotes the dual space of*E,* ·,·denotes the generalized duality
pairing, and ^{} denotes the set of all positive integers. In the sequel, we will denote the
single-valued duality mapping by *j, and considerFT* {x ∈ *C* : *Tx* *x}. When*{x*n*}
is a sequence in *E, then* *x**n* → *x* resp., *x**n* * x,* *x**n* ∗

* x* will denote strong resp.,
weak, weak^{∗} convergence of the sequence {x*n*} to *x. In a Banach space* *E, the following*
resultthe subdiﬀerential inequalityis well known17, Theorem 4.2.1: for all*x, y*∈*E, for all*
*jxy*∈*Jxy, for alljx*∈*J*x,

x^{2}2
*y, jx*

≤*xy*^{2}≤ x^{2}
*y, j*

*xy* *.* 2.2

A real Banach space*E* *is said to be strictly convex if* x*y/2* *<* 1 for all *x, y* ∈ *E* with
xy1 and*x /y. It is said to be uniformly convex if, for all*∈0,2, there exits*δ**>*0
such that

x*y*1 with*x*−*y*≥ implies *xy*

2 *<*1−*δ**.* 2.3
The following results are well known and can be founded in17:

ia uniformly convex Banach space*E*is reflexive and strictly convex17, Theorems
4.2.1 and 4.1.6,

iiif*E*is a strictly convex Banach space and*T* :*E* → *E*is a nonexpansive mapping,
then fixed point set*FT*of*T* is a closed convex subset of*E*17, Theorem 4.5.3.

If a Banach space*E*admits a sequentially continuous duality mapping*J* from weak
topology to weak star topology, then from Lemma 1 of 18, it follows that the duality
mapping*J*is single-valued and also*E*is smooth. In this case, duality mapping*J*is also said
*to be weakly sequentially continuous, that is, for each*{x*n*} ⊂*E*with*x**n** x, thenJx**n** J*^{∗} x
see18,19.

In the sequel, we will denote the single-valued duality mapping by*j. A Banach space*
*Eis said to satisfy Opial’s condition if, for any sequence*{x*n*}in*E,x**n** x*as*n* → ∞implies

lim sup

*n*→ ∞ x*n*−*x<*lim sup

*n*→ ∞

*x**n*−*y* ∀y∈*E*with*x /y.* 2.4

By Theorem 1 of18, we know that if*E*admits a weakly sequentially continuous duality
mapping, then*E*satisfies Opial’s condition and*E*is smooth; for the details, see18.

Now, we present the concept of uniformly asymptotically regular semigroupalso see
20,21. Let *C*be a nonempty closed convex subset of a Banach space*E,*S {Tt : 0 ≤
*t <*∞}a continuous operator semigroup on*C. Then,*S*is said to be uniformly asymptotically*
*regular*in short, u.a.r.on*C*if, for all*h*≥0 and any bounded subset*D*of*C,*

*t→ ∞*limsup

*x∈D* ThTtx−*Ttx*0. 2.5

The nonexpansive semigroup{σ*t* :*t >* 0}defined by the following lemma is an example of
u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in20,
Examples 17 and 18.

**Lemma 2.1**see3, Lemma 2.7. Let*Cbe a nonempty closed convex subset of a uniformly convex*
*Banach spaceE,Da bounded closed convex subset ofC, and*S{Ts: 0≤*s <*∞}*a nonexpansive*
*semigroup onCsuch thatFS/*∅. For each*h >0, setσ**t*x 1/t_{t}

0*T*sxds, then

*t→ ∞*limsup

*x∈D*σ*t*x−*T*hσ*t*x0. 2.6

*Example 2.2. The set*{σ*t*:*t >* 0}defined byLemma 2.1is u.a.r. nonexpansive semigroup. In
fact, it is obvious that{σ*t*:*t >*0}is a nonexpansive semigroup. For each*h >*0, we have

σ*t*x−*σ**h**σ**t*x

*σ**t*x− 1
*h*

_{h}

0

*Tsσ**t*xds

1
*h*

_{h}

0

σ*t*x−*T*sσ*t*xds

≤ 1
*h*

_{h}

0

σ*t*x−*T*sσ*t*xds.

2.7

ApplyingLemma 2.1, we have

*t→ ∞*lim sup

*x∈D*σ*t*x−*σ**h**σ**t*x ≤ 1
*h*

_{h}

0

*t→ ∞*lim sup

*x∈D*σ*t*x−*T*sσ*t*xds0. 2.8
Let*C*be a nonempty closed and convex subset of a Banach space*E*and*D*a nonempty
subset of*C. A mappingQ*:*C* → *D*is said to be sunny if

*QQxtx*−*Qx Qx,* 2.9

whenever*Qxtx−Qx*∈*C*for*x*∈*C*and*t*0. A mapping*Q*:*C* → *D*is called a retraction
if*Qx* *x*for all*x* ∈ *D. Furthermore,Q*is a sunny nonexpansive retraction from*C*onto*D*
if*Q*is a retraction from*C*onto*D*which is also sunny and nonexpansive. A subset*D*of*C*is
called a sunny nonexpansive retraction of*C*if there exists a sunny nonexpansive retraction
from*C*onto*D. The following lemma concerns the sunny nonexpansive retraction.*

**Lemma 2.3**see22,23. Let*Cbe a closed convex subset of a smooth Banach spaceE. LetDbe a*
*nonempty subset ofCandQ* : *C* → *D* *be a retraction. Then,Qis sunny and nonexpansive if and*
*only if*

*u*−*Qu, j*

*y*−*Qu* ≤0 2.10

*for allu*∈*Candy*∈*D.*

**Lemma 2.4**see24, Lemma 2.3. Let{a*n*}*be a sequence of nonnegative real numbers satisfying*
*the property*

*a**n1*≤1−*t**n*a*n**t**n**c**n**b**n**,* ∀n≥0, 2.11
*where*{t*n*},{b*n*}, *and*{c*n*}*satisfy the restrictions*

i_{∞}

*n1**t**n*∞;

ii_{∞}

*n1**b**n**<*∞;

iiilim sup_{n}_{→ ∞}*c**n*≤*0.*

*Then, lim*_{n}_{→ ∞}*a**n**0.*

The following lemma will be frequently used throughout the paper and can be found in25.

**Lemma 2.5**see25, Lemma 2.7. Let*Ebe a real smooth Banach space andF* :*E* → *Ea mapping.*

i*IfFisδ-strongly accretive andλ-strictly pseudocontractive withδλ >* *1, thenI*−*F* *is*
*contractive with constant*

1−*δ/λ.*

i*IfFisδ-strongly accretive andλ-strictly pseudocontractive withδλ >1, then, for any*
*fixed numberτ* ∈0,1,*I*−*τFis contractive with constant 1*−*τ*1−

1−*δ/λ.*

**3. Main Results**

Now, we are in a position to state and prove our main results.

**Theorem 3.1. Let***E* *be a reflexive Banach space which admits a weakly sequentially continuous*
*duality mappingJ. Let*S {Tt : 0 ≤ *t <* ∞}*be a u.a.r. nonexpansive semigroup on* *Esuch*
*that FixS/*∅. Suppose that the real sequences{α*n*} ⊂0,1,{t*n*} ⊂0,∞*satisfy the conditions*

*n→ ∞*lim*α**n*0,

∞
*n0*

*α**n*∞, lim

*n*→ ∞*t**n*∞. 3.1

*LetF* *beδ-strongly accretive andλ-strictly pseudocontractive withδλ >* *1,f* : *E* → *Ea con-*
*traction mapping with coeﬃcientα* ∈0,1, and*γa positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence* {x*n*} *defined by* 1.19 *converges strongly tox, where* *x* *is the*
*unique solution in FixSof the variational inequality*

*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.2

*or equivalentlyx* *Q*FixSI−*Fγfx, whereQ*FixS *is the sunny nonexpansive retraction ofE*
*onto FixS.*

*Proof. Note that FixS*is a nonempty closed convex set. We first show that{x*n*}is bounded.

Let*q*∈FixS. Thus, byLemma 2.5, we have

*x**n1*−*qα**n**γfx**n* I−*α**n**FT*t*n*x*n*−I−*α**n**Fq*−*α**n**Fq*

≤*α**n**γfx**n*−*Fq*I−*α**n**FTt**n*x*n*−*q*

≤*α**n**γfx**n*−*f*

*q* *α**n**γf*

*q* −*Fq*I−*α**n**Fx**n*−*q*

≤*α**n**αγx**n*−*qα**n**γf*

*q* −*Fq*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠*x**n*−*q*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ* −*αγ*

⎞

⎠

⎞

⎠*x**n*−*q*
*α**n*

⎛

⎝1−

1−*δ*
*λ* −*αγ*

⎞

⎠ *γf*

*q* −*Fq*
1−

1−*δ/λ*−*αγ*

≤max

*x**n*−*q,* 1
1−

1−*δ/λ*−*αγγf*

*q* −*Fq*

*,* ∀n≥0.

3.3

By induction, we get
*x**n*−*q*≤max

*x*0−*q,* 1
1−

1−*δ/λ*−*αγ*
*γf*

*q* −*Fq*

*,* *n*≥0. 3.4

This implies that{x*n*}is bounded and, hence, so are{fx*n*}and{FTt*n*x*n*}. This implies
that

*n*lim→ ∞x*n1*−*T*t*n*x*n* lim

*n*→ ∞*α**n**γfx**n*−*FT*t*n*x*n*0. 3.5
Since{Tt}is a u.a.r. nonexpansive semigroup and lim*n*→ ∞*t**n*∞, we have, for all*h >*0,

*n*lim→ ∞ThTt*n*x*n*−*Tt**n*x*n* ≤ lim

*n*→ ∞ sup

*x∈{x**n*}ThTt*n*x−*T*t*n*x0. 3.6
Hence, for all*h >*0,

x*n1*−*T*hx*n1* ≤ x*n1*−*T*t*n*x*n*Tt*n*x*n*−*T*hTt*n*x*n*ThTt*n*x*n*−*T*hx*n1*

≤2x* _{n1}*−

*Tt*

*n*x

*n*Tt

*n*x

*n*−

*ThT*t

*n*x

*n*−→0.

3.7

That is, for all*h >*0,

*n*lim→ ∞x*n*−*T*hx*n*0. 3.8

LetΦ *Q*FixS. Then,ΦI−*F*−*γf*is a contraction on*E. In fact, from*Lemma 2.5i, we have
Φ

*I*−*F*−*γf* *x*−Φ

*I*−*F*−*γf* *y*≤*I*−*F*−*γf* *x*−

*I*−*F*−*γf* *y*

≤I−*Fx*−I−*F*y*γf*x−*f*
*y*

≤

1−*δ*

*λ* *x*−*yαγx*−*y*

⎛

⎝

1−*δ*
*λ* *αγ*

⎞

⎠*x*−*y,* ∀x, y∈*E.*

3.9

Therefore,ΦI−F−γfis a contraction on*E*due to

1−*δ/λαγ*∈0,1. Thus, by Banach
contraction principle,*Q*FixSI−*F*−*γf*has a unique fixed point*x. Then, using* Lemma 2.3,

*x*is the unique solution in FixSof the variational inequality3.2. Next, we show that
lim sup

*n*→ ∞

*γfx*−*Fx, j* x*n*−*x*

≤0. 3.10

Indeed, we can take a subsequence{x*n**k*}of{x*n*}such that

lim sup

*n*→ ∞

*γfx*−*Fx, j* x*n*−*x*
lim

*k*→ ∞

*γfx*−*Fx, jx* *n**k* −*x*

*.* 3.11

We may assume that*x**n**k* * p*∈*E*as*k* → ∞, since a Banach space*E*has a weakly sequentially
continuous duality mapping *J* satisfying Opial’s condition 13. We will prove that *p* ∈
FixS. Suppose the contrary,*p /*∈ FixS, that is,*Th*0*p /p*for some*h*0 *>*0. It follows from
3.8and Opial’s condition that

lim inf

*k*→ ∞ *x**n**k*−*p<*lim inf

*k*→ ∞ *x**n**k*−*Th*0p

≤lim inf

*k*→ ∞

x*n**k*−*Th*0x*n**k**T*h0x*n**k* −*T*h0p

≤lim inf

*k*→ ∞

x*n**k* −*T*h0x*n**k**x**n**k*−*p*
lim inf

*k*→ ∞ *x**n**k*−*p.*

3.12

This is a contradiction, which shows that*p* ∈ *FT*h for all*h >* 0, that is,*p* ∈ FixS. In
view of the variational inequality3.2and the assumption that duality mapping*J*is weakly
sequentially continuous, we conclude

lim sup

*n*→ ∞

*γfx* −*Fx, jx* *n*−*x*
lim

*k*→ ∞

*γfx*−*Fx, j* x*n**k* −*x*

≤

*γfx* −*Fx, j*

*p*−*x* ≤0.

3.13

Finally, we will show that*x**n* → *x. For each* *n*≥0, we have

x*n1*−*x* ^{2}*α**n**γf*x*n* I−*α**n**FT*t*n*x*n*−I−*α**n**Fx*−*α**n**Fx*^{2}

≤*α**n**γf*x*n*−*α**n**Fx* I−*α**n**FTt**n*x*n*−I−*α**n**Fx*^{2}
I−*α**n**FT*t*n*x*n*−I−*α**n**Fx*^{2}2α*n*

*γfx**n*−*Fx, jx* *n1*−*x*

≤

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠

2

x*n*−*x* ^{2}2α*n*

*γfx**n*−*γfx, j* x*n1*−*x*

2α*n*

*γfx* −*Fx, jx* *n1*−*x*
*.*

3.14

On the other hand,

*γfx**n*−*γfx, j* x*n1*−*x*

≤*γαx**n*−*xx* *n1*−*x*

≤*γαx**n*−*x*

⎡

⎢⎢

⎣

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠

2

x*n*−*x* ^{2}2α*n* *γfx**n*−*Fx, j* x* _{n1}*−

*x*

⎤

⎥⎥

⎦

≤*γα*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠x*n*−*x* ^{2}

*γαx**n*−*x* $

2 *γfx**n*−*Fx, j* x*n1*−*x* √
*α**n*

≤*γα*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠x*n*−*x* ^{2}√
*α**n**M*0*,*

3.15
where*M*0is a constant satisfying*M*0≥*γαx**n*−*x* $

2|γfx*n*−*Fx, jx* *n1*−*x|. Substitut-*
ing3.15in3.14, we obtain

x*n1*−*x* ^{2}≤

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠

2

x*n*−*x* ^{2}2α*n**γα*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠

× x*n*−*x* ^{2}2α*n*√

*α**n**M*02α*n*

*γfx*−*Fx, j* x*n1*−*x*

⎛

⎜⎝1−2α*n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠*α*^{2}_{n}

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

2⎞

⎟⎠x*n*−*x* ^{2}

2α*n**γα*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠x*n*−*x* ^{2}

2α*n*√

*α**n**M*02α*n*

*γfx* −*Fx, jx* *n1*−*x*

⎛

⎝1−2α*n*

⎡

⎣

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠−*αγα**n**γα*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎤

⎦

⎞

⎠x*n*−*x* ^{2}

*α**n*

⎡

⎢⎣α*n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

2

x*n*−*x* ^{2}2M0

√*α**n*2

*γfx* −*Fx, jx* *n1*−*x*

⎤

⎥⎦

1−*α**n**γ**n* x*n*−*x* ^{2}*α**n**γ**n*

*β**n*

*γ**n**,*

3.16

where

*γ**n*2

⎡

⎣

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠−*αγα**n**γα*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎤

⎦*,*

*β**n*

⎡

⎢⎣α*n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

2

x*n*−*x* ^{2}2M0√
*α**n*2

*γfx* −*Fx, jx* *n1*−*x*

⎤

⎥⎦.

3.17

It is easily seen that_{∞}

*n1**α**n**γ**n*∞. Since{x*n*}is bounded and lim*n*→ ∞*α**n*0, by3.46, we
obtain lim sup_{n→ ∞}*β**n**/γ**n*≤0, applyingLemma 2.4to3.16to conclude*x**n* → *x*as*n* → ∞.

This completes the proof.

UsingTheorem 3.1, we obtain the following two strong convergence theorems of new
iterative approximation methods for a nonexpansive semigroup{Tt: 0≤*t <*∞}.

**Corollary 3.2. Let***E* *be a reflexive Banach space which admits a weakly sequentially continuous*
*duality mappingJ. Let*S {Tt : 0≤ *t <*∞}*be a u.a.r. nonexpansive semigroup onEsuch that*
FixS*/*∅. Suppose that the real sequences{α*n*} ⊂0,1,{t*n*} ⊂0,∞*satisfy the conditions*

*n→ ∞*lim*α**n*0,

∞
*n0*

*α**n*∞, lim

*n*→ ∞*t**n*∞. 3.18

*LetF* *beδ-strongly accretive andλ-strictly pseudocontractive withδλ >* *1,f* : *E* → *Ea con-*
*traction mapping with coeﬃcientα* ∈0,1, and*γa positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence* {y*n*} *defined by* 1.20 *converges strongly tox, where* *x* *is the*
*unique solution in FixSof the variational inequality*

*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.19

*or equivalentlyx* *Q*FixSI−*Fγfx, whereQ*FixS *is the sunny nonexpansive retraction ofE*
*onto FixS.*

*Proof. Let*{x*n*}be the sequence given by*x*0*y*0and

*x**n1* *α**n**γf*x*n* I−*α**n**FT*t*n*x*n**,* ∀n≥0. 3.20
FormTheorem 3.1,*x**n* → *x. We claim that* *y**n* → *x. Indeed, we estimate*

*x**n1*−*y**n1*

≤*α**n**γf*

*T*t*n*y*n* −*f*x*n*I−*α**n**FTt**n*x*n*−*Tt**n*y*n*

≤*α**n**γαT*t*n*y*n*−*x**n*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠*x**n*−*y**n*

≤*α**n**γαT*t*n*y*n*−*T*t*n**xα**n**γαT*t*n**x*−*x**n*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠*x**n*−*y**n*

≤*α**n**γαy**n*−*xα**n**γαx*−*x**n*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠*x**n*−*y**n*

≤*α**n**γαy**n*−*x**n**α**n**γαx**n*−*x* *α**n**γαx*−*x**n*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠*x**n*−*y**n*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ* −*γα*

⎞

⎠

⎞

⎠*x**n*−*y**n*
*α**n*

⎛

⎝1−

1−*δ*
*λ* −*γα*

⎞

⎠ 2αγ '

1−

1−*δ/λ*−*γα*(*x*−*x**n*.

3.21
It follows from _{∞}

*n1**α**n* ∞, lim*n*→ ∞x*n* −*x* 0, and Lemma 2.4 thatx*n* −*y**n* → 0.

Consequently,*y**n* → *x*as required.

**Corollary 3.3. Let***E* *be a reflexive Banach space which admits a weakly sequentially continuous*
*duality mappingJ. Let*S {Tt : 0≤ *t <*∞}*be a u.a.r. nonexpansive semigroup onEsuch that*
FixS*/*∅. Suppose that the real sequences{α*n*} ⊂0,1,{t*n*} ⊂0,∞*satisfy the conditions*

*n→ ∞*lim*α**n*0,

∞
*n0*

*α**n*∞, lim

*n*→ ∞*t**n*∞. 3.22

*LetF* *beδ-strongly accretive andλ-strictly pseudocontractive withδ* *λ >* *1,f* : *E* → *Eacon-*
*traction mapping with coeﬃcientα* ∈0,1, and*γa positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence*{z*n*}*defined by*1.21*converges strongly tox, where* *xis the unique*
*solution in FixSof the variational inequality*

*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.23

*or equivalentlyx* *Q*FixSI−*Fγfx, whereQ*FixS *is the sunny nonexpansive retraction ofE*
*onto FixS.*

*Proof. Define the sequences*{y*n*}and{β*n*}by

*y**n* *α**n**γf*z*n* I−*α**n**Fz**n**,* *β**n**α**n1* ∀n∈^{}*.* 3.24

Taking*p*∈FixS, we have

*z**n1*−*pTt**n*y*n*−*Tt**n*p≤*y**n*−*p*

*α**n**γf*z*n* I−*α**n**Fz**n*−I−*α**n**Fp*−*α**n**Fp*

≤

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠*z**n*−*pα**n**γfz**n*−*F*
*p*

⎛

⎝1−*α**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠

⎞

⎠*z**n*−*pα**n*

⎛

⎝1−

1−*δ*
*λ*

⎞

⎠*γfz**n*−*F*
' *p*

1−

1−*δ/λ*(*.*
3.25

It follows from induction that
*z**n1*−*p*≤max

*z*0−*p,γfz*0−*F*
*p*
1−

1−*δ/λ*

*,* *n*≥0. 3.26

Thus, both{z*n*}and{y*n*}are bounded. We observe that
*y**n1**α**n1**γf*z*n1* I−*α**n1**Fz**n1**β**n**γf*

*T*t*n*y*n*

*I*−*β**n**F* *T*t*n*y*n**.* 3.27
Thus,Corollary 3.2implies that{y*n*}converges strongly to some point*x. In this case, we also*
have

z*n*−*x ≤* *z**n*−*y**n**y**n*−*xα**n**γfz**n*−*Fz**n**y**n*−*x*−→0. 3.28
Hence, the sequence{z*n*}converges strongly to some point*x. This complete the proof.*

UsingTheorem 3.1,Lemma 2.1, andExample 2.2, we have the following result.

**Corollary 3.4. Let***E* *be a uniformly convex Banach space which admits a weakly sequentially*
*continuous duality mappingJ. Let*S {Tt: 0≤*t <*∞}*be a nonexpansive semigroup onEsuch*
*that FixS/*∅. Suppose that the real sequences{α*n*} ⊂0,1,{t*n*} ⊂0,∞*satisfy the conditions*

*n→ ∞*lim*α**n*0,

∞

*n0**α**n*∞, lim

*n*→ ∞*t**n*∞. 3.29

*LetF* *beδ-strongly accretive andλ-strictly pseudocontractive withδλ >* *1,f* : *E* → *Ea con-*
*traction mapping with coeﬃcientα* ∈0,1, and*γa positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence*{x*n*}*defined by*

*x*0*x*∈*E,*
*x**n1* *α**n**γf*x*n* I−*α**n**F*1

*t**n*

_{t}_{n}

0

*Ttx**n**ds,* *n*≥0

3.30

*converges strongly tox, where* *xis the unique solution in FixSof the variational inequality*
*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.31

*or equivalentlyxQ*FixSI−*Fγfx, whereQ*FixS*is the sunny nonexpansive retraction ofE*
*onto FixS.*

**Corollary 3.5. Let**H*be a real Hilbert space. Let* S {Tt : 0 ≤ *t <* ∞} *be a nonexpansive*
*semigroup onHsuch that FixS/*∅. Suppose that the real sequences{α*n*} ⊂ 0,1,{t*n*} ⊂0,∞
*satisfy the conditions*

*n*lim→ ∞*α**n* 0,
∞
*n0*

*α**n*∞, lim

*n*→ ∞*t**n*∞. 3.32

*Letf* :*E* → *Ebe a contraction mapping with coeﬃcientα*∈0,1*andAa strongly positive bounded*
*linear operator with coeﬃcient* *γ >* 1/2 and 0 *< γ <* 1−$

2−2γ/α. Then, the sequence {x*n*}
*defined by*

*x*0*x*∈*E,*

*x**n1* *α**n**γfx**n* I−*α**n**A*1
*t**n*

_{t}_{n}

0

*Ttx**n**ds,* *n*≥0

3.33

*converges strongly tox, wherexis the unique solution in FixSof the variational inequality*
*A*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.34

*or equivalentlyxQ*FixSI−*Aγfx, whereQ*FixS*is the sunny nonexpansive retraction ofE*
*onto FixS.*

*Proof. SinceA*is a strongly positive bounded linear operator with coeﬃcient*γ, we have*
*Ax*−*Ay, x*−*y*

≥*γx*−*y*^{2}*.* 3.35
Therefore,*A*is*γ-strongly accretive. On the other hand,*

I−*Ax*−I−*Ay*^{2}

*x*−*y* −

*Ax*−*Ay* *,*

*x*−*y* −

*Ax*−*Ay*

*x*−*y, x*−*y*

−2

*Ax*−*Ay, x*−*y*

*Ax*−*Ay, Ax*−*Ay*
*x*−*y*^{2}−2

*Ax*−*Ay, x*−*y*

*Ax*−*Ay*^{2}

≤*x*−*y*^{2}−2

*Ax*−*Ay, x*−*y*

A^{2}*x*−*y*^{2}*.*

3.36

Since*A* is strongly positive if and only if1/AAis strongly positive, we may assume,
without loss of generality, thatA1, so that

*Ax*−*Ay, x*−*y*

≤*x*−*y*^{2}−1

2I−*Ax*−I−*Ay*^{2}
*x*−*y*^{2}−1

2*x*−*y* −

*Ax*−*Ay* ^{2}*.*

3.37

Hence,*A*is 12-strongly pseudocontractive. ApplyingCorollary 3.4, we conclude the result.

**Theorem 3.6. Let***E* *be a reflexive Banach space which admits a weakly sequentially continuous*
*duality mappingJ. Let*S {Tt : 0 *< t <* ∞}*be a u.a.r. nonexpansive semigroup on* *Esuch*
*that FixS/*∅. Let{α*n*}*and*{t*n*}*be sequences of real number satisfying*

0*< α**n**<*1,
∞
*n0*

*α**n* ∞, *t**n**>*0, lim

*n*→ ∞*α**n* lim

*n*→ ∞

*α**n*

*t**n* 0. 3.38

*LetF* *beδ-strongly accretive andλ-strictly pseudocontractive withδλ >* *1,f* : *E* → *Ea con-*
*traction mapping with coeﬃcientα* ∈0,1, and*γa positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence*{x*n*}*defined by*

*x*0*x*∈*E,*

*x*_{n1}*α**n**γfx**n* I−*α**n**FTt**n*x*n**,* *n*≥0

3.39

*converges strongly tox, where* *xis the unique solution in FixSof the variational inequality*
*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.40

*or equivalentlyx* *Q*FixSI−*Fγfx, whereQ*FixS *is the sunny nonexpansive retraction ofE*
*onto FixS.*

*Proof. By the same argument as in the proof of*Theorem 3.1, we can obtain that{x*n*},{fx*n*},
and{FTt*n*x*n*}are bounded and*Q*FixSI−*F*−*γf*is a contraction on*E. Thus, by Banach*
contraction principle,*Q*FixSI−*F*−*γf*has a unique fixed point*x. Then, using* Lemma 2.3,

*x*is the unique solution in FixSof the variational inequality3.40. Next, we show that
lim sup

*n*→ ∞

*γfx*−*Fx, j* x*n*−*x*

≤0. 3.41

Indeed, we can take a subsequence{x*n**k*}of{x*n*}such that
lim sup

*n*→ ∞

*γfx*−*Fx, j* x*n*−*x*
lim

*k*→ ∞

*γfx*−*Fx, jx* *n**k* −*x*

*.* 3.42

We may assume that*x**n**k* * p*∈*E*as*k* → ∞. Now, we show that*p*∈FixS. Put

*x**k**x**n**k**,* *α**k**α**n**k* *s**k**t**n**k* ∀k∈^{}*.* 3.43
Fix*t >*0, then we have

*x**k*−*T*tp^{t/s}^{i}^{−1}

*i0*

Ti1s*k*x*k*−*Tis**k*x*k*

*T*

)* *t*
*s**k*

+
*s**k*

,
*x**k*−*T*

)* *t*
*s**k*

+
*s**k*

,
*p*

*T*

)* *t*
*s**k*

+
*s**k*

,

*p*−*Ttp*

≤

* *t*
*s**k*

+

Ts*k*x*k*−*x**k1**x**k1*−*p*
*T*

)
*t*−

**t*
*s**k*

+
*s**k*

,
*p*−*p*

≤

* *t*
*s**k*

+

*α**k**FTs**k*x*k*−*fx**k**x**k1*−*p*
*T*

)
*t*−

* *t*
*s**k*

+
*s**k*

,
*p*−*p*

≤
)*tα**k*

*s**k*

,*FT*s*k*x*k*−*fx**k**x**k1*−*p*max*T*sp−*p*: 0≤*s*≤*s**k*

*.*
3.44

Thus, for all*k*∈^{}, we obtain
lim sup

*k*→ ∞

*x**k*−*T*tp≤lim sup

*k*→ ∞

*x**k1*−*p*lim sup

*k*→ ∞

*x**k*−*p.* 3.45
Since Banach space *E* has a weakly sequentially continuous duality mapping satisfying
Opial’s condition 13, we can conclude that*T*tp *p*for all*t >* 0, that is,*p* ∈ FixS. In
view of the variational inequality3.2and the assumption that duality mapping*J*is weakly
sequentially continuous, we conclude

lim sup

*n*→ ∞

*γfx* −*Fx, jx* *n*−*x*
lim

*k*→ ∞

*γfx*−*Fx, j* x*n**k* −*x*

≤

*γfx* −*Fx, J*

*p*−*x* ≤0.

3.46

By the same argument as in the proof ofTheorem 3.1, we conclude that*x**n* → *x*as*n* → ∞.

This completes the proof.

Using Theorem 3.6 and the method as in the proof of Corollary 3.7, we have the following result.

**Corollary 3.7. Let***E* *be a reflexive Banach space which admits a weakly sequentially continuous*
*duality mappingJ. Let*S {Tt : 0*< t <* ∞}*be a u.a.r. nonexpansive semigroup onEsuch that*
FixS*/*∅. Let{α*n*}*and*{t*n*}*be sequences of real number satisfying*

0*< α**n* *<*1,
∞
*n0*

*α**n*∞, *t**n**>*0, lim

*n*→ ∞*α**n* lim

*n*→ ∞

*α**n*

*t**n* 0. 3.47

*Let* *F* *be aδ-strongly accretive andλ-strictly pseudocontractive withδ* *λ >* *1,* *f* : *E* → *E* *a*
*contraction mapping with coeﬃcientα*∈0,1, and*γis a positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence*{y*n*}*defined by*

*y*0*y*∈*E,*
*y**n1**α**n**γf*

*T*t*n*y*n* I−*α**n**FTt**n*y*n**,* *n*≥0

3.48

*converges strongly tox, where* *xis the unique solution in FixSof the variational inequality*
*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.49

*or equivalentlyx* *Q*FixSI−*Fγfx, whereQ*FixS *is the sunny nonexpansive retraction ofE*
*onto FixS.*

Using Theorem 3.6 and the method as in the proof of Corollary 3.8, we have the following result.

**Corollary 3.8. Let***E* *be a reflexive Banach space which admits a weakly sequentially continuous*
*duality mappingJ. Let*S {Tt : 0*< t <* ∞}*be a u.a.r. nonexpansive semigroup onEsuch that*
FixS*/*∅. Let{α*n*}*and*{t*n*}*be sequences of real number satisfying*

0*< α**n* *<*1,
∞

*n0**α**n*∞, *t**n**>*0, lim

*n*→ ∞*α**n* lim

*n*→ ∞

*α**n*

*t**n* 0. 3.50

*LetF* *be aδ-strongly accretive andλ-strictly pseudocontractive withδλ >* *1,f* :*E* → *Ea con-*
*traction mapping with coeﬃcientα*∈0,1, and*γis a positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence*{z*n*}*defined by*

*z*0*z*∈*E,*
*z**n1**T*t*n*

*α**n**γf*z*n* I−*α**n**Fz**n* *,* *n*≥0

3.51

*converges strongly tox, where* *xis the unique solution in FixSof the variational inequality*
*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.52

*or equivalentlyx* *Q*FixSI−*Fγfx, whereQ*FixS *is the sunny nonexpansive retraction ofE*
*onto FixS.*

UsingTheorem 3.6,Lemma 2.1, andExample 2.2, we have the following result.

**Corollary 3.9. Let***E* *be a uniformly convex Banach space which admits a weakly sequentially*
*continuous duality mappingJ. Let*S {Tt: 0*< t <*∞}*be a nonexpansive semigroup onEsuch*
*that FixS/*∅. Let{α*n*}*and*{t*n*}*be sequences of real numbers satisfying*

0*< α**n**<*1,
∞
*n0*

*α**n*∞, *t**n**>*0, lim

*n*→ ∞*α**n* lim

*n*→ ∞

*α**n*

*t**n* 0. 3.53

*LetF* *beδ-strongly accretive andλ-strictly pseudocontractive withδλ >* *1,f* : *E* → *Ea con-*
*traction mapping with coeﬃcientα* ∈0,1, and*γa positive real number such thatγ <*1/α1−
1−*δ/λ. Then, the sequence*{x*n*}*defined by*

*x*0*x*∈*E,*

*x**n1**α**n**γfx**n* I−*α**n**F*1
*t**n*

_{t}_{n}

0

*T*tx*n**ds,* *n*≥0

3.54

*converges strongly tox, where* *xis the unique solution in FixSof the variational inequality*
*F*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.55

*or equivalentlyx* *Q*FixSI−*Fγfx, whereQ*FixS *is the sunny nonexpansive retraction ofE*
*onto FixS.*

**Corollary 3.10. Let**H*be a real Hilbert space. Let* S {Tt : 0 ≤ *t <* ∞}*be a nonexpansive*
*semigroup onHsuch that FixS/*∅. Suppose that the real sequences{α*n*} ⊂ 0,1,{t*n*} ⊂0,∞
*satisfy the conditions*

0*< α**n**<*1,
∞
*n0*

*α**n*∞, *t**n**>*0, lim

*n*→ ∞*α**n* lim

*n*→ ∞

*α**n*

*t**n* 0. 3.56

*Letf* :*E* → *Ebe a contraction mapping with coeﬃcientα*∈0,1*andAa strongly positive bounded*
*linear operator with coeﬃcient* *γ >* 1/2 and 0 *< γ <* 1−$

2−2γ/α*. Then, the sequence* {x*n*}
*defined by*

*x*0*x*∈*E,*

*x**n1**α**n**γfx**n* I−*α**n**A*1
*t**n*

_{t}_{n}

0

*T*tx*n**ds,* *n*≥0

3.57

*converges strongly tox, where* *xis the unique solution in FixSof the variational inequality*
*A*−*γf* *x, jx* −*x*

≥0, *x*∈FixS 3.58

*or equivalentlyxQ*FixSI−*Aγfx, whereQ*FixS*is the sunny nonexpansive retraction ofE*
*onto FixS.*

**Acknowledgment**

The project was supported by the “Centre of Excellence in Mathematics” under the Commis- sion on Higher Education, Ministry of Education, Thailand.

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