A new iterative method for equilibrium problems and fixed point problems for infinite family of nonself strictly pseudocontractive mappings

(1)

R E S E A R C H

Open Access

A new iterative method for equilibrium

problems and ﬁxed point problems for

inﬁnite family of nonself strictly

pseudocontractive mappings

Jong Kyu Kim

1*

_{and Nguyen Buong}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Kyungnam University, Changwon, Gyeongnam 631-701, Korea Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to present an iterative method for finding a common element of the set of solutions for an equilibrium problem and the set of common fixed points for a countably infinite family of nonself

λ

i-strictly pseudocontractive

mappings{Ti}∞_i₌₁from a closed convex subsetCinto a Hilbert spaceH.

MSC: 47H17; 47H20

Keywords: regularization; hemicontinuous; lower semicontinuous convex functional; ﬁxed points

1 Introduction and preliminaries

LetHbe a real Hilbert space with the inner product·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofH, and let G:C×C→(–∞, +∞) be a bifunction. The equilibrium problem forGis to ﬁndu∗∈Csuch that

Gu∗,v≥, ∀v∈C. (.) The set of solutions of (.) is denoted byEP(G). The equilibrium problem (.) includes as special cases numerous problems in physics, optimization, economics, transportation, and engineering.

Assume that the bifunctionGsatisﬁes the following standard properties.

Assumption A LetG:C×C→(–∞, +∞) be a bifunction satisfying conditions (A)-(A).

(A) G(u,u) = ,∀u∈C;

(A) G(u,v) +G(v,u)≤,∀(u,v)∈C×C;

(A) For eachu∈C,G(u,·) :C→(–∞, +∞)is lower semicontinuous and convex; (A) lim sup_t_→_+G(( –t)u+tz,v)≤G(u,v),∀(u,z,v)∈C×C×C.

Recall that a mappingTinHis said to beλ-strictly pseudocontractive in the terminology of Browder and Petryshyn [], if there exists a constant ≤λ<  such that

Tx–Ty≤ x–y+λ(I–T)x– (I–T)y

(2)

for allx,y∈D(T), the domain ofT, whereIis the identity operator inH. Clearly, when

λ= ,T is nonexpansive,i.e.,

T(x) –T(y)≤ x–y.

It means that the class ofλ-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.

A single-valued mappingAinHis said to be monotone, if

A(x) –A(y),x–y≥, ∀x,y∈D(A),

andλ-inverse strongly monotone, if there exists a positive constantλsuch that

A(x) –A(y),x–y≥λA(x) –A(y), ∀x,y∈D(A).

Therefore, anyλ-inverse strongly monotone mapping is monotone and Lipschitz contin-uous with the Lipschitz constantLA= /λ. It is well known [] that ifTis a nonexpansive

mapping, thenI–Tis (/)-inverse strongly monotone. It is easy to verify that ifT isλ -strictly pseudocontractive, then it is ( –λ)/-inverse strongly monotone. Hence,I–Tis Lipschitz continuous with the Lipschitz constantL= /( –λ).

Let{Ti}∞i= be a countably inﬁnite family of nonselfλi-strictly pseudocontractive

map-pings fromCintoHwith the set of ﬁxed pointsF(Ti) (i.e.,F(Ti) ={x∈C:x=Tix}). Set

F:=∞_i_=F(Ti). Assume that

S:=F∩EP(G)=∅.

The problem studied in this paper is to ﬁnd an element

u∗∈S. (.)

IfTi≡I,i≥, then problem (.) is reduced to (.) and shown in [] and [] to cover

monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, the Nash equilibria in noncooperative games, vector equilibrium problems, as well as certain ﬁxed point problems (see also []). For ﬁnding approximative solutions of (.), there exist several approaches: the regularization approach [–], the gap-function approach [, –], and iterative procedure approach [–].

In the case thatG≡ andTi=I,i> , (.) is a problem of ﬁnding a ﬁxed point for a λ-strictly pseudocontractive mapping inC(see []).

In the case thatG≡ andTi=I,i>N> , (.) is a problem of ﬁnding a common ﬁxed

point for a ﬁnite family ofλi-strictly pseudocontractive mappingsTiinCstudied in [],

where the following algorithm is constructed:

Letx∈Cand{αn},{βn},{γn}be three sequences in [, ] satisfyingαn+βn+γn=  for

alln≥, and let{un}be a sequence inC. Then the sequence{xn}generated by

(3)

whereTn=TnmodN, is called the implicit iteration process with mean errors for a ﬁnite

family of strictly pseudocontractive mappings{Ti}Ni=. The sequence{xn}converges weakly

to a common ﬁxed point of the maps{Ti}Ni=.

IfTi=I,i> , then (.) is a problem of ﬁnding a ﬁxed point for aλ-strictly

pseudocon-tractive mapping inC, which is an equilibrium point forG(see []).

Theorem .[] Let C be a nonempty closed convex subset of a Hilbert space H.Let G be a bifunction from C×C to(–∞, +∞)satisfying(A)-(A),and let S be a nonexpansive mapping of C into H such that

F(S)∩EP(G)=∅.

Let f be a contraction of H into itself,and let{xn}and{un}be sequences generated by x∈H

and ⎧ ⎨ ⎩

G(un,y) +_r_ny–un,un–xn ≥, ∀y∈C,

xn+=αnf(xn) + ( –αn)Sun

for all n≥,where{αn} ⊂[, ]and{rn} ⊂(,∞)satisfy

lim

n→∞αn= , ∞

n=

αn=∞,

∞

n=

|αn+–αn|<∞,

lim inf

n→∞ rn> ,

∞

n=

|rn+–rn|<∞.

Then{xn}and{un}converge strongly to z∈F(S)∩EP(G),where z=PF(S)∩EP(G)f(z).

IfTi=I,i>N> , then (.) is a problem of finding a common fixed point for a finite

family ofλi-strictly pseudocontractive mappingTifromCintoH, which is an equilibrium

point forG. We have constructed regularization algorithms (see [, , ]). First algo-rithm is deﬁned as follows. The regularization solutionuαis an element being a solution

for the single equilibrium problem

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Gα(uα,v)≥ ∀v∈C,uα∈C,

Gα(u,v) :=

N

i=αμiGi(u,v) +αu,v–u, α> ,

G(u,v) =G(u,v), Gi(u,v) =Ai(u),v–u, i= , . . . ,N, μ=  <μi<μi+< , i= , . . . ,N– ,

(.)

whereαis the regularization parameter.

Theorem .[] For eachα> ,problem(.)has a unique solution uαsuch that

(i) limα→+uα=u∗,u∗∈S.

(ii) u∗ ≤ y,∀y∈S.

(4)

In the second algorithm, the regularization solution is deﬁned on the base of the inertial proximal point algorithm proposed by Alvarez and Attouch [], where the sequence{zn}

is deﬁned by an equilibrium problem

˜

cn

_N

i= αμi

n Fi(zn+,v) +αnzn+,v–zn+

+zn+–zn,v–zn+

–γnzn–zn–,v–zn+ ≥ ∀v∈C,z,z∈C,

and{˜cn}and{γn}are the sequences of positive numbers.

Theorem .[] Assume that the parametersc˜n,γnandαnare chosen such that

(i)  <c<c˜n,≤γn<γ,

(ii) ∞_n_=bn= +∞,bn=c˜nαn/( +c˜nαn),

(iii) ∞_n_=γnb–nzn–zn–< +∞,

(iv) limn→∞αn= ,limn→∞|αnα–nαbnn+|= .

Then the sequence{zn}converges strongly to the element u∗,as n→+∞.

In the case thatG≡, Maingé [] considered the following iteration method:

xn+=αnDxn+

i≥

wi,nTixn, n≥, (.)

where D:C→C is a given contraction with constant ρ∈[, ),x∈C is a starting

point,{αn} ⊂(, ),wi,n≥ for alli≥ andi≥wi,n=  –αnwith the following

con-ditions:

C. ∞_n_=αn=∞.

C. For alli∈NI:={i:Ti=I},

(a) _w i,n| –

αn–

αn | →, or

nwi,n|αn–αn–|<∞, (b) _α

n|

 wi,n–



wi,n–| →, or

n|wi,n–

 wi,n–|<∞,

(c)  wi,nαn

k≥|wk,n–wk,n–| →, or

nwi,n

k≥|wk,n–wk,n–|<∞.

C. αn

wi,n→for alli∈NI.

Then the sequence{xn}given by (.) converges strongly toxthe unique ﬁxed point of

PFoD, wherePFis the metric projection fromHontoF.

For solving (.), Ceng and Yao [] proposed the following algorithm: Letx∈Hbe an arbitrary element and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

G(un,v) +_r_nun–xn,v–un, ∀v∈C,

yn= ( –γn)xn+γnWnun,

xn+= ( –αn–βn)xn+αnf(yn) +βnWnyn,

(5)

where {αn},{βn}, and{γn} are three sequences in (, ) such thatαn+βn≤, and the

mappingWnis deﬁned by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Un,n+=I,

Un,n=λnTnUn,n++ ( –λn)I,

Un,n–=λn–Tn–Un,n+ ( –λn–)I, · · ·

Un,k=λkTnUn,k++ ( –λk)I,

Un,k–=λk–Tk–Un,k+ ( –λk–)I, · · ·

Un,=λTUn,+ ( –λ)I,

Wn=Un,=λTUn,+ ( –λ)I.

Theorem .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let G:C×C→(–∞, +∞)be a bifunction satisfying(A)-(A),and let{Ti}∞i=be a sequence

of nonexpansive self-mappings on C such that

S=

∞

i=

F(Ti)∩EP(G)=∅.

Suppose that{αn},{βn},and{γn}are sequences in(, )such thatαn+βn≤,and{rn} ⊂

(,∞)is a real sequence.Suppose that the following conditions are satisﬁed:

(i) limn→∞αn= and

_∞

n=αn=∞;

(ii)  <lim inf_n_→∞βn≤lim supn→∞βn< ;

(iii)  <lim infn→∞γn≤lim supn→∞γn< andlimn→∞|γn+–γn|= ;

(iv)  <lim infn→∞rnandlimn→∞|rn+–rn|= .

Let f be a contraction of C into itself and given x∈C.Then the sequences{xn}and{un}

generated by(.),where{λn}is a sequence in(,b]for some b∈(, ),converge strongly to

x∗∈∞_i_=F(Ti)∩EP(G),where x∗=P∞_i_=F(Ti)∩EP(G)f(x ∗_).

2 Main results

In this section, for solving problem (.), we present a new iterative method.

Letzbe an arbitrary element in a Hilbert spaceH, the sequence of iterations{zn}is

deﬁned by ﬁndingun∈Csuch that

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

G(un,y) +un–zn,y–un ≥, ∀y∈C,

zn+=PC(zn–βn[zn–un+

∞

i=γiAi(zn) +αnzn])

=PC(zn–βn[

∞

i=γiAi(zn) + ( +αn)zn–un]), n≥,

(.)

wherePCis the metric projection ofHontoC, and{αn},{βn}, and{γi}are three sequences

of positive numbers such that

∞

i= γi

  –λi

(6)

Also, we construct a regularization solutionuαfor (.) by solving the following

varia-tional inequality problem: ﬁnduα∈Csuch that

⎧ ⎨ ⎩

A(uα) +

_∞

i=γiAi(uα) +αuα,v–uα ≥, ∀v∈C,

Ai=I–Ti, i≥,

(.)

where T(x) ={z∈C:G(z,v) +z–x,v–z ≥∀v∈C},{γi}is a sequence of positive

numbers satisfying (, ), andαis a small regularization parameter tending to zero. We need the following important lemmas for the proof of our main results.

Lemma .[] Let C be a nonempty closed convex subset of a Hilbert space H,and let G be a bifunction of C×C into(–∞, +∞)satisfying(A)-(A).Let r> and x∈H.Then there exists z∈C such that

G(z,v) +

rz–x,v–z ≥, ∀v∈C.

Lemma .[] Assume that G:C×C→(–∞, +∞)satisﬁes conditions(A)-(A).For r> and x∈H,deﬁne a mapping Tr:H→C as follows:

Tr(x) =

z∈C:G(z,v) +

rz–x,v–z ≥∀v∈C

. (.)

Then the following statements hold:

(i) Tris single-valued;

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

Tr(x) –Tr(y) 

≤Tr(x) –Tr(y),x–y

;

(iii) F(Tr) =EP(G);

(iv) EP(G)is closed and convex.

It is easy to see thatTris a nonexpansive mapping.

Lemma .[] Let{an},{bn},and{cn}be the sequences of positive numbers satisfying the

conditions:

(i) an+≤( –bn)an+cn,bn< ,

(ii) ∞_n_=bn= +∞,limn→+∞bcnn= . Thenlimn→+∞an= .

Lemma .[, ] Assume that T is aλ-strictly pseudocontractive mapping of a closed convex subset C of a Hilbert space H.Then I–T is demiclosed at zero;that is,whenever

{xn}is a sequence in C weakly converging to some x∈C,and the sequence{(I–T)(xn)}

strongly converges to zero,it follows that(I–T)(x) = .

(7)

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let G:C×C→(–∞, +∞)be a bifunction satisfying(A)-(A),and let{Ti}∞i=be a sequence

of nonselfλi-strictly pseudocontractive mappings from C into H such that

S=

∞

i=

F(Ti)∩EP(G)=∅.

Assume that{γi}satisﬁes condition(.).Then for eachα> ,problem(.)has a unique

solution uαsuch that

(i) limα→uα=u∗,u∗∈Sandu∗ ≤ y,∀y∈S,

(ii) uα–uβ ≤|α–_αβ|u∗.

Proof First, we prove that problem (.) has a unique solution. Set

A:=

∞

i= γiAi.

Letx∗be a common ﬁxed point of{Ti}i≥. Since

γiAi(x)≤γiAi(x) –Ai

x∗+γiAi

x∗≤γi

  –λi

_x_–_x∗

and∞_i_=γi_–λi=γ < +∞, the mappingAis well deﬁned, and ∞

i=γiAi(x) converges

ab-solutely for eachx∈C. It is easy to see thatAis Lipschitz continuous with the Lipschitz constantLA=γ and monotone.

Set

Gi(u,v) =

γiAi(u),v–u

, i≥ and

G(u,v) =

A(u),v–u

.

Then, problem (.) is equivalent to that: ﬁnduα∈Csuch that

Gα(uα,v)≥, ∀v∈C, (.)

where

Gα(u,v) =G˜(u,v) +αu,v–u

and

˜

G(u,v) =

∞

i=

Gi(u,v) =

A(u) +A(u),v–u

.

It is not diﬃcult to verify that for eachi≥,Gi(u,v) is a bifunction. Therefore,G˜(u,v)(u,v)

(8)

(/r) =α>  andx= , we can conclude that problem (.) (consequently (.)) has a unique solutionuαfor eachα> .

(i) Next, we prove that

uα ≤ y, ∀y∈S. (.)

Sincey∈S, we haveAi(y) = ,i≥. Thus,

∞

i=

γiAi(uα),y–uα

+αuα,y–uα ≥, ∀y∈S. (.)

Since

γiAi(uα),uα–y

≥, ∀y∈S,i≥,

from (.), follows (.). Then, there exists a subsequence{uαk}of{uα}that converges

weakly to some elementu∗∈H. Now, we have to prove thatu∗∈S. The ( –λl)/-inverse

strongly monotone property ofAlimplies that

≤γl

 –λl

 Al(uαk)



≤γlAl(uαk),uαk–y

≤

∞

i=

γiAi(uαk),uαk–y

≤–αkuαk,uαk–y

≤αky,y–uαk.

Therefore,

lim

k→∞

Al(uαk)= .

By virtue of Lemma ., we haveu∗∈F(Tl). Since the closed convex setShas only one

element with the minimal norm, using (.) and the weak convergence of {uα}, we can

conclude that all the sequence{uα}converges strongly tou∗asα→.

(ii) By virtue of (.), (.) and the monotone property ofAi, we obtain αuα,uβ–uα+βuβ,uα–uβ ≥

or

uα–uβ ≤ |α–β|

α uβ ≤

|α–β|

α u

∗

for eachα,β> . This completes the proof.

(9)

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let G:C×C→(–∞, +∞)be a bifunction satisfying(A)-(A),and let{Ti}∞i=be a sequence

of nonselfλi-strictly pseudocontractive mappings from C into H such that

S=

∞

i=

F(Ti)∩EP(G)=∅.

Assume that{γi}satisﬁes condition(.),andαn,βnsatisfy the following conditions:

αn,βn> , lim

n→∞αn= , nlim→∞

|αn–αn+| α

nβn

= ,

∞

n=

αnβn=∞, limn→∞βn

( +γ +αn) αn

< .

Then the sequence{zn}given by(.)converges strongly to u∗∈S,that is,

lim

n→∞zn=u

∗_∈_S_.

Proof Clearly, iteration{zn}in (.) has the form

zn+=PC

zn–βn

_∞

i=

γiAi(zn) +αnzn

, z∈C,n≥, (.)

whereγ= .

Letunbe the solution of (.) whenα=αn. Then

un=PC

un–βn

_∞

i=

γiAi(un) +αnun

. (.)

Setn=zn–un. Obviously,

n+=zn+–un+ ≤ zn+–un+un+–un.

From the nonexpansive property ofPC, the monotone and Lipschitz continuous properties

ofAi,i≥, (.), and (.), we have

zn+–un ≤

zn–un–βn

_∞

i= γi

Ai(zn) –Ai(un)

+αn(zn–un)

, and

zn–un–βn

_∞

i= γi

Ai(zn) –Ai(un)

+αn(zn–un)



=zn–un+βn

_∞ i= γi

Ai(zn) –Ai(un)

+αn(zn–un)

(10)

– βn

_∞

i= γi

Ai(zn) –Ai(un)

+αn(zn–un),zn–un

≤ zn–un

 – βnαn+βn

 +

∞

i= γi

  –λi

+αn



.

Thus,

zn+–un ≤n

 – βnαn+βn( +γ+αn)

/

≤n( –βnαn)/ ≤n

 – βnαn

.

Set

bn=

 αnβn,

cn=

αn–αn+ αn

u∗.

It is not diﬃcult to check thatbnandcnsatisfy the conditions in Lemma . for suﬃciently

largen. Hence,limn→+∞n= . Sincelimn→∞un=u∗, we have

lim

n→∞zn=u

∗_∈_S_.

This completes the proof.

Remark The sequencesαn= ( +n)–p,  <p< /, andβn=γαnwith

 <γ<

 ( +γ+α)

satisfy all the necessary conditions in Theorem ..

3 Application

In this section, we show that algorithm (.) can be applied to ﬁnd an element

u∗∈S∩VIC,A, (.)

whereVI(C,A_{) is the set of solutions of the following variational inequality problem:}

A(u),v–u≥, ∀v∈C,u∈C, (.)

involving a monotone hemicontinuous mappingA_{. If}_A_{is a}_λ_{-inverse strongly monotone}

mapping, and{Ti}i≥ is a countably inﬁnite family of nonexpansive self mappings inC,

then problem (.) is recently studied in [] and [].

LetAbe a monotone hemicontinuous mapping. Then it is easy to see thatG(u,v) :=

(11)

construct a nonexpansive mappingT_:_H_→_C_{such that}

T(x) :=z∈C:G(z,v) +z–x,v–z ≥∀v∈C.

Then,F(T_{) =}_VI₍_C_,_A_{). So,}_T _as_T

are nonexpansive. Consequently, both mappings

T_and_T

are (/)-strictly pseudocontractive. Thus, the mappingI–Tis also Lipschitz

continuous with the Lipschitz constantL= . Then, algorithm (.) has the form: Letz be an arbitrary element in a Hilbert spaceH, the sequence of iteration{zn}is

deﬁned by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un∈C:G(un,y) +un–zn,y–un ≥, ∀y∈C,

u_n∈C:G(u_n,y) +u_n–zn,y–un ≥, ∀y∈C,

zn+=PC(zn–βn[zn–un+zn–un+

∞

i=γiAi(zn) +αnzn])

=PC(zn–βn[

∞

i=γiAi(zn) + ( +αn)zn–un–un]), n≥.

(.)

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

C×C→(–∞, +∞)be a bifunction satisfying(A)-(A),let{Ti}∞i=be a sequence of nonself λi-strictly pseudocontractive mappings from C into H,and let Abe a hemicontinuous

monotone mapping from C into H such that

∞

i=

F(Ti)∩EP(G)∩VI

C,A=∅.

Assume that{γi}satisﬁes condition(.),andαn,βnsatisfy the following conditions:

αn,βn> , lim

n→∞αn= , nlim→∞

|αn–αn+| α

nβn

= ,

∞

n=

αnβn=∞, limn→∞βn

( +γ+αn) αn

< .

Then

lim

n→∞zn=u ∗_∈∞

i=

F(Ti)∩EP(G)∩VI

C,A,

where{zn}is deﬁned by(.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by JKK. JKK and NB prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Kyungnam University, Changwon, Gyeongnam 631-701, Korea.}2_{Vietnamese Academy of}

(12)

Acknowledgements

This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the Republic of Korea (2013R1A1A2054617).

Received: 13 April 2013 Accepted: 16 September 2013 Published:08 Nov 2013 References

1. Browder, FE, Petryshyn, WV: Construction of ﬁxed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197-228 (1967)

2. Takahashi, W, Toyota, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.118, 417-428 (2003)

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

4. Oettli, W: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam.22, 215-221 (1997) 5. Göpfert, A, Riahi, H, Tammer, C, Zalinescu, C: Variational Methods in Partially Ordered Spaces. Springer, New York

(2003)

6. Chadli, O, Schaible, S, Yao, JC: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities. J. Optim. Theory Appl.121, 571-596 (2004)

7. Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal.6(1), 117-136 (2005)

8. Konnov, IV, Pinyagina, OV: D-gap functions and descent methods for a class of monotone equilibrium problems. Lobachevskii J. Math.13, 57-65 (2003)

9. Stukalov, AC: Regularization extragradient method for solving equilibrium programming problems in Hilbert spaces. Ž. Vyˇcisl. Mat. Mat. Fiz.45(9), 1538-1554 (2005)

10. Kim, JK, Tuyen, TM: Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl.2011(52), 1-10 (2011).

doi:10.1186/1687-1812-2011-52

11. Konnov, IV, Pinyagina, OV: D-gap functions for a class of monotone equilibrium problems in Banach spaces. Comput. Methods Appl. Math.3, 274-286 (2003)

12. Mastroeni, G: Gap functions for equilibrium problems. J. Glob. Optim.27, 411-426 (2003)

13. Anh, PN, Kim, JK: An interior proximal cutting hyperplane method for equilibrium problems. J. Inequal. Appl.2012, 99 (2012). doi:10.1186/1029-242X-2012-99

14. Antipin, AS: Equilibrium programming: gradient methods. Autom. Remote Control58, 1337-1347 (1997)

15. Antipin, AS: Equilibrium programming: proximal methods. Ž. Vyˇcisl. Mat. Mat. Fiz.37, 1327-1339 (1997) Translation in Comput. Math. Math. Phys.37, 1285-1296 (1997)

16. Antipin, AS: Solution methods for variational inequalities with coupled constraints. Comput. Math. Math. Phys.40, 1239-1254 (2000) Translated from Zh. Vychisl. Mat. Mat. Fiz.40, 1291-1307 (2000)

17. Antipin, AS: Solving variational inequalities with coupling constraints with the use of differential equations. Differ. Equ.36, 1587-1596 (2000) Translated from Differ. Uravn.36, 1443-1451 (2000)

18. Bounkhel, M, Al-Senan, BR: An iterative method for nonconvex equilibrium problems. J. Inequal. Pure Appl. Math. 7(2), Article ID 75 (2006)

19. Buong, N: Approximation methods for equilibrium problems and common solution for a ﬁnite family of inverse-strongly monotone problems in Hilbert spaces. Appl. Math. Sci.2, 735-746 (2008)

20. Chadli, O, Konnov, IV, Yao, JC: Descent methods for equilibrium problems in Banach spaces. Comput. Math. Appl.48, 609-616 (2004)

21. Mastroeni, G: On auxiliary principle for equilibrium problems, Technical Report of the Department of Mathematics of Pisa University, Italy 3.244.1258 (2000)

22. Moudaﬁ, A: Second-order diﬀerential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math.4(1), Article ID 18 (2003)

23. Moudaﬁ, A, Théra, M: Proximal and Dynamical Approaches to Equilibrium Problems. Lecture Notes in Econom. and Math. Systems, vol. 477, pp. 187-201. Springer, Berlin (1999)

24. Noor, MA, Noor, KI: On equilibrium problems. Appl. Math. E-Notes4, 125-132 (2004)

25. Kim, JK, Nam, YM, Sim, JY: Convergence theorem of implicit iterative sequences for a ﬁnite family of asymptotically quasi-nonexpansive type mappings. Nonlinear Anal. TMA71(12), 2839-2848 (2009). doi:10.1016/j.na.2009.06.090 26. Kim, JK, Cho, SY, Qin, X: Some results on generalized equilibrium problems involving strictly pseudocontractive

mappings. Acta Math. Sci.31(5), 2041-2057 (2011)

27. Kim, JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and ﬁxed point problems of the asymptotically quasi-nonexpansive mappings. Fixed Point Theory Appl. (2011).

doi:10.1186/1687-1812-2011-10

28. Kim, JK, Buong, N: An iterative method for common solution of a system of equilibrium problems in Hilbert spaces. Fixed Point Theory Appl.2011, Article ID 780764 (2011). doi:10.1155/2011/780764

29. Kim, JK, Buong, N: Regularization inertial proximal point algorithm for monotone hemicontinuous mappings and inverse strongly monotone mappings in Hilbert spaces. J. Inequal. Appl.2010, Article ID 451916 (2010) doi:10.1155/2010/451916

30. Kim, JK, Anh, PN, Nam, YM: Strong convergence of an extended extragradient method for equilibrium problems and ﬁxed point problems. J. Korean Math. Soc.49(1), 187-200 (2012)

31. Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl.329(2), 336-346 (2007)

32. Wang, G, Peng, J, Lee, HJ: Implicit iteration process with mean errors for common ﬁxed point of a family of strictly pseudocontractive maps. Int. J. Math. Anal.1, 89-99 (2007)

(13)

34. Alvarez, F, Attouch, H: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal.9, 3-11 (2001)

35. Maingé, PE: Approximation methods for common ﬁxed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl.325, 469-479 (2007)

36. Ceng, LC, Yao, JC: Hybrid viscosity approximation schemes for equilibrium problems and ﬁxed point problems of inﬁnitely many nonexpansive mappings. Appl. Math. Comput.198, 729-741 (2008)

37. Vasin, VV, Ageev, AL: Incorrect Problems with Priori Information. Nauka, Ekaterenburg (1993)

38. Osilike, MO, Udomene, A: Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings. J. Math. Anal. Appl.256, 431-445 (2001)

39. Li, G, Kim, JK: Demiclosedness principle and asymptotic behavior for nonexpansive mappings in metric spaces. Appl. Math. Lett.14(5), 645-649 (2001)

40. Wangkeeree, R: An extragradient approximation method for equilibrium problems and ﬁxed point problems of a countable family of nonexpansive mappings. Fixed Point Theory Appl.2008, Article ID 134148 (2008)

41. Yao, Y, Liou, YC, Yao, JC: A new hybrid iterative algorithm for ﬁxed point problems, variational inequality problems, and mixed equilibrium problems. Fixed Point Theory Appl.2008, Article ID 417089 (2008)

10.1186/1687-1812-2013-286