On variational inequality, fixed point and generalized mixed equilibrium problems

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R E S E A R C H

Open Access

On variational inequality, ﬁxed point and

generalized mixed equilibrium problems

Dong Feng Li

1*

_{and Juan Zhao}

2 *_{Correspondence: [email protected]}

1_{School of Information Engineering,}

North China University of Water Resources and Electric Power, Zhengzhou, Henan 450011, China Full list of author information is available at the end of the article

Abstract

In this article, variational inequality, ﬁxed point, and generalized mixed equilibrium problems are investigated based on an extragradient iterative algorithm. Weak convergence of the extragradient iterative algorithm is obtained in Hilbert spaces.

Keywords: ﬁxed point; equilibrium problem; monotone mapping; nonexpansive

mapping; projection

1 Introduction

In this paper, we always assume thatHis a real Hilbert space with the inner product·,· and the norm · , andCis a nonempty, closed, and convex subset ofH.Ris denoted by the set of real numbers. LetFbe a bifunction ofC×CintoR. Consider the problem: ﬁnd apsuch that

F(p,y)≥, ∀y∈C. (.)

In this paper, the solution set of the problem is denoted byEP(F),i.e.,

EP(F) =p∈C:F(p,y)≥,∀y∈C.

The above problem is ﬁrst introduced by Ky Fan []. In the sense of Blum and Oettli [], the Ky Fan problem is also called an equilibrium problem.

Recently, the ‘so-called’ generalized mixed equilibrium problem has been investigated by many authors: The generalized mixed equilibrium problem is to ﬁnd p∈C such that

F(p,y) +Ap,y–p+ϕ(y) –ϕ(p)≥, ∀y∈C, (.)

whereϕ:C→Ris a real valued function andA:C→His mapping. We useGMEP(F, A,ϕ) to denote the solution set of the equilibrium problem. That is,

GMEP(F,A,ϕ) :=p∈C:F(p,y) +Ap,y–p+ϕ(y) –ϕ(z)≥,∀y∈C.

Next, we give some special cases.

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IfA= , then the problem (.) is equivalent to ﬁndp∈Csuch that

F(p,y) +ϕ(y) –ϕ(z)≥, ∀y∈C, (.)

which is called the mixed equilibrium problem.

IfF= , then the problem (.) is equivalent to ﬁndp∈Csuch that

Ap,y–p+ϕ(y) –ϕ(z)≥, ∀y∈C, (.)

which is called the mixed variational inequality of Browder type. Ifϕ= , then the problem (.) is equivalent to ﬁndp∈Csuch that

F(p,y) +Ap,y–p ≥, ∀y∈C, (.)

which is called the generalized equilibrium problem.

IfA=  andϕ= , then the problem (.) is equivalent to (.).

For solving the above equilibrium problems, let us assume that the bifunctionF:C× C→Rsatisﬁes the following conditions:

(A) F(x,x) = ,∀x∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤,∀x,y∈C; (A)

lim sup t↓

Ftz+ ( –t)x,y≤F(x,y), ∀x,y,z∈C;

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

Equilibrium problems have intensively been studied. It has been shown that equilibrium problems cover ﬁxed point problems, variational inequality problems, inclusion problems, saddle problems, complementarity problem, minimization problem, and Nash equilib-rium problem; see [–] and the references therein.

LetS:C→Cbe a mapping. In this paper, we useF(S) to stand for the set of ﬁxed points. Recall that the mappingSis said to be nonexpansive if

Sx–Sy ≤ x–y, ∀x,y∈C.

Sis said to beκ-strictly pseudocontractive if there exists a constantκ∈[, ) such that

Sx–Sy≤ x–y+κx–y–Sx+Sy, ∀x,y∈C.

It is clear that the class ofκ-strictly pseudocontractive includes the class of nonexpansive mappings as a special case. The class ofκ-strictly pseudocontractive mappings was intro-duced by Browder and Petryshyn []; for existence and approximation of ﬁxed points of the class of mappings, see [–] and the references therein.

LetA:C→Hbe a mapping. Recall thatAis said to be monotone if

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Ais said to beκ-inverse strongly monotone if there exists a constantα>  such that

Ax–Ay,x–y ≥κAx–Ay, ∀x,y∈C.

It is clear that theκ-inverse being strongly monotone is monotone and Lipschitz contin-uous.

A set-valued mappingT:H→H_{is said to be monotone if, for all}_x,_y_∈_H,_f _∈_Tx_and

g∈Tyimplyx–y,f –g> . A monotone mappingT:H→H_{is maximal if the graph}

G(T) ofT is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingT is maximal if and only if, for any (x,f)∈H×H, x–y,f–g ≥ for all (y,g)∈G(T) impliesf ∈Tx. The class of monotone operators is one of the most important classes of operators. Within the past several decades, many authors have been devoting their eﬀorts to the studies of the existence and convergence of zero points for maximal monotone operators.

LetF(x,y) =Ax,y–x,∀x,y∈C. We see that the problem (.) is reduced to the follow-ing classical variational inequality. Findx∈Csuch that

Ax,y–x ≥, ∀y∈C. (.)

It is well known thatx∈Cis a solution to (.) if and only ifxis a ﬁxed point of the mapping PC(I–ρA), whereρ>  is a constant, andIis the identity mapping. IfCis bounded, closed,

and convex, then the solution set of the variational inequality (.) is nonempty. In order to prove our main results, we need the following lemmas.

Lemma .[] Let S:C→C be aκ-strictly pseudocontractive mapping.Deﬁne St:C→

C by Stx=tx+ ( –t)Sx for each x∈C.Then,as t∈[κ, ),St is nonexpansive such that

F(St) =F(S).

Lemma .[] Let C be a nonempty,closed,and convex subset of H,and F:C×C→Ra bifunction satisfying(A)-(A).Then,for any r> and x∈H,there exists z∈C such that

F(z,y) +

ry–z,z–x ≥, ∀y∈C.

Further,deﬁne

Trx=

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

for all r> and x∈H.Then the following hold:

(a) Tris single-valued;

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

Trx–Try≤ Trx–Try,x–y;

(c) F(Tr) =EP(F);

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Lemma .[] Let A be a monotone mapping of C into H and NCv the normal cone to C

at v∈C,i.e.,

NCv=

w∈H:v–u,w ≥,∀u∈C

and deﬁne a mapping T on C by

Tv=

⎧ ⎨ ⎩

Av+NCv, v∈C,

∅, v∈/C.

Then T is maximal monotone and∈Tv if and only ifAv,u–v ≥for all u∈C.

Lemma .[] Let{an}∞n=be real numbers in[, ]such that

∞

n=an= .Then we have

the following:

∞

i=

aixi



≤ ∞

i=

aixi

for any given bounded sequence{xn}∞n=in H.

Lemma .[] Let <p≤tn≤q< for all n≥.Suppose that{xn}and{yn}are

se-quences in H such that

lim sup

n→∞ xn ≤d, lim supn→∞ yn ≤d

and

lim

n→∞tnxn+ ( –tn)yn=d

hold for some r≥.Thenlimn→∞xn–yn= .

Lemma .[] Let C be a nonempty,closed,and convex subset of H,and S:C→C a strictly pseudocontractive mapping. If {xn} is a sequence in C such that xnx and limn→∞xn–Sxn= ,then x=Sx.

Lemma .[] Let{an},{bn},and{cn}be three nonnegative sequences satisfying the

fol-lowing condition:

an+≤( +bn)an+cn, ∀n≥n,

where n is some nonnegative integer,

∞

n=bn<∞ and

∞

n=cn<∞. Then the limit limn→∞anexists.

2 Main results

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continuous and monotone mapping.Let Fmbe a bifunction from C×C toRwhich satisﬁes

(A)-(A),Bm:C→H a continuous and monotone mapping,ϕm:C→Ra lower

semicon-tinuous and convex function for each m≥.Assume thatF:=∞_m_=GMEP(Fm,Bm,ϕm)∩ VI(C,A)∩F(S)is not empty.Let{αn},{βn},and{δn,m}be real number sequences in(, ).

Let{λn},{rn,m}be positive real number sequences.Let{xn}be a sequence generated in the

following manner:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈H,

xn+=αnxn+ ( –αn)(βnI+ ( –βn)S)ProjC(

_∞

m=δn,mzn,m–λnAyn), n≥,

yn=ProjC(

∞

m=δn,mzn,m–λnA

∞

m=δn,mzn,m),

where zn,mis such that

Fm(zn,m,z)+Bmzn,m,z–zn,m+ϕm(z)–ϕm(zn,m)+

 rn,m

z–zn,m,zn,m–xn ≥, ∀z∈C.

Assume that{αn},{βn},{δn,m},{λn},{rn,m}satisfy the following restrictions: (a)  <a≤αn≤b< ;

(b) κ≤βn≤c< ;

(c) ∞_m_=δn,m= ,and <d≤δn,m≤;

(d) lim infn→∞rn,m> ande≤λn≤f,wheree,f ∈(, /L).

Then the sequence{xn}weakly converges to some pointx¯∈F.

Proof The proof is split into ﬁve steps.

Step . Show that the sequence{xn}is bounded.

DeﬁneGm(p,y) =Fm(p,y) +Bmp,y–p+ϕm(y) –ϕm(p),∀p,y∈C. Next, we prove that

the bifunctionGmsatisﬁes the conditions (A)-(A). Therefore, generalized mixed

equi-librium problem is equivalent to the following equiequi-librium problem: ﬁndp∈Csuch that Gm(p,y)≥,∀y∈C. It is clear thatGm satisﬁes (A). Next, we proveGm is monotone.

SinceBmis a continuous and monotone operator, we ﬁnd from the deﬁnition ofGthat

Gm(y,z) +Gm(z,y) =Fm(y,z) +Bmy,z–y+ϕm(z) –ϕm(y) +Fm(z,y)

+Bmz,y–z+ϕm(y) –ϕm(z)

=Fm(z,y) +Fm(y,z) +Bmz,y–z+Bmy,z–y

≤ Bmz–Bmy,y–z

≤.

Next, we showGmsatisﬁes (A), that is,

lim sup t↓

Gm

tz+ ( –t)x,y≤Gm(x,y), ∀x,y,z∈C.

SinceBmis continuous andϕmis lower semicontinuous, we have

lim sup t↓

Gm

tz+ ( –t)x,y=lim sup t↓

Fm

tz+ ( –t)x,y

+lim sup t↓

Bm

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+lim sup t↓

ϕm(y) –ϕm

tz+ ( –t)x

≤Fm(x,y) +Bmx,y–x+ϕm(y) –ϕm(x)

=Gm(x,y).

Next, we show that, for eachx∈C,y→Gm(x,y) is a convex and lower semicontinuous.

For eachx∈C, for allt∈(, ) and for ally,z∈C, sinceFmsatisﬁes (A) andϕmis convex,

we have

Gm

x,ty+ ( –t)z

=Fm

x,ty+ ( –t)z+Bmx,ty+ ( –t)z–x

+ϕm

ty+ ( –t)z–ϕm(x)

≤tFm(x,y) +Bmx,y–x+ϕm(y) –ϕm(x)

+ ( –t)Fm(x,z) +Bmx,z–x+ϕm(z) –ϕm(x)

=tGm(x,y) + ( –t)Gm(x,z).

Thus,y→Gm(x,y) is convex. Similarly, we ﬁnd thaty→Gm(x,y) is also lower

semicon-tinuous. Putun=ProjC( N

m=δn,mzn,m–λnAyn) andvn= N

m=δn,mzn,m. Lettingp∈F, we

see that

un–p≤ vn–λnAyn–p–vn–λnAyn–un

=vn–p–vn–un+ λn

Ayn–Ap,p–yn+Ap,p–yn

+Ayn,yn–un

≤ vn–p–vn–yn–yn–un+ vn–λnAyn–yn,un–yn.

Notice thatAisL-Lipschitz continuous andyn=ProjC(vn–λnAvn). It follows that

vn–λnAyn–yn,un–yn ≤λnLvn–ynun–yn.

It follows that

un–p≤ vn–p+

λ_nL– vn–yn. (.)

On the other hand, we have

vn–p≤

∞

m=

δn,mzn,m–p



≤ ∞

m=

δn,mTrn,mxn–p



≤ xn–p, (.)

whereTrn,m={z∈C:Gm(z,y) +



ry–z,z–x ≥,∀y∈C}. Substituting (.) into (.), we

obtain

un–p≤ xn–p+

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PuttingSn=βnI+ ( –βn)S, we ﬁnd from Lemma . thatSnis nonexpansive andF(Sn) =

F(S). It follows that

xn+–p≤αnxn–p+ ( –αn)Snun–p

≤αnxn–p+ ( –αn)un–p

≤αnxn–p+ ( –αn)

xn–p+

λ_nL– vn–yn

≤ xn–p+ ( –αn)

λ_nL– vn–yn

≤ xn–p. (.)

It follows from Lemma . that thelimn→∞xn–pexists. This shows that{xn}is bounded.

Since{xn}is bounded, we may assume that a subsequence{xni}of{xn}converges weakly toξ.

Step . Show thatξ∈VI(C,A)

From (.), we ﬁnd thatβn( –λnL)vn–yn≤ xn–p–xn+–p. In view of the

restrictions (b) and (d), we see thatlim_n_→∞vn–yn= . Sinceyn–un ≤λLvn–yn,

we have thatlimn→∞yn–un= . It follows that

lim

n→∞vn–un= . (.)

Notice that

zn,m–p =Trn,mxn–Trn,mp



≤ Trn,mxn–Trn,mp,xn–p

=  

zn,m–p+xn–p–zn,m–xn

.

This implies thatzn,m–p≤ xn–p–zn,m–xn. Sincevn=

∞

m=δn,mzn,m, where

∞

m=δn,m= , we ﬁnd that

vn–p≤

∞

m=

δn,mzn,m–p

≤ xn–p–

∞

m=

δn,mzn,m–xn.

It follows that

xn+–p≤αnxn–p+ ( –αn)Snun–p

≤αnxn–p+ ( –αn)un–p

≤αnxn–p+ ( –αn)vn–p

≤ xn–p– ( –αn)

∞

m=

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This implies that (–αn)δn,mzn,m–xn≤ xn–p–xn+–p. In view of the restrictions

(a) and (c), we ﬁnd that

lim

n→∞zn,m–xn= . (.)

LetTbe the maximal monotone mapping deﬁned by

Tx=

⎧ ⎨ ⎩

Ax+NCx, x∈C,

∅, x∈/C.

For any given (x,y)∈G(T), we havey–Ax∈NCx. So, we havex–m,y–Ax ≥, for all

m∈C. On the other hand, we haveun=ProjC(vn–λnAyn). We obtain

x–un,

un–vn λn

+Ayn

≥.

In view of the monotonicity ofA, we see that

x–uni,y ≥ x–uni,Ax

≥ x–uni,Ax–

x–uni,

uni–vni

λni

+Ayni

=x–uni,Ax–Auni+x–uni,Auni–Ayni

–

x–uni,

uni–vni

λni

≥ x–uni,Auni–Ayni–

x–uni,

uni–vni

λni

in view ofvn–xn ≤

∞

m=δn,mzn,m–xn. It follows from (.) thatlimn→∞vn–xn= .

Notice thatun–xn ≤ un–vn+vn–xn. It follows that

lim

n→∞un–xn= . (.)

This in turn implies thatuni ξ. It follows thatx–ξ,y ≥. Notice thatT is maximal monotone and hence ∈Tξ. This shows from Lemma . thatξ∈VI(C,A).

Step . Show thatξ∈GMEP(Fm,Bm,ϕm).

It follows from (.) that {zni,m} converges weakly toξ for eachm≥. Sincezn,m= Trn,mxn, we have

Gm(zn,m,z) +

 rn,mz

–zn,m,zn,m–xn ≥, ∀z∈C.

From the assumption (A), we see that

z–zni,m,

zni,m–xni rni,m

≥Gm(z,zni,m), ∀z∈C.

In view of the assumption (A), we ﬁnd from (.) thatGm(z,ξ)≤,∀z∈C. Fortm with

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we haveztm∈C. It follows thatGm(ztm,ξ)≤. Notice that

 =Gm(ztm,ztm)≤tmGm(ztm,z) + ( –tm)Gm(ztm,ξ)≤tmGm(ztm,z),

which yieldsGm(ztm,z)≥,∀z∈C. Lettingtm↓, one sees thatGm(ξ,z)≥,∀z∈C. This implies thatξ∈GMEP(Fm,Bm,ϕm) for eachm≥. This proves thatξ∈

∞

m=GMEP(Fm,

Bm,ϕm).

Step . Show thatξ∈F(S).

Sincelimn→∞xn–pexists, we putlimn→∞xn–p=d> . It follows that

lim

n→∞xn+–p=nlim→∞αn(xn–p) + ( –αn)(Snun–p)=d.

Notice thatlim sup_n_→∞Snun–p ≤d. From Lemma ., we see that

lim

n→∞xn–Snun= . (.)

Since

Snxn–xn ≤ Snxn–Snun+Snun–xn

≤ xn–un+Snun–xn,

we ﬁnd from (.) and (.) that

lim

n→∞xn–Snxn= . (.)

In view ofSxn–xn ≤ Sxn–Snxn+Snxn–xn, we ﬁnd from (.) thatlimn→∞xn–

Sxn= . This implies from Lemma . thatξ∈F(S). This completes the proof thatξ∈F.

Step . Show that the whole sequence{xn}weakly converges toξ.

Let{xnj}be another subsequence of{xn}converging weakly toξ, whereξ=ξ. In the same way, we can show thatξ∈F. Since the spaceHenjoys Opial’s condition, we, there-fore, obtain

d=lim inf

i→∞ xni–ξ<lim infi→∞

xni–ξ

=lim inf j→∞

xj–ξ<lim inf

j→∞ xj–ξ=d.

This is a contradiction. Henceξ=ξ. This completes the proof.

3 Applications

In this section, we consider solutions of the mixed equilibrium problem (.), which in-cludes the Ky Fan inequality as a special case.

The so-called mixed equilibrium problem is to ﬁndp∈Csuch that

F(p,y) +ϕ(y) –ϕ(z)≥, ∀y∈C.

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Theorem . Let C be a nonempty,closed,and convex subset of H,S:C→C aκ-strictly pseudocontractive mapping with a nonempty ﬁxed point set,and A:C→H a L-Lipschitz continuous and monotone mapping.Let Fmbe a bifunction from C×C toRwhich satisﬁes

(A)-(A),andϕm:C→Ra lower semicontinuous and convex function for each m≥.

Assume thatF:=∞_m_=MEP(Fm,ϕm)∩VI(C,A)∩F(S)is not empty.Let{αn},{βn}and

{δn,m}be real number sequences in(, ).Let{λn},{rn,m}be positive real number sequences.

Let{xn}be a sequence generated in the following manner:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈H,

xn+=αnxn+ ( –αn)(βnI+ ( –βn)S)ProjC(

∞

yn=ProjC(

_∞

m=δn,mzn,m),

Fm(zn,m,z) +ϕm(z) –ϕm(zn,m) +

 rn,m

Assume that{αn},{βn},{δn,m},{λn},{rn,m}satisfy the following restrictions: (a)  <a≤αn≤b< ;

(b) κ≤βn≤c< ;

(c) ∞_m_=δn,m= ,and <d≤δn,m≤;

(d) lim infn→∞rn,m> ande≤λn≤f,wheree,f ∈(, /L).

Proof IfBm= , we draw the desired conclusion immediately from Theorem ..

Further, ifSis nonexpansive, we ﬁnd from Theorem . the following result.

Corollary . Let C be a nonempty,closed,and convex subset of H,S:C→C a nonexpan-sive mapping with a nonempty ﬁxed point set,and A:C→H an L-Lipschitz continuous and monotone mapping.Let Fmbe a bifunction from C×C toRwhich satisﬁes(A)-(A),

andϕm:C→Ra lower semicontinuous and convex function for each m≥.Assume that F:=∞_m_=MEP(Fm,ϕm)∩VI(C,A)∩F(S)is not empty.Let{αn},{βn},and{δn,m}be real

number sequences in(, ).Let{λn},{rn,m}be positive real number sequences.Let{xn}be a

sequence generated in the following manner:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈H,

xn+=αnxn+ ( –αn)SProjC(

∞

yn=ProjC(

∞

m=δn,mzn,m),

Fm(zn,m,z) +ϕm(z) –ϕm(zn,m) +

 rn,m

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(b) ∞_m_=δn,m= ,and <d≤δn,m≤;

(c) lim infn→∞rn,m> ande≤λn≤f,wheree,f∈(, /L).

IfA= , we ﬁnd from Theorem . the following result.

Theorem . Let C be a nonempty,closed,and convex subset of H,S:C→C aκ-strictly pseudocontractive mapping with a nonempty ﬁxed point set.Let Fm be a bifunction from

C×C toRwhich satisﬁes(A)-(A),Bm:C→H a continuous and monotone mapping, ϕm:C→Ra lower semicontinuous and convex function for each m≥.Assume that F:=∞_m_=GMEP(Fm,Bm,ϕm)∩F(S)is not empty.Let{αn},{βn},and{δn,m}be real

num-ber sequences in(, ).Let{rn,m}be a positive real number sequence.Let{xn}be a sequence

generated in the following manner:

x∈H, xn+=αnxn+ ( –αn)

βnI+ ( –βn)S

∞

m=

δn,mzn,m, n≥,

Fm(zn,m,z)+Bmzn,m,z–zn,m+ϕm(z)–ϕm(zn,m)+

 rn,mz

–zn,m,zn,m–xn ≥, ∀z∈C.

Assume that{αn},{βn},{δn,m},and{rn,m}satisfy the following restrictions: (a)  <a≤αn≤b< ;

(b) κ≤βn≤c< ;

(c) ∞_m_=δn,m= ,and <d≤δn,m≤; (d) lim infn→∞rn,m> .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this manuscript. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou, Henan}

450011, China.2_{School of Mathematics and Information Science, North China University of Water Resources and Electric}

Power, Zhengzhou, Henan 450011, China.

Acknowledgements

The authors are grateful to the reviewers for useful suggestions which improved the contents of the article.

Received: 14 January 2014 Accepted: 8 May 2014 Published:22 May 2014

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10.1186/1029-242X-2014-203

Cite this article as:Li and Zhao:On variational inequality, ﬁxed point and generalized mixed equilibrium problems.