R E S E A R C H
Open Access
Nonlinear algorithms approach to split
common solution problems
Zhenhua He
1and Wei-Shih Du
2**Correspondence:
2Department 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 introduce some new iterative algorithms for the split common solution problems for equilibrium problems and fixed point problems of nonlinear mappings. Some examples illustrating our results are also given.
MSC: 47J25; 47H09; 65K10
Keywords: fixed point problem; iterative algorithm; equilibrium problem; split common solution problem
1 Introduction
Throughout this paper, we assume thatHis a real Hilbert space with zero vectorθ, whose inner product and norm are denoted by·,·and · , respectively. LetKbe a nonempty subset ofHandTbe a mapping fromKinto itself. The set of fixed points ofTis denoted byF(T). The symbolsNandRare used to denote the sets of positive integers and real numbers, respectively.
LetC andK be nonempty subsets of real Banach spacesE andE, respectively. Let
A:E→Ebe a bounded linear mapping,T a mapping fromCinto itself withF(T)=∅
andf a bi-function fromK×KintoR. The classical equilibrium problem is to findx∈K such that
f(x,y)≥, ∀y∈K. (.)
The symbolEP(f) is used to denote the set of all solutions of the problem (.), that is,
EP(f) =u∈K:f(u,v)≥,∀v∈K.
The equilibrium problem contains optimization problems, variational inequalities prob-lems, saddle point probprob-lems, the Nash equilibrium probprob-lems, fixed point probprob-lems, com-plementary problems, bilevel problems, and semi-infinite problems as special cases and have many applications in mathematical program with equilibrium constraint; for detail, one can refer to [–] and references therein.
In this paper, we study the following split common solution problem (SCSP) for equi-librium problems and fixed point problems of nonlinear mappingsA,T andf:
(SCSP) Findp∈Csuch thatp∈F(T) andu:=Ap∈Kwhich satisfiesf(u,v)≥,∀v∈K. The solution set of (SCSP) is denoted by
=p∈F(T) :Ap∈EP(f).
Many authors had proposed some methods to find the solution of the equilibrium prob-lem (.). As a generalization of the equilibrium probprob-lem (.), finding a common solution for some equilibrium problems and fixed point problems of nonlinear operators, it has been considered in the same subset of the same space; see [–]. However, some equi-librium problems and fixed point problems of nonlinear mappings always belong to dif-ferent subsets of spaces in general. So the split common solution is very important for the research on generalized equilibriums problems and fixed point problems.
Example . LetE=E=R,C:= [, +∞) andK:= (–∞, –]. LetA(x) = –xfor allx∈R
andTx= x
x+ for allx∈C. Letf:K×K→Rbe define byf(u,v) = (u–v) for allu,v∈K.
Clearly,Ais a bounded linear operator,F(T) ={} andA() = –∈EP(f). So={p∈
F(T) :Ap∈EP(f)} =∅.
Example . LetE=Rwith the normα= (a+a)
forα= (a,a)∈RandE=R
with the standard norm| · |. LetC:={α= (a,a)∈R|a+a≤}andK:= [–, ]. Let
Aα= –a forα= (a,a)∈E andTα = (a,a) for allα = (a,a)∈C. Then F(T) =
{(, ), (, ), (, )}andAis a bounded and linear operator fromEintoEwithA= .
Now define a bi-functionf asf(u,v) =v–ufor allu,v∈K. Thenf is a bi-function from K×KintoRwithEP(f) ={–}.
Clearly,p= (, )∈F(T),Ap= –∈EP(f). So={p∈F(T) :Ap∈EP(f)} =∅.
Remark . It is worth to mention that the split common solution problem in Example . lies in two different subsets of the same space and the split common solution problem in Example . lies in two different subsets of the different space. So, Examples . and . also show that the split common solution problem is meaningful.
In this paper, we introduce a weak convergence algorithm and a strong convergence al-gorithm for the split common solution problem when the nonlinear operatorTis a quasi-nonexpansive mapping. Some strong and weak convergence theorems are established. We also give some examples to illustrate our results.
2 Preliminaries
We writexnxto indicate that the sequence{xn}weakly converges toxandxn→xwill symbolize strong convergence as usual.
A Banach space (X, · ) is said to satisfy Opial’s condition, if for each sequence{xn}in Xwhich converges weakly to a pointx∈X, we have
lim inf
n→∞ xn–x<lim infn→∞ xn–y, ∀y∈X,y=x.
It is well known that any Hilbert space satisfies Opial’s condition.
LetKbe a nonempty subset of real Hilbert spacesH. Recall that a mappingT:K→K is said to be nonexpansive ifTx–Ty ≤ x–yfor allx,y∈Kand quasi-nonexpansive if
Example . LetH=Rwith the inner product defined byx,y=xyfor allx,y∈Rand the standard norm| · |. LetC:= [, +∞) andTx=x++x for allx∈C. Obviously,F(T) ={}. It is easy to see that
|Tx– |= x
+x|x– | ≤ |x– | forx∈C
and
T() –T
=
> –
.
Hence,T is a continuous quasi-nonexpansive mapping but not nonexpansive.
Definition .(see []) LetKbe a nonempty closed convex subset of a real Hilbert space H andT a mapping fromKintoK. The mappingT is said to be demiclosed if, for any sequence{xn}which weakly converges toy, and if the sequence{Txn}strongly converges toz, thenTy=z.
Remark . In Definition ., the particular case of demiclosedness at zero is frequently used in some iterative convergence algorithms, which is the particular case whenz=θ, the zero vector ofH; for more detail, one can refer to [].
The following concept of zero-demiclosedness was introduced in [].
Definition .(see []) LetKbe a nonempty, closed, and convex subset of a real Hilbert space andTa mapping fromKintoK. The mappingTis calledzero-demiclosedif{xn}in Ksatisfyingxn–Txn → andxnz∈KimpliesTz=z.
The following result was essentially proved in [], but we give the proof for the sake of completeness.
Proposition . Let K be a nonempty, closed, and convex subset of a real Hilbert space with zero vectorθ and T a mapping from K into K . Then the following statements hold.
(a) T is zero-demiclosed if and only ifI–Tis demiclosed atθ;
(b) IfT is a nonexpansive mappings and there is a bounded sequence{xn} ⊂Hsuch that
xn–Txn →asn→, thenTis zero-demiclosed.
Proof Obviously, the conclusion (a) holds. To see (b), since{xn} is bounded, there is a subsequence{xnk} ⊂ {xn}andz∈H such thatxnk z. One can claimTz=z. Indeed, if
Tz=z, it follows from the Opial’s condition that
lim inf
k→∞ xnk–z< lim infk→∞ xnk–Tz
≤lim inf
k→∞
xnk–Txnk+Txnk–Tz
=lim inf
k→∞ Txnk–Tz ≤lim inf
k→∞ xnk–z,
Example . LetH,C, andTbe the same as in Example .. Let{xn}be a sequence inC. Ifxn→zandxn–Txn→, thenz∈F(T) ={}. Indeed, sinceT is continuous, we have Tz=zandz∈F(T) ={}. Hence,Tis zero-demiclosed.
Example . LetH=Rwith the inner product defined byx,y=xyfor allx,y∈Rand the standard norm| · |. LetC:= [, +∞). LetTbe a mapping fromCintoCdefined by
Tx= ⎧ ⎨ ⎩
x
x+, x∈(, +∞),
, x∈[, ].
ThenT is a discontinuous quasi-nonexpansive mapping but not zero-demiclosed.
Proof Obviously,F(T) ={}, andTis a quasi-nonexpansive operator. On the other hand, let xn= + n for alln∈N, then it is not hard to prove thatxn→,xn–Txn→ and
/∈F(T). SoT is not zero-demiclosed.
LetHandHbe two Hilbert spaces. LetA:H→HandB:H→Hbe two bounded
linear operators.Bis called theadjoint operator(oradjoint) ofA, if for allz∈H,w∈
H,BsatisfiesAz,w=z,Bw. 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 ifBis an adjoint operator ofA, thenA=B.
Example . LetH=R with the normα= (a +a+a)
forα= (a,a,a)∈R
and H=R with the norm γ= (c +c+c +c)
for γ = (c,c,c,c)∈R. Let α,β=ab+ab+ab andγ,η=cd+cd+cd+cddenote the inner
prod-uct ofHandH, respectively, whereα= (a,a,a),β= (b,b,b)∈H,γ = (c,c,c,c),
η= (d,d,d,d)∈H. LetAα= (a,a+a,a–a,a) forα= (a,a,a)∈H. ThenAis
a bounded linear operator fromH into H withA=
√
. For γ = (c,c,c,c)∈H,
let Bγ = (c+c,c –c,c +c). Then B is a bounded linear operator from H into
H with B=
√
. Moreover, for anyα = (a,a,a)∈H andγ = (c,c,c,c)∈H,
Aα,γ=α,Bγ, soBis an adjoint operator ofA.
LetK be a closed and convex subset of a real Hilbert spaceH. For each pointx∈H, there exists a unique nearest point inK, denoted byPKx, such thatx–PKx ≤ x–y,
∀y∈K. The mappingPK is called themetric projectionfromHontoK. It is well known thatPK has the following characterizations:
(i) x–y,PKx–PKy ≥ PKx–PKyfor everyx,y∈H.
(ii) forx∈H, andz∈K,z=PK(x)⇐⇒ x–z,z–y ≥,∀y∈K.
(iii) y–PK(x)+x–PK(x)≤ x–yfor allx∈Handy∈K. The following lemmas are crucial in our proofs.
Lemma .(see []) Let K be a nonempty, closed, and convex subset of H and F be a bi-function of K×K intoRsatisfying the following conditions.
(A) F(x,x) = for allx∈K;
Let r> and x∈H. Then there exists z∈K such that F(z,y) +ry–z,z–x ≥, for all y∈K .
Lemma .(see []) Let K be a nonempty, closed, and 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:
TrF(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 anyr> 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 .(see,e.g., []) Let H be a real Hilbert space. Then the following hold. (a) x+y≤ y+ x,x+yandx–y=x+y– x,yfor allx,y∈H;
(b) αx+ ( –α)y=αx+ ( –α)y–α( –α)x–yfor allx,y∈Hand
α∈[, ].
The following result is simple, but it is very useful in this paper; see also [].
Lemma . Let the mapping TF
r be defined as (.). Then for r,s> and x,y∈H,
TF
r(x) –TsF(y)≤ x–y+
|s–r| s T
F s(y) –y.
In particular,TrF(x) –TrF(y) ≤ x–yfor any r> and x,y∈H, that is TrF is nonexpan-sive for any r> .
Proof Forr,s> andx,y∈H, by (i) of Lemma .,TF
r(x) =z andTsF(y) =zfor some
z,z∈K. By the definition ofTrF, we have
F(z,u) +
ru–z,z–x ≥, ∀u∈K (.)
and
F(z,u) +
su–z,z–y ≥, ∀u∈K. (.)
So, combining (.), (.), and (A), we get
rz–z,z–x+
sz–z,z–y ≥,
or
z–z,
z–x
r
–
z–z,
z–y
s
or
z–z,
s r(z–x)
–z–z,z–y ≥,
or
z–z,z–x–
r s(z–y)
≥,
or
z–z,z–z+z–x–
r s(z–y)
≥,
which implies
z–z≤
z–z,z–x–
r s(z–y)
≤ z–z
z–x–
r s(z–y)
,
and hence
TrF(x) –TsF(y)=z–z
≤z–x–
r s(z–y)
≤ y–x+
–r s
(z–y)
=y–x+|s–r| s T
F s(y) –y.
In particular, the last inequality show that for anyr> ,TF
r is nonexpansive. The proof is
completed.
3 Main results
In this section, we first introduce a weak convergence iterative algorithms for the split common solution problem.
Theorem . Let H and H be two real Hilbert spaces and C ⊂H and K ⊂H be
two nonempty closed convex sets. Let T:C→C be zero-demiclosed quasi-nonexpansive mappings and f :K×K→Rbe bi-functions with={p∈F(T) :Ap∈EP(f)} =∅. Let A:H→Hbe a bounded linear operator with its adjoint B.
Given x∈C andη∈(, ). Let{xn}and{un}be sequences generated by ⎧
⎪ ⎪ ⎨ ⎪ ⎪ ⎩
un=TrfnAxn,
xn+= ( –αn)yn+αnTyn,
yn=PC(xn+εB(Trfn–I)Axn), ∀n∈N,
(.)
Proof Letx∗∈. ThenAx∗∈EP(f). For eachn∈N, by Lemmas . and ., we have
TrfnAxn–TrfnAx
∗
≤TrfnAxn–TrfnAx
∗,Ax n–Ax∗
= T
f
rnAxn–Ax
∗
+Axn–Ax∗–TrfnAxn–Axn
.
So,
TrfnAxn–Ax∗
≤ Axn–Ax∗–TrfnAxn–Axn
for anyn∈N. (.)
By (b) of Lemma . and (.), for eachn∈N, we get
εxn–x∗,B
Trfn–IAxn
= εA(xn–x∗) +
Trfn–IAxn–
Trfn–IAxn,
Trfn–IAxn
= ε
T
f
rnAxn–Ax
∗
+ T
f rn–I
Axn
–
Axn–Ax
∗–Tf rn–I
Axn ≤ε T
f rn–I
Axn
–Trfn–IAxn
= –ε
Trfn–IAxn
. (.)
Note that for anyn∈N,
BTrfn–IAxn
≤ BTrfn–IAxn
, (.)
so it follows from (.), (.), and (.) that
xn+–x∗
= ( –αn)yn–x∗
+αnTyn–x∗
– ( –αn)αnyn–Tyn
≤yn–x∗–ηyn–Tyn
=PC
xn+εB
Trfn–IAxn
–PCx∗
–ηyn–Tyn
≤xn+εB
Trfn–IAxn–x∗
–ηyn–Tyn
=xn–x∗
+εBTrfn–IAxn
+ εxn–x∗,B
Trfn–IAxn
–ηyn–Tyn
≤xn–x∗+εBTrfn–I
Axn–εTrfn–I
Axn–ηyn–Tyn
=xn–x∗
–ε –εBTrfn–IAxn
–ηyn–Tyn. (.)
Sinceε∈(,
B),ε( –εB) > , by (.), we obtain
xn+–x∗≤yn–x∗≤xn–x∗ (.)
and
ηyn–Tyn+ε
–εBTrfn–IAxn
≤xn–x∗
–xn+–x∗
The inequality (.) implies thatlimn→∞xn–x∗exists. Further, from (.) and (.), we get
lim
n→∞xn–x
∗= lim n→∞yn–x
∗, (.)
lim
n→∞yn–Tyn= (.)
and
lim
n→∞T f rn–I
Axn= . (.)
From (.) and (.), we have
yn–xn=PC
xn+εB
Trfn–IAxn
–PCxn
≤εBTrfn–IAxn→ asn→ ∞. (.)
Sincelimn→∞xn–x∗exists,{xn}is bounded and hence{xn}has a weakly convergence subsequence{xnj}. Assume thatxnjpfor somep∈C. ThenAxnj Ap∈K,ynj p
andTrfnjAxnjApby (.) and (.).
We arguep∈. SinceT is a zero-demiclosed mapping, by (.) andynj p, we
ob-tainp∈F(T). Applying Lemma .,EP(f) =F(Trf) for anyr> . We claimTrfAp=Ap. IfTrfAp=Ap, sinceAxn–TrfnAxn= (I–T
f
rn)Axn→ asn→ ∞from (.) and applying
Opial’s condition, we have
lim inf
j→∞ Axnj–Ap
<lim inf
j→∞
Axnj–T
f rAp
=lim inf
j→∞
Axnj–T
f
rnjAxnj+T
f
rnjAxnj–T
f rAp
≤lim inf
j→∞
Axnj–T
f
rnjAxnj+T
f
rnjAxnj–T
f rAp
=lim inf
j→∞ T f
rnjAxnj–T
f rAp
=lim inf
j→∞ T f
rAp–TrfnjAxnj
≤lim inf
j→∞
Axnj–Ap+ |rnj–r|
rnj
TrfnjAxnj–Axnj
(by Lemma .)
=lim inf
j→∞ Axnj–Ap,
which lead to a contradiction. SoAp∈F(Trf) =EP(f), and hence we showp∈.
Now, we prove{xn}converges weakly top∈. Otherwise, if there exists other subse-quence of{xn}which is denoted by{xnl}such thatxnlq∈withq=p. Then, by Opial’s
condition,
lim inf
l→∞ xnl–q<lim infl→∞ xnl–p<lim infl→∞ xnl–q.
Finally, we prove{un} ≡ {TrfnAxn}converges weakly toAp∈EP(f). Sincexnp, we have
AxnApasn→ ∞. Thus, by (.), we obtainunAp∈EP(f) asn→ ∞. The proof is
completed.
Corollary . Let H and H be two real Hilbert spaces. Let T :H→H be a
zero-demiclosed quasi-nonexpansive mapping withF(T)=∅and f :H×H →Rbe a
bi-function with EP(f)=∅. Let A:H→Hbe a bounded linear operator with its adjoint B.
Givenη∈(, ). Let{xn}and{un}be sequences generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈H,
un=TrfnAxn,
xn+= ( –αn)yn+αnTyn,
yn=xn+εB(Trfn–I)Axn, ∀n∈N,
(.)
whereε∈(,B)and{rn} ⊂(, +∞)withlim infn→∞rn> . Suppose ={p∈F(T) : Ap∈EP(f)} =∅and the control coefficient sequence{αn}satisfiesαn∈[η, –η]for n∈N. Then the sequence{xn}converges weakly to an element p∈and{un}weakly to Ap∈EP(f).
Next, we introduce a strong convergence algorithm for the split common solution prob-lem.
Theorem . Let C⊂Hand K⊂Hbe two nonempty, closed, and convex sets, T:C→
C zero-demiclosed quasi-nonexpansive mappings and f :K×K→Ra bi-function with ={p∈F(T) :Ap∈EP(f)} =∅. Let A:H→Hbe a bounded linear operator with the
adjoint B. Given x∈C, C=C andη∈(, ). Let{xn}and{un}be sequences generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
un=TrfnAxn,
yn= ( –αn)zn+αnTzn,
zn=PC(xn+εB(Trfn–I)Axn)),
Cn+={v∈Cn:yn–v ≤ zn–v ≤ xn–v},
xn+=PCn+(x), n∈N,
(.)
where{rn} ⊂(, +∞)withlim infn→∞rn> , PC is a projection operator from H into C
andε∈(,B)is a constant,{αn}satisfiesαn∈[η, –η]for n∈N, then xn→p∈and un→Ap∈EP(f).
Proof First, we claim⊂Cnforn∈N. In fact, letp∈. Following the same argument as in Theorem ., we have
εxn–p,B
Trfn–IAxn
≤–εTrfn–IAxn, (.)
and
BTrfn–IAxn
≤ BTrfn–IAxn
By (.), (.), and (.), we get
yn–p
≤ zn–p– ( –αn)αnzn–Tzn
≤xn+εB
Trfn–IAxn–p
–ηzn–Tzn
=xn–p+εB
Trfn–IAxn
+ εxn–p,B
Trfn–IAxn
–ηzn–Tzn
≤ xn–p+εBTrfn–I
Axn–εTrfn–I
Axn–ηzn–Tzn
≤ xn–p–ε
–εBTrfn–IAxn
–ηzn–Tzn
for anyn∈N. (.)
Noticeε∈(,B),ε( –εB) > . It follows from (.) that
yn–p ≤ zn–p ≤ xn–p for alln∈N,
and hencep∈Cnfor alln∈N. Hence,⊂CnandCn=∅for alln∈N.
Now, we proveCnis a closed convex set for eachn∈N. It is not hard to verify thatCnis closed for eachn∈N, so it suffices to verify thatCnis convex for eachn∈N. Indeed, let w,w∈Cn+. For anyγ ∈(, ), since
yn–
γw+ ( –γ)w
=γ(yn–w) + ( –γ)(yn–w)
=γyn–w+ ( –γ)yn–w–γ( –γ)w–w
≤γzn–w+ ( –γ)zn–w–γ( –γ)w–w
=zn–
γw+ ( –γ)w,
we haveyn– (γw+ ( –γ)w) ≤ zn– (γw+ ( –γ)w). Similarly, we also havezn– (γw+ ( –γ)w) ≤ xn– (γw+ ( –γ)w), which impliesγw+ ( –γ)w∈Cn+. Hence,
we show thatCn+is a convex set for eachn∈N.
Notice thatCn+⊂Cnandxn+=PCn+(x)⊂Cn, thenxn+–x ≤ xn–xforn∈N
withn≥. It follows thatlimn→∞xn–xexists. Hence{xn}is bounded, which yields that{zn}and{yn}are bounded. For anyk,n∈Nwithk>n, fromxk=PCk(x)⊂Cnand the
character (iii) of the projection operatorP, we have
xn–xk+x–xk=xn–PCk(x)
+x–PCk(x)
≤ xn–x. (.)
Sincelimn→∞xn–xexists, by (.), we havelimn→∞xn–xk= , which implies that
{xn}is a Cauchy sequence.
Letxn→p. One claimp∈. Firstly, byxn+=PCn+(x)∈Cn+⊂Cn, from (.) we have
and
zn–xn ≤ zn–xn++xn+–xn ≤xn+–xn → asn→ ∞. (.)
Settingρ=ε( –εB), from (.) again, we have
ρTrfn–IAxn+ηzn–Tzn≤ xn–p–yn–p
≤ xn–yn
xn–p+yn–p
.
So
lim
n→∞Tzn–zn= (.)
and
lim
n→∞T f rn–I
Axn= . (.)
Letr> . Sincexn→pasn→ ∞, Lemma . and equation (.) imply that
TrfAp–Ap≤TrfAp–TrfnAxn+TrfnAxn–Axn+Axn–Ap
≤Axn–Ap+
+|rn–r| rn
TrfnAxn–Axn→ asn→ ∞.
SoTrfAp=Ap, which say thatAp∈F(Trf) =EP(f). On the other hand, sincexn–zn→ by (.) andxn→p, we havezn→p. Notice thatTis zero-demiclosed quasi-nonexpansive mappings, by (.),Tp=p, namely,p∈F(T). Sop∈. From (.), we also have{un} ≡
{TrfnAxn}converges strongly toAp∈EP(f). The proof is completed.
Corollary . Let H and H be two real Hilbert spaces. Let T:H →H be a
zero-demiclosed quasi-nonexpansive mappings with F(T)=∅ and f :H×H →Rbe a bi-function with EP(f)=∅. Let A:H→Hbe a bounded linear operator with the adjoint B.
Given x∈H, C=H, andη∈(, ). Let{xn}and{un}be sequences generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
un=TrfnAxn,
yn= ( –αn)zn+αnTzn,
zn=xn+εB(Trfn–I)Axn,
Cn+={v∈Cn:yn–v ≤ zn–v ≤ xn–v},
xn+=PCn+(x), n∈N,
(.)
where{rn} ⊂(, +∞)withlim infn→∞rn> , andε∈(,B)is a constant. Suppose that
Example . LetH=H=Rwith the inner product defined byx,y=xyfor allx,y∈R
and the standard norm | · |. Let C:= [, +∞) andTx=x+
+x for allx∈C. From Exam-ples . and ., we know thatTis an zero-demiclosed quasi-nonexpansive mapping with F(T) ={}.
LetK:= [–∞, ] andf(x,y) = (y–x)(x+ ) for allx,y∈K, thenf satisfies the condition
(A)-(A) andEP(f) ={–}. LetAx= –xfor allx∈R, thenAis a bounded linear operator withB(the adjoint of A) =AandA=B= .
Obviously,={p∈F(T) :Ap∈EP(f)}={}=F(T), so=∅. Letx∈C,{xn}and{un} be sequences generated by
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
un=TrfnAxn,
xn+= ( –αn)yn+αnTyn,
yn=PC(xn+B(Trfn–I)Axn), ∀n∈N,
(.)
where,rn= andαn∈(, ) for alln∈N,PCis a projection operator fromHintoC. Then
the sequence{xn}converges strongly to ∈and{un}converges strongly toA() = –∈ EP(f).
Proof
(i) Firstly, for givenrn= forn∈N, we prove that for any{xn} ⊂C, there exists a
unique sequence{un}n∈N≡ {–xn– }n∈NinKsuch that
f(un,v) +v–un,un–Axn ≥, ∀v∈K,n∈N. (.)
Because (.) is equivalent with
(v–un)
un+ + (un+ xn)
= (v–un)
un+ + (un–Axn)
≥, ∀v∈K,n∈N, (.)
while (.) is true if and only ifun= –(xn+ )for alln∈N. So the conclusion is
true.
(ii) Secondly, it is not hard to computeB(Trfn–I)Axn=B(un–Axn) = –(xn– )for all n∈N. Hence,
xn+ B
Trfn–IAxn= xn+
∈C for alln∈N.
(iii) By (i) and (ii), forx∈C, we can rewrite the algorithm (.) as follows:
xn+= ( –αn)yn+αnTyn, yn= xn+
(.)
and
As in Example ., we easily obtain|Tyn– | ≤ |yn– |. Hence, from (.) and
(.), we get
|xn+– | ≤( –αn)|yn– |+αn|Tyn– |
≤ |yn– |= |xn– |
≤ · · ·
≤
n
|x– |, ∀n∈N,
which showsxn→∈F(T) =. Sinceun= –(xn+ ),n∈N, we obtain un→– =A()∈EP(f).
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
1Department of Mathematics, Honghe University, Yunnan, 661100, China.2Department of Mathematics, National
Kaohsiung Normal University, Kaohsiung, 824, Taiwan.
Acknowledgements
The authors would like to express their sincere thanks to the anonymous referee for their valuable comments and useful suggestions in improving the paper. The first author was supported by the Natural Science Foundation of Yunnan Province (2010ZC152). The second author was supported partially by Grant no. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.
Received: 21 April 2012 Accepted: 26 July 2012 Published: 6 August 2012
References
1. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63, 123-145 (1994)
2. Moudafi, A, Théra, M: Proximal and dynamical approaches to equilibrium problems. In: Ill-posed Variational Problems and Regularization Techniques. Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187-201. Springer, Berlin (1999)
3. Combettes, PL, Hirstoaga, A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal.6, 117-136 (2005) 4. Flam, SD, Antipin, AS: Equilibrium programming using proximal-link algorithms. Math. Program.78, 29-41 (1997) 5. Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in
Hilbert spaces. J. Math. Anal. Appl.331, 506-515 (2007)
6. Kangtunyakarn, A, Suantai, S: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal.71, 4448-4460 (2009)
7. Qin, X, Cho, YJ, Kang, SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math.225, 20-30 (2009)
8. Saeidi, S: Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings. Nonlinear Anal.70, 4195-4208 (2009)
9. Chang, SS, Lee, JHW, Chan, CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal.70, 3307-3319 (2009)
10. Jung, JS: Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. Appl. Math. Comput.213, 498-505 (2009)
11. Zegeye, H, Ofoedu, EU: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput.12, 3439-3449 (2010) 12. Colao, V, Marino, G, Xu, H-K: An iterative method for finding common solutions of equilibrium and fixed point
problems. J. Math. Anal. Appl.344, 340-352 (2008)
13. Yao, Y, Liou, YC, Yao, JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl.2007, Article ID 64363 (2007)
15. He, Z: A new iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contractive mappings and its application. Math. Commun. (in press)
16. Moudafi, A: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal.74, 4083-4087 (2011)
17. He, Z, Du, W-S: On split common solution problems for nonlinear operators (submitted)
18. He, Z: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl.2012, 162 (2012). doi:10.1186/1029-242X-2012-162
doi:10.1186/1687-1812-2012-130
