The modified general iterative methods for asymptotically nonexpansive semigroups in Banach spaces

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R E S E A R C H

Open Access

The modiﬁed general iterative methods for

asymptotically nonexpansive semigroups in

Banach spaces

Rabian Wangkeeree

1,2*

_{and Pakkapon Preechasilp}

1

*_{Correspondence:} [email protected]

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

Abstract

In this paper, we introduce the modified general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mappingjϕand uniform normal structure. The main result extends various results existing in the current literature.

1 Introduction

LetEbe a normed linear space,Kbe a nonempty, closed and convex subset ofE. LetT be a self-mapping onK. ThenTis said to beasymptotically nonexpansiveif there exists a sequence{kn} ⊂[,∞) withlim_n_→∞kn=  such that

Tnx–Tny≤knx–y, for allx,y∈Kandn≥.

The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [] as an important generalization of the class of nonexpansive maps (i.e., mappingT:K→K such thatTx–Ty ≤ x–y,∀x,y∈K).

A mappingTis said to beuniformly L-Lipschitzian, if there exists a constantL>  such that

Tnx–Tny≤Lx–y, for allx,y∈Kandn≥.

It is clear that every asymptotically nonexpansive is uniformlyL-Lipschitzian with a con-stant L=sup_n_≥_kn ≥. We use F(T) to denote the set of ﬁxed points of T, that is, F(T) ={x∈K:Tx=x}.

A self-mappingf :K→Kis a contraction onKif there exists a constantα∈(, ) such that

f(x) –f(y)≤αx–y, for allx,y∈K. (.)

We useKto denote the collection of all contractions onK. That is,

K={f :f is a contraction onK}.

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A familyS={T(s) : ≤s<∞}of mappings ofK into itself is called anasymptotically nonexpansive semigrouponKif it satisﬁes the following conditions:

(i) T()x=xfor allx∈K;

(ii) T(s+t) =T(s)T(t)for alls,t≥;

(iii) there exists a sequence{kn} ⊂[,∞)withlimn→∞kn= such that for allx,y∈K Tn_(t)x_–_Tn_{(t)y ≤}_knx_–_y_,_∀t_≥__,_∀n_≥__;

(iv) for allx∈K, the mappingt→T(t)xis continuous.

An asymptotically nonexpansive semigroupSis callednonexpansive semigroupifkn=  for alln≥. We denote byF(S) the set of all common ﬁxed points ofS, that is,

F(S) :=x∈K:T(t)x=x, ≤t<∞= t≥

FT(t).

A gauge functionϕis a continuous strictly increasing functionϕ: [,∞)→[,∞) such thatϕ() =  andϕ(t)→ ∞ast→ ∞. LetE∗be the dual space ofE. The duality mapping Jϕ:E→E

∗

associated to a gauge functionϕis deﬁned by

Jϕ(x) =

f∗∈E∗:x,f∗ =xϕx,f∗=ϕx, ∀x∈E.

In particular, the duality mapping with the gauge functionϕ(t) =t, denoted byJ, is referred to as the normalized duality mapping. Clearly, there holds the relationJϕ(x) =ϕ(_xx)J(x) for

allx=  (see []). Set

(t) =

t



ϕ(τ)dτ, ∀t≥,

then

Jϕ(x) =∂

x, ∀x∈E,

where∂denotes the subdiﬀerential in the sense of convex analysis. Furthermore,is a continuous convex and strictly increasing function on [,∞) (see []).

In a Banach spaceEhaving duality mappingJϕwith a gauge functionϕ, an operatorA

is said to bestrongly positive[] if there exists a constantγ¯>  with the property

Ax,Jϕ(x) ≥ ¯γxϕ

x (.)

and

αI–βA= sup

x≤

(αI–βA)x,Jϕ(x) , α∈[, ],β∈[–, ], (.)

whereIis the identity mapping. IfE:=His a real Hilbert space, then the inequality (.) reduces to

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Letu∈C. Then, for eacht∈(, ) and for a nonexpansive mapT, there exists a unique pointxt∈Csatisfying the following condition:

xt= ( –t)Txt+tu,

since the mappingGt(x) = ( –t)Tx+tuis a contraction. WhenHis a Hilbert space andTis a self-map, Browder [] showed that{xt}converges strongly to an element ofF(T), which is nearest touast→+_{. This result was extended to various more general Banach space}

by Morales and Jung [], Takahashi and Ueda [], Reich [] and a host of other authors. Many authors (see,e.g.[, ]) have also shown convergence of the path

xn= ( –αn)Tnxn+αnu

in Banach spaces for asymptotically nonexpansive mapping self-mapTunder some condi-tions onαn. In , motivated and inspired by Moudafi [], Shahzad and Udomene [] introduced and studied the iterative procedures for the approximation of common fixed points of asymptotically nonexpansive mappings in a real Banach space with uniformly Gâteaux differentiable norm and uniform normal structure.

LetS be a nonexpasive semigroup onK. In , Suzuki [] introduced, in Hilbert space, the implicit iteration

un= ( –αn)T(tn)un+αnu, u∈K,n≥, (.)

where{αn}is a sequence in (, ),{tn}is a sequence of positive real numbers. Under certain restrictions to the sequence{αn}and{tn}, Suzuki proved strong convergence of (.) to a member ofF(S) nearest tou. In , Xu [] extended Suzuki []’s result from Hilbert space to a uniformly convex Banach space having a weakly continuous duality mapjϕwith

gauge functionϕ. In , Changet al.[] introduced the following implicit and explicit schemes for an asymptotically nonexpansive semigroup:

yn= ( –αn)Tn(tn)yn+αnu, u∈K,n≥, (.)

and

xn+= ( –βn)Tn(tn)xn+βnu, u∈K,n≥, (.)

whereαn,βn∈(, ) andtn∈R+ in a real Banach space with uniformly Gâteaux diﬀer-entiable norm and uniform normal structure. Suppose, in addition, that limn→∞xn– T(t)xn=  andlimn→∞yn–T(t)yn=  uniformly int∈[,∞). Then the{xn}and{yn} converge strongly to a point ofF(S).

Very recently, motivated and inspired by Moudaﬁ [], Cholumjiak and Suantai [] studied the following implicit and explicit viscosity methods:

yn=αnf(yn) + ( –αn)T(tn)yn, n≥, (.)

and

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They obtained the strong convergence theorems in the frame work of a real reﬂexive strictly convex Banach space which has a uniformly Gâteaux diﬀerentiable norm and ad-mits the duality mappingjϕ. For more related results, see [–].

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∈C 

Ax,x–x,b, (.)

whereCis the ﬁxed point set of a nonexpansive mappingT onHandbis a given point inH. In , motivated and inspired by Marino and Xu [], Liet al.[] introduced the following general iterative procedures for the approximation of common ﬁxed points of a nonexpansive semigroup{T(s) :s≥}on a nonempty, closed and convex subsetKin a Hilbert space:

yn=αnγf(yn) + (I–αnA)  tn

tn



T(s)ynds, n≥, (.)

and

xn+=αnγf(xn) + (I–αnA)  tn

tn



T(s)xnds, n≥, (.)

where{αn}and{tn}are sequences in [, ] and (,∞), respectively,Ais a strongly pos-itive bounded linear operator onCandf is a contraction onC. And their convergence theorems can be proved under some appropriate control conditions on parameter{αn}

and{tn}. Furthermore, by using these results, they obtained two mean ergodic theorems for nonexpansive mappings in a Hilbert space. Many authors extended the Liet al.[]’s results in direction of algorithms and spaces (see [–]).

In this paper, inspired and motivated by Changet al.[], Cholamjiak and Suantai [], Li, Li and Su [], Wangkeeree and Wangkeeree [] and Wangkeereeet al.[], we in-troduce the following iterative approximation methods (.) and (.) for the class of strongly continuous semigroup of asymptotically nonexpansive mappingsS={T(s) : ≤ s<∞}:

yn=αnγf(yn) + (I–αnA)Tn(tn)yn, n≥, (.)

and

xn+=βnγf(xn) + (I–βnA)Tn(tn)xn, n≥, (.)

whereAis a strongly positive bounded linear operator onKandf is a contraction onK. The strong convergence theorems of the iterative approximation methods (.) and (.) in a real Banach space which has a uniformly Gâteaux diﬀerentiable norm and admits the duality mappingjϕare studied. Moreover, we study the strong convergence results of the

following two iterative approximation methods (.) and (.):

wn+=βnγf

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and

zn+=Tn(tn)

βnγf(zn) + (I–βnA)Tn(tn)zn

, n≥. (.)

2 Preliminaries

Throughout this paper, letEbe a real Banach space andE∗be its dual space. We writexn

x(respectivelyxn∗x) to indicate that the sequence{xn}weakly (respectively weak∗) converges tox; as usualxn→xwill symbolize strong convergence. A Banach spaceEis said touniformly convexif, for any∈(, ], there existsδ>  such that, for anyx,y∈U, x–y ≥ impliesx+_y ≤ –δ. It is known that a uniformly convex Banach space is reﬂexive and strictly convex (see also []). LetU={x∈E:x= }. A Banach spaceEis said to besmoothif the limit

lim

t→

x+ty–x t

exists for allx,y∈U. In this case, the norm ofEis said to beGâteaux differentiable. The spaceEis said to have auniformly Gâteaux differentiableif for eachy∈U, the limit at-tained uniformly forx∈U. The spaceEis said to have aFréchet differentiableif for each x∈U, the limit attained uniformly fory∈U anduniformly Fréchet differentiableif, the limit attained uniformly forx,y∈U. It is well known that (uniformly) Gâteaux differen-tiable of the norm ofEimplies (uniformly) Fréchet differentiable.

The following Lemma can be found in [].

Lemma .[, Lemma .] Let E be a Banach space which has a uniformly Gâteaux

diﬀerentiable norm and admits the duality mapping Jϕ,then Jϕ is uniformly continuous

from the norm topology of E to the weak∗topology of E∗on each bounded subset of E.

The next lemma is an immediate consequence of the subdiﬀerential inequality can be found in [].

Lemma .[] Assume that a Banach space E which admits a duality mapping Jϕwith

gaugeϕ.For all x,y∈E,the following inequality holds:

x+y≤x+y,jϕ(x+y), jϕ(x+y)∈Jϕ(x+y).

LetK be a nonempty, bounded, closed and convex subset of a Banach spaceE. The diameter ofKbe deﬁned byd(K) :=sup{x–y:x,y∈K}. For eachx∈K, denoter(x,K) =

sup{x–y:x,y∈K}and denote byr(K) :=inf{r(x,K) :x∈K}the chebyshev radius of K relative to itself.The normal structure coeﬃcient N(E)of Eis deﬁned by

N(E) :=inf

d(K)

r(K) :Kis a bounded, closed and convex ofEwithd(K) > 

.

A Banach spaceEis said to haveuniform normal structureifN(E) > . It is known that every Banach space with a uniform normal structure is reﬂexive. Every uniformly convex and uniformly smooth Banach spaces have uniform normal structure.

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Theorem .[, Theorem ] Suppose E is a Banach space with uniformly normal struc-ture,K is a nonempty bounded subset of E,and T:K→K is a uniformly k-Lipschitzian mapping with k<N(X)/_._{Suppose also there exists a nonempty,}_bounded,_{closed and}

con-vex subset K∗of K with the following property(P):

x∈K∗ implies ωw(x)⊂K∗,

whereωw(x)is the weakω-limit set if T at x,i.e.,the set

y∈E:y=weak–lim

j→∞T

nj_{x for some nj}→ ∞

.

Then T has a ﬁxed point in K∗.

In order to prove our main result, we need the following lemmas and deﬁnitions. Letl∞be the Banach space of all bounded real-valued sequences. LetLIMbe a contin-uous linear functional onl∞ satisfyingLIM=  =LIM(). Then we know thatLIMis mean onNif and only if

inf{an:n∈N} ≤LIM(a)≤sup{an:n∈N}

for every a= (a,a, . . .)∈l∞. Occasionally, we shall useLIMn(an) instead of LIM(a). A meanLIMonNis called a Banach limit if

LIMn(an) =LIMn(an+)

for everya= (a,a, . . .)∈l∞. Using the Hahn-Banach theorem, or the Tychonoﬀ

ﬁxed-point theorem, we can prove the existence of a Banach limit. We know that ifμis a Banach limit, then

lim inf

n→∞ an≤LIMn(an)≤lim supn→∞ an for everya= (a,a, . . .)∈l∞.

Subsequently, the following result was showed in [].

Proposition .[, Proposition .] Let K be a nonempty,closed and convex subset of

a real Banach space E which has a uniformly Gâteaux diﬀerentiable norm and admits the duality mapping Jϕ.Suppose that{xn}is a bounded sequence of K and letLIMnbe a Banach limit and z∈E.Then

LIMn

xn–z=inf

y∈KLIMn

xn–y,

if and only if

LIMn

y–z,jϕ(xn–z) ≤, ∀y∈K.

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Lemma .[, Lemma .] Assume that a Banach space E which admits a duality map-ping Jϕwith gaugeϕ.Let A be a strong positive linear bounded operator on E with coeﬃcient

¯

γ > and <ρ≤ϕ()A–_._Then_I_–_ρ_A_≤_ϕ_{()( –}_ρ_γ_¯_).

In the following, we also need the following lemma that can be found in the existing literature [].

Lemma .[, Lemma .] Let{an}be a sequence of nonnegative real number satisfying

the property

an+≤( –γn)an+γnβn, n≥,

where{γn} ⊆(, )and{βn} ⊆Rsuch that∞_n_=γn=∞andlim supn→∞βn≤.Then{an} converges to zero,as n→ ∞.

Lemma .[] Let C be a nonempty,closed and convex subset of a reﬂexive Banach space E

and f :C→(–∞,∞]a proper lower semicontinuous convex function such that f(xn)→ ∞ asxn → ∞.Then there exists x∈D(f)such that f(x) =infx∈Cf(x).

3 Main theorem

Theorem . Let E be a real Banach space with uniform normal structure which has a

uni-formly Gateaux diﬀerentiable norm and admits the duality mapping Jϕ,K be a nonempty,

bounded,closed and convex subset of E such that K±K⊂K.LetS={T(s) :s≥}be an asymptotically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),limn→∞kn=  andsup_n_≥_kn<N(E)/ _{such that F(S)}₌_∅_._{Let f} _∈

K with coeﬃcientα∈(, ),A a strongly positive bounded linear operator on K with coeﬃcientγ¯> and <γ <ϕ()_αγ¯ and let{αn}and{tn}be sequences of real numbers such that <αn< ,tn> .Then the following hold:

(i) If kn–_α

n <ϕ()γ¯–γ α,∀n≥,then there exists a sequence{yn} ⊂Edeﬁned by

yn=αnγf(yn) + (I–αnA)Tn(tn)yn, n≥. (.)

(ii) Suppose,in addition,limn→∞yn–T(t)yn= uniformly int∈[,∞)and the real sequence{αn}satisﬁeslimn→∞αn= andlimn→∞kn–αn = .

Then{yn}converges strongly as n→ ∞to a common ﬁxed pointx in F(S˜ )which solves the variational inequality:

(A–γf)˜x,Jϕ(˜x–z) ≤, ∀z∈F(S). (.)

Proof We ﬁrst show that the uniqueness of a solution of the variational inequality (.). Suppose bothx˜∈F(S) andx∗∈F(S) are solutions to (.), then

(A–γf)x,˜ Jϕ

˜

x–x∗ ≤ (.)

and

(A–γf)x∗,Jϕ

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Adding (.) and (.), we obtain

(A–γf)x˜– (A–γf)x∗,Jϕ

˜

x–x∗ ≤. (.)

Noticing that for anyx,y∈K,

(A–γf)x– (A–γf)y,Jϕ(x–y)

=A(x–y),Jϕ(x–y) –γ

f(x) –f(y),Jϕ(x–y)

≥ ¯γx–yϕx–y–γf(x) –f(y)Jϕ(x–y)

≥ ¯γ x–y–γ αx–y

= (γ¯–γ α)x–y

≥γ ϕ¯ () –γ αx–y≥. (.)

Therefore, x˜=x∗ and the uniqueness is proved. Below, we usex˜ to denote the unique solution of (.). Sincelimn→∞αn= , we may assume, without loss of generality, that

αn<ϕ()A–. For each integern≥, deﬁne a mappingGn:K→Kby

Gn(y) =αnγf(y) + (I–αnA)Tn(tn)y, ∀y∈K.

We shall show thatGnis a contraction mapping. For anyx,y∈K,

Gn(x) –Gn(y)=αnγf(x) + (I–αnA)Tn(tn)x–αnγf(y) – (I–αnA)Tn(tn)y ds ≤αnγ

f(x) –f(y)+(I–αnA)Tn(tn)x–Tn(tn)y

≤αnγ αx–y+ϕ()( –αnγ¯)knx–y

=αnγ α+ϕ()( –αnγ¯)kn

x–y

≤kn–αnγ α+ϕ()αnγ¯kn

x–y

≤kn–αn

ϕ()γ¯kn–γ αx–y.

Since  < kn–_α

n <ϕ()γ¯–γ α, we have

 <kn– 

αn

<ϕ()γ¯–γ α≤ϕ()γ¯kn–γ α.

It then follows that  < (kn–αn(ϕ()γ¯kn–γ α)) < . We haveGnis a contraction map with coeﬃcient (kn–αn(ϕ()γ¯kn–γ α)). Then, for eachn≥, there exists a uniqueyn∈Ksuch thatGn(yn) =yn, that is,

yn=αnγf(yn) + (I–αnA)Tn(tn)yn, n≥.

Hence, (i) is proved. (ii) Deﬁneμ:K→Rby

μ(y) =LIMn

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whereLIMnis a Banach limit onl∞. Sinceμis continuous and convex andg(z)→ ∞as z → ∞, andEis reﬂexive, by Lemma .,gattains its inﬁmum overE. Letu∈Kbe such that

LIMn

yn–u=inf

y∈ELIMn

yn–y. (.)

Let

K∗:=

z∈E:μ(z) =inf

y∈Kμ(y)

.

We have that K∗ is a nonempty, bounded, closed and convex subset ofK and also has the property (P), indeed, ifx∈K∗andw∈ωw(x),i.e. w= weak –limj→∞Tmjxasj→ ∞. Notice that,limn→∞yn–T(t)yn=  uniformly int∈[,∞), by induction we can prove that for allm≥

lim

n→∞yn–T

m_(t)yn_{= } _{uniformly in}_t_∈_[,_∞). _(.)

From (.) and weakly lower semicontinuous ofμ, and for eachh≥, we have that

μ(w)≤lim inf

j→∞ μ

Tmj_(h)x≤_{lim sup}

m→∞ μ

Tm(h)x

=lim sup

m→∞

LIMnyn–Tm(h)x

≤lim sup

m→∞

LIMnyn–Tm(h)yn+Tm(h)yn–Tm(h)x

=lim sup

m→∞ LIMn

Tm(h)yn–Tm(h)x

≤lim sup

m→∞

LIMn

kmyn–x

=LIMn

yn–x

=μ(x) =inf

y∈Kμ(y),

which implies thatK∗satisﬁes the property (P). By Theorem ., there exists a element z∈Ksuch thatz∈F(S)∩K∗.

SinceK±K⊂K, we havez+γf(z) –Az∈K. By Proposition .,

LIMn

z+γf(z) –Az–z,Jϕ(yn–z) ≤,

it implies that

LIMn

γf(z) –Az,Jϕ(yn–z) ≤. (.)

In fact, since(t) =_tϕ(τ)dτ,∀t≥, andϕ: [,∞)→[,∞) is a gauge function, then for ≥k≥,ϕ(kx)≤ϕ(x) and

(kt) =

kt



ϕ(τ)dτ=k

t



ϕ(kx)dx≤k

t



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It follows from Lemma . that

yn–z

=(I–αnA)Tn(tn)yn– (I–αnA)z

+αn

γf(yn) –γf(z) +γf(z) –Az

≤(I–αnA)Tn(tn)yn– (I–αnA)z+αnγ

f(yn) –f(z)

+αn

γf(z) –Az,Jϕ(yn–z)

≤ϕ()( –αnγ¯)Tn(tn)yn–Tn(tn)z+αnγ αyn–z

+αnγ

f(z) –f(z),Jϕ(yn–z)

≤ϕ()( –αnγ¯)(kn)yn–z+αnγ αyn–z

+αn

≤ϕ()( –αnγ¯)(kn) +αnγ α

yn–z+αn

≤ϕ()( –αnγ¯)kn+αnγ α

yn–z+αn

γf(z) –Az,Jϕ(yn–z).

This implies that

yn–z≤ 

 –ϕ()( –αnγ¯)kn+αnγ α

αn

γf(z) –Az,Jϕ(yn–z),

also

yn–z

≤ 

(ϕ()γ¯–αγ) – ( –αnγ¯)dn

γf(z) –Az,Jϕ(yn–z),

wheredn=kn–_α

n . Thus

LIMn

yn–z≤LIMn



(ϕ()γ¯–αγ) – ( –αnγ¯)dn

≤,

and hence

ϕ()γ¯–γ αLIMn

yn–z≤.

Sinceϕ()γ¯>γ α,LIMn(yn–z) = , and then there exists a subsequence{ynj}of{yn}

such thatynj→zasj→ ∞, we shall denoted by{yj}.

Next, we prove thatzsolves the variational inequality (.). From (.), we have

(A–γf)yn= – 

αn

(I–αnA)

I–Tn(tn)

yn.

On the other hand, note for allx,y∈K,

I–Tn(tn)x–I–Tn(tn)y,Jϕ(x–y)

=x–y,Jϕ(x–y) –

Tn(tn)x–Tn(tn)y,Jϕ(x–y)

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≥x–y–knx–yϕx–y ≥x–y–knx–y

= ( –kn)x–y.

Forp∈F(S), we have

(A–γf)yn,Jϕ(yn–p) = – 

αn

(I–αnA)(I–Tn(tn)yn,Jϕ(yn–p)

= – 

αn

I–Tn(tn)yn–I–Tn(tn)p,Jϕ(yn–p)

+AI–Tn(tn)yn,Jϕ(yn–p)

≤ kn– 

αn

x–y+AI–Tn(tn)yn,Jϕ(yn–p)

≤ kn– 

αn

x–y+Ayn–Tn(tn)ynM,

whereM≥sup_n_≥_ϕ(yn–p). Replacingynwithynj and lettingj→ ∞, note thatyn–

Tn_(t

n)yn → andlimn→∞kn–αn = , we have that

(A–γf)z,Jϕ(z–p) ≤, ∀p∈F(T).

That is,z∈F(S) is a solution of (.). Thenz=x. In summary, we have that each cluster˜ point of{yn}converges strongly tox˜asn→ ∞. This completes the proof.

IfA≡I, the identity mapping onK, andγ = , then Theorem . reduces to the following corollary.

Corollary . Let E be a real Banach space with uniform normal structure which has

a uniformly Gateaux diﬀerentiable norm and admits the duality mapping Jϕ, K be a

nonempty,bounded,closed and convex subset of E.LetS={T(s) :s≥}be an asymp-totically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),limn→∞kn= and

sup_n_≥_kn<N(E)/ _{such that F(S)}₌_∅._{Let f} _∈

K with coeﬃcientα∈(, )and let{αn} and{tn}be sequences of real numbers such that <αn< ,tn> .Then the following hold:

n <  –α,∀n≥,then there exists a sequence{yn} ⊂K

deﬁned by

yn=αnf(yn) + ( –αn)Tn(tn)yn, n≥. (.)

(ii) Suppose,in addition,limn→∞yn–T(t)yn= uniformly int∈[,∞)and the real sequences{αn}satisﬁeslimn→∞αn= andlimn→∞kn–αn = .

Then{yn}converges strongly as n→ ∞to a common ﬁxed pointx in F(S˜ )which solves the variational inequality:

(I–f)˜x,Jϕ(˜x–z) ≤, z∈F(S). (.)

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Corollary . Let E be a real Banach space with uniform normal structure which has a uniformly Gateaux diﬀerentiable norm and admits the duality mapping Jϕ, K be a

nonempty,bounded,closed and convex subset of E.LetS={T(s) :s≥}be an asymp-totically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),limn→∞kn= and

sup_n_≥_kn<N(E)/such that F(S)= ∅.Let{αn}and{tn}be sequences of real numbers such that <αn< ,tn> .Then the following hold:

n < ,∀n≥,then there exists a sequence{yn} ⊂Kdeﬁned by

yn=αnu+ ( –αn)Tn(tn)yn, n≥. (.)

(ii) Suppose,in addition,lim_n_→∞yn–T(t)yn= uniformly int∈[,∞)and the real sequences{αn}satisﬁeslim_n_→∞αn= andlimn→∞kn–αn = .

Then{yn}converges strongly as n→ ∞to a common ﬁxed pointx in F(˜ S),which solves the variational inequality:

˜

x–u,Jϕ(˜x–z) ≤, z∈F(S). (.)

Next, we present the convergence theorem for the explicit scheme.

Theorem . Let E be a real Banach space with uniform normal structure,which has

nonempty,bounded,closed and convex subset of E such that K±K⊂K.LetS={T(s) : s≥}be an asymptotically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),

limn→∞kn=  andsupn≥kn<N(E)/ such that F(S)=∅. Let f ∈K with coeﬃcient

α∈(, ),A a strongly positive bounded linear operator on K with coeﬃcientγ¯ > and  <γ<ϕ()_αγ¯.Let{βn}and{tn}be sequences of real numbers such that <βn< ,tn≥,

(C) limn→∞βn= ; (C) limn→∞kn–βn = ;

(C) ∞_n_=βn=∞.

For any x∈K,let the sequences{xn}be deﬁned by

xn+=βnγf(xn) + (I–βnA)Tn(tn)xnds, n≥. (.)

Suppose,in addition,limn→∞xn–T(t)xn= uniformly in t∈[,∞).Then{xn}converge strongly as n→ ∞to the same pointx in F˜ (S),which solves the variational inequality(.).

Proof By Theorem ., there exists a unique solutionx˜inF(S) which solves the variational inequality (.) andym→ ˜xasm→ ∞. Next, we prove that

lim sup

n→∞

γf(x) –˜ Ax,˜ Jϕ(xn–x)˜ ≤. (.)

For allm≥,n≥, we have

ym–xn=αmγf(ym) + (I–αmA)Tm(tm)ym–xn

=αm

γf(ym) –Aym+Tm(tm)ym–Tm(tm)xn

+Tm(tm)xn–xn+αm

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It follows from Lemma . that

ym–xn=(I–αmA)Tm_(tm_)ym_{– (I}_–_α_mA)xn

+αm

γf(ym) –γf(˜x) +γf(˜x) –A˜x+A˜x–Axn

≤(I–αmA)Tm(tm)xn– (I–αmA)xn

+αmγ

f(xm) –f(˜x)+A˜x–Axn+αn

γf(˜x) –A(˜x),Jϕ(yn–x)˜

≤ϕ()( –αnγ¯)Tn(tn)xn–Tn(tn)˜x+αnγ αxn–x˜

+αnγ

f(x) –˜ f(x),˜ Jϕ(yn–x)˜

≤ϕ()( –αnγ¯)knxn–x˜ +αnγ αxn–x˜

+αn

γf(˜x) –A(˜x),Jϕ(xn–x)˜

≤ϕ()( –αnγ¯)kn+αnγ α

xn–x˜

+αn

≤ϕ()( –αnγ¯)kn+αnγ α

xn–x˜

+αn

γf(x) –˜ A(x),˜ Jϕ(xn–x)˜

≤ –ϕ()αnγ¯

(kn– ) +  –αn

ϕ()γ¯–γ αxn–x˜

+αn

≤ –αn

ϕ()γ¯–γ αxn–x˜

+ –ϕ()αnγ¯

(kn– )M

+αn

γf(˜x) –A(˜x),Jϕ(xn–x)˜ . (.)

It follows that

ym–xnϕym–xn

=αm

γf(ym) –Aym

+Tm_(t

m)ym–Tm(tm)xn

+Tm(tm)xn–xn+αm

Aym–ATm(tm)ym,Jϕ(ym–xn)

=αm

γf(ym) –Aym,Jϕ(ym–xn) +

Tm(tm)ym–Tm(tm)xn,Jϕ(ym–xn)

+Tm(tm)xn–xn,Jϕ(ym–xn) +αm

Aym–ATm(tm)ym,Jϕ(ym–xn)

≤αm

γf(ym) –Aym,Jϕ(ym–xn) +Tm(tm)ym–Tm(tm)xnϕ

ym–xn

+Tm(tm)xn–xnϕ

ym–xn

+αmAym–ATm(tm)ymϕ

ym–xn

≤αm

γf(ym) –Aym,Jϕ(ym–xn) +kmym–xnϕ

ym–xn

+Tm(tm)xn–xnϕym–xn+αmA

ym–Tm(tm)ymϕym–xn.

SinceKis bounded, so that{xn}and{ym}are all bounded, and hence

γf(ym) –Aym,Jϕ(xn–ym)

≤km– 

αm

M₊Tm(tm)xn–xn

αm

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where M is a constant satisfying M ≥ sup_n_,_m_∈_Nϕ(xn – ym). By our hypothesis,

limn→∞xn–T(t)xn= , uniformly int∈[,∞). By induction, we can prove that for allm≥

lim

n→∞xn–T

m_(t)xn_{= ,} _{uniformly in}_t_∈_[,_∞_).

Hence for allm≥, we have

lim

n→∞xn–T

m_(tm)xn_{= ,} _as_n_{→ ∞.} _(.)

Therefore, taking upper limit asn→ ∞in (.), we have

lim sup

n→∞

γf(ym) –Aym,Jϕ(xn–ym)

≤lim sup

n→∞ km– 

αm

M+lim sup

n→∞

Aym–Tm(tm)ymM. (.)

SinceKis bounded, it follows from (C) that

ym–Tm(tm)yn=αnγf(ym) +ATm(tm)ym→ asm→ ∞. (.)

And then, taking upper limit asm→ ∞in (.), by (C) and (.), we get

lim sup

m→∞ lim supn→∞

γf(ym) –Aym,Jϕ(xn–ym) ≤. (.)

On the other hand, sincelimm→∞ym=x˜due to the fact the duality mappingJϕis

norm-to-weak∗uniformly continuous on bounded subset ofE, it implies that

γf(x) –˜ Ax,˜ Jϕ(xn–x)˜ –

γf(ym) –Aym,Jϕ(xn–ym)

=γf(˜x) –A˜x,Jϕ(xn–x) –˜ Jϕ(xn–ym) +

γf(˜x) –γf(ym) +Aym–A˜x,Jϕ(xn–ym)

≤γf(˜x) –A˜x,Jϕ(xn–x) –˜ Jϕ(xn–ym)

+γf(˜x) –γf(ym)+A(ym–x)˜ϕxn–ym→, asm→ ∞.

Therefore, for any givenε> , there exists a positive numberNsuch that for allm≥N

γf(˜x) –A˜x,Jϕ(xn–x)˜ ≤

γf(ym) –Aym,Jϕ(xn–ym) +ε.

It follows from (.) that

lim sup

n→∞

γf(˜x) –A˜x,Jϕ(xn–x)˜

=lim sup

m→∞

lim sup

n→∞

γf(˜x) –A˜x,Jϕ(xn–x)˜

≤lim sup

m→∞

lim sup

n→∞

γf(ym) –Aym,Jϕ(xn–ym) +ε

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Sinceεis arbitrary, we have

lim sup

n→∞

γf(˜x) –A˜x,Jϕ(xn–x)˜ ≤. (.)

Finally, we show thatxn→ ˜xasn→ ∞.

xn+–x˜

=(I–βnA)Tn(tn)xn– (I–βnA)x˜

+βn

γf(xn) –γf(x) +˜ γf(x) –˜ A(x)˜

≤(I–βnA)Tn(tn)xn– (I–βnA)x˜+βnγ

f(xn) –f(x)˜

+βn

γf(x) –˜ A(x),˜ Jϕ(yn–x)˜

≤ϕ()( –βnγ¯)Tn(tn)xn–Tn(tn)˜x+βnγ αxn–x˜

+βnγ

f(˜x) –f(˜x),Jϕ(yn–x)˜

≤ϕ()( –βnγ¯)knxn–x˜ +βnγ αxn–x˜

+βn

≤ϕ()( –βnγ¯)kn+βnγ α

xn–x˜

+βn

≤ϕ()( –βnγ¯)kn+βnγ α

xn–x˜

+βn

≤ –ϕ()βnγ¯

(kn– ) +  –βn

ϕ()γ¯–γ αxn–x˜

+βn

≤ –βn

ϕ()γ¯–γ αxn–x˜ + –ϕ()βnγ¯

(kn– )M

+βn

γf(˜x) –A(˜x),Jϕ(xn–x)˜ , (.)

whereM>  such thatsup_n_≥_(xn–x˜)≤M. Put sn=βn

ϕ()γ¯–γ α

and

σn=

 –ϕ()βnγ¯

ϕ()γ¯–γ α

kn– 

βn

M+ 

(ϕ()γ¯–γ α)

γf(˜x) –A(˜x),Jϕ(xn–x)˜ .

Then (.) is reduced to

xn+–x˜

≤( –sn)xn–x˜ +snσn. (.)

Applying Lemma . to (.), we conclude that(xn+–x˜)→ asn→ ∞; that is,

xn→ ˜xasn→ ∞. This completes the proof.

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Theorem . Let E be a real Banach space with uniform normal structure,which has a uniformly Gateaux diﬀerentiable norm and admits the duality mapping Jϕ, K be a

nonempty,bounded,closed and convex subset of E such that K±K⊂K.LetS={T(s) : s≥}be an asymptotically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),

limn→∞kn=  andsupn≥kn<N(E)/ such that F(S)=∅. Let f ∈K with coeﬃcient

α∈(, ),A a strongly positive bounded linear operator on K with coeﬃcientγ¯ > and  <γ<ϕ()_αγ¯.Let{βn}and{tn}be sequences of real numbers such that <βn< ,tn≥,

(C) ∞_n_=βn=∞.

For any w∈K,let the sequence{wn}be deﬁned by

wn+=βnγf

Tn(tn)wn+ (I–βnA)Tn(tn)wn, n≥. (.)

Then{wn}converges strongly as n→ ∞to a pointx in F(S)˜ which solves the variational inequality(.).

Proof Let{xn}be the sequence given byx=wand

xn+=βnγf(xn) + (I–βnA)Tn(tn)xn, ∀n≥.

By Theorem .,xn→ ˜x. We claim thatwn→ ˜x. We calculate the following:

xn+–wn+=βnγf(xn) –f

Tn(tn)wn+I–βnATn(tn)xn–Tn(tn)wn

≤βnγ αxn–Tn(tn)wn+ϕ()( –βnγ¯)(kn)xn–wn ≤βnγ αxn–Tn(tn)˜x+βnγ αTn(tn)˜x–Tn(tn)wn

+ϕ()( –βnγ¯)(kn)xn–wn

≤βnγ αxn–x˜ +βnγ αkn˜x–wn

≤βnγ αxn–x˜ +βnγ αwn–xn+βnγ αkn˜x–xn

=ϕ()( –βnγ¯)(kn)xn–wn+βnγ αwn–xn

+βn(γ α+kn)xn–x˜

=ϕ()( –βnγ¯)(kn) +βnγ α

xn–wn

≤ –ϕ()βnγ¯

(kn– ) +  –βn

ϕ()γ¯–γ αxn–wn

≤ –βn

ϕ()γ¯–γ αxn–wn+

 –ϕ()βnγ¯

(kn– )M

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whereM>  such thatsup_n_≥_xn–wn ≤M. Put sn=βn

ϕ()γ¯–γ α

and

σn=

 –ϕ()βnγ¯

ϕ()γ¯–γ α

kn– 

βn

M+ (γ α+kn)

(ϕ()γ¯–γ α)xn–x.˜

Then we have that

xn+–wn+ ≤( –sn)xn–wn+snσn. (.)

It follows from (C),lim_n_→∞xn–x˜=  and Lemma . thatxn–wn → asn→ ∞;

that is,wn→ ˜xasn→ ∞.

Theorem . Let E be a real Banach space with uniform normal structure which has a

uni-formly Gâteaux diﬀerentiable norm and admits the duality mapping Jϕ,K be a nonempty,

bounded,closed and convex subset of E such that K±K⊂K.LetS={T(s) :s≥}be an asymptotically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),limn→∞kn=  andsup_n_≥_kn<N(E)/ such that F(S)=∅.Let f ∈K with coeﬃcientα∈(, ),A a strongly positive bounded linear operator on K with coeﬃcientγ¯> and <γ<ϕ()_αγ¯.Let {βn}and{tn}be sequences of real numbers such that <βn< ,tn≥,

(C) ∞_n_=βn=∞.

For any z∈K,let the sequence{zn}be deﬁned by

zn+=Tn(tn)

βnγf(zn) + (I–βnA)zn

, n≥. (.)

Then{zn}converges strongly as n→ ∞to a pointx in F(S)˜ which solves the variational inequality(.).

Proof Deﬁne the sequences{wn}and{σn}by

wn=βnγf(zn) + (I–βnA)zn and σn=βn+, n≥.

We have that

wn+=βn+γf(zn+) + (I–βn+A)zn+=σnγf

Tn(tn)wn+ (I–σnA)Tn(tn)wn.

It follows from Theorem . that{wn}converges strongly tox. Thus, we have˜

zn–x˜ ≤ zn–wn+wn–x˜=βnγf(zn) –Azn+wn–x˜ ≤βnM+wn–x˜ → asn→ ∞,

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IfA≡I, the identity mapping onE, andγ = , then Theorem . reduces to the following corollary.

Corollary . Let E be a real Banach space with uniform normal structure which has

a uniformly Gâteaux diﬀerentiable norm and admits the duality mapping Jϕ, K be a

nonempty,bounded,closed and convex subset of E.LetS={T(s) :s≥}be an asymp-totically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),lim_n_→∞kn= and

sup_n_≥_kn<N(E)/_{such that F(S)}₌_∅._{Let f} _∈

Kwith coeﬃcientα∈(, ).Let{βn}and {tn}be sequences of real numbers such that <βn< ,tn≥,

(C) ∞_n_=βn=∞.

For any x∈K,let the sequence{xn}be deﬁned by

xn+=βnf(xn) + ( –βn)Tn(tn)xnds, n≥. (.)

Suppose,in addition,limn→∞xn–T(t)xn= uniformly in t∈[,∞).Then{xn}converges strongly as n→ ∞to a pointx in F˜ (S),which solves the variational inequality(.).

Iff ≡u, then Corollary . reduces to the following corollary.

Corollary . Let E be a real Banach space with uniform normal structure which has

nonempty,bounded,closed and convex subset of E such that K±K⊂K.LetS={T(s) : s≥}be an asymptotically nonexpansive semigroup on K with a sequence{kn} ⊂[, +∞),

limn→∞kn=  andsupn≥kn<N(E)/ such that F(S)=∅. Let f ∈K with coeﬃcient

α∈(, ).Let{βn}and{tn}be sequences of real numbers such that <βn< ,tn≥, (C) lim_n_→∞βn= ;

(C) limn→∞kn–βn = ;

(C) ∞_n_=βn=∞.

For any x∈K,let the sequence{xn}be deﬁned by

xn+=βnu+ ( –βn)Tn(tn)xnds, n≥. (.)

Suppose,in addition,limn→∞xn–T(t)xn= uniformly in t∈[,∞).Then{xn}converges strongly as n→ ∞to a pointx in F˜ (S)which solves the variational inequality(.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors read and approved the ﬁnal manuscript.

Acknowledgements

The ﬁrst author is supported by the ‘Centre of Excellence in Mathematics’ under the Commission on Higher Education, Ministry of Education, Thailand.

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