Split feasibility and fixed point problems for asymptotically quasi nonexpansive mappings

R E S E A R C H

Open Access

Split feasibility and fixed-point problems for

asymptotically quasi-nonexpansive

mappings

Jitsupa Deepho and Poom Kumam

*

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Abstract

The purpose of this paper is to introduce and analyze a weakly convergent theorem by using the regularized method and the relaxed extragradient method for finding a common element of the solution set

of the split feasibility problem and Fix(T) of fixed points of asymptotically quasi-nonexpansive mappingsTin the setting of infinite-dimensional Hilbert spaces. Consequently, we prove that the sequence generated by the proposed algorithm converges weakly to an element of Fix(T)∩

under mild assumptions.

MSC: 47H09; 47J25; 65K10

Keywords: split feasibility problems; fixed point problems; relaxed extragradient methods; regularization; asymptotically quasi-nonexpansive mappings; maximal monotone mappings

1 Introduction

In , Censor and Elfving [] first introduced the split feasibility problem (SFP) in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase re-trievals and in medical image reconstruction []. It was found that the SFP can also be used to model intensity-modulated radiation therapy (IMRT) (see [–]). Very recently, Xu [] considered the SFP in the framework of infinite-dimensional Hilbert spaces. In this setting, the SFP is formulated as the problem of finding a pointx∗ with the prop-erty

x∗∈C and Ax∗∈Q, (.)

whereCandQare the nonempty closed convex subsets of the infinite-dimensional real Hilbert spacesH andH, respectively. LetA∈B(H,H), whereB(H,H) denotes the

family of all bounded linear operators fromHtoH.

We useto denote the solution set of the SFP,i.e.,

={x∈C:Ax∈Q}.

Assume that the SFP is consistent (i.e., (.) has a solution) so that is closed, convex and nonempty. A special case of the SFP is the following convex constrained linear inverse

problem:

findx∈C such thatAx=b, (.)

which has extensively been investigated by using the Landweber iterative method []:

letxbe arbitrary forn= , , . . . , let

xn+=xn+γAT(b–Axn).

Comparatively, the SFP has received much less attention so far due to the complexity resulting from the setQ. Therefore, whether various versions of the projected Landwe-ber iterative method [] can be extended to solve the SFP remains an interesting open topic.

The original algorithm given in [] involves the computation of the inverseA– (assum-ing the existence of the inverse ofA):

xk+=A–PQ

PA(C)(Axk)

, k≥,

where C,Q⊂Rn_{are closed convex sets, Ais a full rankn×nmatrix andA(C) ={y∈} Rn_{|y=Ax,x∈C}, and thus has not become popular.}

A more popular algorithm that solves the SFP seems to be theCQalgorithm of Byrne [, ] which is found to be a gradient-projection method (GPM) in convex minimization. It is also a special case of the proximal forward-backward splitting method []. TheCQ algo-rithm only involves the computations of the projectionsPCandPQonto the setsCandQ,

respectively, and is therefore implementable in the case wherePCandPQhave closed-form

expressions (for example,CandQare closed balls or half-spaces). It remains, however, a challenge on theCQalgorithm in the case where the projectionPCand/orPQfail to have

closed-form expressions though theoretically we can prove the (weak) convergence of the algorithm.

Recently, Xu [] gave a continuation of the study on theCQalgorithm and its conver-gence. He applied Mann’s algorithm to the SFP and proposed an averagedCQalgorithm, which was proved to be weakly convergent to a solution of the SFP. He derived a weak convergence result, which shows that for suitable choices of iterative parameters (includ-ing the regularization), the sequence of iterative solutions can converge weakly to an exact solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained. Later, Deepho and Kumam [] extended the results of Xu [] by introducing and studying the modified Halpern iterative scheme for solving the split feasibility problem (SFP) in the setting of infinite-dimensional Hilbert spaces.

Throughout this paper, we always assume that the SFP is consistent, that is, the solution setof the SFP is nonempty. Letf:H→Rbe a continuous differentiable function. The

minimization problem

min

x∈Cf(x) :=



Ax–PQAx

is ill-posed. Therefore (see []), consider the following Tikhonov regularized problem:

min

x∈Cfα(x) :=



Ax–PQAx

₊

αx

_{, (.)}

whereα>  is the regularization parameter. We observe that the gradient

∇fα(x) =∇f(x) +αI=A∗(I–PQ)A+αI (.)

is (α+A_{)-Lipschitz continuous andα-strongly monotone.}

Define the Picard iterates

xα_n+=PC

I–γA∗(I–PQ)A+αI

xα_n. (.)

Xu [] showed that if SFP (.) is consistent, then asn→ ∞,xα

n→xαand consequently the stronglimα→xαexists and is the minimum-norm solution of the SFP. Note that (.) is double-step iteration. Xu [] further suggested the following single step regularized method:

xn+=PC(I–γ∇fαn)xn=PC

( –αnγn)xn–γnA∗(I–PQ)Axn

. (.)

He proved that the sequence{xn}generated by (.) converges in norm to the

minimum-norm solution of the SFP provided the parameters{αn}and{γn}satisfy the following

con-ditions:

(i) αn→and <γn<_Aαn_+αn;

(ii) _nαnγn=∞;

(iii) |γn+–γn|+γn|αn+–αn|

(αn+γn+) →.

Motivated by the idea of the relaxed extragradient method and Xu’s regularization, Ceng, Ansari and Yao [] presented the following relaxed extragradient method with reg-ularization for finding a common element of the solution set of the split feasibility problem and the setFix(S) of fixed points of a nonexpansive mappingS:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x=x∈C chosen arbitrarily, yn= ( –βn) +βnPC(xn–λ∇fαn(xn)),

xn+=γnxn+ ( –γn)SPC(yn–λ∇fαn(yn)), ∀n≥.

(.)

They only obtained the weak convergence of iterative algorithm (.).

The purpose of this paper to study and analyze an relaxed extragradient method with regularization for finding a common element of the solution setof the SFP and the set solutions of fixed points for asymptotically quasi-nonexpansive mappings and a Lipschitz continuous mapping in a real Hilbert space. We prove that the sequence generated by the proposed method converges weakly to an elementxˆinFix(T)∩.

2 Preliminaries

and let C be a closed and convex subset of H. LetE be a Banach space. A mapping

T :E→Eis said to bedemi-closed at originif for any sequences{xn} ⊂Ewithxnx∗

and(I–T)xn →,x∗=Tx∗. A Banach spaceEis said to havethe Opial propertyif for

any sequence{xn}withxnx∗,

lim inf

n→∞ xn–x

∗ _{<lim inf}

n→∞ xn–y, ∀y∈Ewithy=x ∗_.

Remark . It is well known that each Hilbert space possesses the Opial property.

Definition . LetH be a real Hilbert space, let Cbe a nonempty and closed convex subset. We denote byFix(T) the set of fixed points ofT, that is,Fix(T) ={x∈C:x=Tx}.

ThenT is said to be

(i) nonexpansive if Tx–Ty ≤ x–yfor allx,y∈C;

(ii) quasi-nonexpansive if Tx–p ≤ x–pfor allx∈Candp∈F(T);

(iii) asymptotically nonexpansiveif there exist a sequencekn≥andlimn→∞kn= 

such that

Tnx–Tny ≤knx–y

for allx,y∈Candn≥;

(iv) asymptotically quasi-nonexpansiveif there exist a sequencekn≥and

limn→∞kn= such that

Tnx–p ≤knx–p

for allx∈C,p∈F(T)andn≥;

(v) uniformlyL-Lipschitzianif there exists a constantL> such that

Tnx–Tny ≤Lx–y

for allx,y∈Candn≥.

Remark . By the above definitions, it is clear that:

(i) a nonexpansive mapping is an asymptotically quasi-nonexpansive mapping; (ii) a quasi-nonexpansive mapping is an asymptotically-quasi nonexpansive mapping; (iii) an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive

mapping.

Proposition .(see []) We have the following assertions.

(i) Tis nonexpansive if and only if the complementI–T is _-ism. (ii) IfT isν-ism andγ > ,thenγTis ν

γ-ism.

(iii) Tis averaged if and only if the complementI–T isν-ism for someν>  .

Indeed,forα∈(, ),Tisα-averaged if and only ifI–Tis __α-ism.

Proposition .(see [, ]) We have the following assertions.

(i) IfT= ( –α)S+αV for someα∈(, ),Sis averaged andVis nonexpansive,thenT

(ii) Tis firmly nonexpansive if and only if the complementI–T is firmly nonexpansive. (iii) IfT= ( –α)S+αVfor someα∈(, ),Sis firmly nonexpansive andVis

nonexpansive,thenT is averaged.

(iv) The composite of finite many averaged mappings is averaged.That is,if each of the mappings{Ti}ni=is averaged,then so is the compositeT◦T◦ · · · ◦TN.In

particular,ifTisα-averaged andTisα-averaged,whereα,α∈(, ),then the

compositeT◦Tisα-averaged,whereα=α+α–αα.

(v) If the mappings{Ti}ni=are averaged and have a common fixed point,then

n

i=

Fix(Ti) =Fix(T· · ·TN).

Lemma .(see [], demiclosedness principle) Let C be a nonempty closed and

con-vex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping with

Fix(S)=∅.If the sequence{xn} ⊆C converges weakly to x and the sequence{(I–S)xn}

con-verges strongly to y,then(I–S)x=y;in particular,if y= ,then x∈Fix(S).

Lemma .(see []) Let the sequences{an}and{un}of real numbers satisfy

an+≤( +un)an, ∀n≥,

where an≥,un≥and

∞

n=un<∞.Then

() limn→∞anexists;

() iflim infn→∞an= ,thenlimn→∞an= .

The following lemma gives some characterizations and useful properties of the metric projectionPCin a Hilbert space.

For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such

that

x–PCx ≤ x–y, ∀y∈C, (.)

wherePCis called themetric projection of HontoC. We know thatPCis a nonexpansive

mapping ofHontoC.

Proposition . For given x∈H and z∈C:

(i) z=PCxif and only ifx–z,y–z ≤for ally∈C.

(ii) z=PCxif and only ifx–z≤ x–y–y–zfor ally∈C.

(iii) For ally∈H,PCx–PCy,x–y ≥ PCx–PCy.

Lemma .(see []) Let H be a real Hilbert space.Then the following equations hold: (i) x–y_=x_–y_{– x–y,yfor allx,y∈H;}

(ii) tx+ ( –t)y_=tx_{+ ( –t)y}_{–t( –t)x–y}_{for allt∈[, ]andx,y∈H.}

LetKbe a nonempty closed convex subset of a real Hilbert spaceHand letF:K→H

that

Fx,y–x ≥, ∀y∈K.

The solution set of the VIP is denoted byVIP(K,F). It is well known that

x∈VI(K,F) ⇔ x=PK(x–λFx), ∀λ> .

A set-valued mappingT:H→H_{is calledmonotoneif for allx,y∈H,f ∈Txandg∈Ty}

imply

x–y,f –g ≥.

A monotone mappingT :H→H _{is calledmaximal if its graphG(T) is not properly}

contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if for (x,f)∈H×H,x–y,f –g ≥ for every (y,g)∈

G(T) impliesf ∈Tx. LetF:K→Hbe a monotone andk-Lipschitz continuous mapping and letNKvbe the normal cone toKatv∈K, that is,

NKv=

w∈H:v–u,w ≥,∀u∈K.

Define

Tv= ⎧ ⎨ ⎩

Fv+NKv ifv∈K,

∅ ifv∈/K.

Then T is maximal monotone and ∈Tvif and only ifv∈VI(K,F); see [] for more details.

We can use fixed point algorithms to solve the SFP on the basis of the following obser-vation.

Letλ>  and assume thatx∗∈. ThenAx∗∈Q, which implies that (I–PQ)Ax∗= , and

thusλA∗(I–PQ)Ax∗= . Hence, we have the fixed point equation (I–λA∗(I–PQ)A)x∗=x∗.

Requiring thatx∗∈C, we consider the fixed point equation

PC(I–λ∇f)x∗=PC

I–λA∗(I–PQ)A

x∗=x∗. (.)

It is proved in [, Proposition .] that the solutions of fixed point equation (.) are exactly the solutions of the SFP; namely, for givenx∗∈H,x∗solves the SFP if and only ifx∗solves

fixed point equation (.).

Proposition .(see []) Given x∗∈H,the following statements are equivalent.

(i) x∗solves the SFP;

(ii) x∗solves fixed point equation(.);

(iii) x∗solves the variational inequality problem(VIP)of findingx∗∈Csuch that

∇fx∗,x–x∗≥, ∀x∈C, (.)

Proof (i)⇔(ii). See the proof in [, Proposition .]. (ii)⇔(iii). Observe that

PC

I–λA∗(I–PQ)A

x∗=x∗ ⇔ I–λA∗(I–PQ)A

x∗–x∗,x–x∗≤, ∀x∈C

⇔ –λA∗(I–PQ)Ax∗,x–x∗

≤, ∀x∈C

⇔ ∇fx∗,x–x∗≥, ∀x∈C,

where∇f =A∗(I–PQ)A.

Remark . It is clear from Proposition . that

:=FixPC(I–λ∇f)

=VI(C,∇f)

for anyλ> , whereFix(PC(I–λ∇f)) andVI(C,∇f) denote the set of fixed points ofPC(I– λ∇f) and the solution set ofVIP.

3 Main result

Theorem . Let C be a nonempty,closed,and convex subset of a real Hilbert space H

and let T:C→C be a uniformly L-Lipschitzian and asymptotically quasi-nonexpansive

mappings withFix(T)∩=∅and{kn} ⊂[,∞)for all n∈Nsuch that

∞

n=(kn– ) <∞.

Let{xn}and{yn}be the sequences in C generated by the following algorithm:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x=x∈C chosen arbitrarily, yn=PC(I–λn∇fαn)xn,

xn+=βnxn+ ( –βn)Tnyn, ∀n≥,

(.)

where∇fαn=∇f+αnI=A∗(I–PQ)A+αnI,and three sequences{αn},{λn},and{βn}satisfy

the conditions: (i) ∞_n=αn<∞,

(ii) {λn} ⊂[a,b]for somea,b∈(,_A)and

_∞

n=|λn+–λn|<∞,

(iii) {βn} ⊂[c,d]for somec,d∈(, ).

Then the sequences{xn}and{yn}converge weakly to an elementxˆ∈Fix(T)∩.

Proof We first show thatPC(I–λ∇fα) isζ-averaged for eachλn∈(,_α+_A), where

ζ = +λ(α+A

₎

 .

Indeed, it is easy to see that∇f =A∗(I–PQ)Ais_A-ism, that is,

∇f(x) –∇f(y),x–y≥ 

A ∇f(x) –∇f(y) 

.

Observe that

α+A∇fα(x) –∇fα(y),x–y

=αx–y+α∇f(x) –∇f(y),x–y

+αAx–y+A∇f(x) –∇f(y),x–y

≥αx–y+ α∇f(x) –∇f(y),x–y+ ∇f(x) –∇f(y) 

= α(x–y) +∇f(x) –∇f(y) 

= ∇f(x) –∇f(y) .

Hence, it follows that∇fα=αI+A∗(I–PQ)Ais_α+_A-ism. Thus,λ∇fαis_λ(α+_A₎-ism. By

Proposition .(iii) the composite (I–λ∇fα) is λ(α+A

₎

 -averaged. Therefore, noting that PCis _-averaged and utilizing Proposition .(iv), we know that for eachλ∈(,_α+_A), PC(I–λ∇fα) isζ-averaged with

ζ = +

λ(α+A₎

 –

 ·

λ(α+A₎

 =

 +λ(α+A₎

 ∈(, ).

This shows thatPC(I–λ∇fα) is nonexpansive. Furthermore, for{λn} ∈[a,b] witha,b∈

(,_A), utilizing the fact thatlimn→∞_αn+_A =_A, we may assume that

 <a≤λn≤b<



A, ∀n≥.

Consequently, it follows that for each integern≥,PC(I–λn∇fαn) isζn-averaged with

ζn=

  +

λn(αn+A)

 –

  ·

λn(αn+A)

 =

 +λn(αn+A)

 ∈(, ).

This immediately implies thatPC(I–λn∇fαn) is nonexpansive for alln≥.

We divide the remainder of the proof into several steps.

Step . We prove that{xn}is bounded. Indeed, we take a fixedp∈Fix(T)∩arbitrarily.

Then we getPC(I–λn∇f)p=pforλn∈(,_A). SincePCand (I–λn∇fαn) are

nonexpan-sive mappings, then we have

yn–p= PC(I–λn∇fαn)xn–PC(I–λn∇f)p

≤ PC(I–λn∇fαn)xn–PC(I–λn∇fαn)p

+ PC(I–λn∇fαn)p–PC(I–λn∇f)p

≤ xn–p+ (I–λn∇fαn)p– (I–λn∇f)p

=xn–p+λn∇fp–λn∇fαnp

=xn–p+λn∇fp–∇fαnp

=xn–p+λn∇fp–∇fp–αnp

≤ xn–p+αnλnp. (.)

Observe that

xn+–p= βnxn+ ( –βn)Tnyn–p

≤βnxn–p+ ( –βn)knyn–p

≤βnxn–p+ ( –βn)kn

xn–p+λnαnp

=βnxn–p+ ( –βn)knxn–p+ ( –βn)knαnλnp

= + (kn– )( –βn)

xn–p+ ( –βn)knαnλnp. (.)

Since∞_n=(kn– ) <∞, according to Lemma . and (i), (ii) and (.), we obtain that

lim

n→∞xn–pexists for eachp∈Fix(T)∩. (.)

This implies that{xn}is bounded and{yn}is also bounded.

It follows that

Tnxn–p ≤knxn–p.

Hence{Tn_x

n–p}is bounded.

Step . We prove that

lim

n→∞yn–Tyn= .

In fact, it follows from (.) that

yn–p =

xn–p+αnλnp



≤ xn–p+ αnλnpxn–p+αnλnp

=xn–p+αn

λnpxn–p+αnλnp

=xn–p+αnM,

whereM=sup_n≥{λnpxn–p+αnλnp}<∞.

It follows that

Tnyn–p 

≤knyn–p



=k_nyn–p

=k_nxn–p+αnknM.

Also, observe that

xn+–p = βnxn+ ( –βn)Tnyn–p 

≤βnxn–p+ ( –βn) Tnyn–p 

–βn( –βn) Tnyn–xn 

≤βnxn–p+ ( –βn)

k_nxn–p+αnknM

–βn( –βn) Tnyn–xn 

=βnxn–p+ ( –βn)knxn–p+ ( –βn)knαnM

–βn( –βn) Tnyn–xn 

=k_n–βn

k_n– xn–p+ ( –βn)knαnM–βn( –βn) Tnyn–xn 

Hence, we have

βn( –βn) Tnyn–xn 

≤k_n–βn

k_n– xn–p–xn+–p+ ( –βn)knαnM. (.)

By the conditions (i), (iii) andlim_n→∞kn= , we can conclude that

lim

n→∞ T n_y

n–xn = . (.)

Consider that sinceyn=PC(xn–λn∇fαnxn) and by Proposition .(ii), we have

yn–p≤ xn–λn∇fαn(xn) –p 

– xn–λn∇fαn(xn) –yn 

=xn–p–xn–yn+ λn

∇fαn(xn),p–yn

=xn–p–xn–yn+ λn

∇fαn(xn) –∇fαn(p),p–xn

+∇fαn(p),p–xn

+∇fαn(xn),xn–yn

≤ xn–p–xn–yn+ λn

∇fαn(p),p–xn

+∇fαn(xn),xn–yn

=xn–p–xn–yn+ λn

(αnI+∇f)p,p–xn

+∇fαn(xn),xn–yn

=xn–p–xn–yn+ λn

αnp,p–xn+

∇fαn(xn),xn–yn

=xn–p–xn–yn+ λnαnp,p–xn+ λn

∇fαn(xn),xn–yn

=xn–p–xn–yn+ λnαnp,p–xn– λn

∇fαn(xn),yn–xn

=xn–p–xn–yn+ λnαnp,p–xn– λn

∇fαn(xn),yn–p+p–xn

=xn–p–xn–yn+ λnαnp,p–xn– λn

∇fαn(xn),yn–p

– λn

∇fαn(xn),p–xn

≤ xn–p–xn–yn+ λnαnpp–xn– λn ∇fαn(xn) yn–p

– λn ∇fαn(xn) p–xn. (.)

Consequently, utilizing Lemma .(ii) and (.), we conclude that

xn+–p = βnxn+ ( –βn)Tnyn–p 

= βnxn+ ( –βn)Tnyn–

βn+ ( –βn)

p 

= βnxn+ ( –βn)Tnyn–βnp– ( –βn)p 

= βn(xn–p) + ( –βn)

Tnyn–p 

=βnxn–p+ ( –βn) Tnyn–p –βn( –βn xn–Tnyn 

≤βnxn–p+ ( –βn)knyn–p–βn( –βn) xn–Tnyn 

=βnxn–p+ ( –βn)kn

xn–p–xn–yn+ λnαnpp–xn

– λn ∇fαn(xn) yn–p– λn ∇fαn(xn) p–xn

=βn+ ( –βn)kn

xn–p– ( –βn)knxn–yn

+ ( –βn)knλnαnpp–xn

– ( –βn)knλn ∇fαn(xn) yn–p

– ( –βn)knλn ∇fαn(xn) p–xn

–βn( –βn) xn–Tnyn 

=k_n–βn

k_n– xn–p– ( –βn)knxn–yn

+ ( –βn)knλnαnpp–xn

– ( –βn)knλn ∇fαn(xn) yn–p

– ( –βn)knλn ∇fαn(xn) p–xn

–βn( –βn) xn–Tnyn 

.

It follows that we get

( –βn)knxn–yn– ( –βn)knλnαnpp–xn

+ ( –βn)knλn ∇fαn(xn) yn–p+p–xn

+βn( –βn) xn–Tnyn 

≤k_n–βn

k_n– xn–p–xn+–p. (.)

So, takingn→ ∞, sincelimn→kn= , (i)-(iii), (.) and (.), we can conclude that

lim

n→yn–xn= . (.)

Consider

xn+–xn= βnxn–xn+ ( –βn)Tnyn

= –( –βn)xn+ ( –βn)Tnyn

≤( –βn) Tnyn–xn . (.)

From (.) we obtain

xn+–xn ≤( –βn) Tnyn–xn → (asn→ ∞). (.)

Observe that

Tnyn–yn = Tnyn–xn+xn–yn

≤ Tnyn–xn +xn–yn.

So, from (.) and (.), we get

lim

n→∞ T n_y

We compute that

yn+–yn= PC(xn+–λn+∇fαn+xn+) –PC(xn–λn∇fαnxn)

= PC(I–λn+∇fαn+)xn+–PC(I–λn∇fαn)xn

≤ PC(I–λn+∇fαn+)xn+–PC(I–λn+∇fαn+)xn

+ PC(I–λn+∇fαn+)xn–PC(I–λn∇fαn)xn

≤ xn+–xn+ (I–λn+∇fαn+)xn– (I–λn∇fαn)xn

=xn+–xn+ xn–λn+∇fαn+xn– (xn–λn∇fαnxn)

=xn+–xn+λn∇fαnxn–λn+∇fαn+xn

=xn+–xn+ λn(∇f+αn)xn–λn+(∇f+αn+)xn

=xn+–xn+ λn∇fxn+λnαnxn– (λn+∇fxn+λn+αn+xn)

=xn+–xn+ (λn–λn+)∇fxn+λnαnxn–λn+αn+xn

=xn+–xn+ (λn–λn+)∇fxn+λnαnxn–λnαn+xn

+λnαn+xn–λn+αn+xn

=xn+–xn+ (λn–λn+)∇fxn+λn(αn–αn+)xn

+ (λn–λn+)αn+xn

≤ xn+–xn+|λn–λn+|∇fxn+λn|αn–αn+|xn

+αn+|λn–λn+|xn.

From the conditions (i), (ii) and (.), we obtain that

yn+–yn → (asn→ ∞). (.)

SinceTis uniformlyL-Lipschitzian continuous, then

yn–Tyn ≤ yn–yn++ yn+–Tn+yn+ + Tn+yn+–Tn+yn + Tn+yn–Tyn

≤ yn–yn++ yn+–Tn+yn+ +Lyn–yn++L Tnyn–yn .

Sincelim_n→∞yn+–yn=  andlimn→∞yn–Tnyn= , it follows that

lim

n→∞yn–Tyn= . (.)

Step . We show thatxˆ∈Fix(T)∩.

Since∇f =A∗(I–PQ)Ais Lipschitz continuous, from (.), we have

lim

n→∞ ∇f(xn) –∇f(yn) = .

Since{xn}is bounded, there is a subsequence{xni}of{xn}that converges weakly to somexˆ.

Put

Sw=

⎧ ⎨ ⎩

∇fw+NCw ifw∈C,

∅ ifw∈/C,

whereNCw={z∈H:w–u,z ≥,∀u∈C}. ThenSis maximal monotone and ∈Sw

if and only ifw∈VI(C,∇f); (see []) for more details. Let (w,z)∈G(S), we have

z∈Sw=∇fw+NCw,

and hence

z–∇fw∈NCw.

So, we have

w–u,z–∇fw ≥, ∀u∈C.

On the other hand, from

yn=PC(I–λn∇fαn)xn and w∈C,

we have

xn–λn∇fαnxn–yn,yn–w ≥,

and

w–yn,

yn–xn λn

+∇fαnxn

≥.

Therefore, fromz–∇fw∈NCwandyni∈C, it follows that

w–yni,z ≥ w–yni,∇fw

≥ w–yni,∇fw–

w–yni,

yni–xni

λni

+∇fα_nixni

=w–yni,∇fw–

w–yni,

yni–xni

λni

+∇fxni

–αniw–yni,xni

=w–yni,∇fw–∇fyni+w–yni,∇fyni–∇fxni

–

w–yni,

yni–xni

λni

–αniw–yni,xni

≥ w–yni,∇fyni–∇fxni–

w–yni,

yni–xni

λni

Hence, we obtain

w–xˆ,z ≥ asi→ ∞.

SinceSis maximal monotone, we havexˆ∈S–_{, and hencexˆ∈VI(C,∇f). Thus, it is clear}

thatxˆ∈.

Next, we show thatxˆ∈Fix(T). Indeed, sinceynixˆandyni–Tyni → by (.) and

Lemma ., we getxˆ∈Fix(T). Therefore, we havexˆ∈Fix(T)∩.

Let{xnj}be another subsequence of{xn}such that{xnj}x¯. Thenx¯∈Fix(T)∩. Let

us show thatxˆ=x¯. Assume thatxˆ=x¯. From the Opial condition [], we have

lim

n→∞xn–xˆ=nlimi→∞

infxni–xˆ

< lim

ni→∞

infxni–x¯

= lim

n→∞xn–x¯

= lim

nj→∞

infxnj–x¯

< lim

nj→∞

infxnj–xˆ

= lim

n→∞xn–xˆ.

This is a contradiction. Thus, we havexˆ=x¯. This implies

xnxˆ∈Fix(T)∩.

Further, fromxn–yn →, it follows thatynxˆ. This shows that both sequences{xn}

and{yn}converge weakly toxˆ∈Fix(T)∩. This completes the proof.

Utilizing Theorem ., we have the following new results in the setting of real Hilbert spaces.

TakeTn_{≡I(identity mappings) in Theorem .. Therefore the conclusion follows.}

Corollary . Let C be a nonempty,closed,and convex subset of a real Hilbert space H.

Suppose that=∅.Let{xn}be a sequence in C generated by the following algorithm:

⎧ ⎨ ⎩

x=x∈C chosen arbitrarily,

xn+=βnxn+ ( –βn)PC(I–λn∇fαn)xn, ∀n≥,

(.)

where∇fαn=∇f+αnI=A∗(I–PQ)A+αnI,and the sequences{αn},{λn},and{βn}satisfy

the conditions: (i) ∞_n=αn<∞,

(ii) {λn} ⊂[a,b]for somea,b∈(,_A)and

∞

n=|λn+–λn|<∞,

(iii) {βn} ⊂[c,d]for somec,d∈(, ).

Then the sequence{xn}converges weakly to an elementxˆ∈.

Corollary . Let C be a nonempty,closed,and convex subset of a real Hilbert space H

and let T:C→C be a uniformly L-Lipschitzian and quasi-nonexpansive mapping with

Fix(T)=∅ and{kn} ⊂[,∞)for all n∈Nsuch that

_∞

n=(kn– ) <∞. Let{xn}be the

sequence in C generated by the following algorithm: ⎧

⎨ ⎩

x=x∈C chosen arbitrarily, xn+=βnxn+ ( –βn)Tnxn, ∀n≥,

(.)

and let the sequence{βn}satisfy the condition{βn} ⊂[c,d]for some c,d∈(, ).Then the

sequence{xn}converges weakly to an elementxˆ∈Fix(T).

Remark . Theorem . improves and extends [, Theorem .] in the following as-pects:

(a) The iterative algorithm [, Theorem .] is extended for developing our relaxed extragradient algorithm with regularization in Theorem ..

(b) The technique of proving weak convergence in Theorem . is different from that in [, Theorem .] because of our technique to use asymptotically

quasi-nonexpansive mappings and the property of maximal monotone mappings. (c) The problem of finding a common element ofFix(T)∩for asymptotically

quasi-nonexpansive mappings which is more general than that for nonexpansive mappings and the problem of finding a solution of the SFP in [, Theorem .].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the referee for comments and suggestions on this manuscript. The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0033/2554) and the King Mongkut’s University of Technology Thonburi (KMUTT). Moreover, this study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (Under Grant No. NRU56000508).

Received: 10 April 2013 Accepted: 21 June 2013 Published: 13 July 2013 References

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doi:10.1186/1029-242X-2013-322