A general iterative method for two maximal monotone operators and 2-generalized hybrid mappings in Hilbert spaces

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R E S E A R C H

Open Access

A general iterative method for two maximal

monotone operators and 2-generalized

hybrid mappings in Hilbert spaces

Rabian Wangkeeree

*

_{and Uraiwan Boonkong}

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

Abstract

LetCbe a closed and convex subset of a real Hilbert spaceH. LetTbe a 2-generalized hybrid mapping ofCinto itself, letAbe an

α

-inverse strongly-monotone mapping of

CintoH, and letBandFbe maximal monotone operators onD(B)⊂CandD(F)⊂C

respectively. The purpose of this paper is to introduce a general iterative scheme for finding a point ofF(T)∩(A+B)–1₀_∩_F–1_{0 which is a unique solution of a hierarchical} variational inequality, whereF(T) is the set of fixed points ofT, (A+B)–1_{0 and}_F–1_{0 are} the sets of zero points ofA+BandF, respectively. A strong convergence theorem is established under appropriate conditions imposed on the parameters. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to mixed equilibrium problems and the set of fixed points of a 2-generalized hybrid mapping in a real Hilbert space.

Keywords: 2-generalized hybrid mapping; inverse strongly monotone mapping; maximal monotone mapping; hierarchical variational inequality

1 Introduction

LetHbe a Hilbert space, and letCbe a nonempty closed convex subset ofH. LetNandR be the sets of all positive integers and real numbers, respectively. Letϕ:C→Rbe a real-valued function, and letf :C×C→Rbe an equilibrium bifunction, that is,f(u,u) =  for eachu∈C. The mixed equilibrium problem is to ﬁndx∈Csuch that

f(x,y) +ϕ(y) –ϕ(x)≥ for ally∈C. (.)

Denote the set of solutions of (.) byMEP(f,ϕ). In particular, ifϕ= , this problem re-duces to the equilibrium problem, which is to ﬁndx∈Csuch that

f(x,y)≥ for ally∈C. (.)

The set of solutions of (.) is denoted byEP(f). The problem (.) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, min-max problems, the Nash equilibrium problems in noncooperative games and others; see, for example, Blum-Oettli [] and Moudaﬁ []. Numerous problems in physics, opti-mization and economics reduce to ﬁnding a solution of the problem (.).

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LetT be a mapping ofCintoC. We denote byF(T) :={x∈C:Tx=x}the set of ﬁxed points ofT. A mappingT:C→Cis said to be nonexpansive ifTx–Ty ≤ x–yfor all x,y∈C. The mappingT:C→Cis said to be ﬁrmly nonexpansive if

Tx–Ty≤ x–y,Tx–Ty for allx,y∈C; (.)

see, for instance, Browder [] and Goebel and Kirk []. The mappingT:C→Cis said to be ﬁrmly nonspreading [] if

Tx–Ty≤ Tx–y+x–Ty (.)

for allx,y∈C. Iemoto and Takahashi [] proved thatT:C→Cis nonspreading if and only if

Tx–Ty≤ x–y+ x–Tx,y–Ty (.)

for allx,y∈C. It is not hard to know that a nonspreading mapping is deduced from a ﬁrmly nonexpansive mapping; see [, ] and a ﬁrmly nonexpansive mapping is a nonexpansive mapping.

In , Kocoureket al.[] introduced a class of nonlinear mappings, say generalized hybrid mappings. A mappingT:C→Cis said to be generalized hybrid if there areα,β∈

Rsuch that

αTx–Ty+ ( –α)x–Ty≤βTx–y+ ( –β)x–y (.) for allx,y∈C. We call such a mapping an (α,β)-generalized hybrid mapping. We ob-serve that the mappings above generalize several well-known mappings. For example, an (α,β)-generalized hybrid mapping is nonexpansive forα=  andβ= , nonspreading for

α=  andβ= , and hybrid forα=_andβ=_.

Recently, Maruyamaet al.[] deﬁned a more general class of nonlinear mappings than the class of generalized hybrid mappings. Such a mapping is a -generalized hybrid map-ping. A mappingTis called -generalized hybrid if there existα,α,β,β∈Rsuch that

αTx–Ty 

+αTx–Ty+ ( –α–α)x–Ty

≤βTx–y+βTx–y+ ( –β–β)x–y (.)

for all x,y∈C; see [] for more details. We call such a mapping an (α,α,β,β

)-generalized hybrid mapping. We can also show that ifTis a -generalized hybrid mapping andx=Tx, then for anyy∈C,

αx–Ty+αx–Ty+ ( –α–α)x–Ty

≤βx–y+βx–y+ ( –β–β)x–y,

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generalize several well-known mappings. For example, a (,α, ,β)-generalized hybrid

mapping is an (α,β)-generalized hybrid mapping in the sense of Kocoureket al.[].

Recall that a linear bounded operatorBis strongly positive if there is a constantγ¯>  with the property

Vx,x ≥ ¯γx for allx∈H. (.)

In general, a nonlinear operatorV :H→H is called strongly monotone if there exists

¯

γ >  such that

x–y,Vx–Vy ≥ ¯γx–y for allx,y∈H. (.) SuchV is calledγ¯-strongly monotone. A nonlinear operatorV:H→His called Lips-chitzian continuous if there existsL>  such that

Vx–Vy ≤Lx–y for allx,y∈H. (.)

SuchVis calledL-Lipschitzian continuous. A mappingA:C→His said to beα -inverse-strongly monotone if x–y,Ax–Ay ≥αAx–Ay_{for all}_x,_y_∈_{C. It is known that}_Ax_–

Ay ≤(_α)x–yfor allx,y∈CifAisα-inverse-strongly monotone; see, for example, [–].

Many studies have been done for structuring the ﬁxed point of a nonexpansive map-pingT. In , Mann [] introduced the iteration as follows: a sequence{xn}deﬁned

by

xn+=αnxn+ ( –αn)Txn, (.)

where the initial guessx∈Cis arbitrary and{αn}is a real sequence in [, ]. It is known

that under appropriate settings the sequence{xn}converges weakly to a ﬁxed point ofT.

However, even in a Hilbert space, Mann iteration may fail to converge strongly; for ex-ample, see []. Some attempts to construct an iteration method guaranteeing the strong convergence have been made. For example, Halpern [] proposed the so-called Halpern iteration

xn+=αnu+ ( –αn)Txn, (.)

whereu,x∈Care arbitrary and{αn}is a real sequence in [, ] which satisﬁesαn→,

∞

n=αn=∞and

∞

n=|αn–αn+|<∞. Then{xn}converges strongly to a ﬁxed point ofT;

see [, ].

In , Baillon [] ﬁrst introduced the nonlinear ergodic theorem in a Hilbert space as follows:

Snx=

 n

n–

k=

Tkx (.)

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Theorem . Let C be a nonempty,closed and convex subset of a Hilbert space H.Let T:

C→C be a-generalized hybrid mapping with F(T)=∅.Suppose that{xn}is a sequence

generated by x=x∈C,u∈C and

xn+=γnu+ ( –γn)

 n

n–

k=

Tkxn, ∀n∈N, (.)

where≤γn≤,limn→∞γn= and

∞

n=γn=∞.Then{xn}converges strongly to PF(T)u.

LetBbe a mapping ofH into H_{. The eﬀective domain of}_B_{is denoted by}_{D(B), that}

is, D(B) ={x∈H:Bx=∅}. A multi-valued mappingBonH is said to be monotone if x–y,u–v ≥ for allx,y∈D(B),u∈Bx, andv∈By. A monotone operatorBonHis said to be maximal if its graph is not properly contained in the graph of any other monotone operator on H. For a maximal monotone operatorBonH andr> , we may deﬁne a single-valued operatorJr= (I+rB)–:H→D(B), which is called the resolvent ofBforr.

We denote byAr=_r(I–Jr) the Yosida approximation ofBforr> . We know [] that

Arx∈BJrx, ∀x∈H,r> . (.)

LetBbe a maximal monotone operator onH, and letB–_{ =}_{_x_∈_H_{: }_∈_Bx_}_{. It is known}

that the resolventJris ﬁrmly nonexpansive andB– =F(Jr) for allr> ,i.e.,

Jrx–Jry ≤ x–y,Jrx–Jry, ∀x,y∈H. (.)

Recently, in the case whenT :C→C is a nonexpansive mapping,A:C→H is an

α-inverse strongly monotone mapping andB∈H ×H is a maximal monotone oper-ator, Takahashiet al. [] proved a strong convergence theorem for ﬁnding a point of F(T)∩(A+B)–_{, where}_F(T_{) is the set of ﬁxed points of}_T _{and (A}₊_B)–_{ is the set}

of zero points ofA+B. In , for ﬁnding a point of the set of ﬁxed points ofT and the set of zero points ofA+Bin a Hilbert space, Manaka and Takahashi [] introduced an iterative scheme as follows:

xn+=βnxn+ ( –βn)T

Jλn(I–λnA)xn

, (.)

whereT is a nonspreading mapping,Ais anα-inverse strongly monotone mapping and Bis a maximal monotone operator such thatJλ= (I–λB)–;{βn}and{λn}are sequences

which satisfy  <c≤βn≤d<  and  <a≤λn≤b< α. Then they proved that{xn}

converges weakly to a pointp=limn→∞PF(T)∩(A+B)–_()x_n.

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scheme: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x=x∈H arbitrarily,

zn=Jλn(I–λnA)xn, yn=_nnk=–Tkzn,

xn+=αnu+ ( –αn)yn for alln∈N,

(.)

where{αn}is an appropriate sequence in [, ]. They obtained strong convergence

theo-rems about a common element of the set of ﬁxed points of a nonspreading mapping and the set of zero points of anα-inverse strongly monotone mapping and a maximal mono-tone operator in a Hilbert space.

On the other hand, iterative methods for nonexpansive mappings have recently been ap-plied to solve convex minimization problems; see,e.g., [–] and the references therein. Convex minimization problems have a great impact and inﬂuence on the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of ﬁxed points a nonexpansive mapping on a real Hilbert space:

θ(x) =min

x∈C



 Vx,x– x,b, (.)

whereV is a linear bounded operator,C is the ﬁxed point set of a nonexpansive map-pingT andbis a given point inH. LetHbe a real Hilbert space. In [], Marino and Xu introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudaﬁ []:

xn+= (I–αnV)Txn+αnγf(xn), n≥, (.)

where V is a strongly positive bounded linear operator on H. They proved that if the sequence{αn}of parameters satisﬁes appropriate conditions, then the sequence{xn}

gen-erated by (.) converges strongly to the unique solution of the variational inequality

(V–γf)x∗,x–x∗≥, x∈C,

which is the optimality condition for the minimization problem

min

x∈C



 Vx,x–h(x), (.)

wherehis a potential function forγf (i.e.,h(x) =γf(x) forx∈H).

Recently, Tian [] introduced the following general iterative scheme based on the vis-cosity approximation method induced by a γ¯-strongly monotone and aL-Lipschitzian continuous operatorV onH

xn+=αnγg(xn) + (I–μαnV)Txn,

for alln∈N, whereμ,γ ∈Rsatisfying  <μ<_Lγ¯,  <γ <μ(γ¯–

L_μ

 )/k,gis ak-contraction

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on the parameters, in [] that{xn}converges strongly to a pointp∈F(T) which is a

unique solution of the variational inequality

(V–γg)p,q–p

≥, ∀q∈F(T).

Very recently, Lin and Takahashi [] obtained the strong convergence theorem for ﬁnd-ing a pointp∈(A+B)–∩F– which is a unique solution of a hierarchical variational

inequality, whereAis anα-inverse strongly-monotone mapping ofCintoH, andBand Fare maximal monotone operators onD(B)⊂CandD(F)⊂C, respectively. More pre-cisely, they introduced the following iterative scheme: Letx=x∈Hand let{xn} ⊂Hbe

a sequence generated

xn+=αnγg(xn) + (I–αnV)Jλn(I–λnA)Trnxn for alln∈N, (.) where{αn} ⊂(, ),{λn} ⊂(,∞) and{rn} ⊂(,∞) satisfy certain appropriate conditions,

Jλ= (I+λB)–andTr= (I+rF)–are the resolvents ofBforλ>  andFforr> ,

respec-tively.

In this paper, motivated by the mentioned results, letCbe a closed and convex subset of a real Hilbert spaceH. LetTbe a -generalized hybrid mapping ofCinto itself, letA be anα-inverse strongly-monotone mapping ofCinto H, and letBandF be maximal monotone operators onD(B)⊂CandD(F)⊂Crespectively. We introduce a new gen-eral iterative scheme for ﬁnding a common element ofF(T)∩(A+B)–__∩_F–_{ which}

is a unique solution of a hierarchical variational inequality, whereF(T) is the set of ﬁxed points ofT, (A+B)–_{ and}_F–_{ are the sets of zero points of}_A₊_B_and_F_{, respectively.}

Then, we prove a strong convergence theorem. Further, we consider the problem for ﬁnd-ing a common element of the set of solutions of a mathematical model related to mixed equilibrium problems and the set of ﬁxed points of a -generalized hybrid mapping in a real Hilbert space.

2 Preliminaries

LetHbe a real Hilbert space with the inner product ·,·and the norm · , respectively. LetCbe a nonempty closed convex subset ofH. The nearest point projection ofHonto Cis denoted byPC, that is,x–PCx ≤ x–yfor allx∈Handy∈C. SuchPCis called

the metric projection ofHontoC. We know that the metric projectionPCis ﬁrmly

non-expansive,i.e.,

PCx–PCy≤ PCx–PCy,x–y (.)

for allx,y∈H. Furthermore, PCx–PCy,x–y ≤ holds for allx∈Handy∈C; see [].

Letα>  be a given constant.

We also know the following lemma from [].

Lemma . Let H be a real Hilbert space, and let B be a maximal monotone operator

on H.For r> and x∈H,deﬁne the resolvent Jrx.Then the following holds:

s–t

s Jsx–Jtx,Jsx–x ≥ Jsx–Jtx

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for all s,t> and x∈H.

From Lemma ., we have that

Jλx–Jμx ≤

|λ–μ|/λx–Jλx (.)

for allλ,μ>  andx∈H; see also [, ]. To prove our main result, we need the following lemmas.

Remark . It is not hard to know that ifAis anα-inverse strongly monotone mapping, then it is_α -Lipschitzian and hence uniformly continuous. Clearly, the class of monotone mappings includes the class ofα-inverse strongly monotone mappings.

Remark . It is well known that ifT:C→Cis a nonexpansive mapping, thenI–Tis



-inverse strongly monotone, whereIis the identity mapping onH; see, for instance, [].

It is known that the resolventJris ﬁrmly nonexpansive andB– =F(Jr) for allr> .

Lemma .[] Let H be a real Hilbert space,and let C be a nonempty closed convex

subset of H.Letα> .Let A be anα-inverse strongly monotone mapping of C into H,and let B be a maximal monotone operator on H such that the domain of B is included in C. Let Jλ= (I+λB)–be the resolvent of B for anyλ> .Then the following hold:

(i) ifu,v∈(A+B)–_()_,_then_Au₌_Av_;

(ii) for anyλ> ,u∈(A+B)–_()_{if and only if}_u₌_J

λ(I–λA)u.

Lemma .[, ] Let{an} be a sequence of nonnegative real numbers satisfying the

property

an+≤( –tn)an+bn+tncn,

where{tn},{bn}and{cn}satisfy the restrictions: (i) ∞_n_=tn=∞;

(ii) ∞_n_=bn<∞; (iii) lim sup_n_→∞cn≤.

Then{an}converges to zero as n→ ∞.

Lemma .[] Let H be a Hilbert space,and let g:H→H be a k-contraction with

 <k< .Let V be aγ¯-strongly monotone and L-Lipschitzian continuous operator on H withγ¯> and L> .Let a real numberγ satisfy <γ <γ_k¯.Then V–γg:H→H is a (γ¯–γk)-strongly monotone and(L+γk)-Lipschitzian continuous mapping.Furthermore, let C be a nonempty closed convex subset of H.Then PC(I–V +γg)has a unique ﬁxed

point zin C.This point z∈C is also a unique solution of the variational inequality

(V–γf)z,q–z

≥, ∀q∈C.

3 Main results

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Theorem . Let H be a real Hilbert space,and let C be a nonempty,closed and convex subset of H.Letα> and A be anα-inverse-strongly monotone mapping of C into H.Let the set-valued maps B:D(B)⊂C→H _{and F}_:_D(F)_⊂_C_→_H _{be maximal monotone.}_Let

Jλ= (I+λB)–and Tr= (I+rF)–be the resolvents of B forλ> and F for r> ,respectively.

Let <k< and let g be a k-contraction of H into itself.Let V be aγ¯-strongly monotone and L-Lipschitzian continuous operator withγ¯> and L> .Let T:C→C be a-generalized hybrid mapping such that:=F(T)∩(A+B)–__∩_F–_₌_∅_._Take_μ_,_γ_∈_R_{as follows:}

 <μ<γ¯

L,  <γ <

¯

γ–L_μ k .

Let the sequence{xn} ⊂H be generated by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x=x∈H arbitrarily,

zn=Jλn(I–λnA)Trnxn, yn=_n

n– k=Tkzn,

xn+=αnγg(xn) + (I–αnV)yn, ∀n= , , . . . ,

(.)

where the sequences{αn},{λn}and{rn}satisfy the following restrictions: (i) {αn} ⊂[, ],limn→∞αn= and

∞

n=αn=∞;

(ii) there exist constantsaandbsuch that <a≤λn≤b< αfor alln∈N; (iii) lim infn→∞rn> .

Then{xn}converges strongly to a point pof,where pis a unique ﬁxed point of P(I–V+

γg).This point p∈is also a unique solution of the hierarchical variational inequality

≥, ∀q∈. (.)

Proof First we prove that{xn}is bounded andlimn→∞xn–pexists for allp∈. Let

p∈, we have thatp=Jλn(I–λnA)pandp=Trnp. Puttingun=Trnxn, we have that

zn–p=Jλn(I–λnA)Trnxn–Jλn(I–λnA)p



≤(Trnxn–Trnp) –λn(ATrnxn–ATrnp)



=Trnxn–Trnp

_{– }_λ

n un–p,Aun–Ap+λnAun–Ap ≤ un–p– λnαAun–Ap+λnAun–Ap

≤ xn–p–λn(α–λn)Aun–Ap

≤ xn–p. (.)

This together with quasi-nonexpansiveness ofTimplies that

yn–p=

 n

n–

k=

Tkzn–p

≤ 

n

n–

k=

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≤ 

n

n–

k=

zn–p

=zn–p ≤ xn–p. (.)

Therefore, we have

xn+–p=αn

γg(xn) –Vp

+ (I–αnV)yn– (I–αnV)p ≤αnγg(xn) –Vp+(I–αnV)yn– (I–αnV)p

≤αnγkxn–p+αnγg(p) –Vp+(I–αnV)yn– (I–αnV)p. (.)

Puttingτ=γ¯–L_μ, we can calculate the following:

(I–αnV)yn– (I–αnV)p =(yn–p) –αn(Vyn–Vp)

=yn–p– αn yn–p,Vyn–Vp+αnVyn–Vp ≤ yn–p– αnγ¯yn–p+αnLyn–p

= – αnγ¯+αnL

yn–p

= – αnτ–αnLμ+αnL

yn–p ≤ – αnτ–αn

Lμ–αnL

+α_nτyn–p ≤ – αnτ+αnτ

yn–p

= ( –αnτ)yn–p. (.)

Since  –αnτ> , we obtain that

(I–αnV)yn– (I–αnV)p≤( –αnτ)yn–p.

Therefore, by (.), we have

xn+–p ≤αnγkxn–p+αnγg(p) –Vp+ ( –αnτ)yn–p ≤αnγkxn–p+αnγg(p) –Vp+ ( –αnτ)xn–p

= –αn(τ–γk)

xn–p+αnγg(p) –Vp

= –αn(τ–γk)

xn–p+αn(τ–γk)

γg(p) –Vp

τ–γk

≤max

xn–p,

γg(p) –Vp

τ–γk

for alln∈N,

which yields that the sequence{xn–p}is bounded, so are{xn},{yn},{Vyn},{g(xn)}and {Tn_z

n}. Using Lemma ., we can take a uniquep∈ of the hierarchical variational

inequality

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We show thatlim sup_n_→∞ (V–γg)p,xn–p ≥. We may assume, without loss of

gen-erality, that there exists a subsequence{xnk}of{xn}converging tow∈C, ask→ ∞, such that

lim sup

n→∞

(V–γg)p,xn–p

= lim

k→∞

(V–γg)p,xnk–p

.

Since {xnk –p} is bounded, there exists a subsequence {xn_ki} of {xnk} such that limi→∞xn_ki–pexists. Now we shall prove thatw∈.

(a) We ﬁrst provew∈F(T). We notice that

xn+–yn=αnγg(xn) + (I–αnV)yn–yn=αnγg(xn) –Vyn.

In particular, replacingnbynkiand takingi→ ∞in the last equality, we have

lim

i→∞xnki+–ynki= ,

so we haveyn_kiw. SinceTis -generalized hybrid, there existα,α,β,β∈Rsuch that

αTx–Ty+αTx–Ty+ ( –α–α)x–Ty

≤βTx–y+βTx–y+ ( –β–β)x–y

for allx,y∈C. For anyn∈Nandk= , , , . . . ,n– , we compute the following:

≤βTTkzn–y 

+βTTkzn–y 

+ ( –β–β)Tkzn–y 

–αTTkzn–Ty–αTTkzn–Ty– ( –α–α)Tkzn–Ty

=βTk+zn–y+βTk+zn–y+ ( –β–β)Tkzn–y

–αTk+zn–Ty 

–αTk+zn–Ty 

– ( –α–α)Tkzn–Ty 

≤βTk+zn–Ty 

+Ty–y+βTk+zn–Ty 

+Ty–y

+ ( –β–β)Tkzn–Ty 

+Ty–y–αTk+zn–Ty 

–αTk+zn–Ty 

– ( –α–α)Tkzn–Ty 

=βTk+zn–Ty 

+Ty–y+ Tk+zn–Ty,Ty–y

+βTk+zn–Ty 

+Ty–y+ Tk+zn–Ty,Ty–y

+ ( –β–β)Tkzn–Ty+Ty–y+ 

Tkzn–Ty,Ty–y

–αTk+zn–Ty–αTk+zn–Ty– ( –α–α)Tkzn–Ty

= (β–α)Tk+zn–Ty 

+ (β–α)Tk+zn–Ty 

+ (α+α–β–β)Tkzn–Ty 

×(β+β+  –β–β)Ty–y+ 

βTk+zn–βTy+βTk+zn–βTy

+ ( –β–β)Tkzn– ( –β–β)Ty,Ty–y

= (β–α)Tk+zn–Ty 

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–(β–α) + (α–β)Tkzn–Ty 

+Ty–y

+ βTk+zn+βTk+zn+ ( –β–β)Tkzn–Ty,Ty–y

= (β–α)Tk+zn–Ty–Tkzn–Ty

+ (β–α)Tk+zn–Ty 

–Tkzn–Ty 

+Ty–y+ βTk+zn+βTk+zn+ ( –β–β)Tkzn–Ty,Ty–y

=Ty–y+ Tkzn–Ty,Ty–y

+ β

Tk+zn–Tkxn

+β

Tk+zn–Tkzn

,Ty–y

+ (β–α)Tk+zn–Ty–Tkzn–Ty

+ (β–α)Tk+zn–Ty–Tkzn–Ty

.

Summing up these inequalities fromk=  ton– , we get

≤

n–

k=

Ty–y+  _n_–

k=

Tkzn–Ty

,Ty–y

+  β n– k=

Tk+zn–Tkzn

+β

n–

k=

Tk+zn–Tkzn

,Ty–y

+ (β–α) n–

k=

_Tk+_z n–Ty



–Tkzn–Ty 

+ (β–α) n–

k=

Tk+zn–Ty 

–Tkzn–Ty 

=nTy–y+  _n_–

k=

Tkzn–nTy,Ty–y

+ β

Tn+zn–Tnzn–zn–Tzn

+β

Tnzn–zn

,Ty–y

+ (β–α)Tn+zn–Ty+Tnzn–Ty–zn–Ty–Tzn–Ty

+ (β–α)Tnzn–Ty–zn–Ty

.

Dividing this inequality byn, we get

≤ Ty–y+ yn–Ty,Ty–y

+ 

 nβ

Tn+zn–Tnzn–zn–Tzn

+ 

nβ

Tnzn–zn

,Ty–y

+ 

n(β–α)T

n+_z n–Ty



+Tnzn–Ty 

–zn–Ty–Tzn–Ty

+ 

n(β–α)T

n_z n–Ty



–zn–Ty

.

Replacingnbynkiand lettingi→ ∞in the last inequality, we have

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In particular, replacingybywin (.), we obtain that

≤ Tw–w+ w–Tw,Tw–w= –Tw–w,

which ensures thatw∈F(T).

(b) We prove thatw∈(A+B)–_{. From (.), (.) and (.), we have}

xn+–p≤(I–αnV)yn– (I–αnV)p 

+ αn

γg(xn) –Vp,xn+–p

≤( –αnτ)yn–p+ αn

≤( –αnτ)zn–p+ αn

≤( –αnτ)

xn–p–λn(α–λn)Aun–Ap

+ αn

= – αnτ+αnτ

xn–p– ( –αnτ)λn(α–λn)Aun–Ap

+ αn

≤ xn–p+αnτxn–p– ( –αnτ)λn(α–λn)Aun–Ap

+ αn

, (.)

and hence

( –αnτ)λn(α–λn)Aun–Ap≤ xn–p–xn+–p+αnτxn–p

+ αn

. (.)

Replacingnbynkiin (.), we have

( –αn_kiτ)λn_ki(α–λn_ki)Aun_ki–Ap

≤ xn_ki–p–xn_ki+–p+αn_kiτxn_ki–p

+ αn_ki

γg(xn) –Vp,xn_ki+–p

.

Sincelimn→∞αn= ,  <a≤λn≤b< αand the existence oflimi→∞xn_ki–p, we have

lim

i→∞Aunki–Ap= . (.)

We also have from (.) that

un–p = Trnxn–Trnp



≤ xn–p,un–p

=xn–p+un–p–un–xn,

and hence

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From (.), (.), (.) and (.), we obtain the following:

+ αn

≤( –αnτ)yn–p+ αn

≤( –αnτ)zn–p+ αn

≤( –αnτ)

un–p–λn(α–λn)Aun–Ap

+ αn

≤( –αnτ)

xn–p–un–xn

– ( –αnτ)λn(α–λn)Aun–Ap

+ αn

≤ – αnτ+αnτ

xn–p– ( –αnτ)un–xn

+ αn

≤ xn–p+αnτxn–p– ( –αnτ)un–xn

+ αn

,

and hence

( –αnτ)un–xn≤ xn–p–xn+–p+αnτxn–p

+ αn

. (.)

Replacingnbynkiin (.), we have

( –αn_kiτ)un_ki–xn_ki≤ xn_ki–p–xn_ki+–p+αn_kiτxn_ki–p

– ( –αn_kiτ)λn_ki(α–λn_ki)Aun_ki–Ap

+ αn_ki

γg(xn_ki) –Vp,xn_ki+–p

.

From (.),limn→∞αn=  and the existence oflimi→∞xn_ki–p, we have

lim

i→∞unki–xnki= . (.)

On the other hand, sinceJλnis ﬁrmly nonexpansive andun=Trnxn, we have that

zn–p=Jλn(I–λnA)un–Jλn(I–λnA)p 

≤zn–p, (I–λnA)un– (I–λnA)p

=  

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–zn–p– (I–λnA)un+ (I–λnA)p 

≤ 



zn–p+un–p–zn–p– (I–λnA)un+ (I–λnA)p

≤ 



zn–p+xn–p–zn–un– λn zn–un,Aun–Ap

–λ_nAun–Ap

,

and hence

zn–p

≤ xn–p–zn–un– λn zn–un,Aun–Ap–λnAun–Ap. (.)

From (.), (.), (.) and (.), we have

+ αn

≤( –αnτ)yn–p+ αn

≤( –αnτ)zn–p+ αn

≤( –αnτ)

xn–z–zn–un– λn zn–un,Aun–Ap

–λ_nAun–Ap

+ αn

≤ xn–p+αnτxn–p– ( –αnτ)zn–un

– ( –αnτ)λn(λn– α)zn–unAun–Ap

– ( –αnτ)λnAun–Ap+ αn

,

and hence

( –αnτ)zn–un

≤ xn–p–xn+–p+αnτxn–p

– ( –αnτ)λn(λn– α)zn–unAun–Ap

– ( –αnτ)λnAun–Ap+ αn

. (.)

Replacingnbynkiin (.), we have

( –αn_kiτ)zn_ki–un_ki

≤ xn_ki–p–xn_ki+–p+αn_kiτxn_ki–p

– ( –αn_kiτ)λn_ki(λn_ki– α)zn_ki–un_kiAun_ki–Ap

– ( –αn_kiτ)λn_kiAun_ki–Ap+ αn_ki

γg(xn_ki) –Vp,xn_ki+–p

.

From (.),limn→∞αn=  and the existence oflimi→∞xn_ki–p, we obtain that

lim

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Sincezn_ki–xn_ki ≤ zn_ki–un_ki+un_ki–xn_ki, by (.) and (.), we obtain that

lim

i→∞znki–xnki= . (.)

SinceAis Lipschitz continuous, we also obtain

lim

i→∞Aznki–Axnki= . (.)

Sincezn=Jλ(I–λA)un, we have that

zn = (I+λnB)–(I–λnA)un

⇔ (I–λnA)un∈(I+λnB)zn=zn+λnBzn ⇔ un–zn–λnAun∈λnBzn

⇔ 

λn

(un–zn–λnAun)∈Bzn.

SinceBis monotone, we have that for (u,v)∈B,

zn–u,



λn

(un–zn–λnAun) –v

≥,

and hence

zn–u,un–zn–λn(Aun+v)

≥. (.)

Replacingnbynkiin (.), we have that

zn_ki–u,un_ki–zn_ki–λn_ki(Aun_ki+v)

≥. (.)

Sincexn_kiwandxn_ki–un_ki→, soun_kiw. From (.), we get thatzn_kiw, together

with (.), we have that

w–u, –Aw–v ≥.

SinceBis maximal monotone, (–Aw)∈Bw, that is,w∈(A+B)–_.

(c) Next, we show that w∈F–_{. Since}_F _{is a maximal monotone operator, we have}

from (.) thatAr_nkixn_ki∈FTr_nkixn_ki, whereAris the Yosida approximation ofFforr> .

Furthermore, we have that for any (u,v)∈F,

u–un_ki,v–

xn_ki–un_ki

rn_ki

≥.

Sincelim infn→∞rn> ,un_kiwandxn_ki–un_ki→, we have

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SinceFis a maximal monotone operator, we have ∈Fw, that is,w∈F–_{. By (a), (b) and}

(c), we conclude that

w∈F(T)∩(A+B)–∩F–.

Using (.), we obtain

lim sup

n→∞

(V–γg)p,xn–p

= lim

k→∞

(V–γg)p,xnk–p)

=(V–γg)p,w–p)

≥.

Finally, we prove thatxn→p. Notice that

xn+–p=αn

γg(xn) –p

+ (I–αnV)yn– (I–αnV)p,

we have

xn+–p≤( –αnτ)yn–p+ αn

γg(xn) –Vp,xn+–p

≤( –αnτ)xn–p+ αn

γg(xn) –Vp,xn+–p

≤( –αnτ)xn–p+ αnγkxn–pxn+–p

+ αn

γg(p) –Vp,xn+–p

≤( –αnτ)xn–p+αnγk

xn–p+xn+–p

+ αn

γg(p) –Vp,xn+–p

≤( –αnτ)+αnγk

xn–p+αnγkxn+–p

+ αn

γg(p) –Vp,xn+–p

,

and hence

xn+–p≤

 – αnτ+ (αnτ)+αnγk

 –αnγk

xn–p

+ αn  –αnγk

γg(p) –Vp,xn+–p

=

 –(τ–γk)αn  –αnγk

xn–p+

(αnτ)

 –αnγk

xn–p

γg(p) –Vp,xn+–p

=

 –(τ–γk)αn  –αnγk

xn–p+

αn·αnτ

 –αnγk

xn–p

γg(p) –Vp,xn+–p

= ( –βn)xn–p

+βn

αnτxn–p

(τ–γk) + 

τ–γk

γg(p) –Vp,xn+–p

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whereβn=(_–τ–αγnkγ)kαn. Since ∞

n=βn=∞, we have from Lemma . and (.) thatxn→p.

This completes the proof.

4 Applications

LetHbe a Hilbert space, and letf be a proper lower semicontinuous convex function of Hinto (–∞,∞]. Then the subdiﬀerential∂f off is deﬁned as follows:

∂f(x) =z∈H:f(x) + z,y–x ≤f(y),y∈H

for allx∈H; see, for instance, []. From Rockafellar [], we know that∂f is maximal monotone. LetCbe a nonempty closed convex subset ofH, and letiC be the indicator

function ofC,i.e.,

iC(x) =

⎧ ⎨ ⎩

, x∈C,

∞, x∈/C.

TheniCis a proper lower semicontinuous convex function ofHinto (–∞,∞], and then the

subdiﬀerential∂iCofiCis a maximal monotone operator. So, we can deﬁne the resolvent Jλof∂iC forλ> ,i.e.,

Jλx= (I+λ∂iC)

–_x

for allx∈H. We have that for anyx∈Handu∈C,

u=Jλx ⇔ x∈u+λ∂iCu

⇔ x∈u+λNCu

⇔ x–u∈λNCu

⇔ 

λ x–u,v–u ≤, ∀v∈C

⇔ x–u,v–u ≤, ∀v∈C

⇔ u=PCx,

whereNCuis the normal cone toCatu,i.e.,

NCu=

x∈H: z,v–u ≤,∀v∈C.

LetCbe a nonempty, closed and convex subset ofH, and letf :C×C→Rbe a bifunc-tion. For solving the equilibrium problem, let us assume that the bifunctionf:C×C→R satisﬁes the following conditions.

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunctionF,ϕand the setC:

(A) f(x,x) = for allx∈C;

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(A) for allx,y,z∈C,

lim sup

t↓

ftz+ ( –t)x,y≤f(x,y);

(A) for allx∈C,f(x,·)is convex and lower semicontinuous;

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

f(z,yx) +ϕ(yx) +



r yx–z,z–x<ϕ(z);

(B) Cis a bounded set.

We know the following lemma which appears implicitly in Blum and Oettli [].

Lemma .[] Let C be a nonempty closed convex subset of H,and let f be a bifunction of C×C intoRsatisfying(A)-(A).Let r> and x∈H.Then there exists a unique z∈C such that

f(z,y) +

r y–z,z–x ≥, ∀y∈C.

By a similar argument as that in [, Lemma .], we have the following result.

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H.Let f : C×C→Rbe a bifunction which satisﬁes conditions(A)-(A),and letϕ:C→R∪ {+∞} be a proper lower semicontinuous and convex function.Assume that either(B)or(B) holds.For r> and x∈H,deﬁne a mapping Tr:H→C as follows:

Tr(x) =

z∈C:f(z,y) +ϕ(y) +

r y–z,z–x ≥ϕ(z),∀y∈C

for all x∈H.Then following conclusions hold:

() For eachx∈H,Tr(x)=∅; () Tris single-valued;

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

Tr(x) –Tr(y)≤

Tr(x) –Tr(y),x–y

;

() Fix(Tr) =MEP(f,ϕ);

() MEP(f,ϕ)is closed and convex.

We call suchTrthe resolvent off forr> . Using Lemmas . and ., Takahashiet al.

[] obtained the following lemma. See [] for a more general result.

Lemma .[] Let H be a Hilbert space,and let C be a nonempty closed convex subset of H.Let f:C×C→Rsatisfy(A)-(A).Let Af be a set-valued mapping of H into itself

deﬁned by

Afx=

⎧ ⎨ ⎩

{z∈H:f(x,y)≥ y–x,z,∀y∈C}, ∀x∈C,

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Then MEP(f) =A–

f and Af is a maximal monotone operator withdomAf ⊂C.

Further-more,for any x∈H and r> ,the resolvent Trof f coincides with the resolvent of Af,i.e.,

Trx= (I+rAf)–x.

Applying the idea of the proof in Lemma ., we have the following results.

Lemma . Let H be a Hilbert space,and let C be a nonempty closed convex subset of H. Let f :C×C→Rsatisfy(A)-(A),and letϕ:C→R∪ {+∞}be a proper lower semicon-tinuous and convex function.Assume that either(B)or(B)hold.Let A(f,ϕ)be a set-valued

mapping of H into itself deﬁned by

A(f,ϕ)x=

⎧ ⎨ ⎩

{z∈H:f(x,y) +ϕ(y) –ϕ(x)≥ y–x,z,∀y∈C}, ∀x∈C,

∅, ∀x∈/C. (.)

Then MEP(f,ϕ) =A–

(f,ϕ)and A(f,ϕ)is a maximal monotone operator withdomA(f,ϕ)⊂C.

Furthermore,for any x∈H and r> ,the resolvent Trof f coincides with the resolvent of

A(f,ϕ),i.e.,

Trx= (I+rA(f,ϕ))–x.

Proof It is obvious thatMEP(f,ϕ) =A–

(f,ϕ). In fact, we have that

z∈MEP(f,ϕ) ⇔ f(z,y) +ϕ(y) –ϕ(z)≥, ∀y∈C

⇔ f(z,y) +ϕ(y) –ϕ(z)≥ y–z, , ∀y∈C

⇔ ∈A(f,ϕ)z

⇔ z∈A–₍_f_,ϕ).

We show thatA(f,ϕ)is monotone. Let (x,z), (x,z)∈A(f,ϕ)be given. Then we have, for all

y∈C,

f(x,y) +ϕ(y) –ϕ(x)≥ y–x,z and f(x,y) +ϕ(y) –ϕ(x)≥ y–x,z,

and hence

f(x,x) +ϕ(x) –ϕ(x)≥ x–x,z and f(x,x) +ϕ(x) –ϕ(x)≥ x–x,z.

It follows from (A) that

≥f(x,x) +f(x,x)≥ x–x,z+ x–x,z= –x–x,z–z.

This implies thatA(f,ϕ) is monotone. We next prove thatA(f,ϕ)is maximal monotone. To

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H for allr> , whereR(I+rA(f,ϕ)) is the range ofI+rA(f,ϕ). Letx∈H andr> . Then,

from Lemma ., there existsz∈Csuch that

f(z,y) +ϕ(y) –ϕ(z) +

r y–z,z–x ≥, ∀y∈C.

So, we have that

f(z,y) +ϕ(y) –ϕ(z)≥

y–z, r(x–z)

, ∀y∈C.

By the deﬁnition ofA(f,ϕ), we get

A(f,ϕ)z

 r(x–z),

and hencex∈z+rA(f,ϕ)z. Therefore,H⊂R(I+rA(f,ϕ)) andR(I+rA(f,ϕ)) =H. Also,x∈

z+rA(f,ϕ)zimplies thatTrx= (I+rA(f,ϕ))–xfor allx∈Handr> .

Using Theorem ., we obtain the following results for an inverse-strongly monotone mapping.

Theorem . Let H be a real Hilbert space,and let C be a nonempty,closed and convex subset of H.Letα> and let A be anα-inverse-strongly monotone mapping of C into H.Let  <k< and let g be a k-contraction of H into itself.Let V be aγ¯-strongly monotone and L-Lipschitzian continuous operator withγ¯> and L> .Let T:C→C be a-generalized hybrid mapping such that:=F(T)∩VI(C,A)= ∅.Takeμ,γ∈Ras follows:

 <μ<γ¯

L,  <γ <

¯

γ–L_μ k .

Let{xn} ⊂H be a sequence generated by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=PC(I–λnA)PCxn,

yn=_n

n–

k=Tkzn, ∀n= , , . . . ,

xn+=αnγg(xn) + (I–αnV)yn for all n∈N,

(.)

where{αn} ⊂(, )and{rn} ⊂(,∞)satisfy

lim

n→∞αn= ,

∞

n=

αn=∞ and lim inf n→∞ rn> .

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Proof PutB=F=∂iCin Theorem .. Then, forλn>  andrn> , we have that

Jλn=Trn=PC.

Furthermore we have, from the proof of [, Theorem ], that

(∂iC)– =C and (A+∂iC)–=VI(C,A).

Thus we obtained the desired results by Theorem ..

Using Theorem ., we ﬁnally prove a strong convergence theorem for inverse-strongly monotone operators and equilibrium problems in a Hilbert space.

Theorem . Let H be a real Hilbert space,and let C be a nonempty,closed and convex subset of H.Letα> and let A be anα-inverse-strongly monotone mapping of C into H. Let B:D(B)⊂C→H _{be maximal monotone.}_{Let J}

λ= (I+λB)–be the resolvent of B forλ> .Let <k< and let g be a k-contraction of H into itself.Let V be aγ¯-strongly monotone and L-Lipschitzian continuous operator withγ¯> and L> .Let f :C×C→R be a bifunction satisfying conditions(A)-(A),and letϕ:C→R∪{+∞}be a proper lower semicontinuous and convex function.Assume that either(B)or(B)holds.Let T:C→ C be a-generalized hybrid mapping with:=F(T)∩(A+B)–__∩_MEP(f_,_ϕ₎₌_∅_._Take

μ,γ ∈Ras follows:

 <μ<γ¯

L,  <γ <

¯

γ–L_μ k .

Let{xn} ⊂H be a sequence generated by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

f(un,y) +ϕ(y) –ϕ(un) +_r_n y–un,un–xn ≥, ∀y∈C,

zn=Jλn(I–λnA)un,

yn=_nkn=–Tkzn, ∀n= , , . . . ,

xn+=αnγg(xn) + (I–αnV)yn, ∀n∈N,

(.)

where{αn} ⊂(, )and{rn} ⊂(,∞)satisfy

lim

n→∞αn= ,

∞

n=

αn=∞ and lim inf n→∞ rn> .

≥, ∀q∈. (.)

Proof Sincef is a bifunction ofC×CintoRsatisfying conditions (A)-(A) andϕ:C→

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Aϕ_f deﬁned by (.) is a maximal monotone operator withdomAϕ_f ⊂C. PutF =Aϕ_f in Theorem .. Then we obtain thatun=Trnxn. Therefore, we arrive at the desired results.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors read and approved the ﬁnal manuscript.

Acknowledgements

The ﬁrst author was partially supported by Naresuan University.

Received: 18 May 2013 Accepted: 20 August 2013 Published:07 Nov 2013 References

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Cite this article as:Wangkeeree and Boonkong:A general iterative method for two maximal monotone operators