New feasible iterative algorithms and strong convergence theorems for bilevel split equilibrium problems

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R E S E A R C H

Open Access

New feasible iterative algorithms and strong

convergence theorems for bilevel split

equilibrium problems

Zhenhua He

1

_{and Wei-Shih Du}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

National Kaohsiung Normal University, Kaohsiung, 824, Taiwan Full list of author information is available at the end of the article

Abstract

In this paper, we first introduce and investigate a bilevel split equilibrium problem (BSEP) which can be regarded as a new development in the field of equilibrium problems. We provide some new feasible iterative algorithms for BSEP and establish strong convergence theorems for these iterative algorithms in different spaces.

MSC: 47J25; 47H09; 65K10

Keywords: metric projection; adjoint operator; equilibrium problem (EP); split equilibrium problem (SEP); bilevel split equilibrium problem (BSEP); bilevel convex optimization problem (BCOP); the common solution of equilibrium problems (CEP); feasible iterative algorithm; strong convergence theorem

1 Introduction and preliminaries

LetKbe a closed convex subset of a real Hilbert spaceH. Let·,·and · denote the inner product ofHand the norm ofH, respectively. For each pointx∈H, there exists a unique nearest point inK, denoted byPKx, such that

x–PKx ≤ x–y for ally∈K.

The mappingPK is called themetric projectionfromHontoK. It is well known thatPK has the following properties:

(i) x–y,PKx–PKy ≥ PKx–PKyfor everyx,y∈H.

(ii) Forx∈Handz∈K,z=PK(x)⇔ x–z,z–y ≥for ally∈K. (iii) Forx∈Handy∈K,

y–PK(x) 

+x–PK(x) 

≤ x–y. (.)

LetHandHbe two Hilbert spaces. LetA:H→HandA∗:H→Hbe two bounded linear operators.A∗is called the adjoint operator (or adjoint) ofAif

Az,w=z,A∗w for allz∈Handw∈H.

It is known that the adjoint operator of a bounded linear operator on a Hilbert space always exists and is bounded linear and unique. Moreover, it is not hard to show that ifA∗is an

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adjoint operator ofA, thenA=A∗. The symbolsNandRare used to denote the sets of positive integers and real numbers, respectively.

Example . ([]) LetH=Rwith the standard norm| · |andH=R with the norm α= (a_+a_) _{for some}_α_{= (}_a_,_a_)∈R_._x_,_y₌_xy_{denotes the inner product of}_H_

for somex,y∈H, andα,β=_i=aibidenotes the inner product ofH for someα= (a,a),β= (b,b)∈H. LetAα=a–aforα= (a,a)∈HandBx= (–x,x) forx∈H, thenBis an adjoint operator ofA.

Example .([]) LetH=Rwith the normα= (a+a)



 _{for some}_α_{= (}_a_,_a_)∈R

andH=Rwith the normγ= (c+c+c)



 _{for some}_γ _{= (}_c_,_c_,_c_)∈R_{. Let}_α,_β₌



i=aibiandγ,η=i=cididenote the inner product ofHandH, respectively, where α= (a,a),β= (b,b)∈H,γ = (c,c,c),η= (d,d,d)∈H. LetAα= (a,a,a–a) forα= (a,a)∈H andBγ = (c+c,c–c) forγ = (c,c,c)∈H. Obviously,Bis an adjoint operator ofA.

Letf be a bi-function fromC×CtoR. The classical equilibrium problem (EP, for short) is deﬁned as follows.

(EP) Findp∈Csuch thatf(p,y)≥,∀y∈C.

The set of such solutions is denoted byEP(f), that is,EP(f) ={u∈C:f(u,v)≥,∀v∈ C}. In fact, equilibrium problem has an important relationship with variational inequality problem. For example, letT:C→Hbe a nonlinear mapping satisfyingTx,y–x ≥ for allx,y∈C. Thenx∈EP(f) if and only ifx∈Cis a solution of the variational inequality Tx,y–x ≥ for ally∈C. It is known that the EP is an important mathematical model for nonlinear analysis and applied sciences which is generalized to many new mathematical models and includes many important problems arising in physics, engineering, science optimization, economics, network, game theory, complementary problems, variational inequalities problems, saddle point problems, ﬁxed point problems and others; for details, one can refer to [–] and references therein. Many authors have proposed some useful methods to solve the EP; see, for instance, [–, –] and references therein.

Recent investigations and developments in equilibrium theory as well as optimization theory have been applied to connect fundamental sciences with the real world. According to our experience, useful methods of real world problems often need to be used to solve several problems arising in diﬀerent spaces. In view of this, recent studies focus on split problems which are more closed in the real world applications; see, for instance, [, – ] and the references therein. Recently, He [] considered the following split equilibrium problem. LetHandHbe two real Hilbert spaces. LetCbe a closed convex subset ofH andKbe a closed convex subset ofH. Letf :C×C→Randg:K×K→Rbe two bi-functions, andA:H→Hbe a bounded linear operator. The split equilibrium problem (SEP, in short) is deﬁned as follows:

(SEP) Findp∈Csuch thatf(p,y)≥,∀y∈C, andu:=Apsatisfyingg(u,v)≥,∀v∈K.

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Motivated and inspired by the works mentioned above, in this paper we shall introduce and investigate the following new problem. LetH,HandHbe three real Hilbert spaces. LetCbe a closed convex subset ofH,Qbe a closed convex subset ofHandKbe a closed convex subset of H. Letf :C×C→R, g:Q×Q→Randh:K×K→Rbe three bi-functions. Let A:H→H andB:H→H be two bounded linear operators with theirs adjoint operatorsA∗ andB∗, respectively. The mathematical model aboutbilevel split equilibrium problem(BSEP, in short) is deﬁned as follows:

(BSEP) Findp∈Candq∈Qsuch that

(i) f(p,x)≥andg(q,y)≥for allx∈Candy∈Q; (ii) Ap=Bq:=u;

(iii) h(u,z)≥for allz∈K.

In fact, BSEP can be regarded as a new development in the field of equilibrium prob-lems and contains several important probprob-lems as special cases. It was profoundly believed that BSEP will motivate and inspire further scientific activities in the fields of equilibrium problems, optimization problems, game problems, complementary problems, variational inequalities problems, fixed point problems and their applications.

Example A LetH, H andH be three real Hilbert spaces. LetC⊂H,Q⊂H and

K⊂Hbe three closed convex sets. Letf∗:C→R,g∗:Q→Randh∗:K→Rbe three convex functions. LetA:H→HandB:H→Hbe two bounded linear operators with their adjoint operatorsA∗andB∗, respectively. Let

f(x,α) =f∗(x) –f∗(α) forx,α∈C,

g(y,β) =g∗(y) –g∗(β) fory,β∈Q,

and

h(z,η) =h∗(z) –h∗(η) forz,η∈K.

Then BSEP reduces the bilevel convex optimization problem (BCOP):

(BCOP) Findp∈Candq∈Qsuch thatu:=Ap=Bq∈K,f∗(x)≥f∗(p),g∗(y)≥g∗(q) and

h∗(z)≥h∗(u) for allx∈C,y∈Qandz∈K.

Example B LetH,HandHbe three real Hilbert spaces. LetC⊂H,Q⊂HandK⊂

H be three closed convex sets. Let T:C→H,S:Q→H andG:K→H be three nonlinear operators. LetA:H→HandB:H→Hbe two bounded linear operators with their adjoint operatorsA∗andB∗, respectively. Iff(p,x) =Tp,x–p,g(q,y) =Sq,y–

qandh(u,z) =Tu,z–u, then BSEP reduces to the bilevel split variational inequality

problem (BSVI):

(BSVI) Findp∈Candq∈Qsuch thatu:=Ap=Bq∈KsatisfyingTp,x–p ≥,Sq,y–

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Example C LetHandHbe two real Hilbert spaces andB:H→Hbe a bounded linear operator with its adjoint operator B∗. LetC⊂H,Q⊂H andK⊂H be three closed convex sets. IfH=H andA=B, then BSEP reduces to the following split equilibrium problem () (SEP()_):

(SEP()) Findp∈Candq∈Qsuch thatu:=Bp=Bq∈Ksatisfyingf(p,x)≥,g(q,y)≥ andh(u,z)≥ for allx∈C,y∈Qandz∈K.

Example D LetH andH be two real Hilbert spaces andA:H →H be a bounded linear operator with its adjoint operatorA∗. LetC⊂H,Q⊂H andK ⊂H be three closed convex sets withQ∩K=∅. IfH=HandB=I(identity operator), then BSEP reduces to the following split equilibrium problem () (SEP()):

(SEP()₎ _Find _p_∈_C _{such that} _u_:=_Ap_∈_Q_∩_K _satisfying_f₍_p_,_x₎_≥_, _g₍_u_,_y₎_≥_{ and}

h(u,z)≥ for eachx∈C,y∈Qandz∈K.

Especially, ifg(p,y)≡ for allp,y∈Q, then (SEP()_{) reduces to ﬁnding}_p_∈_C_{such that}

u:=Ap∈Ksatisfyingf(p,x)≥ andh(u,z)≥ for allx∈Candz∈K, which was studied in [].

Example E LetH andH be two real Hilbert spaces andB:H→H be a bounded linear operator with its adjoint operatorA∗. LetC⊂H, Q⊂H andK⊂H be three closed convex sets with C∩Q=∅. IfH=HandA=I(identity operator), then BSEP reduces to the following split equilibrium problems () (SEP()_):

(SEP()) Find p∈C∩Q such that u:=Bp∈K satisfyingf(p,x)≥,g(p,y)≥ and

h(u,z)≥ for allx∈C,y∈Qandz∈K.

Example F In Example A, ifH=H=H:=H,C=Q=K⊂H andA=B=I(identity operator), then BSEP reduces to the common solution of equilibrium problems (CEP):

(CEP) Findp∈Csuch thatf(p,x)≥,g(p,y)≥ andh(p,z)≥ for eachx,y,z∈C.

The paper is divided into four sections. In Sections  and , we ﬁrst introduce and in-vestigate a bilevel split equilibrium problem (BSEP) and then provide some new feasible iterative algorithms for BSEP and establish strong convergence theorems for these iter-ative algorithms in diﬀerent spaces. In Section , we give the proof of the main result Theorem . in detail. Finally, an example illustrating Theorem . is given in Section .

2 Feasible iterative algorithms for BSEP and their strong convergence theorems

In , Blum and Oettli [] established the following important existence theorem which plays a key role in solving equilibrium problems, variational inequality problems and op-timization problems.

Lemma .(Blum and Oettli []) Let K be a nonempty closed convex subset of H and F be a bi-function of K×K intoRsatisfying the following conditions.

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(A) Fis monotone,that is,F(x,y) +F(y,x)≤for allx,y∈K; (A) for eachx,y,z∈K,

lim sup

t→+ F

tz+ ( –t)x,y≤F(x,y);

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

Let r> and x∈H.Then there exists z∈K such that

F(z,y) +

ry–z,z–x ≥ for all y∈K.

In this paper, we ﬁrst introduce a new iterative algorithm for BSEP and establish a strong convergence theorem for this iterative algorithm. Here, the spaceH×H denotes the product space of two real Hilbert spacesH andH, which is endowed with the usual linear operation and norm, namely, for (x,y), (¯x,y¯)∈H×Handa,b∈R,

a(x,y) +b(¯x,¯y) = (ax+bx¯,ay+by¯)

and

(x,y)=x+y.

Theorem . Let H,H and H be three real Hilbert spaces.Let C be a closed convex

subset of H,Q be a closed convex subset of Hand K be a closed convex subset of H.Let

f :C×C→R,g:Q×Q→Rand h:K×K→Rbe three bi-functions.A:H→H

and B:H→Hare two bounded linear operators with their adjoint operators A∗and B∗,

respectively.Suppose that all the bi-functions f,g and h satisfy conditions(A)-(A).Let x∈C,y∈Q,C=C,Q=Q,{xn},{yn},{un},{vn}and{wn}be sequences generated by

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

un=Trfnxn, vn=T g

rnyn, wn=Trhn(  Aun+

 Bvn),

ln=PC(un–ξA∗(Aun–wn)), kn=PQ(vn–ξB∗(Bvn–wn)),

Cn+×Qn+={(x,y)∈Cn×Qn:ln–x+kn–y

≤ un–x+vn–y≤ xn–x+yn–y},

xn+=PCn+(x),

yn+=PQn+(y), ∀n∈N,

(.)

whereξ ∈(,min(_A,_B))and{rn} ⊂(, +∞)withlim infn→+∞rn> ,PCand PQare

two projection operators from Hinto C and from Hinto Q,respectively.Suppose that

=(p,q)∈EP(f)×EP(g) :Ap=Bq∈EP(h)=∅.

Then there exists(p,q)∈such that

(a) (xn,yn)→(p,q)asn→ ∞;

(b) (un,vn)→(p,q)asn→ ∞;

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The following conclusion is immediate from Theorem . by puttingA=B.

Corollary . Let Hand Hbe two real Hilbert spaces.Let C and Q be two closed convex

subsets of Hand K be a closed convex subset of H.Let f :C×C→R,g:Q×Q→Rand h:

K×K→Rbe three bi-functions.B:H→His a bounded linear operator with its adjoint

operator B∗.Suppose that all the bi-functions f,g and h satisfy conditions(A)-(A).Let x∈C,y∈Q,C=C,Q=Q,{xn},{yn},{un},{vn}and{wn}be sequences generated by

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

un=Trfnxn, vn=T g

rnyn, wn=Trhn(  Bun+

 Bvn),

ln=PC(un–ξB∗(Bun–wn)), kn=PQ(vn–ξB∗(Bvn–wn)),

Cn+×Qn+={(x,y)∈Cn×Qn:ln–x+kn–y≤ un–x+vn–y ≤ xn–x+yn–y},

xn+=PCn+(x),

yn+=PQn+(y), ∀n∈N,

whereξ∈(,_B)and{rn} ⊂(, +∞)withlim infn→+∞rn> ,PCand PQare two

projec-tion operators from Hinto C and from Hinto Q,respectively.Suppose that

=(p,q)∈EP(f)×EP(g) :Bp=Bq∈EP(h)=∅.

(a) (xn,yn)→(p,q)asn→ ∞; (b) (un,vn)→(p,q)asn→ ∞;

(c) wn→w∗:=Bp=Bq∈EP(h)asn→ ∞.

IfH=HandB=I, then Theorem . reduces to the following corollary.

Corollary . Let H and H be two real Hilbert spaces.Let C⊂H and Q,K⊂H be

three closed convex sets.Let f :C×C→R,g:Q×Q→Rand h:K×K→Rbe three bi-functions.A:H→His a bounded linear operator with its adjoint operator A∗.Suppose

that all the bi-functions f,g and h satisfy conditions(A)-(A).Let x∈C,y∈Q,C=C,

Q=Q,{xn},{yn},{un},{vn}and{wn}be sequences generated by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un=Trfnxn, vn=T g

rnyn, wn=Trhn(  Aun+

 vn),

ln=PC(un–ξA∗(Aun–wn)), kn=vn–ξ(vn–wn),

xn+=PCn+(x),

yn+=PQn+(y), ∀n∈N,

whereξ∈(,min{,

A})and{rn} ⊂(, +∞)withlim infn→+∞rn> ,PCand PQare two

projection operators from Hinto C and from Hinto Q,respectively.Suppose that

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Then there exists p∈such that

(a) xn→pasn→ ∞; (b) un→pasn→ ∞;

(c) vn,yn,wn→w∗:=Apasn→ ∞.

IfH=HandA=I, then Theorem . reduces to the following corollary.

Corollary . Let H and H be two real Hilbert spaces.Let C,Q⊂H and K⊂H be

three closed convex sets.Let f :C×C→R,g:Q×Q→Rand h:K×K→Rbe three bi-functions.B:H→His a bounded linear operator with its adjoint operator B∗.Suppose

Q=Q,{xn},{yn},{un},{vn}and{wn}be sequences generated by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un=Trfnxn, vn=T g

rnyn, wn=Trhn(  un+

 Bvn),

ln=un–ξ(un–wn), kn=Pq(vn–ξB∗(Bvn–wn)),

xn+=PCn+(x),

yn+=PQn+(y), ∀n∈N,

whereξ∈(,min{, 

B})and{rn} ⊂(, +∞)withlim infn→+∞rn> ,PCand PQare two

projection operators from Hinto C and from Hinto Q,respectively.Suppose that

=p∈EP(f)∩EP(g) :Ap∈EP(h)=∅.

Then there exists p∈such that

(a) xn,un→pasn→ ∞; (b) yn,vn→pasn→ ∞;

(c) wn→w∗:=Apasn→ ∞.

PuttingA=B=I(identical operator),H=H=H=HandC=Q=K, then we have the following result.

Corollary . Let H be a real Hilbert space.Let C be a closed convex subset of H.Let f,g,h:C×C→Rbe three bi-functions.Suppose that all the bi-functions f,g and h satisfy conditions (A)-(A).Let x,y∈C,C=C, {xn},{yn},{un},{vn}and{wn} be sequences

generated by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un=Trfnxn, vn=Trgnyn, wn=Trhn(  un+

 vn),

ln=un–ξ(un–wn), kn=vn–ξ(vn–wn),

Cn+×Cn+={(x,y)∈Cn×Cn:ln–x+kn–y≤ un–x+vn–y ≤ xn–x+yn–y},

xn+=PCn+(x),

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whereξ∈(, )and{rn} ⊂(, +∞)withlim infn→+∞rn> ,PCis a projection operator from

H into C.Suppose that

=(p,q)∈EP(f)×EP(g) :p=q∈EP(h)=∅.

(c) wn→w∗:=p=q∈EP(h)asn→ ∞.

Remark . In Corollary ., it is obvious that

=(p,q)∈EP(f)×EP(g) :p=q∈EP(h)=∅

implies

=p∈EP(f)∩EP(g)∩EP(h)=∅.

Hence, the problem studied in Corollary . is still the study of a common solution of three equilibrium problems in essence.

IfC,Q,Kare linear subspaces of a real Hilbert space, then we have the following corol-laries from Theorem . and Corollary ..

Corollary . Let H,Hand H be three real Hilbert spaces.Let C⊂H,Q⊂Hand

K⊂Hbe three linear subspaces.Let f :C×C→R,g:Q×Q→Rand h:K×K→R

be three bi-functions.A:H→Hand B:H→Hare two bounded linear operators with

their adjoint operators A∗ and B∗,respectively.Suppose that all the bi-functions f,g and h satisfy conditions(A)-(A).Let x∈C,y∈Q,C=C,Q=Q,{xn},{yn},{un},{vn}and {wn}be sequences generated by

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

un=Trfnxn, vn=Trgnyn, wn=Trhn(  Aun+

 Bvn),

ln=un–ξA∗(Aun–wn), kn=vn–ξB∗(Bvn–wn),

xn+=PCn+(x),

yn+=PQn+(y), ∀n∈N,

whereξ ∈(,min(_A,_B))and{rn} ⊂(, +∞)withlim infn→+∞rn> ,PCand PQare

two projection operators from Hinto C and from Hinto Q,respectively.Suppose that

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(b) (un,vn)→(p,q)asn→ ∞;

(c) wn→w∗:=Ap=Bq∈EP(h)asn→ ∞.

Corollary . Let Hand Hbe two real Hilbert spaces.Let C⊂H,Q⊂Hand K⊂H

be three linear subspaces.Let f :C×C→R,g:Q×Q→Rand h:K×K→Rbe three bi-functions.B:H→His a bounded linear operator with its adjoint operator B∗.Suppose

Q=Q,{xn},{yn},{un},{vn}and{wn}be sequences generated by ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

un=Trfnxn, vn=T g

rnyn, wn=Trhn(  Bun+

 Bvn),

ln=un–ξB∗(Bun–wn), kn=vn–ξB∗(Bvn–wn),

xn+=PCn+(x),

yn+=PQn+(y), ∀n∈N,

whereξ∈(,_B)and{rn} ⊂(, +∞)withlim infn→+∞rn> ,PCand PQare two

projec-tion operators from Hinto C and from Hinto Q,respectively.Suppose that

=(p,q)∈EP(f)×EP(g) :Bp=Bq∈EP(h)=∅.

(c) wn→w∗:=Bp=Bq∈EP(h)asn→ ∞.

Remark . In fact, the problem studied by Corollaries .-. and Corollary . is (SEP).

In order to prove Theorem ., we need the following crucial known results.

Lemma .(see []) Let K be a nonempty closed convex subset of H,and let F be a bi-function of K×K intoRsatisfying(A)-(A).For r> ,deﬁne a mapping TF

r :H→K as

follows:

TF r(x) =

z∈K:F(z,y) +

ry–z,z–x ≥,∀y∈K

(.)

for all x∈H.Then the following hold: (i) TF

r is single-valued andF(TrF) =EP(F)for∀r> andEP(F)is closed and convex; (ii) TF

r is ﬁrmly nonexpansive,that is,for anyx,y∈H, TF

rx–TrFy≤ TrFx–TrFy,x–y.

Lemma .([]) Let FF

r be the same as in Lemma..IfF(TrF) =EP(F)=∅,then,for any

x∈H and x∗∈F(TF

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Lemma .([, ]) Let the mapping TF

r be deﬁned as in Lemma..Then,for r,s> and

x,y∈H,

T_rF(x) –T_sF(y)≤ x–y+|s–r|

s T

F s(y) –y.

In particular,T_rF(x) –T_rF(y) ≤ x–yfor any r> and x,y∈H,that is,T_rFis nonexpan-sive for any r> .

Lemma .(see,e.g., []) Let H be a real Hilbert space.Then the following hold: (a) x–y₌_x₊_y_{– }_x_,_y_{for all}_x_,_y_∈_H_;

(b) αx+ ( –α)y₌_α_x_{+ ( –}_α)_y_–_{α( –}_α)_x_–_y_{for all}_x_,_y_∈_H_and α∈[, ].

3 Proof of Theorem 2.1

Applying Lemmas . and ., we know that{un},{vn}and{wn}are all well deﬁned. It is also easy to verify thatCn,Qn,Cn×Qnare closed convex sets forn∈N.

Now, we claimCn×Qn=∅for alln∈N. Indeed, it suﬃces to prove that⊂Cn×Qn for alln∈N. Let (x∗,y∗)∈. Thenx∗∈EP(f),y∗∈EP(g) and

w∗:=Ax∗=By∗∈EP(h).

Letn∈Nbe given. By Lemma ., we have

un–x∗≤xn–x∗, vn–y∗≤yn–y∗, wn–w∗≤

Aun+Bvn

 –w

∗_,

wn–w∗≤ 

Aun–w

∗ +

Bvn–w

∗

(by Lemma .).

(.)

From (.), (.) and Lemma ., we obtain

ln–x∗=PC

un–ξA∗(Aun–wn)

–x∗≤un–x∗–ξA∗(Aun–wn)

=un–x∗ 

+ξA∗(Aun–wn) 

– ξun–x∗,A∗(Aun–wn)

=un–x∗ 

+ξA∗(Aun–wn) 

– ξAun–Ax∗,Aun–wn

=un–x∗ 

+ξA∗(Aun–wn) 

– ξAun–w∗,Aun–wn

=un–x∗ 

+ξA∗(Aun–wn) 

–ξAun–w∗ 

–ξAun–wn+ξwn–w∗ 

≤un–x∗ 

–ξ –ξA∗Aun–wn

–ξAun–w∗+ξwn–w∗

≤un–x∗–ξ

 –ξA∗Aun–wn–ξAun–w∗

+ξ

Aun–w

∗ +ξ

Bvn–w

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=un–x∗ 

–ξ –ξA∗Aun–wn

–ξ

Aun–w

∗ +ξ

Bvn–w

∗

(.)

and

kn–y∗ 

=PQ

vn–ξB∗(Bvn–wn)

–y∗≤vn–y∗–ξB∗(Bvn–wn) 

=vn–y∗ 

+ξB∗(Bvn–wn) 

– ξvn–y∗,B∗(Bvn–wn)

=vn–y∗ 

+ξB∗(Bvn–wn) 

– ξBvn–By∗,Bvn–wn

=vn–y∗ 

+ξB∗(Bvn–wn) 

– ξBvn–w∗,Bvn–wn

=vn–y∗ 

+ξB∗(Bvn–wn) 

–ξBvn–w∗ 

–ξBvn–wn+ξwn–w∗ 

≤vn–y∗–ξ

 –ξB∗Bvn–wn–ξBvn–w∗+ξwn–w∗

≤vn–y∗–ξ

 –ξB∗Bvn–wn–ξBvn–w∗

+ξ

Aun–w

∗ +ξ

Bvn–w

∗

=vn–y∗ 

–ξ –ξB∗Bvn–wn

–ξ

Bvn–w

∗ +ξ

Aun–w

∗

. (.)

By taking into account inequalities (.), (.) and (.), we obtain ln–x∗+kn–y∗≤un–x∗+vn–y∗–ξ

 –ξA∗Aun–wn

–ξ –ξB∗Bvn–wn

≤xn–x∗+yn–y∗–ξ

 –ξA∗Aun–wn

–ξ –ξB∗Bvn–wn, (.)

which implies

ln–x∗+kn–y∗≤un–x∗+vn–y∗≤xn–x∗+yn–y∗. (.)

Inequality (.) shows that (x∗,y∗)∈Cn×Qn. Hence⊂Cn×QnandCn×Qn=∅for all

n∈N. For eachn∈N, since⊂Cn×Qn,Cn+⊂Cn, we have

xn+=PCn+(x)⊂Cn.

Similarly, sinceQn+⊂Qn, we have

yn+=PQn+(y)⊂Qn.

So, for any (x∗,y∗)∈, we get

(12)

and

yn+–y ≤y∗–y.

The last inequalities deduce that{xn}and{yn} are bounded and hence show that{kn},

{ln},{un}and{vn}are all bounded. For somen∈Nwithn> , fromxn=PCn(x)⊂Cn,

yn=PQn(y)⊂Qnand (.), we have

xn+–xn+x–xn=xn+–PCn(x) 

+x–PCn(x) 

≤ xn+–x, yn+–yn+y–yn=yn+–PCn(y)



+y–PCn(y) _≤

yn+–y,

which yields that

x–xn ≤ xn+–x, y–yn ≤ yn+–y.

Together with the boundedness of {xn} and {yn}, we know limn→∞xn – x and

limn→∞yn–y exist. For some k,n∈N with k>n> , due to xk =PCk(x)⊂Cn,

yk=PQk(y)⊂Qnand (.), we have

xk–xn+x–xn=xk–PCn(x) 

+x–PCn(x) 

≤ xk–x,

yk–yn+y–yn=yk–PQn(y) 

+y–PQn(y) 

≤ yk–y.

(.)

By (.), we havelimn→∞xn–xk=  andlimn→∞yn–yk= . Hence{xn}and{yn}are all Cauchy sequences. Letxn→pandyn→qfor some (p,q)∈C×Q. We want to prove that (p,q)∈. For anyn∈N, since

(xn+,yn+)∈Cn+×Qn+⊂Cn×Qn,

from (.), we have

ln–xn++kn–yn+≤ un–xn++vn–yn+

≤ xn–xn++yn–yn+. (.)

By taking the limit from both sides of (.), we obtain

lim

n→∞ln–xn+=nlim→∞kn–yn+= ,

lim

n→∞un–xn+=nlim→∞vn–yn+= .

(.)

Moreover, by (.), we get

lim

n→∞ln–un=nlim→∞un–xn=nlim→∞ln–xn= ,

lim

n→∞kn–vn=nlim→∞vn–yn=nlim→∞kn–yn= .

(.)

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Now, we claimp∈EP(f) andq∈EP(g). In fact, forr> , by Lemma ., we have

T_rfp–p=T_rfp–T_rf_nxn+Trfnxn–xn+xn–p

≤ xn–p+|

rn–r|

rn

T_rf_nxn–xn+Trfnxn–xn+xn–p

=xn–p+ |rn–r|

rn

un–xn+un–xn+xn–p →

and

T_rgq–q≤T_rgq–T_rg_nyn+Trgnyn–yn+yn–q

≤ yn–q+|

rn–r|

rn

T_rg_nyn–yn+Trgnyn–yn+yn–q

=yn–q+ |rn–r|

rn

vn–yn+vn–yn+yn–q →.

So,p∈EP(f) andq∈EP(g).

Finally, we proveAp=Bq∈EP(h). Setting

θ=minξ –ξA∗,ξ –ξB∗.

Then, for anyn∈N, by (.) and (.), we have

θAun–wn+θBvn–wn

≤xn–x∗ 

+yn–y∗ 

–ln–x∗ 

–kn–y∗ 

=xn–x∗–ln–x∗xn–x∗+ln–x∗

+yn–y∗–kn–y∗yn–y∗+kn–y∗

≤ ln–xnxn–x∗+ln–x∗+kn–ynyn–y∗+kn–y∗

→. (.)

Hence (.) implies

lim

n→∞Aun–wn=nlim→∞Bvn–wn= , and nlim→∞Aun–Bvn= . (.)

SinceAun→Ap,Bvn→Bqand (.), we obtainAp=Bqandwn→w∗, wherew∗:=Ap=

Bq. On the other hand, forr> , by Lemma . again, we have

T_rhw∗–w∗

=T_rhw∗–T_rh_nAun+Bvn

 +T

h rn

Aun+Bvn

 –

Aun+Bvn

 +

Aun+Bvn

 –w

∗

≤Aun+Bvn

 –w

∗₊|rn–r|

rn

T_rh_nAun+Bvn

 –

Aun+Bvn 

+T_rh_nAun+Bvn

 –

Aun+Bvn 

+Aun+Bvn

 –w

(14)

= Aun+Bvn

 –w

∗₊|rn–r|

rn wn–

Aun+Bvn 

+wn–

Aun+Bvn 

→.

Hencew∗∈EP(h), namelyAp=Bq∈EP(h). Therefore, conclusions (a), (b) and (c) are all proved. The proof is completed.

4 An example of Theorem 2.1

In this section, we give an example illustrating Theorem ..

Example . LetH=R,H=RandH=Rbe three real Hilbert spaces with the stan-dard norm and inner product. For eachα= (α,α)∈Randν= (z,z,z,z)∈R, deﬁne

Aα= (α,α,α+α,α–α)

and

A∗ν= (z+z+z,z+z–z).

ThenAis a bounded linear operator fromR_into_R_with_A₌√_{, and}_A∗_{is an adjoint} operator ofAwithA∗=√. For eachβ= (β,β,β)∈Randν= (z,z,z,z)∈R, let

Bβ= (β,β,β,β–β)

and

B∗ν= (z+z,z–z,z).

ThenBis a bounded linear operator fromR_into_R_with_B₌√_{, and}_B∗_{is an adjoint} operator ofBwithB∗=√. Put

C:=α= (α,α)∈R: –≤α≤, ≤α≤,

Q:=β= (β,β,β)∈R: –≤β≤, ≤β≤, ≤β≤

and

K:=z= (z,z,z,z)∈R: ≤z≤, ≤z≤, ≤z≤, –≤z≤–

.

For eachα= (α,α)∈C,β= (β,β,β)∈Qandz= (z,z,z,z)∈K, deﬁne

f∗(α) =α_+α_,

g∗(β) =β_+β_+β_

and

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For eachα,x∈C, let

f(α,x) =f∗(x) –f∗(α).

For eachβ,y∈Q, let

g(β,y) =g∗(y) –g∗(β).

For eachη,z∈K, let

h(η,z) =h∗(z) –h∗(η).

It is not hard to verify thatf,gandhsatisfy conditions (A)-(A) withEP(f) ={p= (, )}, EP(g) ={q= (, , )},EP(h) ={(, , , –)}and

LetC=C,Q=Q,ξ = _ andrn≡ forn∈N. Thus, for eachx¯= (a,b)∈Candy¯= (c,d,e)∈Qwithc> , we have the following:

• u= (a_, ) =Trfnx¯, • v= (c

, , ) =T g rny¯, • w= (a+c_ , , , –) =Th

rn(  Ax¯+

 B¯y), • l=PC(u–_A∗(Au–w)) = (a+c_ , ), • k=PQ(v–_B∗(Bv–w)) = (c+a_ ,  +_c, ).

Forx= (a,b)∈Candy= (c,d,e)∈Qwith c≥a> , a≥candd>  +_c, we obtain the following:

• u= (a_, ), • v= (c_, , ), • w= (a_+c, , , –), • l= (a_+c, ), • k= (c_+a,  +c_, ),

• C={α= (α,α)∈C: –≤α≤a_+c, ≤α≤b_+}, • x=PC(x) = (

a+c

 , b+

 ) := (a,b),

• Q={β= (β,β,β)∈Q: –≤β≤c_+a,  +_c ≤β≤d_+, ≤β≤e_+}, • y=PQ(y) = (

c+a

 , d+

 , e+

 ) := (c,d,e).

Fromx,y, we haveu= (_a, ),v= (_c, , ) andw= (_(a+b), , , –). Since

d>  +

c ,

c≥a> ,

and

a≥c,

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• C={α= (α,α)∈C: –≤α≤a_+c, ≤α≤_(b+ )}, • x= (a_+c,_(b+ )) := (a,b),

• Q={β= (β,β,β)∈Q: –≤β≤c_+a,  +_c ≤β≤d_+, ≤β≤e_+}, • y= (c_+a,d_+,e_+) := (c,d,e).

Similarly, forn∈Nwithn> , we obtain

• Cn+={α= (α,α)∈Cn: –≤α≤an–_+cn–, ≤α≤_(bn–+ )}, • xn+= (an–_+cn–,_(bn–+ )),

• Qn+={β= (β,β,β)∈Qn: –≤β≤cn–_+an–,  +cn_– ≤β≤dn–_+, ≤β≤ en–+

 },

• yn+= (cn–_+an–,dn–_+,en–_+), • un+= (_an–, ),

• vn+= (_cn–, , ), • wn+= (cn–+a_n–, , , –).

By mathematical induction, we know that{an},{bn},{cn},{dn}and{en}all are decreasing sequences. Moreover,an→,bn→,cn→,dn→ anden→ asn→ ∞. So, we have

un→(, ),vn→(, , ),wn→(, , , –),xn→(, ) andyn→(, , ) asn→ ∞.

5 Conclusion

In this paper, we first introduce and investigate BSEP which can be regarded as a new development in the field of equilibrium problems. We provide some new iterative algo-rithms for BSEP and establish strong convergence theorems for these iterative algoalgo-rithms in different spaces. An example illustrating Theorem . is also given.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and signiﬁcantly in writing this paper. Both authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Honghe University, Yunnan, 661100, China.}2_{Department of Mathematics, National}

Kaohsiung Normal University, Kaohsiung, 824, Taiwan.

Acknowledgements

The ﬁrst author was supported by the Natural Science Foundation of Yunnan Province (201401CA00262) and the Candidate Foundation of Youth Academic Experts at Honghe University (2014HB0206); the second author was supported by Grant No. MOST 103-2115-M-017-001 of the Ministry of Science and Technology of the Republic of China.

Received: 1 June 2014 Accepted: 18 August 2014 Published:02 Sep 2014 References

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