An explicit algorithm for solving the optimize hierarchical problems

(1)

R E S E A R C H

Open Access

An explicit algorithm for solving the optimize

hierarchical problems

Poom Kumam

1,2

_{, Thanyarat Jitpeera}

3*

_{and Wararat Yarangkham}

4

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Faculty of Science and Agriculture, Rajamangala University of Technology Lanna, Phan, Chiangrai, 57120, Thailand

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

Abstract

In this paper, we consider the variational inequality problem over the generalized mixed equilibrium problem which has a hierarchical structure. Strong convergence of the algorithm to the unique solution is guaranteed under some assumptions.

MSC: 47H09; 47H10; 47J20; 49J40; 65J15

Keywords: nonexpansive; strong convergence; variational inequality; ﬁxed point; hierarchical problem

1 Introduction

LetCbe a closed convex subset of a real Hilbert spaceHwith the inner product·,·and the norm · . We denote weak convergence and strong convergence by the notations

and→, respectively. LetA:C→Hbe a nonlinear mapping and letFbe a bifunction of C×CintoR, whereRis the set of real numbers.

Consider thegeneralized mixed equilibrium problemwhich is to ﬁndx∈Csuch that

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

The solution set of (.) is denoted byGMEP(F,ϕ,A). See [–].

Ifϕ≡, the problem (.) is reduced to thegeneralized equilibrium problemwhich is to ﬁndx∈Csuch that

F(x,y) +Ax,y–x ≥, ∀y∈C. (.)

The set of solutions of (.) is denoted byGEP(F,A).

IfA≡ andϕ≡, the problem (.) is reduced to theequilibrium problem[] which is to ﬁndx∈Csuch that

F(x,y)≥, ∀y∈C. (.)

The solution set of (.) is denoted byEP(F).

IfF≡ andϕ≡, the problem (.) is reduced to theHartmann-Stampacchia varia-tional inequality[] which is to ﬁndx∈Csuch that

Ax,y–x ≥, ∀y∈C. (.)

The solution set of (.) is denoted byVI(C,A).

(2)

A mappingT:C→Cis callednonexpansiveifTx–Ty ≤ x–yfor allx,y∈C. IfC is bounded closed convex andT is a nonexpansive mapping ofCinto itself, thenF(T) is nonempty []. A pointx∈His aﬁxed pointofT providedTx=x. Denote byF(T) the set of ﬁxed points ofT; that is,F(T) ={x∈H:Tx=x}.

We discuss the following variational inequality problem over the generalized mixed equilibrium problem, which is called thehierarchical problem over the generalized mixed equilibrium problem, which is to ﬁnd a pointx∈GMEP(F,ϕ,B) such that

Ax,y–x ≥, ∀y∈GMEP(F,ϕ,B),

whereA,Bare two monotone operators. See [, ].

A mappingA:C→Cis calledα-strongly monotoneif there exists a positive real number

αsuch thatAx–Ay,x–y ≥αx–yfor allx,y∈C. A mappingA:C→Cis called L-Lipschitz continuousif there exists a positive real numberLsuch thatAx–Ay ≤Lx–y for allx,y∈C. A linear bounded operatorAis calledstrongly positiveonHif there exists a constantγ¯>  with the propertyAx,x ≥ ¯γx_{for all}_x_∈_{H. A mapping}_f _:_C_→_H_is

called aρ-contractionif there exists a constantρ∈[, ) such thatf(x) –f(y) ≤ρx–y for allx,y∈C.

In , Yaoet al.[] considered the hierarchical problem over the generalized equi-librium problem,xs,tbeing deﬁned by implicit algorithms:

xs,t=s

tf(xs,t) + ( –t)(xs,t–λAxs,t)

+ ( –s)Tr(xs,t–rBxs,t), s,t∈(, ), (.)

for each (s,t)∈(, )_{. The net}_xs

,thierarchically converges to the unique solutionx∗of the problem of the variational inequality which is to ﬁnd a pointx∗∈GEP(F,B) such that

Ax∗,x–x∗≥, ∀x∈GEP(F,B), (.)

whereA,Bare two monotone operators. The solution set of (.) is denoted by. Fur-thermore,x∗also solves the following variational inequality:

x∗∈, (I–f)x∗,x–x∗≥, ∀x∈.

In , Yaoet al.[] studied the hierarchical problem over the ﬁxed point set. Let the sequence{xn}be generated by two algorithms as follows.

Implicit Algorithm:xt=TPC[I–t(A–γf)]xt,∀t∈(, )and

Explicit Algorithm:xn+=βnxn+ ( –βn)TPC[I–αn(A–γf)]xn,∀n≥.

They showed that these two algorithms converge strongly to the unique solution of the problem of the variational inequality which is to ﬁndx∗∈F(T) such that

(A–γf)x∗,x–x∗≥, ∀x∈F(T),

where A:C→H is a strongly positive linear bounded operator, f :C →H is a ρ -contraction, andT:C→Cis a nonexpansive mapping.

(3)

forx∈C,

xn+=αn

βnxn+ ( –βn)PCI–λn(A–γf)xn+ ( –αn)Trn(I–rnB)xn, (.)

where {αn},{βn},{λn} ⊂[, ], andrn∈(, β) satisfy some conditions. Then {xn} con-verges strongly tox∗∈GMEP(F,ϕ,B), which is the unique solution of the variational in-equality:

(A–γf)x∗,x–x∗≥, ∀x∈GMEP(F,ϕ,B). (.)

Our results improve the results of Yaoet al.[], Yaoet al.[] and some other authors.

2 Preliminaries

LetCbe a nonempty closed convex subset ofH. We have the following inequality in an inner product space:x+y_≤_x_{+ }_y,_x₊_y_,_∀_x,_y_∈_{H. For every point}_x_∈_{H, there}

exists a uniquenearest pointinC, denoted byPCx, such that

x–PCx ≤ x–y, for ally∈C.

PCis called themetric projectionofHontoC. It is well known thatPCis a nonexpansive mapping ofHontoCand satisﬁes

x–y,PCx–PCy ≥ PCx–PCy,

for everyx,y∈H. Moreover,PCxis characterized by the following properties:PCx∈C and

x–PCx,y–PCx ≤, (.)

x–y≥ x–PCx+y–PCx,

for allx∈H,y∈C. LetBbe a monotone mapping ofCintoH. In the context of the vari-ational inequality problem the characterization of projection (.) implies the following:

u∈VI(C,B) ⇔ u=PC(u–λBu), λ> .

It is also well known thatHsatisﬁes the Opial condition [],i.e., for any sequence{xn} ⊂H withxnx, the inequalitylim infn→∞xn–x<lim infn→∞xn–y, holds for everyy∈H

withx=y.

For solving the generalized mixed equilibrium problem and the mixed equilibrium prob-lem, let us give the following assumptions for the bifunctionF,ϕ, and the setC:

(A) F(x,x) = for allx∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachy∈C,x→F(x,y)is weakly upper semicontinuous; (A) for eachx∈C,y→F(x,y)is convex;

(4)

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

F(z,yx) +ϕ(yx) –ϕ(z) +

ryx–z,z–x< ; (.)

(B) Cis a bounded set.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let F be a bifunction from C×C toRsatisfying(A)-(A)and letϕ:C→Rbe a proper lower semicontinuous and convex function.For r> and x∈H,deﬁne a mapping Tr:H→C as follows.

Tr(x) =

z∈C:F(z,y) +ϕ(y) –ϕ(z) +

ry–z,z–x ≥,∀y∈C (.)

for all x∈H.Assume that either(B)or(B)holds.Then the following results hold: () for eachx∈H,Tr(x)=∅;

() Tris single-valued;

() Tris ﬁrmly nonexpansive,i.e.,for anyx,y∈H,Trx–Try≤ Trx–Try,x–y; () F(Tr) =MEP(F,ϕ);

() MEP(F,ϕ)is closed and convex.

Lemma .[] Let C be a closed convex subset of a real Hilbert space H and let T:C→C be a nonexpansive mapping.Then I–T is demiclosed at zero,that is,xnx,xn–Txn→ implies x=Tx.

Lemma .[] Assume A is a self adjoint and strongly positive linear bounded operator on a Hilbert space H with coeﬃcientγ¯> and <ρ≤ A–_,_then_I_–_ρ_A_≤_{ –}_ρ_γ_¯_.

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

an+≤( –γn)an+δn, ∀n≥,

where{γn} ⊂(, )and{δn}are sequences inRsuch that (i) ∞_n_=γn=∞,

(ii) lim sup_n_→∞δn γn ≤or

∞

n=|δn|<∞. Thenlimn→∞an= .

3 Strong convergence theorems

In this section, we introduce an explicit algorithm for solving some hierarchical problem over the set of ﬁxed points of a nonexpansive and the generalized mixed equilibrium prob-lem.

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

(5)

and lower semicontinuous with either(B)or(B).Let{xn}be a sequence generated by the following algorithm for arbitrary x∈C:

xn+=αn

βnxn+ ( –βn)PCI–λn(A–γf)xn+ ( –αn)Trn(I–rnB)xn, (.)

where{αn},{βn},{λn} ⊂[, ],αn≤λn,and rn∈(, β)satisfy the following conditions: (C) ∞_n_=|αn–αn–|<∞;

(C) ∞_n_=|βn–βn–|<∞;

(C) ∞_n_=|λn–λn–|<∞,∞n=λn=∞,limn→∞λn= ; (C) ∞_n_=|rn–rn–|<∞,lim infn→∞rn> .

Then {xn}converges strongly to x∗∈GMEP(F,ϕ,B), which is the unique solution of the variational inequality:

(A–γf)x∗,x–x∗≥, ∀x∈GMEP(F,ϕ,B). (.)

Proof We will divide the proof into ﬁve steps.

Step . We will show{xn}is bounded. For anyq∈GMEP(F,ϕ,B) and takingyn=PC[I–

λn(A–γf)]xn, we note that

yn–q =PCI–λn(A–γf)xn–PCq

≤I–λn(A–γf)xn–q

≤λnγf(xn) –γf(q)+λnγf(q) –Aq+|I–λnA|xn–q

≤λnγρxn–q+λnγf(q) –Aq+ ( –λnγ¯)xn–q = – (γ¯–γρ)λn

xn–q+λnγf(q) –Aq. (.)

From (.), we have

xn+–q=αn

βnxn+ ( –βn)yn+ ( –αn)Trn(I–rnB)xn–q

≤αnβnxn+ ( –βn)yn–q+ ( –αn)Trn(I–rnB)xn–q

≤αnβnxn–q+αn( –βn)yn–q

+ ( –αn)Trn(I–rnB)xn–Trn(I–rnB)q

≤αnβnxn–q+αn( –βn)

 – (γ¯–γρ)λn

xn–q+λnγf(q) –Aq + ( –αn)xn–q

=αnβnxn–q+αn( –βn)

 – (γ¯–γρ)λn

xn–q

+αn( –βn)λnγf(q) –Aq+ ( –αn)xn–q = –αn( –βn)xn–q+αn( –βn) – (γ¯–γρ)λn

xn–q

+αn( –βn)λnγf(q) –Aq = –αn( –βn)

 – – (γ¯–γρ)λn

xn–q

(6)

= –αn( –βn)λn(γ¯–γρ)xn–q+αn( –βn)λnγf(q) –Aq = –αn( –βn)λn(γ¯–γρ)xn–q

+αn( –βn)λn(γ¯–γρ)γf(q) –Aq

¯

γ –γρ .

It follows by induction that

xn–q ≤max

x–q,

γf(q) –Aq

¯

γ –γρ , ∀n≥.

Therefore{xn}is bounded and so are{yn},{Axn}, and{f(xn)}.

Step . We show thatlimn→∞xn+–xn= . Settingvn= [I–λn(A–γf)]xnand we

observe that

yn+–yn=PCvn+–PCvn ≤I–λn+(A–γf)

xn+–

I–λn(A–γf)xn =λn+γ

f(xn+) –f(xn)

+ (λn+–λn)γf(xn) + (I–λn+A)(xn+–xn)

+(λn+–λn)Axn

≤λn+γf(xn+) –f(xn)+ ( –λn+γ¯)xn+–xn

+|λn+–λn|γf(xn)+Axn

≤λn+γρxn+–xn+ ( –λn+γ¯)xn+–xn

+|λn+–λn|γf(xn)+Axn

= – (γ¯–γρ)λn+

xn+–xn+|λn+–λn|M, (.)

whereM=sup{γf(xn)+Axn:n∈N}. Settingzn=βnxn+ ( –βn)ynfor alln≥. We

observes that

zn+–zn=βn+xn++ ( –βn+)yn+–

βnxn+ ( –βn)yn

≤βn+xn+–xn+|βn+–βn|xn–yn+| –βn+|yn+–yn. (.)

Substituting (.) into (.) it follows that

zn+–zn ≤βn+xn+–xn+|βn+–βn|xn–yn

+| –βn+|

 – (γ¯–γρ)λn+

xn+–xn+|λn+–λn|M

≤βn+xn+–xn+|βn+–βn|xn–yn

+ –βn+– ( –βn+)(γ¯–γρ)λn+

xn+–xn+|λn+–λn|M

= – ( –βn+)(γ¯–γρ)λn+

xn+–xn+|βn+–βn|M

(7)

whereM=sup{xn–yn:n∈N}. On the other hand, fromun–=Trn–(xn––rn–Bxn–)

andun=Trn(xn–rnBxn) it follows that

F(un–,y) +Bxn–,y–un–+ϕ(y) –ϕ(un–) +

 rn–

y–un–,un––xn– ≥,

∀y∈C, (.)

and

F(un,y) +Bxn,y–un+ϕ(y) –ϕ(un) +  rn

y–un,un–xn ≥, ∀y∈C. (.)

Substitutingy=uninto (.) andy=un–into (.), we have

F(un–,un) +Bxn–,un–un–+ϕ(un) –ϕ(un–) +

 rn–

un–un–,un––xn– ≥

and

F(un,un–) +Bxn,un––un+ϕ(un–) –ϕ(un) + 

rnun––un,un–xn ≥.

From (A), we have

un–un–,Bxn––Bxn+

un––xn–

rn–

–un–xn rn

≥,

and then

un–un–,rn–(Bxn––Bxn) +un––xn––

rn–

rn (un–xn)

≥,

so

un–un–,rn–Bxn––rn–Bxn+un––un+un–xn––

rn–

rn (un–xn)

≥.

It follows that

un–un–, (I–rn–B)xn– (I–rn–B)xn–+un––un+un–xn– rn–

rn (un–xn)

≥,

un–un–,un––un+

un–un–,xn–xn–+

 –rn– rn

(un–xn)

≥.

Without loss of generality, let us assume that there exists a real numbercsuch thatrn–>

c> , for alln∈N. Then we have

un–un–≤

un–un–,xn–xn–+

 –rn– rn

(un–xn)

≤ un–un–

xn–xn–+

 –rn–

rn

(8)

and hence

un–un– ≤ xn–xn–+



rn|rn–rn–|un–xn

≤ xn–xn–+

M

c |rn–rn–|, (.)

whereM=sup{un–xn:n∈N}. From (.), we have

xn+–xn=αnzn+ ( –αn)un–αn–zn–– ( –αn–)un–

≤αnzn–zn–+|αn–αn–|zn––un–+| –αn|un–un–

=αnzn–zn–+|αn–αn–|M+| –αn|un–un–, (.)

whereM=sup{zn–un:n∈N}. Substituting (.) and (.) into (.)

xn+–xn ≤αn

 – ( –βn)(γ¯–γρ)λn

xn–xn–

+|βn–βn–|M+|λn–λn–|M

+|αn–αn–|M+| –αn|

xn–xn–+

M

c |rn–rn–|

≤ – ( –βn)(γ¯–γρ)αnλn

xn–xn–+αn|βn–βn–|M

+αn|λn–λn–|M+|αn–αn–|M+

M

c |rn–rn–|, (.)

from (C)-(C) and the boundedness of {xn}, {yn}, {zn}, {f(xn)}, and {Axn}. Applying Lemma ., we obtain

lim

n→∞xn+–xn= . (.)

Step . We show thatlimn→∞xn–un= . For eachq∈GMEP(F,ϕ,B), note thatTrn is ﬁrmly nonexpansive, then we have

un–q =Trn(xn–rnBxn) –Trn(q–rnBq)



≤Trn(xn–rnBxn) –Trn(q–rnBq),un–q =(xn–rnBxn) – (q–rnBq),un–q

= 

(xn–rnBxn) – (q–rnBq)



+un–q

–(xn–rnBxn) – (q–rnBq) – (un–q)

≤ 



xn–q+un–q–xn–un–rn(Bxn–Bq)

 ≤ 



xn–q+un–q–xn–un

+ rnxn–un,Bxn–Bq–rnBxn–Bq

, (.)

which implies that

(9)

From (.), we get

yn–xn=PCI–λn(A–γf)xn–PCxn

≤I–λn(A–γf)

xn–xn

≤λn(A–γf)xn.

By (C), we have

lim

n→∞yn–xn= . (.)

Settingwn= [I–λn(A–γf)]xn. It follows that

wn–xn=I–λn(A–γf)xn–xn

≤I–λn(A–γf)xn–xn

≤λn(A–γf)xn.

By using (C) again, we get

lim

n→∞wn–xn= . (.)

Fromyn=PC[I–λn(A–γf)]xn, we compute

yn–q =PCI–λn(A–γf)xn–PCq

≤I–λn(A–γf)xn–q

=wn–q. (.)

It follows from (.) that

xn–q ≤ wn–q. (.)

Then we get

wn–q≤

I–λn(A–γf)

xn–q,wn–q

=λnγfxn–Aq,wn–q+

(I–λnA)(xn–q),wn–q

≤λnγfxn–Aq,wn–q+ ( –λnγ¯)xn–qwn–q

≤( –λnγ¯)wn–q+λnγfxn–Aq,wn–q.

It follows that

wn–q≤ 

¯

γγfxn–Aq,wn–q

= 

¯ γ

γfxn–fq,wn–q+γfq–Aq,wn–q

≤  ¯ γ

(10)

that is,

wn–q≤ 

¯ γ –γρ

(A–γf)q,q–wn. (.)

On the other hand, we note that

un–q =Trn(xn–rnBxn) –Trn(q–rnBq)



≤(xn–rnBxn) – (q–rnBq)



=(xn–q) –rn(Bxn–Bq)



≤ xn–q– rnxn–q,Bxn–Bq+rnBxn–Bq

≤ xn–q– rnβBxn–Bq+r_nBxn–Bq. (.)

Using (.), (.), (.), and (.), we note that

xn+–q≤αnβnxn–q+αn( –βn)yn–q+ ( –αn)un–q

≤αnβnwn–q+αn( –βn)wn–q+ ( –αn)un–q

=αnwn–q+ ( –αn)un–q

≤ αn

¯ γ–γρ

(A–γf)q,q–wn

+ ( –αn)xn–q– rnβBxn–Bq+r_nBxn–Bq

= αn

¯ γ–γρ

(A–γf)q,q–wn

+ ( –αn)xn–q+rn(rn– β)Bxn–Bq

≤ αn

¯ γ–γρ

(A–γf)q,q–wn+xn–q

+ ( –αn)rn(rn– β)Bxn–Bq. (.)

Then we have

( –αn)c(β–d)Bxn–Bq

≤ αn

¯ γ–γρ

(A–γf)q,q–wn+xn–q–xn+–q ≤ αn

¯ γ–γρ

(A–γf)q,q–wn+xn–xn+

xn–q+xn+–q

.

From (C),{rn} ⊂[c,d]⊂(, β), and (.), we obtain

lim

n→∞Bxn–Bq= . (.)

Substituting (.) into (.), we have

xn+–q≤αnwn–q+ ( –αn)un–q

≤αnwn–q+ ( –αn)

(11)

+ rnxn–unBxn–Bq

≤αnwn–q+xn–q– ( –αn)xn–un

+ rn( –αn)xn–unBxn–Bq,

and it follows that

( –αn)xn–un≤αnwn–q+xn–q–xn+–q

+ rn( –αn)xn–unBxn–Bq

≤αnwn–q+xn–xn+

xn–q+xn+–q

+ rn( –αn)xn–unBxn–Bq.

Since we have (C), (.), and (.),

lim

n→∞xn–un= . (.)

By (C), we obtain

lim

n→∞

xn–un

rn

= lim

n→∞ 

rnxn–un= . (.)

Step . Next, we will show that

lim sup

n→∞

(γf –A)x∗,xn–x∗

≤.

Indeed, we choose a subsequence{xni}of{xn}such that

lim sup

n→∞

(γf –A)x∗,xn–x∗=lim

i→∞

(γf–A)x∗,xni–x∗.

Since{xn_i}is bounded, there exists a subsequence{xn_ij}of{xn_i}which converges weakly to z∈C. We notice thatwn–xn ≤λn(A–γf)xn →. Hence, we getlim supn→∞(γf –

A)x∗,xn–x∗ ≤. Next, we will show thatz∈GMEP(F,ϕ,B). Sinceun=Trn(xn–rnBxn), we have

F(un,y) +Bxn,y–un+ϕ(y) –ϕ(un) + 

rny–un,un–xn ≥, ∀y∈C.

From (A), we also have

Bxn,y–un+ϕ(y) –ϕ(un) + 

rny–un,un–xn ≥F(y,un), ∀y∈C,

and hence

Bxni,y–uni+ϕ(y) –ϕ(uni) +

y–uni,uni–xni rni

(12)

Fortwith  <t≤ andy∈C, letyt=ty+ ( –t)z. Sincey∈Candz∈C, we haveyt∈C. So, from (.), we have

yt–uni,Byt ≥ yt–uni,Byt–ϕ(yt) +ϕ(uni) –yt–uni,Bxni

–

yt–uni,uni–xni rni

+F(yt,uni)

=yt–un_i,Byt–Bun_i+yt–un_i,Bun_i–Bxn_i–ϕ(yt) +ϕ(un_i)

–

yt–un_i,uni–xni rni

+F(yt,un_i).

Sinceuni–xni →, we haveBuni–Bxni →. Further, from the inverse strongly mono-tonicity ofB, we haveyt–un_i,Byt–Bun_i ≥. So, from (A), (A), and the weak lower semicontinuity ofϕ,uni_r–xni

ni → anduniw, we have in the limit

yt–w,Byt ≥–ϕ(yt) +ϕ(w) +F(yt,w) (.)

asi→ ∞. From (A), (A) and (.), we also get

 = F(yt,yt) +ϕ(yt) –ϕ(yt)

≤tF(yt,y) + ( –t)F(yt,z) +tϕ(y) – ( –t)ϕ(z) –ϕ(yt)

=tF(yt,y) +ϕ(y) –ϕ(yt)+ ( –t)F(yt,z) +ϕ(z) –ϕ(yt)

≤tF(yt,y) +ϕ(y) –ϕ(yt)+ ( –t)yt–z,Byt =tF(yt,y) +ϕ(y) –ϕ(yt)+ ( –t)ty–z,Byt, ≤F(yt,y) +ϕ(y) –ϕ(yt) + ( –t)y–z,Byt.

Lettingt→, we have, for eachy∈C,

F(z,y) +ϕ(y) –ϕ(z) +y–z,Bz ≥.

This implies thatz∈GMEP(F,ϕ,B). It is easy to see thatPGMEP(F,ϕ,B)(I–A+γf)(x∗) is a

contraction ofHinto itself. HenceHis complete, there exists a unique ﬁxed pointx∗∈H, such thatx∗=PGMEP(F,ϕ,B)(I–A+γf)(x∗).

Step . Next, we will provexn→x∗∈GMEP(F,ϕ,B), which solves the variational in-equality (.). It follows from (.) that

xn+–x∗ 

=αnβn

xn–x∗,xn+–x∗

+αn( –βn)PCI–λn(A–γf)xn–PCI–λn(A–γf)x∗,xn+–x∗

+αn( –βn)PCI–λn(A–γf)x∗–x∗,xn+–x∗

+ ( –αn)Trn(I–rnB)xn–Trn(I–rnB)x∗,xn+–x∗

≤αnβnxn–x∗xn+–x∗

+αn( –βn)I–λn(A–γf)xn–I–λn(A–γf)x∗,xn+–x∗

(13)

+αn( –βn)I–λn(A–γf)x∗–x∗,xn+–x∗

+ ( –αn)Trn(I–rnB)xn–Trn(I–rnB)x∗xn+–x∗ ≤αnβnxn–x∗xn+–x∗

+αn( –βn)

 – (γ¯–γρ)λnxn–x∗ +λnγf

x∗–Ax∗xn+–x∗

–αn( –βn)λn

(A–γf)x∗,xn+–x∗

+ ( –αn)xn–x∗xn+–x∗

=αnβnxn–x∗xn+–x∗+ ( –αn)xn–x∗xn+–x∗

+αn( –βn) – (γ¯–γρ)λnxn–x∗xn+–x∗

+αn( –βn)λnγf

= –αn( –βn)xn–x∗xn+–x∗

+αn( –βn)

 – (γ¯–γρ)λnxn–x∗xn+–x∗

= –αn( –βn) –  + (γ¯–γρ)λnxn–x∗xn+–x∗

= –αn( –βn)(γ¯–γρ)λnxn–x∗xn+–x∗

+αn( –βn)λnγf

≤ –αn( –βn)(γ¯–γρ)λn

 xn–x

∗

+xn+–x∗ 

–αn( –βn)λn

≤ – ( –βn)(γ¯–γρ)αnλn

 xn–x

∗

+

xn+–x

∗

+ ( –βn)αnλnγf

– ( –βn)αnλn

,

which implies that

_xn_+–x∗≤ – ( –βn)(γ¯–γρ)αnλnxn–x∗



+ ( –βn)αnλnγf

– ( –βn)αnλn

(14)

Since{xn},{f(xn)}, and{Axn}are all bounded, we can choose a constantM>  such that

sup 

¯ γ –γρ

γfx∗–Ax∗xn+–x∗+ 

≤M.

It follows that

xn+–x∗ 

≤ – ( –βn)(γ¯–γρ)αnλnxn–x∗



+ ( –βn)αnλnM.

By (C), we conclude thatxn→x∗, asn→ ∞. This completes the proof.

4 An example

Next, the following example shows that all conditions of Theorem . are satisﬁed.

Example . For instance, letαn= _nn+_+,βn=_n_+ ,λn=_(_n_+), andrn=_nn_+. Then clearly the sequences{αn},{λn}satisfy the following condition:

n+  n_{+ }≤

 (n+ ).

We will show that the condition (C) is fulﬁlled. Indeed, we have

∞

n=

|αn–αn–|= ∞

n=

_nn+ _{+ }–

n (n– )_{+ }

= ∞ n=

(n+ )(n_(n_{+ )(n}– n+ ) –_{– n}_{+ )}n(n+ )

= ∞ n=

_n_{– n} +_{+ n}n–n_{– n} _{+ }

.

The sequence{αn}satisﬁes the condition (C) by ap-series.

Next, we will show that the condition (C) is fulﬁlled. We compute

∞

n=

|βn–βn–|= ∞

n=

_n_{+ }– n =  –   +  –   +  –   +· · · = .

The sequence{βn}satisﬁes the condition (C).

Next, we will show that the condition (C) is fulﬁlled. We compute

∞

n=

|λn–λn–|= ∞

n=

(15)

lim

n→∞λn=nlim→∞  (n+ )= ,

and

∞

n= λn=

∞

n=



(n+ )=∞.

The sequence{λn}satisﬁes the condition (C).

Finally, we will show that the condition (C) is fulﬁlled. We compute

∞

n=

|rn–rn–|= ∞

n=

_nn_{+ }– n–  (n– ) + 

=

∞

n=

n(n) – (n_(n_{+ )n}– )(n+ )

=

∞

n=

n_(n–_{+ )n}n+ 

=

∞

n=

_n(n_{+ )}

and

lim inf

n→∞ rn=lim infn→∞

n n+ = .

The sequence{rn}satisﬁes the condition (C).

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let

A:H→H be a strongly positive linear bounded operator,f :C→H beρ-contraction,γ

be a positive real number such thatγ¯_ρ–<γ <γ_ρ¯ and T:C→C be a nonexpansive mapping with F(T)=∅.Let{xn} be a sequence generated by the following algorithm for arbitrary x∈C:

xn+=βnxn+ ( –βn)TPC

I–λn(A–γf)xn, (.)

where{βn},{λn} ⊂[, ]satisfy the following conditions:

(C) ∞_n_=|βn+–βn|<∞, <lim infn→∞βn<lim supn→∞βn< ; (C) ∞_n_=|λn+–λn|<∞,

∞

n=λn=∞,limn→∞λn= .

Then{xn}converges strongly to x∗∈F(T),which is the unique solution of the variational inequality:

(A–γf)x∗,x–x∗≥, ∀x∈F(T). (.)

Proof Setting{αn} ≡ andT to be a nonexpansive mapping in Theorem ., we obtain

the desired conclusion immediately.

(16)

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let A: H→H be a strongly positive linear bounded operator,B:C→H be aβ-inverse-strongly monotone and F be a bifunction from C×C→Rsatisfying(A)-(A)and letϕ:C→R be convex and lower semicontinuous with either(B)or(B).Suppose GMEP(F,ϕ,B)=∅. Let{xn}be a sequence by the following algorithm for arbitrary x∈C:

xn+=αn

βnxn+ ( –βn)[I–λnA]xn+ ( –αn)Trn(I–rnB)xn, (.)

where{αn},{βn},{λn} ⊂[, ],αn≤λn,and rn∈(, β)satisfy the following conditions: (C) ∞_n_=|αn–αn–|<∞;

(C) ∞_n_=|βn–βn–|<∞;

(C) ∞_n_=|λn–λn–|<∞,∞n=λn=∞,limn→∞λn= ; (C) ∞_n_=|rn–rn–|<∞,lim infn→∞rn> .

Then {xn}converges strongly to x∗∈GMEP(F,ϕ,B), which is the unique solution of the variational inequality:

Ax∗,x–x∗≥, ∀x∈GMEP(F,ϕ,B). (.) Proof SettingT,PCto be the identity andf ≡ in Theorem ., we obtain the desired

conclusion immediately.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let

A:H→H be a strongly positive linear bounded operator,f :C→H beρ-contraction,B: C→H beβ-inverse-strongly monotone and F be a bifunction from C×C→Rsatisfying (A)-(A)and letϕ:C→Rbe convex and lower semicontinuous with either(B)or(B). Let{xn}be a sequence generated by the following algorithm for arbitrary x∈C:

xn+=αn

λn( –βn)f(xn) +I–λn( –βn)Axn+ ( –αn)Trn(I–rnB)xn, (.) where{αn},{βn},{λn} ⊂[, ],λn≤βn,and rn∈(, β)satisfy the following conditions:

(C) ∞_n_=|αn+–αn|<∞; (C) ∞_n_=|βn+–βn|<∞;

(C) ∞_n_=|λn+–λn|<∞,∞n=λn=∞,limn→∞λn= ; (C) ∞_n_=|rn+–rn|<∞,lim infn→∞rn> .

Then {xn}converges strongly to x∗∈GMEP(F,ϕ,B), which is the unique solution of the variational inequality

(A–f)x∗,x–x∗≥, ∀x∈GMEP(F,ϕ,B). (.) Proof SettingT,PCto be the identity andγ ≡ in Theorem ., we obtain the desired

conclusion immediately.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

(17)

Author details

1_{Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, 126 Pracha u-thit}

Rd., Bang Mod, Thrung Khru, Bangmod, Bangkok, 10140, Thailand.2_{Centre of Excellence in Mathematics, CHE, Si}

Ayutthaya Road, Bangkok, 10400, Thailand.3_{Department of Mathematics, Faculty of Science and Agriculture,}

Rajamangala University of Technology Lanna, Phan, Chiangrai, 57120, Thailand.4_{Department of Logistic Engineering,}

Faculty of Engineering, Rajamangala University of Technology Lanna, Phan, Chiangrai, 57120, Thailand. Acknowledgements

This project was partially supported by Centre of Excellence in Mathematics, the Commission on Higher Education, Ministry of Education, Thailand. The second author and third author were supported by Innovation park, RMUTL Hands-on Researcher Project, Rajamangala University of Technology Lanna Chiangrai under Grant no. 57HRG-10 during the preparation of this paper.

Received: 28 April 2014 Accepted: 5 September 2014 Published:16 Oct 2014

References

1. Al-Mazeooei, AE, Latif, A, Yao, JC: Solving generalized mixed equilibria, variational inequalities, and constrained convex minimization. Abstr. Appl. Anal.2014, Article ID 587865 (2014)

2. Ceng, LC, Chen, CM, Pang, CT: Hybrid extragradient-like viscosity methods for generalized mixed equilibrium problems, variational inclusions, and optimization problems. Abstr. Appl. Anal.2014, Article ID 120172 (2014) 3. Ceng, LC, Chen, CM, Wen, CF, Pang, CT: Relaxed iterative algorithms for generalized mixed equilibrium problems with

constraints of variational inequalities and variational inclusions. Abstr. Appl. Anal.2014, Article ID 345212 (2014) 4. Ceng, LC, Ho, JL: Hybrid extragradient method with regularization for convex minimization, generalized mixed

equilibrium, variational inequality and ﬁxed point problems. Abstr. Appl. Anal.2014, Article ID 436069 (2014) 5. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63, 123-145

(1994)

6. Hartman, P, Stampacchia, G: On some nonlinear elliptic diﬀerential functional equations. Acta Math.115, 271-310 (1966)

7. Kirk, WA: Fixed point theorem for mappings which do not increase distance. Am. Math. Mon.72, 1004-1006 (1965) 8. Yao, YH, Cho, YJ, Yang, PX: An iterative algorithm for a hierarchical problem. J. Appl. Math.2012, Article ID 320421

(2012)

9. Yao, YH, Kang, JI, Cho, YJ, Liou, YC: Composite schemes for variational inequalities over equilibrium problems and variational inclusions. J. Inequal. Appl.2013, Article ID 414 (2013)

10. Yao, Y, Liou, Y-C, Chen, C-P: Hierarchical convergence of a double-net algorithm for equilibrium problems and variational inequality problems. Fixed Point Theory Appl.2010, Article ID 642584 (2010). doi:10.1155/2010/642584 11. Yao, Y, Liou, Y-C, Kang, SM: Algorithms construction for variational inequalities. Fixed Point Theory Appl.2011, Article

ID 794203 (2011). doi:10.1155/2011/794203

12. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc.73, 595-597 (1967)

13. Peng, JW, Yao, JC: A new hybrid-extragradient method for generalized mixed equilibrium problems and ﬁxed point problems and variational inequality problems. Taiwan. J. Math.12(6), 1401-1432 (2008)

14. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math.18, 78-81 (1976)

15. Marino, G, Xu, HK: A general iterative method for nonexpansive mapping in Hilbert space. J. Math. Anal. Appl.318, 43-52 (2006)

16. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.66, 240-256 (2002)

10.1186/1029-242X-2014-405