A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces

(1)

R E S E A R C H

Open Access

A new iterative algorithm of

pseudomonotone mappings for equilibrium

problems in Hilbert spaces

Jong Kyu Kim

*

_{and Won Hee Lim}

*_{Correspondence:} [email protected] Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, Korea

Abstract

In this paper, we introduce a new algorithm for finding a common element of the set of fixed points ofNstrict pseudocontractions and the set of solutions of equilibrium problems with a pseudomonotone and Lipschitz-type continuous bifunction. The scheme is motivated by the idea of extragradient methods and fixed point iteration methods. We show that the iterative sequences generated by this algorithm converge strongly to the above mentioned common element under some suitable conditions on algorithm parameters in a real Hilbert space. And also, we consider the variational inequality problems as an application.

MSC: 46H09; 47H10; 47J25; 65K10

Keywords: strict pseudocontractions; pseudomonotone; Lipschitz-type continuous; equilibrium problems; ﬁxed points

1 Introduction

LetCbe a nonempty closed convex subset of a real Hilbert spaceHwith the inner product

·,·and the norm · , and letf be a bifunction fromC×CintoRsuch thatf(x,x) =  for allx∈C. We consider the equilibrium problem in the sense of Blum and Oettli []: Find x*_∈_C_{such that}

fx*,y≥ EP(f)

for ally∈C.

We denote bySol(EP(f)) the set of solutions of the equilibrium problemEP(f). We know that the problemEP(f,C) covers many important problems in optimization and nonlinear analysis. It has also found many applications in economics, transportation and engineering (see [, ] and the references quoted therein). Theory and methods for solving this problem have been developed by many authors [–]. Alternatively, the prob-lem of finding a common fixed point of a sequence of finite self-mappings{Si}Ni=(N≥) is described as follows: Findx*_∈_C_{such that}

x*∈

N

i=

F(Si), (FP)

(2)

whereF(Si) is the set of ﬁxed points of the mappingsSi(i= , . . . ,N) onC. This problem

has now become a mature subject in nonlinear analysis. The theory and solution methods of this problem can be found in many research papers and monographs (see [–]).

We are interested in the problem of ﬁnding a common element of the set of solutions of the equilibrium problemEP(f) and the set of solutions of the ﬁxed problem (FP), namely: Findx*_∈_C_{such that}

x*∈

N

i=

F(Si)∩Sol

EP(f). (.)

A special case of problem (.) is thatf(x,y) =F(x),y–x, and this problem is reduced to ﬁnding a common element of the set of solutions of variational inequalities,i.e., ﬁnd x*_∈_C_{such that}

Fx*,x–x*≥, ∀x∈C, VI(F)

and the set solutions of a ﬁxed point problem (see [–]).

In this paper, we introduce a new iterative scheme for solving problem (.). This method can be considered to be an improvement of the viscosity approximation method in [, , ] and the iterative method in [] via an improvement of the extragradient methods [, , –].

The paper is organized as follows. Section  recalls some concepts in equilibrium prob-lems and ﬁxed point probprob-lems that are used in the sequel and an iterative algorithm for solving problem (.). In Section , we prove the convergence theorems for the algorithms which are deﬁned in Section  as the main results of this paper. In Section , we consider the variational inequality problems as an application of the main theorem.

2 Preliminaries

We ﬁrst recall the following deﬁnitions that will be used for the main theorem.

Deﬁnition . LetCbe a nonempty closed convex subset of a real Hilbert spaceH. A bi-functionf :C×C→Ris said to be

(a) monotoneonCiff(x,y) +f(y,x)≤,∀x,y∈C;

(b) pseudomonotoneonCiff(x,y)≥impliesf(y,x)≤,∀x,y∈C; (c) Lipschitz-type continuousonCwith two constantsc> andc> if

f(x,y) +f(y,z)≥f(x,z) –cx–y–cy–z, ∀x,y,z∈C. (.)

We know that every monotone bifunctionf is pseudomonotone, but the converse is not true (see []).

Deﬁnition . LetC be a nonempty closed convex subset of a real Hilbert space H. A mappingS:C→Cis said to be astrict pseudocontractionif there exists a constant ≤L<  such that

S(x) –S(y)≤ x–y+L(I–S)(x) – (I–S)(y), ∀x,y∈C,

(3)

Now, we deﬁne the projection onC, denoted byPrC(·),i.e.,

Pr_C(x) =argminy–x:y∈C , ∀x∈H.

And we use the symbolsand→to denote weak convergence and strong convergence, respectively. The following proposition gives some useful properties for strict pseudocon-tractions.

Proposition .[] Let C be a nonempty closed convex subset of a real Hilbert space H, let S:C→C be an L-strict pseudocontraction,and for each i= , . . . ,N,let Si:C→C be

an Li-strict pseudocontraction for some≤Li< .Then we have the following. (a) Ssatisﬁes the following Lipschitz condition:

S(x) –S(y)≤ +L

 –Lx–y, ∀x,y∈C;

(b) (I–S)is demiclosed at zero.That is,if the sequence{xk_}_{is in}_C_{such that}_xk_x_¯_and

(I–S)(xk₎_→__,_then_(I_–_S)(_{x) = }_¯ _; (c) The setF(S)is closed and convex;

(d) Ifλi> (i= , . . . ,N)andNi=λi= ,thenNi=λiSiis anL¯-strict pseudocontraction,

whereL¯:=max{Li|≤i≤N};

(e) Ifλiis the same as in(d)and{Si|i= , . . . ,N}has a common ﬁxed point,then

F

_N

i=

λiSi

=

N

i= F(Si).

Many authors studied the problem of finding a common fixed point of a finite family of mappings. For instance, Marino and Xu [] constructed an iterative algorithm for finding a common fixed point ofN strict pseudocontractionsSi(i= , . . . ,N). They defined the

sequence{xk_}_{starting from}_x_∈_H_{and taking}

xk+=αkxk+ ( –αk) N

i=

λk,iSi

xk, (.)

where the control sequence of parameters{λk}was made in order to get the guarantee for

the convergence of the iterative sequence{xk}. And they proved that the sequence{xk} converges weakly to the pointx¯∈N_i_=F(Si).

Recently, Chenet al.[] introduced a new iterative scheme for ﬁnding a common el-ement of the set of common ﬁxed points of a sequence of strict pseudocontractions{¯Si}

and the set of solutions of the equilibrium problemEP(f) in a real Hilbert spaceH. Given a starting pointx_∈_{H, three iterative sequences}_{_xk_}_,_{_yk_}_and_{_zk_}_{are generated as the}

following scheme:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Computeyk=αkxk+ ( –αk)S¯k(xk);

Findzk_∈_C_{such that}_f_(zk_,_{y) +} 

rky–z

k_,_zk_–_yk_≥_, _∀_y_∈_C;

Computexk+=PrCk(x

_{), where}_C

k:={v∈C|zk–v ≤ xk–v}.

(4)

Here, two sequences{αk}and{rk}are given as control parameters. The authors proved

that the sequences{xk_}_,_{_yk_}_and_{_zk_}_{converged strongly to the same point}_x*_{, under} cer-tain conditions on{αk}and{rk}, such that

x*∈PrSol(EP(f))∩F(S)

x,

whereSis a nonexpansive mapping ofCinto itself deﬁned by

S(x) = lim

j→∞S¯j(x)

for allx∈C.

The methods for ﬁnding a common element of the setsSol(EP(f)) andN_i_=F(Si) in a

real Hilbert space have been studied in many research papers (see [, , , , –]). We need the following assumptions for the main theorems.

Assumption . The bifunctionf satisﬁes the following conditions:

(i) f is pseudomonotone and weakly continuous onC; (ii) f is Lipschitz-type continuous onC;

(iii) for eachx∈C,f(x,·)is convex and subdiﬀerentiable onC.

Assumption . EverySiis anLi-strict pseudocontraction for some ≤Li< .

Assumption . The solution set of (.) is nonempty,i.e.,

N

i=

F(Si)∩Sol

EP(f)=∅.

Note that ifC⊆ri(dom(f(x,·))), whereri(dom(f(x,·))) is the set of relative interior points of the domain off(x,·), then Assumption .(iii) is satisﬁed. Now we construct the new algorithms as follows.

Algorithm .

Initialization: Choose positive sequences{λk},{αk},{βk},{γk}and{λk,i}satisfying the

following conditions:

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

αk+βk≤, ∀k≥,

lim infk→∞βk∈(, ),

lim infk→∞αk+αkβk ∈(L, ),¯ whereL¯:=max{Li|≤i≤N},

lim infk→∞(γk+ ( –γk)(αk+βk)) > , {γk} ⊂(, ),

{λk} ⊂[a,b] for somea,b∈(,_L), whereL:=max{c, c},

p

i=λk,i=  for allk≥.

Take an initial pointx_∈_C_{and set}_k_{:= .}

(5)

• Step . Solve two strongly convex programs:

⎧ ⎨ ⎩

yk:=argmin{λkf(xk,y) +_y–xk|y∈C},

tk_:=_argmin_{_λ

kf(yk,y) +_y–xk|y∈C}.

• Step . Compute the iterations

⎧ ⎨ ⎩ ¯

yk:= ( –γk)xk+γktk,

zk_{:= ( –}_α

k–βk)y¯k+αktk+βkNi=λk,iSi(tk).

• Step . Set

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Ck:={z∈C| zk–z≤ xk–z–βk( α_k

αk+βk –L)¯ ¯Sk(t

k_{) –}_tk_}_,

whereS¯k:=

N

i=λk,iSi(xk),

Qk:={z∈C| xk–z,x–xk ≥}.

Computexk+:=PrCk∩Qk(x ₎_.

Increasekby one and go back to Step.

3 Convergence of the algorithms

In this section, we study the convergence of Algorithm .. We need the following useful lemmas for the main theorems.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H,and let g:C→Rbe subdiﬀerentiable on C.Then x*_{is a solution of the following convex problem:}

ming(x)|x∈C

if and only if

∈∂gx*+NC

x*,

where∂g(·)denotes the subdiﬀerential of g and NC(x*)is the(outward)normal cone of C

at x*_∈_C.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H and x∈H.Let{xk}be a bounded sequence such that every weakly cluster pointx of¯ {xk}belongs to C and

xk–x≤x–Pr_Cx, ∀k≥.

Then{xk}converges strongly toPrC(x)as k→ ∞.

(6)

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Suppose that Assumptions.-.are satisﬁed.Then the sequences{xk_}_,_{_yk_}_and_{_zk_}_{generated by}

Algorithm.converge strongly to the same point x*∈N_i_=F(Si)∩Sol(EP(f)),where

x*=PrN

i=F(Si)∩Sol(EP(f))

x. (.)

Proof The proof of this theorem is divided into several steps. Step . Suppose thatx*_∈N

i=F(Si)∩Sol(EP(f)). Then we have

tk–x*≤xk–x*– ( – λkc)tk–yk 

– ( – λkc)xk–yk, ∀k≥. (.)

Sincef(x,·) is convex onCfor eachx∈C, by Lemma ., we see that

tk=argmin

 t–x

k +λkf

yk,t t∈C

if and only if

∈∂

λkf

yk,y+ y–x

k tk+NC

tk, (.)

whereNC(x) is the (outward) normal cone ofCatx∈C.

Sincef(yk_,_·_{) is subdiﬀerentiable on}_{C, by the well-known Moreau-Rockafellar theorem}

(see []), there existsw∈∂f(yk_,_tk_{) such that}

fyk,t–fyk,tk≥w,t–tk, ∀t∈C.

Substitutingt=x*_{into this inequality, we obtain}

fyk,x*–fyk,tk≥w,x*–tk. (.)

And also, it follows from (.) that  =λkw+tk–xk+w, where¯ w∈∂f(yk,tk) andw¯ ∈

NC(tk). By the deﬁnition of the normal coneNC, we have

tk–xk,t–tk≥λk

w,tk–t, ∀t∈C. (.)

Substitutingt=x*_∈_C_{into the last inequality, we obtain}

tk–xk,x*–tk≥λk

w,tk–x*. (.)

Combining (.) and (.), we have

fyk,tk–fyk,x*. (.)

Sincex*_∈_Sol_(EP(f_)),_f_(x*_,_y)_≥_{ for all}_y_∈_{C, and}_f _{is pseudomonotone on}_{C, we have} f(yk_,_x*₎_≤_{. Hence, (.) implies that}

tk–xk,x*–tk≥λkf

(7)

From Lipschitz condition (.) forf withx=xk_,_y₌_yk_and_z₌_tk_{, we have}

fyk,tk≥fxk,tk–fxk,yk–cyk–xk–ctk–yk. (.)

Combining (.) and (.), we get

fxk,tk–fxk,yk–cyk–xk 

–ctk–yk 

. (.)

Similarly, sinceykis the unique solution of the strongly convex program

min

 y–x

k +λkf

xk,y y∈C

,

we have

λk

fxk,y–fxk,yk≥yk–xk,yk–y, ∀y∈C.

Substitutingy=tk∈Cinto the last inequality, we have

λk

fxk,tk–fxk,yk≥yk–xk,yk–tk. (.)

Since

tk–xk,x*–tk=xk–x*–tk–xk–tk–x*,

from (.), (.), we have

_xk_–_x*_–_tk_–_xk_–_tk_–_x*_≥__yk_–_xk_,_yk_–_tk_{– }_λ

kcxk–yk



– λkctk–yk 

.

Hence, we have

tk–x*≤xk–x*–tk–xk– yk–xk,yk–tk

+ λkcxk–yk



+ λkctk–yk



=xk–x*–tk–yk+yk–xk– yk–xk,yk–tk

+ λkcxk–yk



+ λkctk–yk



≤xk_–_x*

–tk_–_yk_–_xk_–_yk_{+ }_λ

kcxk–yk 

+ λkctk–yk 

=xk–x*– ( – λkc)xk–yk 

– ( – λkc)yk–tk 

.

The implies that the inequality (.) holds. Step . Next, we show that

N

i=

F(Si)∩Sol

EP(f)⊆Ck

(8)

Using Step  andy¯k_{= ( –}_γ

k)xk+γktk, we have

y_¯k–x*=( –γk)

xk–x*+γk

tk–x*

≤( –γk)xk–x*+γktk–x*

≤( –γk)xk–x*+γkxk–x*– ( – λkc)xk–yk

– ( – λkc)yk–tk

=xk–x*–γk( – λkc)xk–yk 

–γk( – λkc)yk–tk 

, (.)

wherex*_∈_Sol_(EP(f_)). Set

¯

Sk:= N

i=

λk,iSi.

Letx*_∈p

i=Fix(Si)∩Sol(EP(f)), using Proposition .(d), (.) and the relation

λx+ ( –λ)y=λx+ ( –λ)y–λ( –λ)x–y, ∀x,y∈H,λ∈[, ],

andzk= ( –αk–βk)y¯k+αktk+βkS¯k(tk), we have

zk–x*

=( –αk–βk)

¯

yk–x*+ (αk+βk)



αk+βk

αk

tk–x*+βk

_¯

Sk

tk–x* 

≤( –αk–βk)y¯k–x*



+ (αk+βk)

αk

αk+βk

tk–x*+ βk

αk+βk

_¯

Sk

tk–x* 

= ( –αk–βk)y¯k–x*



+αktk–x*

 +βkS¯k

tk–x*– αkβk

αk+βk

S¯k

tk–tk



+αktk–x*



+βktk–x*



+L¯(I–S¯k)

tk– (I–S¯k)

x*– αkβk

αk+βk

S¯k

tk–tk



+ (αk+βk)tk–x*

 +

βkL¯–

αkβk

αk+βk

_¯

Sk

tk–tk

≤( –αk–βk)xk–x*



–γk( – λkc)xk–yk



–γk( – λkc)yk–tk



+ (αk+βk)xk–x*



– ( – λkc)xk–yk



– ( – λkc)yk–tk



+

βkL¯–

αkβk

αk+βk

_¯

Sk

tk–tk

≤xk–x*–mk( – λkc)xk–yk–mk( – λkc)yk–tk

–βk

αk

αk+βk

–L¯S¯k

tk–tk

≤xk–x*–βk

αk

αk+βk

–L¯S¯k

(9)

wheremk=γk+ ( –γk)(αk+βk). This means thatx*∈Ck. Hence

N

i=

F(Si)∩Sol

EP(f)⊆Ck, ∀k≥.

Step . Now, we have to prove that

N

i=

F(Si)∩Sol

EP(f)⊆Ck∩Qk

for allk≥.

We show this assertion by mathematical induction. Fork=  we haveQ=C. Hence by Step , we obtain

N

i=

F(Si)∩Sol

EP(f)⊆P∩Q.

Assume that for somek≥,

N

i=

F(Si)∩Sol

EP(f)⊆Ck∩Qk. (.)

Fromxk+₌_Pr

Ck∩Qk(x

_{) it follows that}

xk+–x,x–xk+≥, ∀x∈Ck∩Qk.

Using this and (.), we have

xk+–x,x–xk+≥, ∀x∈

N

i=

F(Si)∩Sol

EP(f).

Hence we have

N

i=

F(Si)∩Sol

EP(f)⊆Qk+.

Then it follows from Step  that

N

i=

F(Si)∩Sol

EP(f)⊆Ck+∩Qk+.

Consequently, we have

N

i=

F(Si)∩Sol

(10)

Step . Next, we claim that

lim

k→∞x

k+_–_xk₌ _lim k→∞x

k_–_zk

= lim

k→∞x k_–_yk

= lim

k→∞

xk–tk

= lim

k→∞

S¯k

tk–tk

= .

It follows from Step  andxk+=PrCk∩Qk(x _{) that}

xk+–x≤PrN

x–x, ∀k≥. (.)

Hence, we get that{xk}is bounded. By Step , also the sequences{tk}and{zk}are bounded. Otherwise, we have

xk–x,x–xk≥, ∀x∈Qk,

and hencexk₌_Pr Qk(x

_{). Using this and}_xk+_∈_C

k∩Qk⊆Qk, we have

xk–x≤xk+–x, ∀k≥.

Therefore, there exists

A= lim

k→∞x

k_–_x_. _(.)

Usingxk₌_Pr Qk(x

_),_xk+_∈_Q

kand the property of projections

PrQk(x) –x _≤

x–y–PrQk(x) –y 

, ∀x∈H,y∈Qk,

we have

xk+–xk≤xk+–x–xk–x, ∀k≥.

Combining this and (.), we get

lim

k→∞

xk+–xk= . (.)

It follows fromxk+₌_Pr

Ck∩Qk(x

_{) that}_xk+_∈_C

k,i.e.,

zk–xk+≤xk–xk+.

Hence

(11)

Then, by (.), we have

lim

k→∞

xk–zk= . (.)

Step  and (.) imply that {tk_} _{is bounded, and hence}_{¯_S

k(tk) –tk} and{zk} are also

bounded.

By (.), we have

βk

αk

αk+βk

–L¯S¯k

tk–tk≤xk–x*–zk–x*

=xk–x*–zk–x*xk–x*+zk–x*

≤xk–zkxk–x*+zk–x*.

From this and (.), we obtain

lim

k→∞S¯k

tk_–_tk_{= .} _(.)

Using (.), we also have

mk( – λkc)xk–yk 

≤xk–zkxk–x*+zk–x*,

and hence

lim

k→∞

xk–yk= . (.)

Similarly, we have

lim

k→∞

tk–yk= . (.)

Combining (.), (.) andxk_–_tk_≤_xk_–_yk₊_yk_–_tk_{, we have}

lim

k→∞

xk–tk= . (.)

This completes the proof of Step .

In Step  and Step  of this theorem, we consider weakly clusters of{xk_}_{. It follows from}

(.) that the sequence{xk_}_{is bounded, and hence there exists a subsequence}_{_xkj} con-verging weakly tox¯asj→ ∞. By Step , also the sequences{ykj}_,{_tkj}_and{_zkj}_converge weakly tox.¯

Step . Claim thatx¯∈N_i_=F(Si).

For eachi= , . . . ,N, we suppose that{λkj,i}convergesλ¯iasj→ ∞such that

p

i=λ¯i= .

Then we have

Skj(x)→S(x) :=

N

i=

¯

(12)

SinceN_i_=λ¯i= , from Step  and

tkj_–_S_tkj≤_tkj_–_S¯

kj

tkj₊_S¯

kj

tkj_–_S_tkj

=tkj_–_S¯

kj

tkj₊

N i=

λkj,iSi

tkj_–

N

i=

¯

λiSi

tkj

=tkj_–_S¯

kj

tkj₊

N i=

(λkj,i–λ¯i)Si

tkj

≤tkj_–_S¯

kj

tkj₊

N

i=

|λkj,i–λ¯i|Si

tkj_,

we obtain thatlim_k_→∞tkj_–_S(tkj₎_{= . By Proposition .(b), we have}

¯

x∈F(S) =F

_N

i=

¯

λiSi

.

Then, it implies thatx¯∈N_i_=F(Si) from Proposition .(e).

Step . Now we prove that ifxkj_x¯_as_j→ ∞_{, then we have}_x¯∈_Sol_(EP(f_)). Sinceykis the unique strongly convex problem

min

 x–x

k

+fxk,y y∈C

,

from Lemma ., we have

∈∂

λkf

xk,y+ y–x

k yk+NC

yk.

It follows that

 =λkw+yk–xk+w,¯

wherew∈∂f(xk_,_yk_{) and}_w_¯_∈_N

C(yk). The deﬁnition of the normal coneNCimplies that

yk–xk,y–yk≥λk

w,yk–y, ∀y∈C. (.)

On the other hand, sincef(xk_,_·_{) is subdiﬀerentiable on}_{C, by the Moreau-Rockafellar}

the-orem [], there existsw∈∂f(xk_,_yk_{) such that}

fxk,y–fxk,yk≥w,y–yk, ∀y∈C.

Combining this with (.), we have

λk

fxk,y–fxk,yk≥yk–xk,yk–y, ∀y∈C.

Hence

λkj

(13)

Then, using{λk} ⊂[a,b]⊂(,_L), Step ,xkj x¯asj→ ∞and weak continuity off, we

have

f(x,¯ y)≥, ∀y∈C.

This means thatx¯∈Sol(EP(f)).

Step . Finally, we claim that the sequences{xk_}_,_{_yk_}_,_{_zk_}_and_{_tk_}_{converge strongly to}

the same pointx*, where

x*=PrN

x.

From Step  and Step  it follows that for every weakly cluster pointx¯of the sequence

{xk_}_,

¯

x∈

N

i=

F(Si)∩Sol

EP(f).

On the other hand, using the deﬁnition ofQk, we have

xk=Pr_Q_kx.

Combining this with (.), we obtain

x–xk≤x–x

for allx∈N_i_=F(Si,C)∩Sol(EP(f)). Forx=x*, we have

x–xk≤x–x*.

By Lemma ., we know that the sequence{xk_}_{converges strongly to}_x*_as_k_{→ ∞}_{, where}

x*=PrN

i=F(Si,C)∩Sol(EP(f))

x.

We also have thatyk_,_zk_,_tk_→_x*_as_k_{→ ∞}_{by Step .}

4 Applications

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetFbe a function fromCintoH. In this section, we consider the variational inequality problem which is presented as follows:

Findx*_∈_C_{such that}

Fx*,x–x*≥, ∀x∈C. VI(F)

Letf :C×C→Rbe deﬁned byf(x,y) :=F(x),y–x. Then problemEP(f) can be written inVI(F). The set of solutions ofVI(F) is denoted bySol(VI(F)).

(14)

• strongly monotoneonCwithβ> if

F(x) –F(y),x–y≥βx–y, ∀x,y∈C;

• monotoneonCif

F(x) –F(y),x–y≥, ∀x,y∈C;

• pseudomonotoneonCif

F(y),x–y≥ ⇒ F(x),x–y≥, ∀x,y∈C;

• Lipschitz continuousonCwith constantsL> if

F(x) –F(y)≤Lx–y, ∀x,y∈C.

Since

yk=argmin

λkf

xk,y+ y–x

k

y∈C

=argmin

λk

Fxk,y–xk+ y–x

k_y_∈_C

=PrC

xk–λkF

xk,

from Algorithm ., we obtain the algorithm for ﬁnding a common element of the set of ﬁxed points ofpstrict pseudocontractions and the solution set of variational inequality problemVI(F).

Algorithm .

Initialization: Choose positive sequences{λk},{αk},{βk},{γk}and{λk,i}satisfying the

conditions:

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

αk+βk≤, ∀k≥,

lim infk→∞βk∈(, ),

lim infk→∞αkα+kβk ∈(L, ),¯ whereL¯:=max{Li|≤i≤N}, lim infk→∞(γk+ ( –γk)(αk+βk)) > , {γk} ⊂(, ), {λk} ⊂[a,b] for somea,b∈(,_L),

N

i=λk,i=  for allk≥.

Find an initial pointx_∈_C.

Iterationk: Perform the three steps below.

• Step . Solve two strongly convex programs: ⎧

⎨ ⎩

yk_:=_Pr

C(xk–λkF(xk)),

tk_:=_Pr

(15)

• Step . Compute the iterations ⎧

⎨ ⎩ ¯

yk_{:= ( –}_γ

k)xk+γktk,

zk:= ( –αk–βk)y¯k+αktk+βk

N

i=λk,iSi(tk).

• Step . Set ⎧ ⎨ ⎩

Ck:={z∈C| zk–z≤ xk–z–βk( α_k

αk+βk –L)¯ ¯Sk(t

k_{) –}_tk_}_,

Qk:={z∈C| xk–z,x–xk ≥}.

Computexk+_:=_Pr

Ck∩Qk(x ₎_.

Increasekby one and go back to Step.

Now, we can prove the following convergence theorem with respect toVI(F) from The-orem ..

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let F be a function from C into H such that F is pseudomonotone,weakly continuous and L-Lipschitz continuous on C.If each i= , . . . ,N,Si:C→C is Li-strict pseudocontraction

for some≤Li< and

N

i=

F(Si)∩Sol

VI(F)=∅,

then the sequences{xk},{yk}and{zk}generated by Algorithm.converge strongly to the same point x*_∈p

i=Fix(Si)∩Sol(F,C),where

x*=PrN

i=F(Si)∩Sol(VI(F))

x.

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 WHL prepared the manuscript initially and performed all the steps of proof in this research. Both authors read and approved the ﬁnal manuscript.

Acknowledgements

This work was supported by the Kyungnam University Foundation Grant 2011.

Received: 28 November 2012 Accepted: 8 March 2013 Published: 26 March 2013 References

1. Blum, E, Oettli, W: From optimization and variational inequality to equilibrium problems. Math. Stud.63, 127-149 (1994)

2. Daniele, P, Giannessi, F, Maugeri, A: Equilibrium Problems and Variational Models. Kluwer Academic, Dordrecht (2003) 3. Anh, PN: A logarithmic quadratic regularization method for solving pseudo-monotone equilibrium problems. Acta

Math. Vietnam.34, 183-200 (2009)

4. Anh, PN: An LQP regularization method for equilibrium problems on polyhedral. Vietnam J. Math.36, 209-228 (2008) 5. Chang, SS, Cho, YJ, Kim, JK: Approximation methods of solutions for equilibrium problem in Hilbert spaces. Dyn. Syst.

Appl.17, 503-508 (2008)

6. Mastroeni, G: Gap function for equilibrium problems. J. Glob. Optim.27, 411-426 (2004)

(16)

8. Goebel, K, Kirk, WA: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

9. Kim, JK, Sahu, DR, Nam, YM: Convergence theorem for ﬁxed points of nearly uniformlyL-Lipschitzian asymptotically generalizedφ-hemicontractive mappings. Nonlinear Anal. TMA71, 2833-2838 (2009). doi:10.1016/j.na.2009.06.091 10. 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, 2839-2848 (2009). doi:10.1016/j.na.2009.06.090 11. Chang, SS, Lee, HWJ, Chan, CK, Kim, JK: Approximating solutions of variational inequalities for asymptotically

nonexpansive mappings. Appl. Math. Comput.212, 51-59 (2009)

12. 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).

doi:10.1186/1687-1812-2011-10

13. Kim, JK, Cho, SY, Qin, X: Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Math. Sci., Ser. B31(5), 985-996 (2011)

14. 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)

15. Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and ﬁxed point problems in Hilbert spaces. J. Math. Anal. Appl.331, 506-515 (2007)

16. Takahashi, S, Toyoda, M: Weakly convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.118, 417-428 (2003)

17. Zeng, LC, Yao, JC: Strong convergence theorem by an extragradient method for ﬁxed point problems and variational inequality problems. Taiwan. J. Math.10, 1293-1303 (2010)

18. Li, XS, Huang, J, Kim, JK: General viscosity approximation methods for common ﬁxed points of nonexpansive semigroup in Hilbert spaces. Fixed Point Theory Appl.2011, Article ID 783502 (2011). doi:10.1155/2011/783502 19. Li, XS, Kim, JK, Huang, NJ: Viscosity approximation of common ﬁxed points forL-Lipschitzian semigroup of

pseudocontractive mappings in Banach spaces. J. Inequal. Appl.2009, Article ID 936121 (2009). doi:10.1155/2009/936121

20. Chen, R, Shen, X, Cui, S: Weak and strong convergence theorems for equilibrium problems and countable strict pseudocontractions mappings in Hilbert space. J. Inequal. Appl. (2010). doi:10.1155/2010/474813

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

22. Kim, JK, Anh, PN, Nam, YM: Strong convergence of an extended extragradient method for equilibrium problems and ﬁxed point problems. J. Korean Math. Soc.47, 187-200 (2011)

23. 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, Article ID 52 (2011).

doi:10.1186/1687-1812-2011-52

24. Schaible, S, Karamardian, S, Crouzeix, JP: Characterizations of generalized monotone maps. J. Optim. Theory Appl.76, 399-413 (1993)

25. Acedo, GL, Xu, HK: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal.67, 2258-2271 (2007)

26. Marino-Yanes, C, Xu, HK: Strong convergence of the CQ method for ﬁxed point processes. Nonlinear Anal.64, 2400-2411 (2006)

27. Ceng, LC, Petrusel, A, Lee, C, Wong, MM: Two extragradient approximation methods for variational inequalities and ﬁxed point problems of strict pseudo-contractions. Taiwan. J. Math.13, 607-632 (2009)

28. Wang, S, Cho, YJ, Qin, X: A new iterative method for solving equilibrium problems and ﬁxed point problems for inﬁnite family of nonexpansive mappings. Fixed Point Theory Appl. (2010). doi:10.1155/2010/165098 29. Wang, S, Guo, B: New iterative scheme with nonexpansive mappings for equilibrium problems and variational

inequality problems in Hilbert spaces. J. Comput. Appl. Math.233, 2620-2630 (2010)

30. Yao, Y, Liou, YC, Wu, YJ: An extragradient method for mixed equilibrium problems and ﬁxed point problems. Fixed Point Theory Appl. (2009). doi:10.1155/2009/632819

31. Rockafellar, RT: Convex Analysis. Princeton University Press, Princeton (1970)

32. Kim, JK, Cho, SY, Qin, X: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J. Inequal. Appl.2010, Article ID 312602 (2010). doi:10.1155/2010/312602 doi:10.1186/1029-242X-2013-128