Hybrid methods for common solutions in Hilbert spaces with applications

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R E S E A R C H

Open Access

Hybrid methods for common solutions in

Hilbert spaces with applications

Lijuan Sun

*

*_{Correspondence: [email protected]}

Kaifeng Vocational College of Culture and Arts, Kaifeng, Henan, China

Abstract

In this paper, hybrid methods are investigated for treating common solutions of nonlinear problems. A strong convergence theorem is established in the framework of real Hilbert spaces.

Keywords: equilibrium problem; ﬁxed point; projection; variational inequality; zero point

1 Introduction and preliminaries

Common solutions to variational inclusion, equilibrium and ﬁxed point problems have been recently extensively investigated based on iterative methods; see [–] and the ref-erences therein. The motivation for this subject is mainly to its possible applications to mathematical modeling of concrete complex problems, which use more than one con-straint. The aim of this paper is to investigate a common solution of variational inclusion, equilibrium and ﬁxed point problems. The organization of this paper is as follows. In Sec-tion , we provide some necessary preliminaries. In SecSec-tion , a hybrid method is intro-duced and analyzed. Strong convergence theorems are established in the framework of Hilbert spaces. In Section , applications of the main results are discussed.

In what follows, we always assume thatHis a real Hilbert space with the inner product

·,·and the norm · . LetCbe a nonempty, closed, and convex subset ofHand letPC be the metric projection fromHontoC. LetS:C→Cbe a mapping.F(S) stands for the ﬁxed point set ofS; that is,F(S) :={x∈C:x=Sx}.

Recall thatSis said to becontractiveiﬀ there exists a constantα∈[, ) such that

Sx–Sy ≤αx–y, ∀x,y∈C.

Ifα= , thenSis said to benonexpansive. LetA:C→Hbe a mapping. IfCis nonempty closed and convex, then the ﬁxed point set ofSis nonempty.

Recall thatAis said to bemonotoneiﬀ

Ax–Ay,x–y ≥, ∀x,y∈C.

Recall thatAis said to bestrongly monotoneiﬀ there exists a constantα>  such that

Ax–Ay,x–y ≥αx–y, ∀x,y∈C.

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For such a case,Ais also said to beα-strongly monotone. Recall thatAis said to be inverse-strongly monotoneiﬀ there exists a constantα>  such that

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.

For such a case,Ais also said to beα-inverse-strongly monotone.

Recall that a set-valued mappingM:H⇒His said to bemonotoneiff, for allx,y∈H, f ∈Mx, andg∈Myimplyx–y,f –g> .Mismaximaliff the graphGraph(M) ofRis not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingMis maximal if and only if, for any (x,f)∈H×H,x–y,f–g ≥, for all (y,g)∈Graph(M) impliesf ∈Rx. For a maximal monotone operatorMonH, and r> , we may define the single-valued resolventJr:H→D(M), whereD(M) denote the domain ofM. It is well known thatJris firmly nonexpansive, andM–() =F(Jr), where F(Jr) :={x∈D(M) :x=Jrx}, andM–_{() :=}_{_x_∈_H_{: }_∈_Mx_}_.

LetA:C→Hbe a inverse-strongly monotone mapping, and letFbe a bifunction ofC× CintoR, whereRdenotes the set of real numbers. We consider the following generalized equilibrium problem.

Findx∈Csuch thatF(x,y) +Ax,y–x ≥, ∀y∈C. (.)

In this paper, the set of such anx∈Cis denoted byEP(F,A).

To study the equilibrium problems (.), we may assume thatFsatisﬁes the following conditions:

(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 semicontinuous.

In order to prove our main results, we also need the following lemmas.

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

() ∞_n_=γn=∞;

() lim sup_n_→∞δn/γn≤or _∞

n=|δn|<∞. Thenlimn→∞αn= .

Lemma .[] 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) +

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Deﬁne a mapping Tr:H→C as follows:

Trx=

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

, x∈H,

then the following conclusions hold:

() Tris single-valued;

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

Trx–Try≤ Trx–Try,x–y;

() F(Tr) =EP(F);

() EP(F)is closed and convex.

Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings and{γi}be a nonneg-ative real sequence with ≤γi< ,∀i≥. Forn≥ deﬁne a mappingWn:C→Cas follows:

Un,n+=I,

Un,n=γnSnUn,n++ ( –γn)I,

Un,n–=γn–Sn–Un,n+ ( –γn–)I,

.. .

Un,k=γkSkUn,k++ ( –γk)I,

Un,k–=γk–Sk–Un,k+ ( –γk–)I,

.. .

Un,=γSUn,+ ( –γ)I,

Wn=Un,=γSUn,+ ( –γ)I.

(.)

Such a mappingWnis nonexpansive fromCtoCand it is called aW-mapping generated bySn,Sn–, . . . ,Sandγn,γn–, . . . ,γ.

Lemma .[] Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings with a nonempty common ﬁxed point set and let{γi}be a real sequence such that <γi≤l< , where l is some real number,∀i≥.Then

() Wnis nonexpansive andF(Wn) = ∞

i=F(Si),for eachn≥;

() for eachx∈Cand for each positive integerk,the limitlimn→∞Un,kexists;

() the mappingW:C→Cdeﬁned by

Wx:= lim

n→∞Wnx=nlim→∞Un,x, x∈C, (.)

is a nonexpansive mapping satisfyingF(W) =∞_i_=F(Si)and it is called the

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Lemma .[] Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings with a nonempty common ﬁxed point set and let{γi}be a real sequence such that <γi≤l< ,

∀i≥.If K is any bounded subset of C,then

lim

n→∞sup_x_∈_KWx–Wnx= .

Throughout this paper, we always assume that  <γi≤l< ,∀i≥.

Lemma .[] Let B:C→H be a mapping and let M:H⇒H be a maximal monotone operator.Then F(Jr(I–sB)) = (B+M)–_().

Lemma .[] Let{xn}and{yn}be bounded sequences in H and let{βn}be a sequence in(, )with <lim inf_n_→∞βn≤lim supn→∞βn< .Suppose that xn+= ( –βn)yn+βnxn for all n≥and

lim sup

n→∞

yn+–yn–xn+–xn

≤.

Thenlimn→∞yn–xn= .

Lemma .[] Let A:C→H a Lipschitz monotone mapping and let NCx be the normal cone to C at x∈C;that is,NCx={y∈H:x–u,y,∀u∈C}.Deﬁne

Dx= ⎧ ⎨ ⎩

Ax+NCx, x∈C,

∅ x∈/C.

Then D is maximal monotone and∈Dx if and only if x∈VI(C,A).

2 Main results

Theorem . Let C be a nonempty closed convex subset of a Hilbert space H and F a bifunction from C×C toRwhich satisﬁes(A)-(A).Let A:C→H be aδ

-inverse-strongly monotone mapping,A:C→H be aδ-inverse-strongly monotone mapping,A:

C→H be aδ-inverse-strongly monotone mapping,M :H⇒H a maximal monotone

operator such thatDom(M)⊂C and M:H⇒H a maximal monotone operator such that Dom(M)⊂C.Let{Si:C→C}be a family of inﬁnitely nonexpansive mappings.Assume that:=∞_i_=F(Si)∩EP(F,A)∩(A+M)–()∩(A+M)–()= ∅.Let x∈C and{xn} be a sequence generated by

⎧ ⎨ ⎩

zn=Jsn(un–snAun),

xn+=αnu+βnxn+γnWnJrn(zn–rnAzn), ∀n≥,

where u is a ﬁxed element in C,unis such that

F(un,y) +Axn,y–un+

 λn

y–un,un–xn ≥, ∀y∈C,

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(a)  <a≤λn≤b< δ, <a≤rn≤b< δ, <a¯≤sn≤ ¯b< δ;

(b) limn→∞αn= and ∞

n=αn=∞;

(c)  <lim infn→∞βn≤lim supn→∞βn< ;

Then the sequence{xn}converges strongly tox¯∈,wherex¯=Pu.

Proof First, we show that the mappingI–rnA,I–snA, andI–λnAare nonexpansive.

Indeed, we ﬁnd from the restriction (a) that (I–rnA)x– (I–rnA)y



=x–y– rnx–y,Ax–Ay+rnAx–Ay

≤ x–y– rnδAx–Ay+rnAx–Ay

=x–y+rn(rn– δ)Ax–Ay

≤ x–y, ∀x,y∈C,

which implies that the mappingI–rnAis nonexpansive. In the same way, we ﬁndI–snA

andI–λnAare also nonexpansive. Putyn=Jrn(zn–rnAzn). Fixingx∗∈, we ﬁnd

yn–x∗=Jrn(zn–rnAzn) –Jrn

x∗–rnAx∗ ≤zn–x∗

=Jsn(un–snAun) –Jsn

x∗–snAx∗ ≤Tλn(I–λnA)xn–Tλn(I–λnA)x∗ ≤xn–x∗.

It follows that

xn+–x∗=αnu+βnxn+γnWnyn–x∗

≤αnu–x∗+βnxn–x∗+γnWnyn–x∗

≤αnu–x∗+ ( –αn)xn–x∗.

This implies that{xn}is bounded, and so are{yn},{zn}, and{un}. Without loss of generality, we can assume that there exists a bounded setK⊂Csuch thatxn,yn,zn,un∈K. Notice that

F(un+,y) +

 λn+

y–un+,un+– (I–rn+A)xn+

≥, ∀y∈C, (.)

and

F(un,y) +  λn

y–un,un– (I–rnA)xn

≥, ∀y∈C. (.)

Lety=unin (.) andy=un+in (.). By adding these two inequalities, we obtain

un+–un,

un– (I–λnA)xn

λn

–un+– (I–λn+A)xn+ λn+

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It follows that

un+–un≤

un+–un, (I–λn+A)xn+– (I–λnA)xn

+

 – λn λn+

un+– (I–λn+A)xn+

≤ un+–un

(I–λn+A)xn+– (I–λnA)xn

+ – λn λn+

un+– (I–λn+A)xn+)

.

It follows that

un+–un ≤(I–λn+A)xn+– (I–λnA)xn

+|λn+–λn| λn+

=(I–λn+A)xn+– (I–λn+A)xn+ (I–λn+A)xn– (I–λnA)xn

+|λn+–λn| λn+

≤ xn+–xn+|λn+–λn|M, (.)

whereMis an appropriate constant such that

M=sup

n≥

Axn+

un+– (I–λn+A)xn+ a

.

SinceJsnis ﬁrmly nonexpansive, we ﬁnd that

zn+–zn

≤un+–sn+Aun+– (un–snAun)

=(I–sn+A)un+– (I–sn+A)un+ (sn–sn+)Aun

≤ un+–un+|sn–sn+|Aun. (.)

Combining (.) with (.) yields

zn+–zn ≤ xn+–xn+|λn+–λn|M+|sn–sn+|Aun. (.)

SinceJrnis also ﬁrmly nonexpansive, we ﬁnd that

yn+–yn ≤ zn+–zn+|rn–rn+|Azn. (.)

Substituting (.) into (.), we see that

yn+–yn ≤ xn+–xn+

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whereMis an appropriate constant such that

M=max

sup

n≥

Azn

,sup

n≥

Aun

,M

.

SinceWnis nonexpansive, we ﬁnd that

Wn+yn+–Wnyn

=Wn+yn+–Wyn++Wyn+–Wyn+Wyn–Wnyn

≤ Wn+yn+–Wyn++Wyn+–Wyn+Wyn–Wnyn ≤sup

x∈K

Wn+x–Wx+Wx–Wnx

+yn+–yn, (.)

whereKis the bounded subset ofCdeﬁned as above. Substituting (.) into (.), we ﬁnd that

Wn+yn+–Wnyn ≤ sup

x∈K

+xn+–xn

+|rn+–rn|+|λn–λn+|+|sn–sn+|

M. (.)

Letting

xn+= ( –βn)vn+βnxn,

we see that

vn+–vn=

αn+u+γn+Wn+yn+

 –βn+

–αnu+γnWnyn  –βn

= αn+  –βn+

u+ –αn+–βn+  –βn+

Wn+yn+

–

αn  –βn

u+ –αn–βn  –βn

Wnyn

= αn+  –βn+

(u–Wn+yn+) –

αn  –βn

(u–Wnyn)

+Wn+yn+–Wnyn.

Hence, we have

vn+–vn ≤ αn+

 –βn+

u–Wn+yn++ αn  –βn

u–Wnyn

+Wn+yn+–Wnyn. (.)

Substituting (.) into (.), we ﬁnd that

vn+–vn–xn+–xn ≤

αn+

 –βn+

u–Wn+yn++

αn  –βn

u–Wnyn

+sup

x∈K

+|rn+–rn|+|λn–λn+|+|sn–sn+|

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It follows from Lemma . that

lim sup

n→∞

vn+–vn–xn+–xn

≤.

In view of Lemma ., we ﬁnd thatlimn→∞vn–xn= . It follows that

lim

n→∞xn+–xn= . (.)

For anyx∗∈, we see that

xn+–x∗ 

≤αnu–x∗



+βnxn–x∗



+γnyn–x∗



. (.)

Since

yn–x∗



≤(I–rnA)zn– (I–rnA)x∗ 

=zn–x∗– rnzn–x∗,Azn–Ax∗

+r_nAzn–Ax∗

≤zn–x∗– rnδAzn–Ax∗+rnAzn–Ax∗

=xn–x∗+rn(rn– δ)Azn–Ax∗,

we ﬁnd that

γnrn(δ–rn)Azn–Ax∗

≤αnu–x∗+xn–x∗+xn+–x∗xn–xn+.

Using the restrictions (a) and (b), we obtain

lim

n→∞Azn–Ax

∗_{= .} _(.)

It follows from (.) that

xn+–x∗ 

≤αnu–x∗



+βnxn–x∗



+γnzn–x∗



.

Since

zn–x∗=Jrn(un–snAun) –x

∗

≤(I–snA)un– (I–snA)x∗ 

=un–x∗– snun–x∗,Aun–Ax∗

+s_nAun–Ax∗ 

≤un–x∗



– snδAun–Ax∗ 

+s_nAun–Ax∗ 

=un–x∗



+sn(sn– δ)Aun–Ax∗ 

,

we have

xn+–x∗ 

≤αnf(xn) –x∗



+xn–x∗+γnsn(sn– δ)Aun–Ax∗ 

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which implies that

γnsn(δ–sn)Aun–Ax∗

≤αnf(xn) –x∗+xn–x∗+xn+–x∗xn–xn+.

lim

n→∞Aun–Ax

∗_{= .} _(.)

Note that

xn+–x∗ 

≤αnu–x∗



+βnxn–x∗



+γnxn–x∗–rn

Axn–Ax∗ 

≤αnu–x∗



+βnxn–x∗



+γnxn–x∗



+λ_nAxn–Ax∗ 

– λn

Axn–Ax∗,xn–x∗

≤αnu–x∗



+βnxn–x∗



+γnxn–x∗



–λnγn(δ–λn)Axn–Ax∗.

This implies that

λnγn(δ–λn)Axn–Ax∗ 

≤αnu–x∗



+xn–x∗+xn+–x∗xn–xn+.

Using the restrictions (a) and (b), we see that

lim

n→∞Axn–Ax

∗_{= .} _(.)

SinceTλnis ﬁrmly nonexpansive, we ﬁnd that

un–x∗≤(I–λnA)xn– (I–λnA)x∗,un–x∗

≤ 

xn–x

∗

+un–x∗



–xn–un–λnAxn–Ax∗ 

+ λn

Axn–Ax∗,xn–un

.

This in turn implies that

γnxn–un≤αnf(xn) –x∗+xn–x∗+xn+–x∗xn–xn+

+ λnAxn–Ax∗xn–un.

Using the restrictions (a) and (b), we see that

lim

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SinceJsnis also ﬁrmly nonexpansive mapping, we see that

zn–x∗≤(I–snA)un– (I–snA)x∗,zn–x∗

≤ 

un–x

∗

+zn–x∗–un–zn–snAun–Ax∗ 

≤ 

un–x

∗

+zn–x∗–un–zn

+ snun–znAun–Ax∗–snAun–Ax∗

,

which implies that

zn–x∗



≤xn–x∗



–un–zn+ snun–znAun–Ax∗.

It follows that

γnun–zn≤αnf(xn) –x∗+xn–x∗+xn+–x∗xn–xn+

+ snun–znAun–Ax∗.

lim

n→∞un–zn=  (.)

and

lim

n→∞yn–zn= . (.)

Note that

( –βn)Wnyn–xn ≤ xn–xn++αnu–Wnyn.

Using the restrictions (b) and (c), we obtain

lim

n→∞Wnyn–xn= . (.)

On the other hand, one has

Wnyn–yn ≤ yn–zn+zn–un+un–xn+xn–Wnyn.

Using (.), (.), (.), and (.), we ﬁnd that

lim

n→∞Wnyn–yn= . (.)

Next, we prove that

lim sup

n→∞

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To see this, we choose a subsequence{xni}of{xn}such that

lim sup

n→∞

u–x,¯ xn–x¯= lim

i→∞u–x,¯ xni–x¯. (.)

Since{xni}is bounded, there exists a subsequence{xn_ij}of{xni}which converges weakly to w. Without loss of generality, we may assume that xni q. Therefore, we see that yniq. We also havezniq.

Next, we show thatq∈∞_i_=F(Si). Suppose the contrary,q∈/ CFPS,i.e., Wq=q. Since yniq, we see from Opial’s condition that

lim inf

i→∞ yni–q<lim infi→∞ yni–Wq

≤lim inf

i→∞

yni–Wyni+Wyni–Wq

≤lim inf

i→∞

yni–Wyni+yni–q

. (.)

On the other hand, we have

Wyn–yn ≤ Wyn–Wnyn+Wnyn–yn

≤sup

x∈K

Wx–Wnx+Wnyn–yn.

In view of Lemma ., we obtain thatlimn→∞Wyn–yn= . This implies from (.)

thatlim infi→∞yni–q<lim infi→∞yni–q. This is a contradiction. Thus, we haveq∈ ∞

i=F(Si).

Now, we are in a position to prove thatq∈(A+M)–(). Notice that zn_r–_nyn –Azn∈

Myn. Letμ∈Mν. SinceMis monotone, we ﬁnd that

zn–yn

rn –Azn–μ,yn–ν

≥.

This implies that–Aq–μ,q–ν ≥. This implies that –Aq∈Mq, that is,q∈(A+

M)–().

Now, we prove thatq∈(A+M)–(). Notice thatun_s–_nzn–Aun∈Mzn. Letμ∈Mν.

SinceMis monotone, we ﬁnd that

un–zn

sn –Aun–μ

_,_z

n–ν

≥.

This implies that–Aq–μ,q–ν ≥. This implies that –Aq∈Mq, that is,q∈(A+

M)–().

Next, we show thatq∈EP(F,A). Sinceun=Tλn(I–λnA)xn, for anyy∈C, we have

 λn

y–un,un–xn ≥.

Replacingnbyni, we ﬁnd from (A) that

Axni,y–uni+

y–uni,

uni–xni λni

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Puttingyt=ty+ ( –t)qfor anyt∈(, ] andy∈C, we see thatyt∈C. It follows that

yt–uni,Ayt

≥ yt–uni,Ayt–Axni,yt–uni–

yt–uni,

uni–xni λni

+F(yt,uni)

=yt–uni,Ayt–Auni+yt–uni,Auni–Axni–

yt–uni,

uni–xni λn_i

+F(yt,uni).

In view of the monotonicity ofA, and the restriction (a), we obtain from (A) that

yt–q,Ayt ≥F(yt,q).

From (A) and (A), we see that

 =F(yt,yt)≤tF(yt,y) + ( –t)F(yt,q)

≤tF(yt,y) + ( –t)yt–q,Ayt

=tF(yt,y) + ( –t)ty–q,Ayt.

It follows that

≤F(yt,y) + ( –t)y–w,Ayt, ∀y∈C.

It follows from (A) thatq∈EP(F,A). Hence,

lim sup

n→∞

u–x,¯ xn–x¯ ≤.

Finally, we show thatxn→ ¯x. Note that

xn+–x¯

≤αnu–x,¯ xn+–x¯+βnxn–x¯xn+–x¯+γnyn–x¯xn+–x¯

≤αnu–x,¯ xn+–x¯+

 –αn 

xn–x¯+xn+–x¯

.

This implies that

xn+–x¯≤αnu–x,¯ xn+–x¯+ ( –αn)xn–x¯.

Using Lemma ., we ﬁnd thatlimn→∞xn–x¯= . This completes the proof.

3 Applications

In this section, we consider some applications of the main results.

Recall that the classical variational inequality is to ﬁnd anx∈Csuch that

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In this paper, we useVI(C,A) to denote the solution set of the inequality. It is well known thatx∈Cis a solution of the inequality iﬀxis a ﬁxed point of the mappingPC(I–rA), where r>  is a constant,I stands for the identity mapping. IfAisα-inverse-strongly monotone andr∈(, α], then the mappingI–rAis nonexpansive. It follows thatVI(C,A) is closed and convex.

Letg:H→(–∞, +∞] be a proper convex lower semicontinuous function. Then the subdiﬀerential∂gofgis deﬁned as follows:

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

From Rockafellar [], we know that∂gis maximal monotone. It is not hard to verify that ∈∂g(x) if and only ifg(x) =miny∈Hg(y).

LetICbe the indicator function ofC,i.e.,

IC(x) = ⎧ ⎨ ⎩

, x∈C,

+∞, x∈/C.

SinceIC is a proper lower semicontinuous convex function onH, we see that the subd-iﬀerential∂ICofIC is a maximal monotone operator. It is clearly thatJrx=PCx,∀x∈H, (A+∂IC)–() =VI(C,A) and (A+∂IC)–() =VI(C,A).

Theorem . Let C be a nonempty closed convex subset of a Hilbert space H and F a bi-function from C×C toRwhich satisﬁes(A)-(A).Let A:C→H be aδ-inverse-strongly

monotone mapping,A:C→H be aδ-inverse-strongly monotone mapping,A:C→H

be aδ-inverse-strongly monotone mapping,and{Si:C→C}be a family of inﬁnitely

non-expansive mappings.Assume that:=∞_i_=F(Si)∩EP(F,A)∩VI(C,A)∩VI(C,A)=∅.

Let x∈C and{xn}be a sequence generated by

⎧ ⎨ ⎩

zn=PC(un–snAun),

xn+=αnf(xn) +βnxn+γnWnPC(zn–rnAzn), ∀n≥,

where unis such that

F(un,y) +Axn,y–un+  λn

y–un,un–xn ≥, ∀y∈C,

{Wn:C→C}is the sequence generated in(.),{αn},{βn},and{γn}are sequences in(, ) such thatαn+βn+γn= for each n≥and{rn},{sn},and{λn}are positive number se-quences.Assume that the above control sequences satisfy the following restrictions:

(a)  <a≤λn≤b< δ, <a≤rn≤b< δ, <a¯≤sn≤ ¯b< δ; (b) lim_n_→∞αn= and

∞

n=αn=∞;

(c)  <lim infn→∞βn≤lim sup_n_→∞βn< ;

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Recall that a mappingT:C→Cis said to be ak-strict pseudo-contraction if there exists a constantk∈[, ) such that

Tx–Ty≤ x–y+k(I–T)x– (I–T)y, ∀x,y∈C.

PuttingA=I–T, whereT:C→Cis ak-strict pseudo-contraction, we ﬁnd thatAis

–k

 -inverse-strongly monotone.

Next, we consider ﬁxed points of strict pseudo-contractions.

Theorem . Let C be a nonempty closed convex subset of a Hilbert space H and F a bifunction from C×C toRwhich satisﬁes(A)-(A).Let T:C→H be a k-strict

pseudo-contraction,T:C→H be a k-strict pseudo-contraction, A:C→H be a

δ-inverse-strongly monotone mapping,and{Si:C→C}be a family of inﬁnitely nonexpansive map-pings.Assume that:=∞_i_=F(Si)∩EP(F,A)∩F(T)∩F(T)=∅.Let x∈C and{xn}be

a sequence generated by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

zn= ( –sn)un+snTun,

yn= ( –rn)un+rnTun,

xn+=αnf(xn) +βnxn+γnWnyn, ∀n≥,

where unis such that

 λn

y–un,un–xn ≥, ∀y∈C,

{Wn:C→C}is the sequence generated in(.),{αn},{βn},and{γn}are sequences in(, ) such thatαn+βn+γn= for each n≥and{rn},{sn},and{λn}are positive number se-quences.Assume that the above control sequences satisfy the following restrictions:

(a)  <a≤λn≤b< δ, <a≤rn≤b<  –k, <a¯≤sn≤ ¯b<  –k;

(b) limn→∞αn= and ∞

n=αn=∞;

(c)  <lim infn→∞βn≤lim supn→∞βn< ;

(d) lim_n_→∞|λn–λn+|=limn→∞|sn–sn+|=limn→∞|rn–rn+|= . Then the sequence{xn}converges strongly tox¯=Pu.

Proof TakingAi=I–Ti, wee see thatAi:C→H is aδi-strict pseudo-contraction with δi=–_ki andF(Ti) =VI(C,Ai) fori= , . In view of Theorem ., we ﬁnd the desired

conclusion immediately.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author is very grateful to the reviewers for useful suggestions which improved the contents of this paper.

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References

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

2013, 199 (2013)

2. 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)

3. 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)

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

5. Cho, SY, Qin, X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput.235, 430-438 (2014)

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. 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) 8. Cho, SY, Kang, SM: Approximation of ﬁxed points of pseudocontraction semigroups based on a viscosity iterative

process. Appl. Math. Lett.24, 224-228 (2011)

9. 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)

10. 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)

11. Zhang, Q: Iterative approximation of solutions of monotone quasi-variational inequalities via nonlinear mappings. J. Fixed Point Theory2014, Article ID 1 (2014)

12. Jeong, JU: Fixed point solutions of generalized mixed equilibrium problems and variational inclusion problems for nonexpansive semigroups. Fixed Point Theory Appl.2014, Article ID 57 (2014)

13. Cho, SY, Qin, X, Wang, L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl.2014, 94 (2014)

14. Qin, X, Cho, SY: Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions. Appl. Math. Lett.23, 1252-1255 (2010)

15. Chen, JH: Iterations for equilibrium and ﬁxed point problems. J. Nonlinear Funct. Anal.2013, Article ID 4 (2013) 16. Qing, Y, Cho, SY: Proximal point algorithms for zero points of nonlinear operators. Fixed Point Theory Appl.2014, 42

(2014)

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

18. Wu, C, Lv, S: Bregman projection methods for zeros of monotone operators. J. Fixed Point Theory2013, Article ID 7 (2013)

19. Lv, S, Hao, Y: Some results on continuous pseudo-contractions in a reﬂexive Banach space. J. Inequal. Appl.2013, 538 (2013)

20. 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)

21. Qin, X, Agarwal, RP: Shrinking projection methods for a pair of asymptotically quasi-φ-nonexpansive mapping. Numer. Funct. Anal. Optim.31, 1072-1089 (2010)

22. Kim, KS, Kim, JK, Lim, WH: Convergence theorems for common solutions of various problems with nonlinear mapping. J. Inequal. Appl.2014, 2 (2014)

23. Lv, S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl.2013, 19 (2013) 24. 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)

25. Qin, X, Shang, M, Su, Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model.48, 1033-1046 (2008)

26. Qin, X, Shang, M, Zhou, H: Strong convergence of a general iterative method for variational inequality problems and ﬁxed point problems in Hilbert spaces. Appl. Math. Comput.200, 242-253 (2008)

27. Chang, SS, Lee, HWJ, Chan, CK: A new method for solving equilibrium problem ﬁxed point problem and variational inequality problem with application to optimization. Nonlinear Anal.70, 3307-3319 (2009)

28. Qin, X, Cho, SY, Wang, L: Iterative algorithms with errors for zero points ofm-accretive operators. Fixed Point Theory Appl.2013, 148 (2013)

29. Song, J, Chen, M: On generalized asymptotically quasi-φ-nonexpansive mappings and a Ky Fan inequality. Fixed Point Theory Appl.2013, 237 (2013)

30. Zhang, QN: Common solutions of equilibrium and ﬁxed point problems. J. Inequal. Appl.2013, 425 (2013) 31. Qin, X, Cho, SY, Wang, L: A regularization method for treating zero points of the sum of two monotone operators.

Fixed Point Theory Appl.2014, 75 (2014)

32. Nadezhkina, N, Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.128, 191-201 (2006)

33. Qin, X, Cho, SY, Kang, SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim.49, 679-693 (2011)

34. Liu, LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl.194, 114-125 (1995)

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

36. Shimoji, K, Takahashi, W: Strong convergence to common ﬁxed points of inﬁnite nonexpansive mappings and applications. Taiwan. J. Math.5, 387-404 (2001)

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

(16)

38. Suzuki, T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals. J. Math. Anal. Appl.305, 227-239 (2005)

39. Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc.149, 75-88 (1970)

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