Strong convergence theorems based on the viscosity approximation method for a countable family of nonexpansive mappings

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R E S E A R C H

Open Access

Strong convergence theorems based on the

viscosity approximation method for a

countable family of nonexpansive mappings

Mozhgan Bagherboum

1

_{, Abdolrahman Razani}

2

_{and Choonkil Park}

3*

*_{Correspondence:}

[email protected]

3_{Research Institute for Natural}

Sciences, Hanyang University, Seoul, 133-791, Korea

Full list of author information is available at the end of the article

Abstract

In a real Hilbert space, an iterative scheme is considered to obtain a common fixed point for a countable family of nonexpansive mappings. In addition, strong convergence to the common fixed point of this sequence is investigated. As an application, an equilibrium problem is solved. We also state more applications of this procedure to obtain a common fixed point ofW-mappings.

MSC: 47H09; 47H10; 47J20

Keywords: viscosity approximation method; nonexpansive mapping; equilibrium problem;W-mapping; Hilbert space

1 Introduction

LetHbe a real Hilbert space,Cbe a nonempty closed convex subset ofH, andIbe an identity mapping onH. The strong (weak) convergence of{xn}toxis written byxn→x (xnx) asn→ ∞.

It is well known thatHsatisﬁes Opial’s condition []; for any sequence{xn}withxnx, the inequality

lim inf

n→∞xn–x<lim infn→∞xn–y

holds for everyy∈Hwithx=y.

A metric (nearest point) projectionPCfrom a Hilbert spaceHto a closed convex subset

CofHis deﬁned as follows.

For any pointx∈H, there exists a uniquePCx∈Csuch that

x–PCx ≤ x–y,

for all y∈C. It is well known that PC is a nonexpansive mapping fromH onto Cand satisﬁes the following:

x–y,PCx–PCy ≥ PCx–PCy,

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for all x,y∈H. Furthermore,PCxis characterized by the following properties:PCx∈C and

x–PCx,PCx–y ≥,

x–y≥ x–PCx+y–PCx,

(.)

for ally∈C.

LetAbe a mapping ofCintoH. The variational inequality problem is to ﬁnd anx∈C

such that

Ax,y–x ≥, ∀y∈C. (.)

We shall denote the set of solutions of the variational inequality problem (.) byVI(C,A). Then we have

x∈VI(C,A) ⇐⇒ x=PC(x–λAx), ∀λ> . (.) A mappingS fromC into itself is called nonexpansive ifSx–Sy ≤ x–y, for all

x,y∈C.Fix(S) :={x∈C:Sx=x}is the set of ﬁxed point ofS. Note thatFix(S) is closed and convex ifSis nonexpansive. A mappingf fromCintoCis said to be contraction, if there exists a constantk∈[, ) such thatf(x) –f(y) ≤kx–y, for allx,y∈C.

In , Moudaﬁ [] introduced the following viscosity approximation methods:x∈C and

xn+=αnf(xn) + ( –αn)Sxn, n∈N,

wheref is a contraction on closed convex subset of a real Hilbert space. It was shown in [] (also see Xu []) that such a sequence converges strongly to the unique solution of the variational inequality problem. In , Chenet al.[] suggested the following iterative scheme:

xn+=αnf(xn) + ( –αn)SPC(I–λnA)xn, n∈N,

wherex∈C,Sis a nonexpansive self-mapping andAanα-inverse strongly monotone mapping. They proved that the sequence{xn}converges strongly to a common ﬁxed point of a nonexpansive mapping which solves the corresponding variational inequality. Re-cently, Kumam and Plubtieng [] used the following viscosity iterative method for a count-able family of nonexpansive mappings:x∈Cand

xn+=αnf(xn) + ( –αn)SnPC(I–λnA)xn, n∈N.

They proved the generated sequence{xn}converges strongly to a common element of the set of common ﬁxed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality.

On the other hand, in , Yaoet al.[] considered a new sequence that is generated byx∈Cand

xn+=αnxn+ ( –αn)SPC( –λn)xn, n∈N,

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It is worth pointing out that many authors have extended the results in Hilbert space to the more general uniformly convex and uniformly smooth Banach space (see, for instance, [, –]).

In this work, motivated and inspired by the above results, an iterative scheme based on the viscosity approximation method is utilized to find a common element of the set of com-mon fixed points of a countable family of nonexpansive mappings. Moreover, a strong con-vergence theorem with different conditions on the parameters is studied. As an applica-tion, an equilibrium problem is solved. In addiapplica-tion, a common fixed point forW-mappings is obtained.

The following lemmas will be useful in the sequel.

Lemma .([]) Let{xn}and{yn}be bounded sequences in a Banach space X and{βn}be

a sequence in[, ]with <lim infn→∞βn≤lim supn→∞βn< .Suppose xn+= ( –βn)yn+

βnxnfor all integers n≥andlim supn→∞(yn+–yn–xn+–xn)≤.Thenlimn→∞yn–

xn= .

Lemma .([]) Let{an}be a sequence of nonnegative real numbers satisfying the

fol-lowing relation:

an+≤( –αn)an+αnσn+γn, n≥,

where

() {αn} ⊂[, ],

_∞

n=αn=∞;

() lim sup_n_→∞σn≤;

() γn≥,for alln∈Nand

∞

n=γn<∞.

Thenlimn→∞an= .

Lemma .([]) Let C be a nonempty closed subset of a Banach space and {Sn}be a

sequence of nonexpansive mappings from C into itself.Suppose∞_n_=sup{Sn+x–Snx:x∈

C}<∞.Then,for each x∈C,{Snx}converges strongly to some point of C.If S is a mapping

from C into itself which is deﬁned by Sx:=limn→∞Snx,for all x∈C,thenlimn→∞sup{Snx–

Sx:x∈C}= .

2 Strong convergence theorem

In this section, we use the viscosity approximation method to ﬁnd a common element of the set of common ﬁxed points of a countable family of nonexpansive mappings.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Assume that{Sn}is a sequence of nonexpansive mappings from C into itself such that

_∞

n=Fix(Sn)=

∅,and f is a contraction from H into C with constant k≤_.Suppose that{αn},{βn},{γn},

and{λn}are real sequences in(, ).Set x∈C and let{xn}be the iterative sequence deﬁned

by

yn:=PC( –λn)xn,

xn+:=αnxn+βnf(xn) +γnSnyn, n∈N,

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() αn+βn+γn= ,

() lim_n_→∞βn= ,

∞

n=βn=∞,

()  <lim infn→∞γn≤lim supn→∞γn< ,

() limn→∞λn= ,

∞

n=λn=∞,

∞

n=|λn+–λn|<∞,

() ∞_n_=sup{Sn+x–Snx:x∈B}<∞,for any bounded subsetBofC.

Let S be a mapping from C into itself deﬁned by Sx:=limn→∞Snx for all x∈C andFix(S) :=

_∞

n=Fix(Sn).Then{xn}converges strongly to an elementω∈Fix(S),whereω=PFix(S)f(ω).

Proof Fix(S) is a closed convex set, thenPFix(S)is well deﬁned andPFix(S)is nonexpansive. In addition,

PFix(S)f(x) –PFix(S)f(y)≤f(x) –f(y)≤kx–y,

for allx,y∈H. This shows thatPFix(S)fis a contraction fromHintoC. SinceHis complete, there exists a unique element ofω∈Fix(S)⊂Hsuch thatω=PFix(S)f(ω).

Letx∈Fix(S), we note that

xn+–x ≤αnxn–x+βnf(xn) –x+γnSnyn–x

≤αnxn–x+kβnxn–x+βnf(x) –x+γnyn–x

≤αnxn–x+kβnxn–x+βnf(x) –x+γn( –λn)xn–x

≤αnxn–x+kβnxn–x+βnf(x) –x

+γn( –λn)xn–x+γnλnx

= (αn+kβn+γn–γnλn)xn–x+βnf(x) –x+γnλnx

≤( –βn+kβn–γnλn)xn–x

+ ( –k)βnf (x) –x

 –k +γnλnx

≤max

xn–x,x,

f(x) –x  –k

.. .

≤max

x–x,x,

f(x) –x

 –k

.

Therefore{xn}is bounded. Hence,{f(xn)},{yn}, and{Snyn}are bounded. Also

Sn+yn+–Snyn ≤ Sn+yn+–Sn+yn+Sn+yn–Snyn

≤ yn+–yn+sup

Sn+x–Snx:x∈ {yn}

≤( –λn+)xn+– ( –λn)xn

+supSn+x–Snx:x∈ {yn}

≤ xn+–xn+λn+xn+–xn+|λn+–λn|xn

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Now, we deﬁnexn+=αnxn+ ( –αn)wn, for alln∈N. One can observe that

wn+–wn=

βn+f(xn+) +γn+Sn+yn+  –αn+

–βnf(xn) +γnSnyn  –αn

= βn+  –αn+

f(xn+) +

 –αn+–βn+  –αn+

Sn+yn+

– βn  –αn

f(xn) – –αn–βn  –αn

Snyn

= βn+  –αn+

f(xn+) –Sn+yn+

+ βn  –αn

Snyn–f(xn)

+Sn+yn+–Snyn. (.)

Substituting (.) into (.), it follows that

wn+–wn–xn+–xn

≤ βn+  –αn+

f(xn+) –Sn+yn++

βn  –αn

Snyn–f(xn)

+λn+xn+–xn+|λn+–λn|xn

+supSn+x–Snx:x∈ {yn} .

Therefore,

lim sup

n→∞ wn+–wn–xn+–xn ≤.

In view of Lemma ., we obtainlimn→∞wn–xn= , which implies that

lim

n→∞xn+–xn=nlim→∞( –αn)wn–xn= . On the other hand, one has

xn+–xn=βn

f(xn) –Snyn

+ ( –αn)(Snyn–xn).

It follows that

( –αn)Snyn–xn ≤ xn+–xn+βnSnyn–f(xn).

Hence,limn→∞xn–Snyn= . Also, fromyn–Snyn ≤ xn–Snyn+λnxn, we obtain

lim

n→∞yn–Snyn= .

Now, we prove

lim sup

n→∞

f(ω) –ω,Snyn–ω

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whereω=PFix(S)f(ω). Indeed, since{Snyn}is bounded, one can ﬁnd a subsequence{Sniyni}

of{Snyn}such that

lim sup

n→∞

=lim

i→∞

f(ω) –ω,Sniyni–ω

.

{yni}is bounded, there exists a subsequence{yn_ij}of{yni}which converges weakly toz.

Without loss of generality, assume thatyniz.yniis a sequence inCandCis closed and

convex, soz∈C. Now, using the fact thatSnyn–yn →, we obtainSniyniz. Next we

showz∈Fix(S).

Assume thatz∈/Fix(S). From Opial’s condition and Lemma ., we have

lim inf

i→∞ yni–z <lim infi→∞ yni–Sz =lim inf

i→∞ yni–Sniyni+Sniyni–Syni+Syni–Sz

≤lim inf

i→∞ Syni–Sz

≤lim inf

i→∞ yni–z.

This is a contradiction. Thus,z∈Fix(S).

Also, we note thatω=PFix(S)f(ω) and so, by (.), we have

lim sup

n→∞

= lim

i→∞

f(ω) –ω,Sniyni–ω

=f(ω) –ω,z–ω≤.

To complete the proof, we show{xn}converges strongly toω∈F(S). For this, by con-vexity of · , we have

Snyn–ω≤ yn–ω≤( –λn)xn–ω≤( –λn)xn–ω+λnω.

Hence,

xn+–ω =αn(xn–ω) +βn

f(xn) –ω+γn(Snyn–ω)

≤αn(xn–ω) +βn

f(xn) –ω+γ_nSnyn–ω

+ γn

αn(xn–ω) +βn

f(xn) –ω

,Snyn–ω

=αnxn–ω+βnf(xn) –ω 

+γ_nSnyn–ω

+ αnγnxn–ω,Snyn–ω + βnγn

f(xn) –ω,Snyn–ω

≤α_nxn–ω+βnf(xn) –ω 

+γ_nSnyn–ω

+ αnβnxn–ωf(xn) –ω+ αnγnxn–ωSnyn–ω

+ βnγn

f(xn) –f(ω),Snyn–ω

+ βnγn

≤α_nxn–ω+βnf(xn) –ω 

+γ_nSnyn–ω

+αnβn

xn–ω+f(xn) –ω 

+αnγn

xn–ω+Snyn–ω

+ kβnγnxn–ωSnyn–ω+ βnγn

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≤α_n+αnβn+αnγn

xn–ω+

β_n+αnβnf(xn) –ω 

+γ_n+αnγn

Snyn–ω+kβnγn

xn–ω+Snyn–ω

+ βnγn

≤(αn+kβnγn)xn–ω+βn( –γn)f(xn) –ω 

+γ_n+αnγn+kβnγn

Snyn–ω+ βnγn

≤(αn+kβnγn)xn–ω+βn( –γn)f(xn) –ω 

+γ_n+αnγn+kβnγn

( –λn)xn–ω+λnω

+ βnγn

≤αn+kβnγn+

γ_n+αnγn+kβnγn

( –λn)xn–ω

+βn( –γn)f(xn) –ω+γ_n+αnγn+βnγn

λnω

+ βnγn

.

Now, supposeL=sup{xn–ω,f(xn) –ω,ω}. Then

xn+–ω≤( –βnγn)xn–ω

+βnγn

f(ω) –ω,Snyn–ω

+λn

βn

ω+ –γn

γn

f(xn) –ω

+αn+kβnγn+ (γ 

n+αnγn+kβnγn)( –λn) +βnγn– 

βnγn xn

–ω

≤( –βnγn)xn–ω+βnγn

f(ω) –ω,Snyn–ω

+

λnγn+βn–βnγn+αn+kβnγn+ (γn( –βn) +kβnγn)( –λn)

βnγn

+βnγn– 

βnγn

L

= ( –δn)xn–ω+δnσn,

where

δn=βnγn,

σn= 

+

λnγn+βn–βnγn+αn+kβnγn+ (γn( –βn) +kβnγn)( –λn)

βnγn

+βnγn– 

βnγn

L

= f(ω) –ω,Snyn–ω

+

λnγn+kβnγn–γn+γn( –βn)( –λn) +kβnγn( –λn)

βnγn

L

= f(ω) –ω,Snyn–ω

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+

λn+kβn–  + ( –βn)( –λn) +kβn( –λn)

βn

L

≤f(ω) –ω,Snyn–ω

+(k– ) + ( –k)λn

L.

It is easy to see that{δn} ⊂[, ],

∞

n=δn=∞andlim supn→∞σn≤. Hence, by Lem-ma ., we ﬁnd that{xn}strongly converges toω∈Fix(S), whereω=PFix(S)f(ω). This

com-pletes the proof of this theorem.

The following example shows that this theorem is not a special case of [, Theorem .].

Example . LetC= [–, ]⊂H=Rwithαn=_n_n–_–,βn=_n, andλn=__n. Setf(x) =_x andSn(x) =_nx. Thenf is a_-contraction andSnis a sequence of nonexpansive mappings. It readily follows that the sequence{xn}generated by

yn:=PC( –λn)xn=_n–_n xn,

xn+:=αnxn+βnf(xn) +γnSnyn= n _–

n_–_nxn+n _–_n_+ n_–_n yn,

with initial valuex∈C, converges strongly to an element (zero) ofFix(S) =

∞

n=Fix(Sn) andPFix(S)f() = .

3 Applications

In this section, we consider the equilibrium problems andW-mappings.

3.1 Equilibrium problems

Equilibrium theory plays a central role in various applied sciences such as physics, me-chanics, chemistry, and biology. In addition, it represents an important area of the math-ematical sciences such as optimization, operations research, game theory, and ﬁnancial mathematics. Equilibrium problems include ﬁxed point problems, optimization problems, variational inequalities, Nash equilibria problems, and complementary problems as spe-cial cases.

Let ϕ:C→Rbe a real-valued function and A:C→H a nonlinear mapping. Also supposeF:C×C→Ris a bifunction. The generalized mixed equilibrium problem is to ﬁndx∈C(see []) such that

F(x,y) +ϕ(y) –ϕ(x) +Ax,y–x ≥, (.)

for ally∈C.

We shall denote the set of solutions of this generalized mixed equilibrium problem by GMEP; that is

GMEP :=x∈C:F(x,y) +ϕ(y) –ϕ(x) +Ax,y–x ≥,∀y∈C .

We now discuss several special cases of GMEP as follows:

. Ifϕ= , then the problem (.) is reduced to generalized equilibrium problem,i.e., ﬁndingx∈Csuch that

F(x,y) +Ax,y–x ≥,

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. IfA= , then the problem (.) is reduced to the mixed equilibrium problem, that is, to ﬁndx∈Csuch that

F(x,y) +ϕ(y) –ϕ(x)≥,

for ally∈C. We shall write the set of solutions of the mixed equilibrium problem by

MEP.

. Ifϕ= ,A= , then the problem (.) is reduced to the equilibrium problem, which is to ﬁndx∈Csuch that

F(x,y)≥,

for ally∈C.

. Ifϕ= ,F= , then the problem (.) is reduced to the variational inequality problem (.).

Now letϕ:C→Rbe a real-valued function. To solve the generalized mixed equilibrium problem for a bifunctionF:C×C→R, let us assume thatF,ϕ, andCsatisfy 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,limt→+F(tz+ ( –t)x,y)≤F(x,y);

(A) for eachx∈C,y→F(x,y)is convex and lower semicontinuous;

(B) for eachx∈Handr> , there exist a bounded subsetDx⊆Candyx∈Csuch that

for eachz∈C\Dx,

F(z,yx) +ϕ(yx) –ϕ(z) + 

ryx–z,z–x < ;

(B) Cis a bounded set.

In what follows we state some lemmas which are useful to prove our convergence results.

Lemma .([]) Assume that F:C×C→Rsatisﬁes(A)-(A),and letϕ:C→Rbe a

lower semicontinuous and convex function.Assume that either(B)or(B)holds.For r> 

and x∈H,deﬁne a mapping Tr(F,ϕ):H→C as follows:

T_r(F,ϕ)(x) :=

z∈C:F(z,y) +ϕ(y) –ϕ(z) +

ry–z,z–x ≥,∀y∈C

,

for all x∈H.Then the following assertions hold:

() For eachx∈H,Tr(F,ϕ)=∅;

() Tr(F,ϕ)is single-valued;

() Tr(F,ϕ)is ﬁrmly nonexpansive,i.e.,for anyx,y∈H,

T_r(F,ϕ)x–T_r(F,ϕ)y≤T_r(F,ϕ)x–T_r(F,ϕ)y,x–y;

() Fix(Tr(F,ϕ)) = MEP;

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Lemma .([]) Let C be a nonempty closed convex subset of a real Hilbert space H.

Assume that Sis a nonexpansive mapping from C into H and Sa ﬁrmly nonexpansive

mapping from H into C such thatFix(S)∩Fix(S)=∅.Then SSis a nonexpansive

map-ping from H into itself andFix(SS) =Fix(S)∩Fix(S).

Lemma .([, ]) Let C be a nonempty closed convex subset of a real Hilbert space H.

Let F be a bifunction from C×C intoRsatisfying(A)-(A)andϕ:C→Rbe a lower

semicontinuous and convex function.Assume that either(B)or(B)holds.Let{rn}be a

se-quence in(,∞),such thatinf{rn:n∈N}> ,

∞

n=|rn+–rn|<∞and Tr(nF,ϕ)be a mapping deﬁned as in Lemma..Then

() ∞_n_=sup{Tr(nF+,ϕ)x–T (F,ϕ)

rn x:x∈B}<∞,for any bounded subsetBofC; () Fix(Tr(F,ϕ)) =

∞

n=Fix(T (F,ϕ)

rn ),whereT

(F,ϕ)

r is a mapping deﬁned by

Tr(F,ϕ)x:=limn→∞Tr(Fn,ϕ)x,for allx∈C.Moreover,limn→∞T

(F,ϕ) rn x–T

(F,ϕ) r x= .

Now letCbe a nonempty closed convex subset of a real Hilbert spaceH. A mappingA:

C→His calledmonotoneifAx–Ay,x–y ≥ for allx,y∈C. It is calledα-inverse strongly monotoneif there exists a positive real numberαsuch thatAx–Ay,x–y ≥αAx–Ay_, for allx,y∈C. Anα-inverse strongly monotone mapping is sometimes calledα-cocoercive. A mappingAis said to berelaxedα-cocoerciveif there existsα>  such that

Ax–Ay,x–y ≥–αAx–Ay,

for allx,y∈C. The mappingAis said to berelaxed(α,λ)-cocoerciveif there existα,λ>  such that

Ax–Ay,x–y ≥–αAx–Ay+λx–y,

for allx,y∈C. A mappingA:H→H is said to beμ-Lipschitzianif there existsμ≥ such that

Ax–Ay ≤μx–y,

for allx,y∈H. It is clear that eachα-inverse strongly monotone mapping is monotone and_α-Lipschitzian and that eachμ-Lipschitzian, relaxed (α,λ)-cocoercive mapping with

αμ≤λis monotone. Also, ifAis anα-inverse strongly monotone, thenI–λAis a non-expansive mapping fromCtoH, provided thatλ≤α(see []).

Now we have the following theorem.

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

Let F be a bifunction from C ×C intoRsatisfying (A)-(A)and ϕ:C→R a lower

semicontinuous and convex function. Assume that either(B)or(B)holds. Let A be a

μ-Lipschitzian,relaxed(α,λ)-cocoercive mapping from C into H,B aβ-inverse strongly monotone mapping from C into H and f a contraction from H into C with constant k≤ _.Suppose that Snis a sequence of nonexpansive mappings from H into C such that

∞

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(, )and{rn},{sn} ⊂(,∞).Let{yn},{un},and{xn}be generated by x∈C,

⎧ ⎪ ⎨ ⎪ ⎩

yn:=PC( –λn)xn,

un:=T(F,

ϕ)

rn (yn–rnByn),

xn+:=αnxn+βnf(xn) +γnSnPC(un–snAun), n∈N.

Suppose that the following conditions are satisﬁed:

() αn+βn+γn= ,

() lim_n_→∞βn= ,

∞

n=βn=∞,

() limn→∞λn= ,

∞

n=λn=∞,

∞

n=|λn+–λn|<∞,

()  <a≤rn≤β,

∞

n=|rn+–rn|<∞,

()  <sn≤(λ–αμ ₎

μ ,

∞

n=|sn+–sn|<∞,

() ∞_n_=sup{Sn+x–Snx:x∈E}<∞,for any bounded subsetEofC.

Then{xn} converges strongly to an elementω∈

_∞

n=Fix(Sn)∩VI(C,A)∩GMEP,where

ω=P∞

n=Fix(S)∩VI(C,A)∩GMEPf(ω).

Proof For allx,y∈Candsn∈[,(λ–αμ ₎

μ ], we obtain

(I–snA)x– (I–snA)y 

=x–y–sn(Ax–Ay) 

=x–y– snx–y,Ax–Ay +snAx–Ay

≤ x–y– sn

–αAx–Ay+λx–y+s_nAx–Ay

≤ x–y+ snμαx–y– snλx–y+μsnx–y 

= + snμα– snλ+μsn

x–y≤ x–y.

This shows thatI–snAis nonexpansive for eachn∈N. By Lemma ., it implies that

∞

n=

supT(F,ϕ) rn+ x–T

(F,ϕ)

rn x:x∈E <∞,

for any bounded subset E of C. In addition, the mapping Tr(F,ϕ), deﬁned by Tr(F,ϕ)x:=

limn→∞Tr(nF,ϕ)xfor allx∈C, satisﬁesFix(T

(F,ϕ) r ) =

_∞

n=Fix(T (F,ϕ)

rn ) = MEP.

PutTn:=SnPC(I–snA)Tr(nF,ϕ)(I–rnB) =SnPC(I–snA)Un. Then, by Lemmas ., ., and

(.), we ﬁnd thatTnis a nonexpansive mapping fromCinto itself andFix(Tn) =Fix(Sn)∩

VI(C,A)∩Fix(TrFn,ϕ(I–rnB)) =Fix(Sn)∩VI(C,A)∩GMEP, for alln∈N, and so

∞

n=

Fix(Tn) =

∞

n=

Fix(Sn)∩VI(C,A)∩GMEP.

Also we note that

Tn+x–Tnx=Sn+PC(I–sn+A)Un+x–SnPC(I–snA)Unx

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+SnPC(I–sn+A)Un+x–SnPC(I–snA)Unx

≤ Sn+vn–Snvn+(I–sn+A)Un+x– (I–snA)Unx

≤ Sn+vn–Snvn+(I–sn+A)Un+x– (I–sn+A)Unx

+(I–sn+A)Unx– (I–snA)Unx

≤ Sn+vn–Snvn+Tr(Fn+,ϕ)(I–rn+B)x–T (F,ϕ)

rn (I–rnB)x

+|sn+–sn|AUnx

≤ Sn+vn–Snvn+Tr(Fn+,ϕ)(I–rn+B)x–T (F,ϕ)

rn (I–rn+B)x

+T_r(_nF,ϕ)(I–rn+B)x–Tr(Fn,ϕ)(I–rnB)x

+|sn+–sn|AUnx

≤ Sn+vn–Snvn+Tr(Fn+,ϕ)wn–T (F,ϕ) rn wn

+|rn+–rn|Bx+|sn+–sn|AUnx,

wherevn=PC(I–sn+A)Un+xandwn= (I–rn+B)x. Moreover, for any bounded subsetEof

C,F={PC(I–sn+A)Un+x:x∈E,n∈N}andG={(I–rn+B)x:x∈E,n∈N}are bounded and

∞

n=

supTn+x–Tnx:x∈E

≤ ∞

n=

supSn+y–Sny:y∈F +

∞

n=

supT_r(_nF_+,ϕ)z–T_r(_nF,ϕ)z:z∈G

+

∞

n=

|rn+–rn|sup

Bx:x∈E +

∞

n=

|sn+–sn|sup

AUnx:x∈E <∞.

Therefore, by Theorem ., {xn} converges strongly to an element ω∈

∞

n=Fix(Sn)∩

VI(C,A)∩GMEP, whereω=P∞

n=Fix(Sn)∩VI(C,A)∩GMEPf(ω). This completes the proof.

3.2 W-Mappings

The concept ofW-mappings was introduced in [, ]. It is now one of the main tools in studying convergence of iterative methods to approach a common ﬁxed point of nonlinear mapping; more recent progress can be found in [] and the references cited therein.

Let{Sn}be a countable family of nonexpansive mappingsSn:H→Handδ,δ, . . . be real numbers such that ≤δn≤ for everyn∈N. We consider the mappingWndeﬁned by

Un,n+:=I,

Un,n:=δnSnUn,n++ ( –δn)I,

Un,n–:=δn–Sn–Un,n+ ( –δn–)I,

.. .

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Un,k–:=δk–Sk–Un,k+ ( –δk–)I,

.. .

Un,:=δSUn,+ ( –δ)I,

Wn:=Un,=δSUn,+ ( –δ)I.

One can ﬁnd the proof of the following lemma in [].

Lemma . Let H be a real Hilbert space.Let{Sn}be a sequence of nonexpansive mappings

from H into itself such that∞_n_=Fix(Sn)=∅.Letδ,δ, . . .be real numbers such that <δn≤

b< for all n∈N.Then

() Wnis nonexpansive andFix(Wn) =

n

i=Fix(Si)for alln∈N;

() limn→∞Un,kxexists,for allx∈Handk∈N;

() the mappingW:C→Cdeﬁned byWx:=limn→∞Wnx=limn→∞Un,x,for allx∈C is a nonexpansive mapping satisfyingFix(W) =∞_n_=Fix(Sn);and it is called

W-mapping generated byS,S, . . . ,Sn,andδ,δ, . . . ,δn.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Assume that{Sn}is a sequence of nonexpansive mappings from C into itself such that

∞

n=Fix(Sn)=

∅,and f a contraction from H into C with constant k≤_.Let{αn},{βn},{γn}and{λn}be

real sequences in(, ).Also,suppose Wnare the W -mappings from C into itself generated

by S,S, . . . ,Sn,andδ,δ, . . . ,δnsuch that <δn≤b< for every n∈N.Set x∈C and let

{xn}be the iterative sequence deﬁned by

yn:=PC( –λn)xn,

xn+:=αnxn+βnf(xn) +γnWnyn, n∈N,

satisfying the following conditions:

() αn+βn+γn= ,

() lim_n_→∞βn= ,

∞

n=βn=∞,

() limn→∞λn= ,

∞

n=λn=∞,

∞

n=|λn+–λn|<∞.

Let W be a mapping from C into itself deﬁned by Wx:=limn→∞Wnx for all x∈C.Then

{xn}converges strongly to an elementω∈

∞

n=Fix(Sn),whereω=P∞_n_=Fix(Sn)f(ω). Proof SinceSiandUn,iare nonexpansive, by (.), we deduce that, for eachn∈N,

Wn+x–Wnx=δSUn+,x–δSUn,x

≤δUn+,x–Un,x

=δδSUn+,x–δSUn,x

≤δδUn+,x–Un,x

≤ · · ·

≤δδ· · ·δnUn+,n+x–Un,n+x ≤M n

i=

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whereM>  is a constant such thatsup{Un+,n+x–Un,n+x:x∈B} ≤M, for any bounded subsetBofC. Then

∞

n=

supWn+x–Wnx:x∈B <∞.

Now, by settingSn:=Wnin Theorem . and using Lemma ., we obtain the result.

Applying Lemma . and Theorem ., we obtain the following result.

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

Let F be a bifunction from C ×C intoRsatisfying (A)-(A)and ϕ:C→R a lower

semicontinuous and convex function. Assume that either(B)or(B)holds. Let A be a

μ-Lipschitzian,relaxed(α,λ)-cocoercive mapping from C into H,B aβ-inverse strongly monotone mapping from C into H and f a contraction from H into C with constant k≤ _.Suppose that Snis a sequence of nonexpansive mappings from H into C such that

∞

n=Fix(Sn)∩VI(C,A)∩GMEP=∅.Let{yn},{un},and{xn}be generated by x∈C,

⎧ ⎪ ⎨ ⎪ ⎩

yn:=PC( –λn)xn,

un:=Tr(Fn,ϕ)(yn–rnByn),

xn+:=αnxn+βnf(xn) +γnWnPC(un–snAun), n∈N,

where{αn},{βn},{γn},and{λn}are real sequences in(, )and{rn},{sn} ⊂(,∞)satisfying

the following conditions:

() αn+βn+γn= ,

() lim_n_→∞βn= ,

∞

n=βn=∞,

() limn→∞λn= ,

∞

n=λn=∞,

∞

n=|λn+–λn|<∞,

()  <a≤rn≤β,

∞

n=|rn+–rn|<∞,

()  <sn≤(λ–αμ ₎

μ ,

∞

n=|sn+–sn|<∞.

Then {xn} converges strongly to an elementω∈

∞

n=Fix(Sn)∩VI(C,A)∩GMEP,where

ω=P∞

n=Fix(Sn)∩VI(C,A)∩GMEPf(ω).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, College of Basic Sciences, Karaj Branch, Islamic Azad University, Alborz, Karaj, Iran.} 2_{Department of Mathematics, Faculty of Science, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin,}

Iran.3_{Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea.}

Acknowledgements

The authors express their gratitude to the referees for reading this paper carefully.

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