An algorithm for treating asymptotically strict pseudocontractions and monotone operators

(1)

R E S E A R C H

Open Access

An algorithm for treating asymptotically strict

pseudocontractions and monotone operators

Mingliang Zhang

*

*_{Correspondence:} [email protected] School of Mathematics and Information Science, Henan University, Kaifeng, 475000, China

Abstract

In this paper, an algorithm for treating asymptotically

κ

-strict pseudocontractions and monotone operators is proposed. Convergence analysis of the algorithm is

investigated in the framework of Hilbert spaces.

Keywords: asymptotically

κ

-strict pseudocontraction; inverse-strongly monotone mapping; maximal monotone operator; ﬁxed point

1 Introduction-preliminaries

In this paper, we are concerned with the problem of ﬁnding a common element in the intersectionF(T)∩(A+B)–(), whereF(T) denotes the ﬁxed point set of the mappingT and (A+B)–_{() denotes the zero point set of the sum of the operator}_A_{and the operator}_B.

The motivation for the common element problem is mainly due to its possible appli-cations to mathematical modeling of concrete complex problems. The common element problems include mini-max problems, complementarity problems, equilibrium problems, common ﬁxed point problems and variational inequalities as special cases; see, for exam-ple, [–] and the references therein.

Throughout the article, we always assume thatHis a real Hilbert space with the inner product·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofH, and letProj_Cbe the metric projection fromHontoC.

LetA:C→Hbe a mapping.A–_{() stands for the zero point set of}_{A; that is,}_A–_{() :=} {x∈C:Ax= }. Recall thatAis said to be monotone iﬀ

Ax–Ay,x–y ≥, ∀x,y∈C.

Ais said to beα-strongly monotone iﬀ there exists a constantα>  such that

Ax–Ay,x–y ≥αx–y, ∀x,y∈C.

Ais said to beα-inverse-strongly monotone iﬀ there exists a constantα>  such that

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.

(2)

It is not hard to see thatα-inverse-strongly monotone mappings are Lipschitz continuous. Indeed, we have

αAx–Ay≤ Ax–Ay,x–y ≤ Ax–Ayx–y.

This shows thatAx–Ay ≤ _αx–y.

Recall that the classical variational inequality, denoted byVI(C,A), is to ﬁndu∈Csuch that

Au,v–u ≥, ∀v∈C. (.)

One can see that the variational inequality (.) is equivalent to a fixed point problem of the mappingProj_C(I–rA), whereIis the identity andris some positive real number. The elementu∈Cis a solution of the variational inequality (.) iffu∈Csatisfies the equation u=PC(u–rAu). This alternative equivalent formulation has played a significant role in the studies of variational inequalities and related optimization problems.

A multivalued operatorB:H→H_{with the domain}_{D(B) =}_{_x_∈_H_:_Bx₌_∅}_{and the} rangeR(B) ={Bx:x∈D(B)}is said to be monotone if forx∈D(B),x∈D(B),y∈Bxand

y∈Bx, we havex–x,y–y ≥. A monotone operatorBis said to be maximal if its

graphG(B) ={(x,y) :y∈Bx}is not properly contained in the graph of any other monotone operator. LetIdenote the identity operator onHandB:H→Hbe a maximal monotone operator. Then we can deﬁne, for eachr> , a nonexpansive single-valued mappingJr: H→HbyJr= (I+rB)–. It is called the resolvent ofB. We know thatB– =F(Jr) for all r>  andJris ﬁrmly nonexpansive.

LetT:C→Cbe a mapping. In this paper, we useF(T) to denote the ﬁxed point set ofT; that is,F(T) :={x∈C:x=Tx}. Recall thatT is said to be nonexpansive iﬀ

Tx–Ty ≤ x–y, ∀x,y∈C.

T is said to be asymptotically nonexpansive iﬀ there exists a sequence{kn} ⊂[,∞) such that

_Tn_x_–_Tn_y_≤_kn_x_–_y_, _∀_x,_y_∈_C,_n_≥_.

Tis said to be aκ-strict pseudocontraction iﬀ there exists a constantκ∈[, ) such that

Tx–Ty≤ x–y+κ(x–Tx) – (y–Ty), ∀x,y∈C.

Note that the class ofκ-strict pseudocontractions strictly includes the class of nonexpan-sive mappings as a special case. That is,T is nonexpansive iff the coefficientκ= .T is said to be an asymptoticallyκ-strict pseudocontraction iff there exist a constantκ∈[, ) and a sequence{kn}in [,∞) such that

Tn_x_–_Tn_y_≤_k

nx–y+κx–Tnx

–y–Tn_y_, _∀_x,_y_∈_C,_n_≥_.

(3)

In [], Kamimura and Takahashi investigated the problem of ﬁnding zero points of a maximal monotone operator based on the following iterative algorithm:

x∈H, xn+=αnx+ ( –αn)Jrnxn, ∀n≥, (.)

where{αn}is a sequence in (, ),{rn}is a positive real number sequence,B:H→His maximal monotone andJrn= (I+rnB)–. It is proved that the sequence{xn}generated in

(.) converges strongly to somez∈B–_{() provided that the control sequence satisﬁes}

some restrictions. Further, using this result, they also investigated the case that B=∂f, wheref :H→(–∞,∞] is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces; for more details, see []. Recently, Takahashiet al.studied zero point problems of the sum of two monotone map-pings and ﬁxed point problems of a nonexpansive mapping based on the following iterative algorithm:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈C,

yn=αnx+ ( –αn)Jrn(xn–rnAxn),

xn+=βnxn+ ( –βn)Tyn, ∀n≥,

(.)

where {αn} and {βn} are real number sequences in (, ), {rn} is a positive sequence, T :C→Cis a nonexpansive mapping andA:C→H is an inverse-strongly monotone mapping. It is proved that the sequence{xn}generated in (.) converges strongly to some z∈(A+B)–_()_∩_{F(S) provided that the control sequence satisﬁes some restrictions; for}

more details, see [].

Motivated by the above results, we investigate ﬁxed point problems of asymptotically strict pseudocontractions and zero point problems of the sum of two monotone mappings.

In order to state our main results, we need the following tools.

Recall that a space is said to satisfy Opial’s condition [] if, for any sequence{xn} ⊂H withxnx, wheredenotes the weak convergence, the inequality

lim inf

n→∞ xn–x<lim infn→∞ xn–y

holds for everyy∈Hwithy=x. Indeed, the above inequality is equivalent to the following:

lim sup

n→∞ xn–x<lim supn→∞ xn–y.

Lemma .[] Let{xn}and{yn}be bounded sequences in a Banach space X,and letβn be a sequence in [, ]with <lim infn→∞βn≤lim supn→∞βn< .Suppose that xn+=

( –βn)yn+βnxnfor all integers n≥and

lim sup

n→∞

yn+–yn–xn+–xn

≤.

Thenlim_n_→∞yn–xn= .

(4)

Lemma .[] Assume that{αn}is a sequence of nonnegative real numbers such that

αn+≤( –γn)αn+δn,

where{γn}is a sequence in(, )and{δn}is a sequence such that (i) ∞_n_=γn=∞;

(ii) lim sup_n_→∞δn/γn≤or ∞n=|δn|<∞. Thenlimn→∞αn= .

Lemma .[] Let C be a nonempty,closed and convex subset of H.Let T:C→C be an asymptotically strict pseudocontraction.Then T is Lipschitz continuous and I–T is demiclosed at zero.

2 Main results

Theorem . Let C be a nonempty closed convex subset of H.Let T:C→C be an asymp-totically κ-strict pseudocontraction. Let A:C→H be an α-inverse-strongly monotone mapping, and let B be a maximal monotone operator on H.Assume that F(T)∩(A+ B)–()=∅.Let{αn},{βn}and{γn}be real number sequences in(, ).Let Jrn= (I+rnB)–,

where{rn}is a positive real number sequence.Let{xn}be a sequence in C generated by: x∈C is chosen arbitrarily and

⎧ ⎨ ⎩

zn=Proj_C(αnu+ ( –αn)Jrn(xn–rnAxn)),

xn+=βnxn+ ( –βn)(γnzn+ ( –γn)Tnzn), ∀n≥.

Assume that the sequences{αn},{βn},{γn}and{rn}satisfy the following restrictions: (a)  <a≤rn≤b< α,limn→∞|rn+–rn|= ;

(b) limn→∞αn= , ∞n=αn=∞; (c)  <c≤βn≤d< ;

(d) κ≤γn≤e< ,limn→∞|γn+–γn|= ,

where a,b,c,d and e are some real numbers.If T is asymptotically regular,then the se-quence{xn}converges strongly to some pointx,¯ wherex¯=PF(T)∩(A+B)–_()u.

Proof First, we show that the mappingI–rnAis nonexpansive. Indeed, we have

(I–rnA)x– (I–rnA)y



=x–y– rnx–y,Ax–Ay+rnAx–Ay

≤ x–y–rn(α–rn)Ax–Ay.

It follows from Restriction (a) thatI–rnAis nonexpansive. Putyn=γnzn+ ( –γn)Tnzn and ﬁxp∈F(T)∩(A+B)–_{(). It follows from Lemma . that}

zn–p=Proj_Cαnu+ ( –αn)Jrn(xn–rnAxn)

–p

≤αnu–p+ ( –αn)Jrn(xn–rnAxn) –p

(5)

In view of Restriction (d), we ﬁnd that

yn–p≤γn(zn–p) + ( –γn)

Tnzn–Tnp



=γnzn–p+ ( –γn)Tnzn–Tnp



–γn( –γn)(zn–p) –Tnzn–Tnp

≤γnzn–p+ ( –γn)

zn–p+κ(zn–p) –Tnzn–Tnp

–γn( –γn)(zn–p) –Tnzn–Tnp

=zn–p– ( –γn)(γn–κ)(zn–p) –

Tnzn–Tnp

≤ zn–p. (.)

Substituting (.) into (.), we obtain that

xn+–p ≤βnxn–p+ ( –βn)yn–p

≤βnxn–p+ ( –βn)

αnu–p+ ( –αn)xn–p

≤ –αn( –βn)

xn–p+αn( –βn)u–p.

PuttingM=max{x–p,u–p}, we ﬁnd thatxn–p ≤Mfor alln≥. Indeed, it is

clear thatx–p ≤M. Suppose thatxm–p ≤Mfor some positive integerm. It follows

that

xm+–p ≤

 –αm( –βm)xm–p+αm( –βm)u–p

≤ –αm( –βm)

M+αm( –βn)M

=M.

This ﬁnds that{xn}is bounded. Puttingρn=Jrn(xn–rnAxn), we ﬁnd that

ρn+–ρn ≤Jrn+(xn+–rn+Axn+) –Jrn+(xn–rnAxn)

+Jrn+(xn–rnAxn) –Jrn(xn–rnAxn)

≤(xn+–rn+Axn+) – (xn–rnAxn)

+Jrn+(xn–rnAxn) –Jrn(xn–rnAxn)

≤ xn+–xn+|rn+–rn|Axn

+Jrn+(xn–rnAxn) –Jrn(xn–rnAxn). (.)

On the other hand, we have

zn+–zn ≤αn+u+ ( –αn+)ρn+

–αnu+ ( –αn)ρn

(6)

Substituting (.) into (.), we ﬁnd that

zn+–zn ≤αn+u+ ( –αn+)ρn+

–αnu+ ( –αn)ρn

≤( –αn+)xn+–xn+|rn+–rn|Axn+|αn+–αn|ρn–u

+ ( –αn+)Jrn+(xn–rnAxn) –Jrn(xn–rnAxn). (.)

Putξn=xn–rnAxn. SinceBis monotone, we ﬁnd that

Jrn+ξn–Jrnξn,

ξn–Jrn+ξn rn+

–ξn–Jrnξn rn

≥.

It follows thatJrn+ξn–Jrnξn, ( –

rn+

rn )(ξn–Jrnξn) ≥ Jrn+ξn–Jrnξn

_{. This yields that}_|_rn +–

rn|ξn–Jrnξn ≥rnJrn+ξn–Jrnξn. This combines with (.) to yield that

zn+–zn ≤( –αn+)xn+–xn+|rn+–rn|Axn

+|αn+–αn|ρn–u+|

rn+–rn|

rn ξn–Jrnξn. (.)

yn+–yn ≤γn+zn+–zn+|γn+–γn|zn–Tnzn

+ ( –γn+)Tn+zn+–Tnzn. (.)

Substituting (.) into (.), we ﬁnd that

≤ |rn+–rn|Axn+|αn+–αn|ρn–u

+|rn+–rn| rn

ξn–Jrnξn+|γn+–γn|zn–T

n_zn

+ ( –γn+)Tn+zn+–Tnzn.

It follows from Restrictions (a), (c) and (d) that

lim sup

n→∞

≤.

It follows from Lemma . thatlim_n_→∞yn–xn= . Sincexn+–xn= ( –βn)(yn–xn), we ﬁnd thatlimn→∞xn+–xn= . Notice that

Jrn(I–rnA)xn–Jrn(I–rnA)p 

=(xn–p) –rn(Axn–Ap)

=xn–p– rnxn–p,Axn–Ap+rnAxn–Ap

(7)

Since the norm is convex, we see from (.) and (.) that

xn+–p≤βnxn–p+ ( –βn)yn–p

≤βnxn–p+ ( –βn)zn–p

≤βnxn–p+ ( –βn)αn(u–p) + ( –αn)

Jrn(xn–rnAxn) –p 

≤βnxn–p+αn( –βn)u–p

+ ( –αn)( –βn)Jrn(xn–rnAxn) –p 

≤ xn–p+αnu–p–rn(α–rn)( –αn)( –βn)Axn–Ap. (.)

This yields that

rn(α–rn)( –αn)( –βn)Axn–Ap

≤ xn–p+αnu–p–xn+–p ≤xn–p+xn+–p

xn+–xn+αnu–p.

In view of Restrictions (a), (b) and (c), we obtain that

lim

n→∞Axn–Ap= . (.)

Notice that

ρn–p=Jrn(xn–rnAxn) –Jrn(p–rnAp) 

≤(xn–rnAxn) – (p–rnAp),ρn–p

= 

(xn–rnAxn) – (p–rnAp)



+ρn–p

–(xn–rnAxn) – (p–rnAp) – (ρn–p)



≤ 



xn–p+ρn–p–xn–ρn–rn(Axn–Ap)



≤ 



xn–p+ρn–p–xn–ρn–rnAxn–Ap

+ rnxn–ρnAxn–Ap

≤ 



xn–p+ρn–p–xn–ρn+ rnxn–ρnAxn–Ap

.

It follows that

ρn–p≤ xn–p–xn–ρn+ rnxn–ρnAxn–Ap. (.)

This yields that

zn–p≤αnu–p+ ( –αn)ρn–p

(8)

It follows from (.) that

xn+–p≤βnxn–p+ ( –βn)yn–p

≤βnxn–p+ ( –βn)zn–p

≤ xn–p+αnu–p– ( –αn)( –βn)xn–ρn

+ rn( –βn)xn–ρnAxn–Ap.

We therefore obtain that

( –αn)( –βn)xn–ρn

≤ xn–p+αnu–p–xn+–p

+ rn( –βn)xn–ρnAxn–Ap

≤xn–p+xn+–p

xn–xn++αnu–p

+ rn( –βn)xn–ρnAxn–Ap.

In view of Restrictions (a), (b) and (c), we ﬁnd from (.) that

lim

n→∞xn–ρn= . (.)

Next, we show thatlim sup_n_→∞u–x,¯ ρn–x¯ ≤, wherex¯=PF(T)∩(A+B)–_()u. To show it, we can choose a subsequence{ρni}of{ρn}such that

lim sup

n→∞

u–x,¯ ρn–x¯=lim

i→∞u–x,¯ ρni–x¯.

Sinceρniis bounded, we can choose a subsequence{ρn_ij}of{ρni}which converges weakly

to some pointx. We may assume, without loss of generality, thatρniconverges weakly tox.

Sinceρn=Jrn(xn–rnAxn), we ﬁnd that

xn–ρn

rn –Axn∈Bρn. SinceBis monotone, we get, for

any (μ,ν)∈B, thatρn–μ,xn–_r_nρn–Axn–ν ≥. Replacingnbyniand lettingi→ ∞, we obtain from (.) thatx–μ, –Ax–ν ≥. This means –Ax∈Bx, that is, ∈(A+B)(x). Hence we getx∈(A+B)–_{(). Next, we show that}_x_∈_F(T_{). Notice that}

zn–xn ≤Proj_Cαnu+ ( –αn)Jrn(xn–rnAxn)

–xn

≤αnu–xn+ ( –αn)Jrn(xn–rnAxn) –xn.

In view of Restriction (a), we ﬁnd from (.) thatlim_n_→∞zn–xn= . Note that

γnxn+ ( –γn)Tnxn

–xn

≤γnxn+ ( –γn)Tnxn

–γnzn+ ( –γn)Tnzn

+γnzn+ ( –γn)Tnzn–xn

≤γnxn–zn+ ( –γn)Tnxn–Tnzn+γnzn+ ( –γn)Tnzn

–xn

(9)

It follows from (.) that

lim

n→∞γnxn+ ( –γn)T

n_x n

–xn= . (.)

Note that

Tnxn–xn≤Tnxn–γnxn+ ( –γn)Tnxn

+γnxn+ ( –γn)Tnxn–xn

≤γnTnxn–xn+γnxn+ ( –γn)Tnxn

–xn.

It follows that ( –γn)Tnxn–xn ≤ (γnxn+ ( –γn)Tnxn) –xn. This implies from Restric-tion (d) and (.) thatlimn→∞Tnxn–xn= . SinceTis uniformlyL-Lipschitz continu-ous, we can obtain thatlimn→∞Txn–xn= . In view of Lemma ., we ﬁnd thatx∈F(T). This implies that

lim sup

n→∞

u–x,¯ ρn–x¯=u–x,¯ x–x¯ ≤.

xn+–x¯≤βnxn–x¯+ ( –βn)yn–x¯

≤βnxn–x+ ( –βn)zn–x¯

≤βnxn–x+ ( –βn)αn(un–x) + ( –¯ αn)(ρn–x)¯



≤βnxn–x+ ( –αn)( –βn)ρn–x

+ αn( –βn)u–x,¯ ρn–x

≤ –αn( –βn)xn–x¯+ αn( –βn)u–x,¯ ρn–x¯.

From Lemma ., we ﬁnd thatlimn→∞xn–x¯= . This completes the proof. IfT is asymptotically nonexpansive, then we ﬁnd the following result.

Corollary . Let C be a nonempty closed convex subset of H.Let T:C→C be an asymp-totically nonexpansive mapping.Let A:C→H be anα-inverse-strongly monotone map-ping,and let B be a maximal monotone operator on H.Assume that F(T)∩(A+B)–()=∅. Let{αn}and{βn}be real number sequences in(, ).Let Jrn= (I+rnB)–,where{rn}is a

positive real number sequence.Let{xn}be a sequence in C generated by:x∈C is chosen

arbitrarily and

⎧ ⎨ ⎩

zn=PC(αnu+ ( –αn)Jrn(xn–rnAxn)),

xn+=βnxn+ ( –βn)Tnzn, ∀n≥.

Assume that the sequences{αn},{γn}and{rn}satisfy the following restrictions: (a)  <a≤rn≤b< α,limn→∞|rn+–rn|= ;

(10)

where a,b,c and d are some real numbers.If T is asymptotically regular,then the sequence

{xn}converges strongly to some pointx,¯ wherex¯=PF(T)∩(A+B)–_()u.

3 Applications

In this section, we shall consider equilibrium problems and variational inequalities. LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers. Recall the following equilibrium problem:

Findx∈Csuch thatF(x,y)≥, ∀y∈C.

In this work, we useEP(F) to denote the solution set of the equilibrium problem. To study the equilibrium problems, we may assume thatFsatisﬁes the following condi-tions:

(A) F(x,x) = for allx∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,

lim sup

t↓

Ftz+ ( –t)x,y≤F(x,y);

(A) for eachx∈C,y→F(x,y)is convex and weakly lower semi-continuous.

PuttingF(x,y) =Ax,y–xfor everyx,y∈C, we see that the equilibrium problem is reduced to the variational inequality (.).

The following lemma can be found in [].

Lemma . Let C be a nonempty closed convex subset of H,and let F:C×C→Rbe a bifunction satisfying(A)-(A).Then,for any r> and x∈H,there exists z∈C such that

F(z,y) +

ry–z,z–x ≥, ∀y∈C.

Further,deﬁne

Trx=

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

(.)

for all r> and x∈H.Then the following hold: (a) Tris single-valued;

(b) Tris ﬁrmly nonexpansive,i.e.,for anyx,y∈H,

Trx–Try≤ Trx–Try,x–y;

(c) F(Tr) =EP(F);

(d) EP(F)is closed and convex.

(11)

mapping of H into itself deﬁned by

AFx=

⎧ ⎨ ⎩

{z∈H:F(x,y)≥ y–x,z,∀y∈C}, x∈C,

∅, x∈/C. (.)

Then AFis a maximal monotone operator with the domain D(AF)⊂C,EP(F) =A–

F ()and

Trx= (I+rAF)–x, ∀x∈H,r> ,

where Tris deﬁned as in(.).

The following result is not derived based on Theorem . and Lemma ..

Theorem . Let C be a nonempty closed convex subset of H.Let T:C→C be an asymp-toticallyκ-strict pseudocontraction.Let F be a bifunction from C×C toRwhich satisﬁes (A)-(A).Assume that F(T)∩EP(F)=∅.Let{αn},{βn}and{γn}be real number sequences in(, ).Let{xn}be a sequence in C generated by:x∈C is chosen arbitrarily and

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

wn∈C such that FB(wn,u) +_r_nu–wn,wn–xn ≥, ∀u∈C,

zn=PC(αnu+ ( –αn)wn),

yn=γnzn+ ( –γn)Tn_zn,

xn+=βnxn+ ( –βn)yn, ∀n≥.

(d) κ≤γn≤e< ,limn→∞|γn+–γn|= ,

where a,b,c,d and e are some real numbers.If T is asymptotically regular,then the se-quence{xn}converges strongly to some pointx,¯ wherex¯=PF(T)∩EP(F)u.

IfT is asymptotically nonexpansive, then Theorem . is reduced to the following.

Corollary . Let C be a nonempty closed convex subset of H.Let T:C→C be an asymp-totically nonexpansive mapping.Let F be a bifunction from C×C toRwhich satisﬁes (A)-(A).Assume that F(T)∩EP(F)=∅.Let{αn}and{βn}be real number sequences in(, ). Let{xn}be a sequence in C generated by:x∈C is chosen arbitrarily and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

wn∈C such that FB(wn,u) +_r_nu–wn,wn–xn ≥, ∀u∈C,

zn=PC(αnu+ ( –αn)wn),

xn+=βnxn+ ( –βn)Tnzn, ∀n≥.

(12)

where a,b,c and d are some real numbers.If T is asymptotically regular,then the sequence

{xn}converges strongly to some pointx,¯ wherex¯=PF(T)∩EP(F)u.

LetHbe a Hilbert space andf :H→(–∞, +∞] be a proper convex lower semicontin-uous function. Then the subdiﬀerential∂f off is deﬁned as follows:

∂f(x) =y∈H:f(z)≥f(x) +z–x,y,z∈H, ∀x∈H.

From Rockafellar [], we ﬁnd that∂f is maximal monotone. It is easy to verify that ∈

∂f(x) if and only iff(x) =miny∈Hf(y). LetICbe the indicator function ofC,i.e.,

IC(x) =

⎧ ⎨ ⎩

, x∈C,

+∞, x∈/C.

SinceICis a proper lower semicontinuous convex function onH, we see that the subdif-ferential∂ICofICis a maximal monotone operator.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H,Proj_C

the metric projection from H onto C,∂ICthe subdiﬀerential of IC,where ICis deﬁned above and Jλ= (I+λ∂IC)–.Then

y=Jλx ⇐⇒ y=ProjCx, x∈H,y∈C. Now, we consider a variation inequality problem.

Theorem . Let C be a nonempty closed convex subset of H.Let T:C→C be an asymp-totically κ-strict pseudocontraction. Let A:C→H be an α-inverse-strongly monotone mapping.Assume that F(T)∩VI(C,A)=∅.Let{αn},{βn} and{γn}be real number se-quences in(, ).Let{xn}be a sequence in C generated by:x∈C is chosen arbitrarily and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

zn=PC(αnu+ ( –αn)PC(xn–rnAxn)),

yn=γnzn+ ( –γn)Tnzn,

xn+=βnxn+ ( –βn)yn, ∀n≥.

(d) κ≤γn≤e< ,limn→∞|γn+–γn|= ,

where a,b,c,d and e are some real numbers.If T is asymptotically regular,then the se-quence{xn}converges strongly to some pointx,¯ wherex¯=PF(T)∩VI(C,A)u.

Proof PutBx=∂IC. Next, we show thatVI(C,A) = (A+∂IC)–_{(). Notice that}

x∈(A+∂IC)–() ⇐⇒ ∈Ax+∂ICx

(13)

⇐⇒ Ax,y–x ≥

⇐⇒ x∈VI(C,A).

From Lemma ., we can conclude the desired conclusion immediately.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author is grateful to the referees for useful suggestions which improved the contents of the article.

Received: 16 December 2013 Accepted: 13 February 2014 Published:27 Feb 2014

References

1. Cho, SY, Qin, X, Kang, SM: Iterative processes for common ﬁxed points of two diﬀerent families of mappings with applications. J. Glob. Optim.57, 1429-1446 (2013)

2. Takahashi, S, Takahashi, W, Toyoda, M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl.147, 27-41 (2010)

3. Park, S: A review of the KKM theory onφA-space or GFC-spaces. Adv. Fixed Point Theory3, 355-382 (2013)

4. Zegeye, H, Shahzad, N: Strong convergence theorem for a common point of solution of variational inequality and ﬁxed point problem. Adv. Fixed Point Theory2, 374-397 (2012)

5. Kim, JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and ﬁxed point problems of the asymptotically quasi-φ-nonexpansive mappings. Fixed Point Theory Appl.2011, Article ID 10 (2011) 6. Zhang, M: Iterative algorithms for common elements in ﬁxed point sets and zero point sets with applications. Fixed

Point Theory Appl.2012, Article ID 21 (2012)

7. Zhang, M: Strong convergence of a viscosity iterative algorithm in Hilbert spaces. J. Nonlinear Funct. Anal.2014, Article ID 1 (2014)

8. Qin, X, Cho, YJ, Kang, SM: Convergence theorems of common elements for equilibrium problems and ﬁxed point problems in Banach spaces. J. Comput. Appl. Math.225, 20-30 (2009)

9. Wang, Z, Lou, W: A new iterative algorithm of common solutions to quasi-variational inclusion and ﬁxed point problems. J. Math. Comput. Sci.3, 57-72 (2013)

10. Cho, SY, Li, W, Kang, SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl.

2013, Article ID 199 (2013)

11. Wu, C, Liu, A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl.2012, Article ID 90 (2012)

12. Noor, MA, Noor, KI, Waseem, M: Decomposition method for solving system of linear equations. Eng. Math. Lett.2, 34-41 (2012)

13. Cho, SY, Kang, SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci.32, 1607-1618 (2012)

14. He, RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory2, 47-57 (2012)

15. Qin, X, Su, Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Anal.

67, 1958-1965 (2007)

16. Wu, C: Mann iteration for zero theorems of accretive operators. J. Fixed Point Theory2013, Article ID 3 (2013) 17. Hecai, Y: On weak convergence of an iterative algorithm for common solutions of inclusion problems and ﬁxed point

problems in Hilbert spaces. Fixed Point Theory Appl.2013, Article ID 155 (2013)

18. Cho, SY, Kang, SM: Approximation of ﬁxed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett.24, 224-228 (2011)

19. Lv, S: Generalized systems of variational inclusions involving (A,η)-monotone mappings. Adv. Fixed Point Theory1, 1-14 (2011)

20. Qin, X, Cho, SY, Kang, SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim.48, 423-445 (2010)

21. Hecai, Y: On solutions of inclusion problems and ﬁxed point problems. Fixed Point Theory Appl.2013, Article ID 11 (2013)

22. Cheng, P, Wu, H: Strong convergence theorems for common solutions of a family of nonexpansive mappings and an accretive operator. Fixed Point Theory Appl.2013, Article ID 172 (2013)

23. Qin, X, Cho, SY, Kang, SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math.233, 231-240 (2009)

24. Kamimura, S, Takahashi, W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory106, 226-240 (2000)

25. Yang, S: Zero theorems of accretive operators in reﬂexive Banach spaces. J. Nonlinear Funct. Anal.2013, Article ID 2 (2013)

26. Cho, SY, Qin, X, Kang, SM: Hybrid projection algorithms for treating common ﬁxed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett.25, 854-857 (2012)

(14)

28. Cho, SY, Kang, SM: Zero point theorems form-accretive operators in a Banach space. Fixed Point Theory13, 49-58 (2012)

29. Wu, C: Convergence of algorithms for an inﬁnite family nonexpansive mappings and relaxed cocoercive mappings in Hilbert spaces. Adv. Fixed Point Theory4, 125-139 (2014)

30. Kangtunyakarn, A: Iterative algorithms for ﬁnding a common solution of system of the set of variational inclusion problems and the set of ﬁxed point problems. Fixed Point Theory Appl.2011, Article ID 38 (2011)

31. Yang, S: Zero theorems of accretive operators in reﬂexive Banach spaces. J. Nonlinear Funct. Anal.2013, Article ID 2 (2013)

32. Kim, JK, Chandok, S: Coupled common ﬁxed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces. Fixed Point Theory Appl.2013, Article ID 307 (2013) 33. Qin, X, Agarwal, RP: Shrinking projection methods for a pair of asymptotically quasi-φ-nonexpansive mappings.

Numer. Funct. Anal. Optim.31, 1072-1089 (2010)

34. Luo, H, Wang, Y: Iterative approximation for the common solutions of a inﬁnite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci.2, 1660-1670 (2012)

35. Cho, SY: Approximation of solutions of a generalized variational inequality problem based on iterative methods. Commun. Korean Math. Soc.25, 207-214 (2010)

36. Opial, Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. Am. Math. Soc.73, 591-597 (1967)

37. Cho, SY: Strong convergence of an iterative algorithm for sums of two monotone operators. J. Fixed Point Theory

2013, Article ID 6 (2013)

38. Liu, L: Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involvingm-accretive operators in Banach spaces. Nonlinear Anal.34, 307-317 (1998)

39. Sahu, DR, Xu, HK, Yao, JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal.70, 3502-3511 (2009)

40. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63, 123-145 (1994)

41. Rockafellar, RT: Characterization of the subdiﬀerentials of convex functions. Pac. J. Math.17, 497-510 (1966)

10.1186/1687-1812-2014-52