Strong convergence theorems of general split equality problems for quasi nonexpansive mappings

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R E S E A R C H

Open Access

Strong convergence theorems of general split

equality problems for quasi-nonexpansive

mappings

Shih-sen Chang

1

_{and Ravi P Agarwal}

2*

*_{Correspondence:}

Agarwal@tamuk.edu

2_{Department of Mathematics, Texas} A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to introduce and study the general split equality problem and general split equality fixed point problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converges strongly to a solution of the general split equality fixed point problem and the general split equality problem for

quasi-nonexpansive mappings in Hilbert spaces. As an application, we shall utilize our results to study the null point problem of maximal monotone operators, the split feasibility problem, and the equality equilibrium problem. The results presented in the paper extend and improve the corresponding results announced by Moudaﬁ

et al.(Nonlinear Anal. 79:117-121, 2013; Trans. Math. Program. Appl. 1:1-11, 2013), Eslamian and Latif (Abstr. Appl. Anal. 2013:805104, 2013) and Chenet al.(Fixed Point Theory Appl. 2014:35, 2014), Censor and Elfving (Numer. Algorithms 8:221-239, 1994), Censor and Segal (J. Convex Anal. 16:587-600, 2009) and some others.

Keywords: general split equality problem; general split equality ﬁxed point problem; quasi-nonexpansive mapping; split feasibility problem

1 Introduction

LetCandQbe nonempty closed convex subsets of real Hilbert spacesHandH, respec-tively. Thesplit feasibility problem(SFP) is formulated as

to ﬁndingx∗∈CandAx∗∈Q, (.)

whereA:H→His a bounded linear operator. In , Censor and Elfving [] first intro-duced theSFPin finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction []. It has been found that theSFPcan also be used in various disciplines such as image restoration, and com-puter tomograph and radiation therapy treatment planning [–]. TheSFPin an infinite-dimensional real Hilbert space can be found in [, , –].

Assuming that theSFPis consistent, it is not hard to see thatx∗∈CsolvesSFPif and only if it solves the ﬁxed point equation

x∗=PCI–γA∗(I–PQ)Ax∗,

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wherePCandPQare the metric projection fromHontoCand fromHontoQ, respec-tively,γ >  is a positive constant andA∗is the adjoint ofA.

A popular algorithm to be used to solveSFP(.) is due to Byrne’sCQ-algorithm[]:

xk+=PCI–γA∗(I–PQ)Axk, k≥,

whereγ ∈(, /λ) withλbeing the spectral radius of the operatorA∗A. Recently, Moudaﬁ [] introduced the followingsplit equality problem(SEP):

to ﬁndx∈C,y∈Qsuch thatAx=By, (.)

whereA:H→HandB:H→Hare two bounded linear operators. Obviously, ifB=I (identity mapping onH) andH=H, then (.) reduces to (.). This kind of split equality problem (.) allows asymmetric and partial relations between the variablesxandy. The interest is to cover many situations, such as decomposition methods for PDEs, applications in game theory, and intensity-modulated radiation therapy.

In order to solve the split equality problem (.), Moudaﬁ [] introduced the following relaxed alternatingCQ-algorithm:

xk+=PC_k(xk–γA∗(Axk–Byk)),

yk+=PQ_k(yk+βB∗(Axk–Byk)), (.)

where

Ck=x∈H;c(xk) +ξk,x–xk ≤, ξk∈∂c(xk), Qk=

y∈H;q(yk) +ηk,y–yk ≤

, ηk∈∂q(yk),

(.)

andc:H→R(respectivelyq:H→R) is a convex and subdiﬀerentiable function. Under suitable conditions, he proved that the sequence{xn}deﬁned by (.) converges weakly to a solution of the split equality problem (.).

Each nonempty closed convex subset of a Hilbert space can be regarded as a set of fixed points of a projection. In [], Moudafi and Al-Shemas introduced the followingsplit equality fixed point problem:

ﬁndx∈C:=F(S),y∈Q:=F(T) such thatAx=By, (.)

whereS:H→H andT:H→H are two ﬁrmly quasi-nonexpansive mappings,F(S) andF(T) denote the ﬁxed point sets ofSandT, respectively.

To solve the split equality fixed point problem (.) for firmly quasi-nonexpansive map-pings, Moudafiet al.[–] proposed the following iteration algorithm:

xk+=S(xk–γkA∗(Axk–Byk)),

yk+=T(yk+γkB∗(Axk–Byk)). (.)

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Motivated by the above works, the purpose of this paper is to introduce the following general split equality ﬁxed point problem:

(GSEFP) to ﬁndx∈C:= ∞

i=

F(Si),y∈Q:= ∞

i=

F(Ti) such thatAx=By, (.)

and thegeneral split equality problem:

(GSEP) to ﬁndx∈Cy∈Qsuch thatAx=By. (.)

For solving the GSEFP (.) and GSEP (.), in Sections  and , we propose an algorithm for ﬁnding the solutions of the general split equality ﬁxed point problem and general split equality problem in a Hilbert space. We establish the strong convergence of the proposed algorithms to a solution of GSEFP and GSEP. As applications, in Section  we utilize our results to study the split feasibility problem, the null point problem of maximal monotone operators, and the equality equilibrium problem.

2 Preliminaries

LetHbe a real Hilbert space andCbe a nonempty closed convex subset ofH. In the sequel, denote byF(T) the set of ﬁxed points of a mappingT and byxn→x∗ andxnx∗, the strong convergence and weak convergence of a sequence{xn}to a pointx∗, respectively.

Recall that a mappingT:H→His said to be nonexpansive, ifTx–Ty ≤ x–y, ∀x,y∈H. A typical example of a nonexpansive mapping is the metric projectionPCfrom H ontoC⊆H deﬁned byx–PCx=infy∈Cx–y. The metric projectionPC isﬁrmly nonexpansive,i.e.,

PCx–PCy≤ x–y,PCx–PCy, ∀x,y∈H, (.)

and it can be characterized by the fact that

PC(x)∈C and y–PC(x),x–PC(x) ≤, ∀x∈H,y∈C. (.)

Deﬁnition . A mappingT:H→His said to bequasi-nonexpansive, ifF(T)=∅, and

Tx–p ≤ x–p for eachx∈Handp∈F(T).

Lemma .[] Let H be a real Hilbert space,and{xn}be a sequence in H.Then,for any given sequence{λn}of positive numbers with

∞

i=λn= such that for any positive integers i,j with i<j,the following holds:

∞

i=

λnxn

 ≤

∞

i=

λnxn–λiλjxi–xj.

Lemma .[] Let H be a real Hilbert space.For any x,y∈H,the following inequality holds:

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Lemma .[] Let{tn}be a sequence of real numbers.If there exists a subsequence{ni}of {n}such that tni<tni+for all i≥,then there exists a nondecreasing sequence{τ(n)}with

τ(n)→ ∞ such that for all(suﬃciently large)positive integer numbers n, the following holds:

tτ(n)≤tτ(n)+, tn≤tτ(n)+.

In fact,

τ(n) =max{k≤n:tk≤tk+}.

Deﬁnition .(Demiclosedness principle) LetCbe a nonempty closed convex subset of

a real Hilbert spaceH, andT:C→Cbe a mapping withF(T)=∅. ThenI–T is said to bedemi-closed at zero, if for any sequence{xn} ⊂Cwithxnxandxn–Txn →, then x=Tx.

Remark . It is well known that ifT:C→Cis a nonexpansive mapping, thenI–Tis demi-closed at zero.

Lemma . Let{an},{bn},and{cn}be sequences of positive real numbers satisfying an+≤ ( –bn)an+cnfor all n≥.If the following conditions are satisﬁed:

() bn∈(, )and∞_n=bn=∞, () ∞_n=cn<∞,orlim sup_n_→∞cn

bn ≤,

thenlim_n_→∞an= .

3 Strong convergence theorem for general split equality ﬁxed point problem

Throughout this section we always assume that

() H,H,Hare real Hilbert spaces;

() {Si}∞_i=:H→Hand{Ti}∞_i=:H→Hare two families of one-to-one and quasi-nonexpansive mappings;

() A:H→HandB:H→Hare two bounded linear operators; () f =_ff, wherefi,i= , is ak-contractive mapping onHiwithk∈(, );

() C:=∞_i=F(Si),Q:=∞_i=F(Ti),is the set of solutions of GSEFP (.),

P=

PC PQ

, Ki=

Si Ti

, G= [A –B], G∗G=

A∗A –A∗B –B∗A B∗B

;

() for any givenw∈H×H, the iterative sequence{wn} ⊂H×His generated by

wn+=P

αnwn+βnf(wn) + ∞

i=

γn,i

KiI–λn,iG∗Gwn

, n≥, (.)

where{αn},{βn},{γn,i}are the sequences of nonnegative numbers with

αn+βn+ ∞

i=

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We are now in a position to give the following main result.

Lemma . Let w∗= (x∗,y∗)be a point in C×Q,i.e.,x∗∈C=∞_i=F(Si)and y∗∈Q=

_∞

i=F(Ti).Then the following statements are equivalent:

(i) w∗is a solution to GSEFP(.);

(ii) w∗=Ki(w∗)for eachi≥andG(w∗) = ;

(iii) for eachi≥and for eachλ> ,w∗solves the ﬁxed point equations:

w∗=Kiw∗ and w∗=KiI–λG∗Gw∗. (.) Proof (i)⇒(ii). Ifw∗∈C×Qis a solution to GSEFP (.), then for eachi≥,w∗=Kiw∗, andAx∗=By∗. This implies that for eachi≥,w∗=Kiw∗, and

Gw∗= [A –B]

x∗ y∗

=Ax∗–By∗= .

(ii)⇒(iii). Ifw∗=Ki(w∗),∀i≥ andG(w∗) = , it is easy to see that (.) holds. (iii)⇒(i). From (.), for eachi≥ we haveKiw∗=Ki(I–λG∗G)w∗. SinceSiandTi both are one-to-one, so isKi. Hence we havew∗– (I–λG∗G)w∗= , for anyλ> . This implies thatG∗G(w∗) = , and so

 =G∗Gw∗,w∗ =Gw∗,Gw∗ =Gw∗,

i.e.,G(w∗) =Ax∗–By∗= .

This completes the proof of Lemma ..

Lemma . Ifλ∈(,

L),where L=G

_,_then_(I_–_λ_G∗_{G) :}_H

×H→H×His a non-expansive mapping.

Proof In fact for anyw,u∈H×H, we have

I–λG∗Gu–I–λG∗Gw

=(u–w) –λG∗G(u–w)

=u–w+λG∗G(u–w)– λu–w,G∗G(u–w) ≤ u–w+λLG(u–w)– λG(u–w),G(u–w) =u–w+λLG(u–w)– λG(u–w) =u–w–λ( –λL)G(u–w)

≤ u–w.

This completes the proof.

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(i) αn+βn+

∞

i=γn,i= ,for eachn≥;

(ii) limn→∞βn= ,and

∞

n=βn=∞;

(iii) lim inf_n_→∞αnγn,i> for eachi≥;

(iv) {λn,i} ⊂(,_L)for eachi≥,whereL=G;

(v) for eachi≥,the mappingI–Ki(I–λn,iG∗G)is demi-closed at zero,

then the sequence{wn}converges strongly to w∗=Pf(w∗)which is a solution of GSEFP (.).

Proof (I)First we prove that the sequence{wn}is bounded. In fact, for any givenz∈, it follows from Lemma . that

G(z) = , Kiz=z and z=KiI–λn,iG∗Gz for eachi≥.

By the assumptions and Lemma ., for eachλ∈(,_L), (I–λG∗G) :H×H→H×H is nonexpansive, and for eachi≥,Ki=

Si

Ti

is quasi-nonexpansive, hence we have

wn+–z=

P

i=

γn,iKiI–λn,iG∗Gwn

–P(z)

≤αnwn–z+βnf(wn) –z+ ∞

i=

γn,iKi

I–λn,iG∗G

wn–z

i=

γn,iI–λn,iG∗G

wn–z

i=

γn,iI–λn,iG∗G

wn–

I–λn,iG∗G

z

i=

γn,iwn–z = ( –βn)wn–z+βnf(wn) –z

≤( –βn)wn–z+βnf(wn) –f(z)+βnf(z) –z ≤( –βn)wn–z+kβnwn–z+βnf(z) –z = – ( –k)βn

wn–z+ ( –k)βn 

 –kf(z) –z

≤max

wn–z, 

 –kf(z) –z

.

By induction, we can prove that

wn–z ≤max

w–z, 

 –kf(z) –z

.

This shows that{wn}is bounded, and so is{f(wn)}. (II)Now we prove that the following inequality holds:

αnγn,iwn–Ki

I–λn,iG∗Gwn

≤ wn–z–wn+–z+βnf(wn) –z 

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Indeed, it follows from (.) and Lemma . that for eachi≥

wn+–z =

P

αn(wn–z) +βn

f(wn) –z+ ∞

i=

γn,i

KiI–λn,iG∗Gwn

–z 

≤αnwn–z+βnf(wn) –z+ ∞

i=

γn,iKi

I–λn,iG∗Gwn–z

–αnγn,iwn–Ki

I–λn,iG∗Gwn ≤αnwn–z+βnf(wn) –z

 +

∞

i=

γn,iwn–z

I–λn,iG∗Gwn = ( –βn)wn–z+βnf(wn) –z



I–λn,iG∗Gwn. This implies that for eachi≥

αnγn,iwn–Ki

I–λn,iG∗Gwn≤ wn–z–wn+–z+βnf(wn) –z. Inequality (.) is proved.

It is easy to see that the solution setof GSEFP (.) is a nonempty closed and convex subset inC×Q, hence the metric projectionPis well deﬁned. In addition, sincePf : H×H→H×His a contractive mapping, there exists aw∗∈such that

w∗=Pf

w∗. (.)

(III)Now we prove that wn→w∗. For this purpose, we consider two cases.

Case I. Suppose that the sequence {wn–w∗} is monotone. Since {wn–w∗} is bounded, {wn–w∗}is convergent. Since w∗ ∈, in (.) taking z=w∗ and letting n→ ∞, in view of conditions (ii) and (iii), we have

lim

n→∞wn–Ki

I–λn,iG∗Gwn=  for eachi≥. (.) On the other hand, by Lemma . and (.), we have

wn+–w∗ =

P

i=

γn,iKiI–λn,iG∗Gwn

–w∗



≤αn

wn–w∗+βn

f(wn) –w∗

+ ∞ i= γn,i

KiI–λn,iG∗Gwn–w∗



≤αn

wn–w∗+ ∞

i=

γn,i

KiI–λn,iG∗Gwn–w∗

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+ βn

f(wn) –w∗,wn+–w∗ (by Lemma .)

≤

αnwn–w∗+ ∞

i=

γn,iwn–w∗



+ βn

f(wn) –fw∗,wn+–w∗ + βn

fw∗–w∗,wn+–w∗

= ( –βn)wn–w∗+ βnkwn–w∗wn+–w∗ + βn

fw∗–w∗,wn+–w∗

≤( –βn)wn–w∗+βnkwn–w∗+wn+–w∗ + βn

fw∗–w∗,wn+–w∗.

Simplifying we have

wn+–w∗ 

≤( –βn)+βnk

 –βnk wn–w ∗

+ βn  –βnk

fw∗–w∗,wn+–w∗

=  – βn+βnk

 –βnk wn–w ∗

+ β  n

 –βnkwn–w ∗

+ βn  –βnk

fw∗–w∗,wn+–w∗

=

 –( –k)βn  –βnk

wn–w∗

+( –k)βn  –βnk

βnM ( –k)+

  –k

fw∗–w∗,wn+–w∗

= ( –ηn)wn–w∗+ηnδn, (.)

whereηn=(–k)_–_β_nβ_kn,δn=_(–k)βnM +_–k f(w∗) –w∗,wn+–w∗,M:=supn≥wn–w∗. By condition (ii),lim_n_→∞βn=  and

∞

n=βn=∞, and so

∞

n=ηn=∞. Next we prove that

lim sup

n→∞ δn≤. (.)

In fact, since{wn}is bounded inC×Q, there exists a subsequence{wnk} ⊂ {wn}with

wn_kv∗(some point inC×Q), andλnk,i→λi∈(,



L) such that

lim

n→∞

fw∗–w∗,wnk–w

∗ ₌_{lim sup} n→∞

fw∗–w∗,wn–w∗.

In view of (.)

wnk–Ki

I–λnk,iG

∗_G_wn

k→ for eachi≥.

Again by the assumption that for eachi≥, the mappingI–Ki(I–λn,iG∗G) is demi-closed at zero, hence we have

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By Lemma ., this implies thatv∗∈. In addition, sincew∗=Pf(w∗), we have

lim sup

n→∞

fw∗–w∗,wn–w∗ = lim

n→∞

fw∗–w∗,wnk–w

∗

=fw∗–w∗,v∗–w∗ ≤.

This shows that (.) is true. Takingan=wn–w∗_,_bn₌_η_{n, and}_cn₌_δ

nηnin Lemma ., all conditions in Lemma . are satisﬁed. We havewn→w∗.

Case II. If the sequence{wn–w∗}is not monotone, by Lemma ., there exists a se-quence of positive integers:{τ(n)},n≥n(wherenis large enough) such that

τ(n) =maxk≤n:wk–w∗≤wk+–w∗. (.)

Clearly{τ(n)}is nondecreasing,τ(n)→ ∞asn→ ∞, and for alln≥n

wτ(n)–w∗≤wτ(n)+–w∗; wn–w∗≤wτ(n)+–w∗. (.)

Therefore{wτ(n)–w∗}is a nondecreasing sequence. According to Case I,limn→∞wτ(n)– w∗=  andlimn→∞wτ(n)+–w∗= . Hence we have

≤wn–w∗≤maxwn–w∗,wτ(n)–w∗≤wτ(n)+–w∗→, asn→ ∞.

This implies thatwn→w∗andw∗=Pf(w∗) is a solution of GSEFP (.).

This completes the proof of Theorem ..

Remark . Theorem . extends and improves the main results in Moudaﬁet al.[–]

in the following aspects:

(a) For the mappings, we extend the mappings from firmly quasi-nonexpansive mappings to an infinite family of one-to-one quasi-nonexpansive mappings. (b) For the algorithms, we propose new iterative algorithms which are different from

ones given in [–].

(c) For the convergence, the iterative sequence proposed by our algorithm converges strongly to a solution of GSEFP (.). But the iterative sequences proposed in [–] are only of weak convergence to a solution of the split equality problem.

4 Strong convergence theorem for general split equality problem

Throughout this section we always assume that

() H,H,Hare real Hilbert spaces;{Ci}_i=∞ ⊂Hand{Qi}∞_i=⊂Hare two families of nonempty closed and convex subsets withC=∞_i=Ci=∅andQ=∞_i=Qi=∅; () PCi(resp.PQi) is the metric projection fromHontoCi(resp.HontoQi), and

Pi:=_PPCi

Qi

,i= , , . . ., andP:=_PP_QC;

() A:H→HandB:H→Hare two bounded linear operators; () f,G,G∗Gare the same as in Theorem ..

The so-calledgeneral split equality problem(GSEP) is

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Lemma . Let H,H,H,P,{Pi},A,B,f,C,Q,G,G∗G be the same as above.Then a point w∗= (x∗,y∗)is a solution to GSEP(.),if and only if for each i≥and for eachλ> , w∗solves the following ﬁxed point equations:

w∗=Piw∗ and w∗=PiI–λG∗Gw∗. (.) Proof In fact, a pointw∗= (x∗,y∗) is a solution of GSEP (.)

⇔ w∗=x∗,y∗∈C×Q and Ax∗=By∗ ⇔ for eachi≥, x∗=PCi

x∗, y∗=PQi

y∗ and Ax∗=By∗

⇔ w∗=Piw∗ and Ax∗=By∗

⇔

Ax∗=PA(Ci)∩B(Qi)By∗,

By∗=PB(Qi)∩A(Ci)Ax∗

⇔

Ax∗–PB(Qi)By∗,Au–Ax∗ ≥, ∀u∈Ci,

By∗–PA(C_i)Ax∗,Bv–By∗ ≥, ∀v∈Qi

⇔

Ax∗–By∗,Au–Ax∗ ≥, ∀u∈Ci, By∗–Ax∗,Bv–By∗ ≥, ∀v∈Qi

⇔

γA∗(Ax∗–By∗),u–x∗ ≥, ∀u∈Ci,γ > , γB∗(By∗–Ax∗),v–y∗ ≥, ∀v∈Qi,γ > 

⇔

x∗– (x∗–γA∗(Ax∗–By∗)),u–x∗ ≥, ∀u∈Ci,γ > , y∗– (y∗–γB∗(By∗–Ax∗)),v–y∗ ≥, ∀v∈Qi,γ > 

⇔

x∗=PCi(x∗–γA∗(Ax∗–By∗)),

y∗=PQi(y∗–γB∗(By∗–Ax∗))

⇔ w∗=PiI–γG∗Gw∗ and w∗=Piw∗.

This completes the proof of Lemma ..

The metric projectionsPCiandPQiare nonexpansive withF(PCi) =CiandF(PQi) =Qi,

i≥. This implies that the metric projectionsPCiandPQi all are quasi-nonexpansive. In

addition, by Lemma ., for eachi≥ and eachλ∈(,_L), the mappingPi(I–λG∗G) : H×H→Ci×Qiis nonexpansive. By Remark ., for eachi≥ and eachλ∈(,_L), the mapping (I–Pi(I–λG∗G)) is demi-closed at zero.

Consequently, we have the following.

Theorem . Let H,H,H,P,{Pi},A,B,f,C,Q,G,G∗G be the same as above.Let{wn} be the sequence generated by w∈H×H

wn+=P

i=

γn,iPiI–λn,iG∗Gwn

, n≥. (.)

If the solution setof GSEP(.)is nonempty and the following conditions are satisﬁed:

(i) αn+βn+

∞

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∞

n=βn=∞;

(iii) lim infn→∞αnγn,i> for eachi≥;

(iv) {λn,i} ⊂(,_L)for eachi≥,whereL=G,

then the sequence{wn}deﬁned by(.)converges strongly to a solution w∗of GSEP(.) and w∗=Pf(w∗).

Proof TakingSi=PCi,Ti=PQi, and Ki=Pi,i= , , . . . in Theorem ., we know that

SiandTiboth are nonexpansive withF(Si) =Ci andF(Ti) =Qi and so they are quasi-nonexpansive mappings, and C =∞_i=F(Si) and Q=∞_i=F(Ti). Therefore all condi-tions in Theorem . are satisﬁed. The conclusion of Theorem . can be obtained from

Lemma . and Theorem . immediately.

Remark . Theorem . extends and improves the corresponding results in Censor and

Elfving [], Moudaﬁet al.[, ], Eslamian and Latif [], Chenet al.[], Censor and Segal [].

5 Applications

In this section we shall utilize the results presented in the paper to give some applications.

5.1 Application to split feasibility problem

LetC⊂H andQ⊂Hbe two nonempty closed convex subsets andA:H→Hbe a bounded linear operator. The so-calledsplit feasibility problem(SFP) [] is to ﬁnd

x∈C,y∈Qsuch thatAx=y. (.)

LetPCandPQbe the metric projection fromHontoCandHontoQ, respectively. Thus F(PC) =CandF(PQ) =Q. From Theorem . we have the following.

Theorem . Let H,H be two real Hilbert spaces, A:H →H be a bounded linear

operator and I be the identity mapping on H.Let C⊂Hand Q⊂Hbe nonempty closed convex subsets and PCand PQare the metric projections from Honto C and Honto Q, respectively.Let{wn}be the sequence generated by w∈H×H:

wn+=Pαnwn+βnf(wn) +γnPI–λnU∗Uwn, n≥, (.) where f is the mapping as given in Theorem.and

U= [A –I], P=

PC PQ

, U∗U=

A∗A –A∗ –A I

. (.)

If the solution setof SFP(.)is nonempty and the following conditions are satisﬁed:

(i) αn+βn+γn= ,for eachn≥;

(ii) limn→∞βn= ,and∞n=βn=∞;

(iii) lim inf_n_→∞αnγn> ;

(iv) {λn} ⊂(,_L),whereL=U,

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Proof In Theorem . takingH=H,B=I,G=U,{Ci}={C}, and{Qi}={Q}, the con-clusions of Theorem . can be obtained from Theorem . immediately.

Remark Theorem . generalizes and extends the main results of Censor and Elfving []

and Censor and Segal [] from weak convergence to strong convergence.

5.2 Application to null point problem of maximal monotone operators

LetH,H,H,A,B, be the same as in Theorem .. LetM:H→H, andN:H→Hbe two strictly maximal monotone operators. It is well known that the associated resolvent mappingsJM

μ(x) := (I+μM)–andJμN(x) := (I+μN)–ofMandN, respectively, are one-to-one nonexpansive mappings, and

x∈M–() ⇔ x∈FJμM

; y∈N–() ⇔ y∈FJμN

. (.)

DenoteS=JμM,T=JμN,C=M–() =F(JμM), andQ=N–() =F(JμN), then the general split equality ﬁxed point problem (.) is reduced to the followingnull point problem related to strictly maximal monotone operators M and N (NPP(M,N)):

to ﬁndx∗∈M–(),y∗∈N–() such thatAx∗=By∗. (.)

From Theorem . we can obtain the following.

Theorem . Let H,H,H,A,B,f,G,be the same as in Theorem..Let C,Q,S,and T be the same as above.Let{wn}be the sequence generated by w∈H×H

wn+=Pαnwn+βnf(wn) +γnKI–λnG∗Gwn, n≥, (.) where P=PC

PQ

, K=_TS.If the solution set  ofNPP(M,N) (.)is nonempty and the following conditions are satisﬁed:

∞

n=βn=∞;

(iii) lim infn→∞αnγn> ;

(iv) {λn} ⊂(,_L),whereL=G,

then the sequence{wn}deﬁned by(.)converges strongly to w∗=Pf(w∗),which is a so-lution ofNPP(M,N) (.).

Proof SinceS =JμM and T =JμN both are one-to-one nonexpansive with F(S)=∅ and F(T)=∅. Hence they are one-to-one quasi-nonexpansive mappings andI–K(I–λnG∗G) is demi-closed at zero. Therefore all conditions in Theorem . are satisﬁed. The conclu-sions of Theorem . can be obtained from Theorem . immediately.

5.3 Application to equality equilibrium problem

LetDbe a nonempty closed and convex subset of a real HilbertH. A bifunctiong:D× D→(–∞, +∞) is said to be aequilibrium function, if it satisﬁes the following conditions:

(A) g(x,x) = , for allx∈D;

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(A) for eachx∈D,y→g(x,y)is convex and lower semi-continuous.

The so-calledequilibrium problem with respective to the equilibrium functions g and Dis

to ﬁndx∗∈Dsuch thatgx∗,y≥, ∀y∈D. (.)

Its solution set is denoted byEP(g,D).

For givenλ>  andx∈H, theresolvent of the equilibrium function g is the operator Rλ,g:H→Ddeﬁned by

Rλ,g(x) :=

z∈D:g(z,y) + 

λy–z,z–x ≥,∀y∈D

. (.)

It is well known that the resolventRλ,g of the equilibrium functionghas the following properties []:

() Rλ,gis single-valued;

() F(Rλ,g) =EP(g,D)andF(Rλ,g)is a nonempty closed and convex subset ofD;

() Rλ,gis a nonexpansive mapping, and so it is quasi-nonexpansive.

Deﬁnition . Leth,j:D×D→(–∞, +∞) be two equilibrium functions and, for given

λ> , letRλ,h andRλ,jbe the resolvents ofhandj(deﬁned by (.)), respectively. Denote byS=Rλ,h,T=Rλ,j,C:=F(Rλ,h), andQ:=F(Rλ,j). Then theequality equilibrium problem with respective to the equilibrium functions h,j,and Dis

EEP(h,j,D) to ﬁndx∗∈F(Rλ,h),y∗∈F(Rλ,j) such thath

x∗,u≥,

∀u∈D,jy∗,v≥,∀v∈DandAx∗=By∗, (.)

whereA,B:H→Hare two linear and bounded operators.

The following theorem can be obtained from Theorem . immediately.

Theorem . Let H be a real Hilbert space,D be a nonempty and closed convex subset

of H.Let G,f be the same as in Theorem..For givenλ> ,let h,j,Rλ,h,Rλ,j,S,T,C,Q be the same as above.Let{wn}be the sequence generated by w∈H×H:

wn+=Pαnwn+βnf(wn) +γnKI–λnG∗Gwn, n≥, (.) where P=PC

PQ

,K=_TS.If the solution setofEEP(h,j,D) (.)is nonempty and the following conditions are satisﬁed:

(ii) lim_n_→∞βn= ,and

∞

n=βn=∞;

(iii) lim infn→∞αnγn> ;

(iv) {λn} ⊂(,_L),whereL=G,

then the sequence{wn}converges strongly to w∗=Pf(w∗),which is a solution ofEEP(h, j,D) (.).

Competing interests

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Authors’ contributions

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

Author details

1_{College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221,} P.R. China.2_{Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202,} USA.

Received: 14 May 2014 Accepted: 12 September 2014 Published:25 Sep 2014

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