An iterative approach to mixed equilibrium problems and fixed points problems

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R E S E A R C H

Open Access

An iterative approach to mixed equilibrium

problems and ﬁxed points problems

Yonghong Yao

1

_{, Yeong-Cheng Liou}

2

_{and Shin Min Kang}

3*

*_{Correspondence:}

[email protected]

3_{Department of Mathematics and}

the RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract

In the present paper, an iterative algorithm for solving mixed equilibrium problems and fixed points problems has been constructed. It is shown that under some mild conditions, the sequence generated by the presented algorithm converges strongly to the common solution of mixed equilibrium problems and fixed points problems. As an application, we can find the minimum norm element without involving projection.

MSC: 47J05; 47J25; 47H09

Keywords: mixed equilibrium problem; ﬁxed point problem; minimization problem; strictly pseudo-contractive mapping

1 Introduction

LetHbe a real Hilbert space with the inner product·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofH. For a nonlinear mappingA:C→Hand a bifunctionF:C×C→R, the mixed equilibrium problem is to ﬁndz∈Csuch that

F(z,y) +Az,y–z ≥, ∀y∈C. (.)

The solution set of (.) is denoted byMEP. IfA= , then (.) reduces to the following equilibrium problem of ﬁndingz∈Csuch that

F(z,y)≥, ∀y∈C. (.)

The solution set of (.) is denoted byEP. IfF= , then (.) reduces to the variational inequality problem of ﬁndingz∈Csuch that

Az,y–z ≥, ∀y∈C. (.)

The solution set of (.) is denoted byVI. Problem (.) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax prob-lems, Nash equilibrium problem in noncooperative games and others. See,e.g., [–].

For solving mixed equilibrium problem (.), Moudaﬁ [] introduced an iterative algo-rithm and proved a weak convergence theorem. Further, Takahashi and Takahashi []

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introduced the following iterative algorithm for ﬁnding an element ofF(S)∩MEP:

⎧ ⎨ ⎩

F(zn,y) +Axn,y–zn+λny–zn,zn–xn ≥, ∀y∈C,

xn+=βnxn+S(αnu+ ( –βn)zn)

(.)

for alln≥, whereS:C→Cis a nonexpansive mapping. They proved that the sequence {xn}generated by (.) converges strongly toz=ProjF(S)∩MEP(u).

Recently, Yao and Shahzad [] gave the following iteration process for nonexpansive mappings with perturbation:x∈Cand

xn+= ( –βn)xn+βnProjC

αnun+ ( –αn)Txn

, n≥,

where{αn}and{βn}are sequences in [, ], and the sequence{un}inHis a small

perturba-tion for then-step iteration satisfyingun → asn→ ∞. In fact, there are perturbations

always occurring in the iterative processes because the manipulations are inaccurate. Using the ideas in [], Chuanget al.[] introduced the following iteration process for ﬁnding a common element of the set of solutions of the equilibrium problem and the set of ﬁxed points for a quasi-nonexpansive mapping with perturbation:q∈Hand

⎧ ⎨ ⎩

xn∈Csuch thatF(xn,y) +λny–xn,xn–qn ≥, ∀y∈C,

qn+=αnun+ ( –αn)(βnxn+ ( –βn)Sxn)

for alln≥. They showed that the sequence{qn}converges strongly toProjF(S)∩EP.

Motivated and inspired by the above works, in the present paper, we construct an it-erative algorithm for solving mixed equilibrium problems and ﬁxed points problems. It is shown that under some mild conditions the sequence{xn}generated by the presented

algorithm converges strongly to the common solution of mixed equilibrium problems and ﬁxed points problems. As an application, we can ﬁnd the minimum norm element without involving projection.

2 Preliminaries

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Recall that a mapping A:C→His calledα-inverse-strongly monotoneif there exists a positive real numberα>  such that

Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.

It is clear that anyα-inverse-strongly monotone mapping is monotone and _α-Lipschitz continuous. A mappingS:C→Cis said to benonexpansiveifSx–Sy ≤ x–yfor all x,y∈C. And a mappingS:C→Cis said to bestrictly pseudo-contractiveif there exists a constant ≤κ<  such that

Sx–Sy≤ x–y+κ(I–S)x– (I–S)y, ∀x,y∈C.

For such a case, we also say thatSis aκ-strictly pseudo-contractive mapping.

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(H) F(x,x) = for allx∈C;

(H) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (H) for eachx,y,z∈C,limt↓F(tz+ ( –t)x,y)≤F(x,y);

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

We need the following lemmas for proving our main results.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let F:C×C→R be a bifunction which satisﬁes conditions(H)-(H).Let r> and x∈H. Then there exists z∈C such that

F(z,y) +

ry–z,z–x ≥, ∀y∈C.

Further,if Tr(x) ={z∈C:F(z,y) +_ry–z,z–x ≥,∀y∈C},then we have (i) Tris single-valued andTris ﬁrmly nonexpansive,i.e.,for anyx,y∈H,

Trx–Try≤ Trx–Try,x–y; (ii) EPis closed and convex andEP=F(Tr).

Lemma .[] Let C,H,F and Trx be as in Lemma..Then we have

Tsx–Ttx≤

s–t

s Tsx–Ttx,Tsx–x

for all s,t> and x∈H.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let the mapping A:C→H beα-inverse strongly monotone and r> be a constant.Then we have

(I–rA)x– (I–rA)y≤ x–y+r(r– α)Ax–Ay, ∀x,y∈C.

In particular,if≤r≤α,then I–rA is nonexpansive.

Lemma .[] Let{xn}and{yn}be bounded sequences in a Banach space X and let{βn}

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

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

xn= .

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let S:C→C be aλ-strict pseudo-contraction.Then we have

(i) F(S) ={x:Sx=x}is closed convex; (ii) κI+ ( –κ)Sforκ∈[λ, )is nonexpansive.

Lemma .[] Let C be a nonempty closed and convex of a real Hilbert space H.Let S:C→C be aκ-strictly pseudo-contractive mapping.Then I–S is demi-closed at,i.e., if xnx∈C and xn–Sxn→,then x=Sx.

Lemma .[] Assume that{an}is a sequence of nonnegative real numbers such that

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

() lim sup_n_→∞δn≤or

∞

n=|δnγn|<∞.

Thenlimn→∞an= .

3 Main results

In this section, we prove our main results.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H and let F: C×C→R be a bifunction satisfying conditions(H)-(H).Let A:C→H be anα -inverse-strongly monotone mapping and let S:C→C be aκ-strictly pseudo-contractive mapping. Suppose that F(S)∩MEP=∅.Let x∈C,{zn}and{xn}be sequences in C generated by

⎧ ⎨ ⎩

F(zn,y) +Axn,y–zn+_λ_ny–zn,zn– (αnun+ ( –αn)xn) ≥, ∀y∈C,

xn+=βnxn+ ( –βn)γzn+ ( –βn)( –γ)Szn,

(.)

for all n≥,where{λn} ⊂(, α),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) limn→∞un=ufor someu∈H;

(r) limn→∞αn= and

∞

n=αn=∞; (r)  <c≤βn≤d< andγ ∈[κ, );

(r) a( –αn)≤λn≤b( –αn),where[a,b]⊂(, α)andlimn→∞(λn+–λn) = .

Then{xn}generated by(.)converges strongly toProjF(S)∩MEP(u).

Proof Note thatzncan be rewritten aszn=Tλn(αnun+ ( –αn)xn–λnAxn) for eachn. Take

z∈F(S)∩MEP. It is obvious thatz=Tλn(z–λnAz) =Tλn(αnz+ ( –αn)(z–

λnAz

–αn)) for all

n≥. By using the nonexpansivity ofTλnand the convexity of · , we derive

zn–z

=Tλn

αnun+ ( –αn)xn–λnAxn

–Tλn(z–λnAz) 

=Tλn αnun+ ( –αn) xn–

λnAxn

 –αn

–Tλn αnz+ ( –αn) z–

λnAz

 –αn



≤ αnun+ ( –αn) xn–

λnAxn

 –αn

– αnz+ ( –αn) z–

λnAz

 –αn



=( –αn)

xn–

λnAxn

 –αn

– z– λnAz  –αn

+αn(un–z) 

≤( –αn) xn–

λnAxn

 –αn

+αnun–z.

SinceAisα-inverse strongly monotone, we know from Lemma . that

xn–

λnAxn

 –αn



≤ xn–z+

λn(λn– ( –αn)α)

( –αn) Axn

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It follows that

zn–z≤( –αn) xn–z+

λn(λn– ( –αn)α)

( –αn)

Axn–Az

+αnun–z

≤( –αn)xn–z+αnun–z. (.)

So, we have

xn+–z=βn(xn–z) + ( –βn)

γI+ ( –γ)Szn–z 

≤βnxn–z+ ( –βn)zn–z

≤βnxn–z+ ( –βn)

( –αn)xn–z+αnun–z

= – ( –βn)αn

xn–z+ ( –βn)αnun–z

≤maxxn–z,un–z

.

Sincelimn→∞un=u,{un} is bounded. Therefore, by induction, we deduce that{xn}is

bounded. Hence,{Axn},{zn}and{Szn}are also bounded.

Puttingyn=αnun+ ( –αn)xn–λnAxnfor alln, we have

zn+–zn=Tλn+yn+–Tλn+yn+Tλn+yn–Tλnyn.

It follows that

zn+–zn ≤ Tλn+yn+–Tλn+yn+Tλn+yn–Tλnyn

≤ yn+–yn+Tλn+yn–Tλnyn. (.)

From Lemma ., we know thatI–λAis nonexpansive for allλ∈(, α). Thus, we have I– λn+

–αn+Ais nonexpansive for allndue to the fact that

λn+

–αn+∈(, α). Then we get yn+–yn=αn+un++ ( –αn+)xn+–λn+Axn+–

≤( –αn+) xn+–

λn+

 –αn+

Axn+

– ( –αn) xn–

λn

 –αn

Axn

+αn+un++αnun

≤( –αn+)

I– λn+  –αn+

A

xn+– I–

λn+

 –αn+

A

xn

+( –αn+) xn–

λn+

 –αn+

Axn

– ( –αn) xn–

λn

 –αn

Axn

+αn+un++αnun

≤ xn+–xn+|αn+–αn|xn+|λn+–λn|Axn

+αn+un++αnun. (.)

By Lemma ., we have

Tλn+yn–Tλnyn ≤ |

λn+–λn|

λn+

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From (.)-(.), we obtain

zn+–zn ≤ xn+–xn+|αn+–αn|xn+|λn+–λn|Axn

+|λn+–λn| λn+ T

λn+yn–yn+αn+un++αnun. Then

γI+ ( –γ)Szn+–

γI+ ( –γ)Szn

≤ zn+–zn

≤ xn+–xn+|αn+–αn|xn+|λn+–λn|Axn

+|λn+–λn| λn+

Tλn+yn–yn+αn+un++αnun.

Therefore,

γI+ ( –γ)Szn+–

γI+ ( –γ)Szn–xn+–xn

≤ |αn+–αn|xn+|λn+–λn|Axn+αn+un++αnun

+|λn+–λn| λn+

Tλn+yn–yn.

Sinceαn→,λn+–λn→ andlim infn→∞λn> , we obtain

lim sup

n→∞

γI+ ( –γ)Szn+–

γI+ ( –γ)Szn–xn+–xn

≤.

This together with Lemma . implies that

lim

n→∞γI+ ( –γ)S

zn–xn= . (.)

Consequently, we obtain

lim

n→∞xn+–xn=nlim→∞( –βn)γI+ ( –γ)S

zn–xn= .

From (.) and (.), we have

xn+–z≤( –βn)γI+ ( –γ)S

Tλn

–z

+βnxn–z

≤( –βn)

( –αn) xn–z+

λn

( –αn)

λn– ( –αn)α

Axn–Az

+αnun–z

+βnxn–z

= – ( –βn)αn

xn–z+

( –βn)λn

 –αn

λn– ( –αn)α

Axn–Az

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≤ xn–z+

( –βn)λn

 –αn

λn– ( –αn)α

Axn–Az

+ ( –βn)αnun–z.

Then we obtain

( –βn)λn

 –αn

( –αn)α–λn

Axn–Az

≤ xn–z–xn+–z+ ( –βn)αnun–z

≤xn–z–xn+–z

xn+–xn+ ( –βn)αnun–z.

Sincelimn→∞αn= ,limn→∞xn+–xn=  andlim infn→∞(–_–βαn)nλn(( –αn)α–λn) > ,

we have

lim

n→∞Axn–Az= . (.)

Next, we showxn–zn=xn–Tλnyn →. By using the ﬁrm nonexpansivity ofTλn, we

have

Tλnyn–z =Tλnyn–Tλn(z–λnAz) 

≤yn– (z–λnAz),Tλnyn–z

= 

yn– (z–λnAz)



+Tλnyn–z 

–αnun+ ( –αn)xn–λn(Axn–λnAz) –Tλnyn 

.

We note that

yn– (z–λnAz) 

≤( –αn)xn–z+αnun–z.

Thus,

Tλnyn–z _≤ 



( –αn)xn–z+αnun–z+Tλnyn–z 

–αnun+ ( –αn)xn–Tλnyn–λn(Axn–λnAz) 

.

That is,

Tλnyn–z≤( –αn)xn–z+αnun–z

–αnun+ ( –αn)xn–Tλnyn–λn(Axn–λnAz) 

= ( –αn)xn–z+αnun–z–αnun+ ( –αn)xn–Tλnyn 

+ λn

αnun+ ( –αn)xn–Tλnyn,Axn–Az

–λ_nAxn–Az

≤( –αn)xn–z+αnun–z–αnun+ ( –αn)xn–Tλnyn 

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It follows that

xn+–z≤βnxn–z+ ( –βn)( –αn)xn–z+ ( –βn)αnun–z

– ( –βn)αnun+ ( –αn)xn–Tλnyn 

+ λn( –βn)αnun+ ( –αn)xn–TλnynAxn–Az

= – ( –βn)αn

xn–z+ ( –βn)αnun–z

– ( –βn)αnun+ ( –αn)xn–Tλnyn 

+ λn( –βn)αnun+ ( –αn)xn–TλnynAxn–Az.

Hence,

( –βn)αnun+ ( –αn)xn–Tλnyn 

≤ xn–z–xn+–z– ( –βn)αnxn–z

+( –βn)αnun–z+ λn( –βn)αnun+ ( –αn)xn–TλnynAxn–Az

≤xn–z+xn+–z

xn+–xn+ ( –βn)αnun–z

+λn( –βn)αnun+ ( –αn)xn–TλnynAxn–Az.

Sincelim sup_n_→∞βn< ,xn+–xn →,αn→ andAxn–Az →, we deduce

lim

n→∞αnun+ ( –αn)xn–Tλnyn= .

This implies that

lim

n→∞xn–zn=xn–Tλnyn= . (.)

Putx˜=Proj_F₍_S₎_∩_MEP(u). We will ﬁnally show thatxn→ ˜x.

Settingvn=xn–_–λαnn(Axn–Ax) for all˜ n. Takingz=x˜in (.) to getAxn–Ax˜ →.

First, we provelim sup_n_→∞u–x,˜ vn–x ≤˜ . We take a subsequence{vni}of{vn}such that

lim sup

n→∞ u

–x,˜ vn–x˜ =lim

i→∞u–x,˜ vni–x.˜

It is clear that{vni}is bounded due to the boundedness of{xn}andAxn–Ax˜ →. Then

there exists a subsequence {vn_ij} of{vni} which converges weakly to some pointw∈C.

Hence,{xn_ij}also converges weakly tow. At the same time, from (.) and (.), we have

lim

j→∞ _x_n

ij–

γI+ ( –γ)Sxn_ij= . (.)

By the demi-closedness principle (see Lemma .) and (.), we deducew∈F(S). Further, we show thatwis also inMEP. From (.), we have

F(zn,y) +Axn,y–zn+

 λn

y–zn,zn–

αnun+ ( –αn)xn

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From (H), we have

Axn,y–zn+

 λn

y–zn,zn–

≥F(y,zn). (.)

Putxt=ty+ ( –t)wfor allt∈(,  –_λα) andy∈C. Then we havext∈C. So, from (.),

we have

xt–zn,Axt ≥ xt–zn,Axt–xt–zn,Axn

–  λn

xt–zn,zn–

+F(xt,zn)

=xt–zn,Axt–Azn+xt–zn,Azn–Axn

–  λn

xt–zn,zn–

+F(xt,zn).

Sincezn–xn →, we haveAzn–Axn →. Further, from monotonicity ofA, we have

xt–zn,Axt–Azn ≥. So, from (H), we have

xt–w,Axt ≥F(xt,w), asn→ ∞. (.)

From (H), (H) and (.), we also have

 = F(xt,xt)

≤tF(xt,y) + ( –t)F(xt,w)

≤tF(xt,y) + ( –t)xt–w,Axt

=tF(xt,y) + ( –t)ty–w,Axt

and hence

≤F(xt,y) + ( –t)y–w,Axt.

Lettingt→, we have, for eachy∈C,

≤F(w,y) +y–w,Aw.

This impliesw∈MEP. Hence, we havew∈F(S)∩MEP. This implies that

lim sup

n→∞ u

–x,˜ vn–x˜ =lim

j→∞u–x,˜ vnij–x˜ =u–x,˜ w–x.˜

Note thatx˜=Proj_F₍_S₎_∩_MEP(u). Thenu–x,˜ w–x ≤˜ ,w∈F(S)∩MEP. Therefore,

lim sup

n→∞ u

–x,˜ vn–x ≤˜ .

Sinceun→u, we have

lim sup

n→∞

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From (.), we have

xn+–x˜ 

≤βnxn–x˜ + ( –βn)γI+ ( –γ)S

Tλnyn–x˜ 

≤βnxn–x˜+ ( –βn)Tλnyn–x˜

=βnxn–x˜+ ( –βn)Tλnyn–Tλn(x˜–λnAx)˜ 

≤βnxn–x˜+ ( –βn)yn– (x˜–λnAx)˜ 

=βnxn–x˜ + ( –βn)αnun+ ( –αn)xn–λnAxn– (x˜–λnA˜x)

= ( –βn) ( –αn)

xn–

λn

 –αn

Axn

– x˜– λn  –αnA˜

x

+αn(un–x)˜ 

+βnxn–x˜ 

= ( –βn) ( –αn) xn–

λn

 –αn

Axn

– x˜– λn  –αnA˜

x



+ αn( –αn)

un–x,˜ xn–

λn

 –αn

Axn

– x˜– λn  –αn

Ax˜

+α_nun–x˜ 

+βnxn–x˜ 

≤βnxn–x˜ + ( –βn) ( –αn)xn–x˜ 

+ αn( –αn)

un–x,˜ xn–

λn

 –αn

(Axn–A˜x) –x˜

+α_nun–x˜ 

≤ – ( –βn)αn

xn–x˜

+ ( –βn)αn

( –αn)un–x,˜ vn–x˜ +αnun–x˜ 

.

It is clear that ∞_n_=( –βn)αn=∞andlim supn→∞(( –αn)un–x,˜ vn–x˜+αnun–

˜

x₎_≤_{. We can therefore apply Lemma . to conclude that}_x

n→ ˜x. This completes the

proof.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let F: C×C→R be a bifunction satisfying conditions(H)-(H).Let A:C→H be anα -inverse-strongly monotone mapping and let S:C→C be a nonexpansive mapping.Suppose that F(S)∩MEP=∅.Let x∈C,{zn}and{xn}be sequences in C generated by

⎧ ⎨ ⎩

F(zn,y) +Axn,y–zn+λny–zn,zn– (αnun+ ( –αn)xn) ≥, ∀y∈C,

xn+=βnxn+ ( –βn)γzn+ ( –βn)( –γ)Szn

(.)

for all n≥,where{λn} ⊂(, α),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) limn→∞un=ufor someu∈H;

(r) lim_n_→∞αn= and

∞

n=αn=∞; (r)  <c≤βn≤d< andγ ∈(, );

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Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let F:C×C→R be a bifunction satisfying conditions(H)-(H).Let S:C→C be aκ -strictly pseudo-contractive mapping.Suppose that F(S)∩EP=∅.Let x∈C,{zn}and{xn}

be sequences in C generated by

⎧ ⎨ ⎩

F(zn,y) +λny–zn,zn– (αnun+ ( –αn)xn) ≥, ∀y∈C,

(.)

for all n≥,where{λn} ⊂(, ),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) limn→∞un=ufor someu∈H;

∞

n=αn=∞; (r)  <c≤βn≤d< andγ ∈[κ, );

(r) a( –αn)≤λn≤b( –αn),where[a,b]⊂(, )andlimn→∞(λn+–λn) = .

Then{xn}generated by(.)converges strongly toProjF(S)∩EP(u).

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let F :C×C→R be a bifunction satisfying conditions(H)-(H). Let S:C→C be a nonexpansive mapping.Suppose that F(S)∩EP=∅.Let x∈C,{zn}and{xn}be sequences

in C generated by

⎧ ⎨ ⎩

F(zn,y) +λny–zn,zn– (αnun+ ( –αn)xn) ≥, ∀y∈C,

(.)

for all n≥,where{λn} ⊂(, ),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) lim_n_→∞un=ufor someu∈H;

∞

n=αn=∞; (r)  <c≤βn≤d< andγ ∈(, );

Then{xn}generated by(.)converges strongly toProjF(S)∩EP(u).

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let F: C×C→R be a bifunction satisfying conditions(H)-(H).Let A:C→H be anα -inverse-strongly monotone mapping and let S:C→C be aκ-strictly pseudo-contractive mapping. Suppose that F(S)∩MEP=∅.Let x∈C,{zn}and{xn}be sequences in C generated by

⎧ ⎨ ⎩

F(zn,y) +Axn,y–zn+λny–zn,zn– ( –αn)xn ≥, ∀y∈C,

(.)

for all n≥,where{λn} ⊂(, α),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) limn→∞αn= and

∞

n=αn=∞; (r)  <c≤βn≤d< andγ ∈[κ, );

Then{xn}generated by(.)converges strongly toProjF(S)∩MEP(),which is the minimum

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Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let F:C×C→R be a bifunction satisfying conditions(H)-(H).Let S:C→C be aκ -strictly pseudo-contractive mapping.Suppose that F(S)∩EP=∅.Let x∈C,{zn}and{xn}

be sequences in C generated by

⎧ ⎨ ⎩

F(zn,y) +λny–zn,zn– ( –αn)xn ≥, ∀y∈C,

(.)

for all n≥,where{λn} ⊂(, ),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) limn→∞αn= and∞n=αn=∞;

(r)  <c≤βn≤d< andγ ∈[κ, );

Then{xn}generated by(.)converges strongly toProjF(S)∩EP(),which is the minimum

norm element in F(S)∩EP.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let F:C×C→R be a bifunction satisfying conditions(H)-(H).Let A:C→H be an α-inverse-strongly monotone mapping.Suppose that MEP=∅.Let x∈C,{zn}and{xn}be

sequences in C generated by

⎧ ⎨ ⎩

F(zn,y) +Axn,y–zn+λny–zn,zn– ( –αn)xn ≥, ∀y∈C,

xn+=βnxn+ ( –βn)zn

(.)

for all n≥,where{λn} ⊂(, α),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) limn→∞αn= and∞n=αn=∞;

(r)  <c≤βn≤d< ;

Then{xn}generated by(.)converges strongly toProjMEP(),which is the minimum norm

element in MEP.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and let F:C×C→R be a bifunction satisfying conditions(H)-(H).Suppose EP=∅.Let x∈C,

{zn}and{xn}be sequences in C generated by ⎧

⎨ ⎩

F(zn,y) +λny–zn,zn– ( –αn)xn ≥, ∀y∈C,

xn+=βnxn+ ( –βn)zn

(.)

for all n≥,where{λn} ⊂(, ),{αn} ⊂(, )and{βn} ⊂(, )satisfy (r) limn→∞αn= and

∞

n=αn=∞; (r)  <c≤βn≤d< ;

Then{xn}generated by(.)converges strongly toProjEP(),which is the minimum norm

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China.}2_{Department of Information}

Management, Cheng Shiu University, Kaohsiung, 833, Taiwan.3_{Department of Mathematics and the RINS, Gyeongsang}

National University, Jinju, 660-701, Korea.

Received: 25 February 2013 Accepted: 28 June 2013 Published: 12 July 2013

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doi:10.1186/1687-1812-2013-183