R E S E A R C H
Open Access
Moudafi’s open question and simultaneous
iterative algorithm for general split equality
variational inclusion problems and general
split equality optimization problems
Shih-sen Chang
*, Lin Wang, Yong Kun Tang and Gang Wang
*Correspondence:
[email protected] College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P.R. China
Abstract
The purpose of this paper is first to introduce and studythe general split equality variational inclusion problemsandthe general split equality optimization problemsin the setting of infinite-dimensional Hilbert spaces and then propose a new simultaneous iterative algorithm. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm converging strongly to a solution for these two kinds of problems are proved. As special cases, we shall utilize our results to study the split feasibility problems, the split equality equilibrium problems, and the split optimization problems. The results presented in the paper not only extend and improve the corresponding recent results announced by many authors, but they also provide an affirmative answer to an open question raised by Moudafi in his recent work.
Keywords: general split equality variational inclusion problem; general split equality optimization problem; split feasibility problem; split equality equilibrium problem; split optimization problem
1 Introduction
LetCandQbe nonempty closed convex subsets of real Hilbert spacesHandH, 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 introduced the (SFP) in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction []. It has been found that the (SFP) can also be used in various disciplines such as image restoration, and computer tomograph and radiation therapy treatment planning [–]. The (SFP) in an infinite-dimensional real Hilbert space can be found in [, , –].
Assuming that the (SFP) is consistent, it is not hard to see thatx∗∈Csolves (SFP) if and only if it solves the fixed-point equation
x∗=PCI–γA∗(I–PQ)Ax∗,
wherePCandPQare the metric projection fromHontoCand fromHontoQ, respec-tively,γ > is a positive constant andA∗is the adjoint ofA.
A popular algorithm to be used to solves theSFP(.) 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 following split equality feasibility problem
(SEFP):
to findx∈C,y∈Qsuch thatAx=By, (.)
whereA:H→HandB:H→Hare two bounded linear operators. Obviously, ifB=I
(identity mapping onH) andH=H, then (.) reduces to (.). The kind of split
equal-ity problems (.) allows asymmetric and partial relations between the variablesxandy. The interest is to cover many situations, such as decomposition methods for PDEs, and applications in game theory and intensity-modulated radiation therapy.
In order to solve the split equality feasibility problem (.), Moudafi [] introduced the following simultaneous iterative method:
xk+=PC(xk–γA∗(Axk–Byk)),
yk+=PQ(yk+βB∗(Axk+–Byk)), (.)
and under suitable conditions he proved the weak convergence of the sequence{(xn,yn)}
to a solution of (.) in Hilbert spaces.
At the same time, he raised the following open question.
Moudafi’s Open Question . Is there any strong convergence theorem of an alternating algorithm for the split equality feasibility problem (.) in real Hilbert spaces?
More recently, Eslamian and Latif [], Chenet al.[], Chuang [] and Chang and Wang [] introduced and studied some kinds ofgeneral split feasibility problem,general split equality problem,andsplit variational inclusion problemin real Hilbert spaces. Under suitable conditions some strong convergence theorems are proved. Also a comprehensive survey and update bibliography on split feasibility problems are given in Ansari and Rehan [].
Motivated by the above works and related literature, in this paper, we continue to con-sider the problem (.). We obtain some strongly convergent theorems to a solution of the problem (.) which provide an affirmative answer to Moudafi’s open question.
For the purpose we first introduce and consider the following more general problems. (I) General split equality variational inclusion problem:
(GSEVIP) to findx∗∈Handy∗∈Hsuch that
∈
∞
i=
Uix∗, ∈
∞
i=
whereH,H andH are three real Hilbert spaces,Ui:H→H andKi:H→H,i= , , . . . are two families of set-valued maximal monotone mappings,A:H→H andB:
H→Hare two linear and bounded operators.
(II) General split equality optimization problem:
(GSEOP) to findx∗∈Handy∗∈Hsuch that for eachi≥
hix∗=min x∈H
hi(x), giy∗=min y∈H
gi(y) and Ax∗=By∗, (.)
whereH,H, andHare three real Hilbert spaces,A:H→HandB:H→Hare two
linear and bounded operators,hi:H→Randgi:H→Rare two countable families of proper, convex, and lower semicontinuous functions.
The following problems are special cases of Problem I and II. (III) Split equality feasibility problems.
Let C⊂H andQ⊂H be two nonempty closed convex subsets and A:H→H,
B:H→H be two bounded linear operators. As mentioned above the so-called ‘split
equality feasibility problem’ (SEFP) is to find
x∗∈C,y∗∈Qsuch thatAx∗=By∗. (.)**
LetiCandiQbe the indicator functions ofCandQ, respectively,i.e.,
iC(x) =
, ifx∈C,
+∞, ifx∈/C; iQ(y) =
, ify∈C,
+∞, ify∈/Q. (.)
Denote byNC(x) andNQ(y) thenormal cones of C and Q at x and y, respectively:
NC(x) =
z∈H:z,v–x ≤,∀v∈C
,
NQ(y) =z∈H:z,v–y ≤,∀v∈Q.
It is easy to know thatiCandiQboth are proper convex and lower semicontinuous func-tions onHandH, respectively, and the sub-differentials∂iCand∂iQboth are maximal monotone operators. We define the resolvent operatorJ∂iC
β ofiCby
J∂iC
β (x) = (I+β∂iC)–(x), β> ,x∈H.
Here
∂iC(x) =z∈H:iC(x) +z,u–x ≤iC(u),∀u∈H
=z∈H:z,u–x ≤,∀u∈C=NC(x), x∈C. Hence we have
u=J∂iC
β (x) ⇔ x–u∈βNC(u)
⇔ x–u,y–u ≤, ∀y∈C ⇔ u=PC(x).
This implies thatJ∂iC
β =PCfor anyβ> . Similarly, we also have∂iQ(y) =NQ(y), andJ ∂iQ
β =
optimization problem,i.e., to findx∗∈H, andy∗∈Hsuch that
iC
x∗=min x∈H
iC(x), iQ
y∗=min y∈H
iQ(y) and Ax∗=By∗;
⇔ ∈∂iCx∗, ∈∂iQy∗ and Ax∗=By∗.
(IV)Split equality equilibrium problem.
LetDbe a nonempty closed and convex subset of a real Hilbert spaceH. A bifunction
g:D×D→(–∞, +∞) is said to be aequilibrium function, if it satisfies the following conditions:
(A) g(x,x) = , for allx∈D;
(A) gis monotone,i.e.,g(x,y) +g(y,x)≤for allx,y∈D; (A) lim supt↓g(tz+ ( –t)x,y)≤g(x,y)for allx,y,z∈D;
(A) for eachx∈D,y→g(x,y)is convex and lower semicontinuous. The so-calledequilibrium problem with respect to the equilibrium function gis
to findx∗∈Dsuch thatgx∗,y≥, ∀y∈D. (.)
Its solution set is denoted byEP(g).
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 . (.)
Proposition .[] The resolvent operator Rλ,gof the equilibrium function g has the fol-lowing properties:
() Rλ,g is single-valued;
() F(Rλ,g) =EP(g)andEP(g)is a nonempty closed and convex subset ofD;
() Rλ,g is a firmly nonexpansive mapping.
Leth,g:D×D→(–∞, +∞) be two equilibrium functions. For givenλ> , letRλ,hand Rλ,gbe the resolvent ofhandg(defined by (.)), respectively.
The so-called split equality equilibrium problem with respective to h, g, and D
(SEEP(h,g,D)) is to findx∗∈D,y∗∈Dsuch that
hx∗,u≥, ∀u∈D, gy∗,v≥, ∀v∈D and Ax∗=By∗, (.)
whereA,B:D→Dare two linear and bounded operators.
By Proposition ., the (SEEP(h,g,D)) (.) is equivalent to findx∗∈D,y∗∈Dsuch that for eachλ>
x∗∈EP(h,D), y∗∈EP(g,D) and Ax∗=By∗
⇔ x∗∈F(Rλh), y∗∈F(Rλg) and Ax∗=By∗.
LettingC=F(Rλh),Q=F(Rλg), by Proposition .,CandQboth are nonempty closed and
feasibility problem:
to findx∗∈C,y∗∈Qsuch thatAx∗=By∗. (.)
(V)Split optimization problem.
LetHandHbe two real Hilbert spaces,A:H→Hbe a linear and bounded opera-tors,h:H→Randg:H→Rbe two proper convex and lower semicontinuous
func-tions. Thesplit optimization problem(SOP) is to findx∗∈H,Ax∗∈Hsuch that
hx∗=min x∈H
hi(x) and gAx∗=min z∈H
g(z). (.)
Denote byU=∂handK=∂g, then the (SOP) (.) is equivalent to the followingsplit variational inclusion problem (SVIP): to findx∗∈Hsuch that
∈Ux∗, ∈KAx∗. (.)
For solving (GSEVIP) (.) and (GSEOP) (.), in Sections and , we propose a new si-multaneous type iterative algorithm. Under suitable conditions some strong convergence theorems for the sequences generated by the algorithm are proved in the setting of infinite-dimensional Hilbert spaces. As special cases, we shall utilize our results to study the split feasibility problem, split equality equilibrium problem and the split optimization problem. By the way, we obtain a strongly convergent iterative sequence to a solution of the prob-lem (.), which provides an affirmative answer to the open question raised by Moudafi []. The results presented in the paper extend and improve the corresponding results an-nounced by Moudafiet al.[, , ], Eslamian and Latif [], Chenet al.[], Censoret al.[, –, ], Chuang [], Naraghirad [], Chang and Wang [], Ansari and Rehan [], and some others.
2 Preliminaries
We first recall some definitions, notations, and conclusions.
Throughout this paper, we assume thatHis a real Hilbert space andCis a nonempty closed convex subset ofH. In the sequel, we 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 nonexpansive mapping is the metric projectionPC from
H ontoC⊆H defined byx–PCx=infy∈Cx–y. The metric projectionPC isfirmly nonexpansive, if
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. (.)
A mappingT:H→His said to bequasinonexpansive, ifF(T)=∅, and
It is easy to see that ifT is a quasi-nonexpansive mapping, thenF(T) is a closed and convex subset ofC. Besides,Tis said to be afirmly nonexpansive, if
Tx–Ty≤ x–y,Tx–Ty ∀x,y∈C
⇔ Tx–Ty≤ x–y–(I–T)x– (I–T)y ∀x,y∈C.
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= 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:
x+y≤ x+ y,x+y .
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)}
withτ(n)→ ∞such that for all(sufficiently large)positive integer number 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–Tis said to be
demiclosed at zero, if for any sequence{xn} ⊂Cwithxnxandxn–Txn →,x=Tx.
Remark .[] It is well known that ifT:C→Cis a nonexpansive mapping, thenI–T
is demiclosed 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 supn→∞cn
bn≤,
thenlimn→∞an= .
Lemma . [] Let H be a real Hilbert space,B:H→H be a set-valued maximal monotone mapping,β> ,and let JB
β be the resolvent mapping of B defined by JβB:= (I+
βB)–,then
(ii) D(JB
β) =HandF(JβB) =B–();
(iii) (I–JB
β)is a firmly nonexpansive mapping for eachβ> ;
(iv) suppose thatB–()=∅,then for eachx∈H,eachx∗∈B–()and eachβ>
x–JβBx
+JβBx–x∗≤x–x∗
;
(v) suppose thatB–()=∅.Thenx–JβBx,JβBx–w ≥for eachx∈H,eachw∈B–(),
and eachβ> .
Lemma . Let H,Hbe two real Hilbert spaces,A:H→Hbe a linear bounded op-erator and A∗be the adjoint of A.Let B:H→H be a set-valued maximal monotone
mapping,β> ,and let JB
β be the resolvent mapping of B,then
(i) (I–JB
β)Ax– (I–JβB)Ay≤ (I–JβB)Ax– (I–JβB)Ay,Ax–Ay ;
(ii) A∗(I–JB
β)Ax–A∗(I–JβB)Ay≤ A(I–JβB)Ax– (I–JβB)Ay,Ax–Ay ;
(iii) ifρ∈(,A),then(I–ρA∗(I–JβB)A)is a nonexpansive mapping.
Proof By Lemma .(iii), the mapping (I–JβB) is firmly nonexpansive, hence the
conclu-sions (i) and (ii) are obvious. Now we prove the conclusion (iii).
In fact, for anyx,y∈H, it follows from the conclusions (i) and (ii) that
I–ρA∗I–JB
β
Ax–I–ρA∗I–JB
β
Ay
=x–y– ρx–y,A∗I–JB
β
Ax–A∗I–JB
β
Ay
+ρA∗I–JβBAx–A∗I–JβBAy
≤ x–y– ρAx–Ay,I–JβBAx–I–JβBAy
+ρAI–JβBAx–I–JβBAy
≤ x–y–ρ –ρAI–JβB
Ax–I–JβB
Ay
≤ x–y sinceρ –ρA≥.
This completes the proof of Lemma ..
3 General split equality variational inclusion problem and strong convergence theorems
Throughout this section we assume that () H,H,Hare three real Hilbert spaces;
() {Ui}∞i=:H→Hand{Ki}∞i=:H→Hare two families of set-valued maximal
monotone mappings,β> andγ > are given positive numbers;
() A:H→HandB:H→Hare two bounded linear operators andA∗,B∗are the
adjoint ofAandB, respectively; () f =f
f
, wherefi,i= , is ak-contractive mapping onHiwithk∈(, ); () the set of solutions of (GSEVIP) (.) =∅,
J(Ui,Ki)
μi :=
JUi
μi
JKi
μi
, G= [A–B], G∗=
A∗
–B∗
, G∗G=
A∗A –A∗B
–B∗A B∗B
() for any givenw∈H×H, the iterative sequence{wn} ⊂H×His generated by
wn+=αnwn+βnf(wn) + ∞
i=
γn,i
J(Ui,Ki)
μi
I–λn,iG∗G
wn
, n≥, (.)
or its equivalent form:
xn+=αnxn+βnf(xn) +∞i=γn,i(JμUii(xn–λn,i(A∗(Axn–Byn)))),
yn+=αnyn+βnf(yn) +∞i=γn,i(JμKii(yn+λn,i(B∗(Axn–Byn)))),
(.)
where{αn},{βn},{γn,i}are the sequences of nonnegative numbers satisfying
αn+βn+ ∞
i=
γn,i= , for eachn≥.
We are now in a position to give the following results.
Lemma . Let H,H,H,A,B,A∗,B∗,{Ui},{Ki},J(Ui,Ki)
μi ,G,G∗be the same as above.
If =∅(the solution set of(GSEVIP) (.)),then w∗:= (x∗,y∗)∈H×His a solution of
(GSEVIP) (.)if and only if for each i≥,and for any givenγ> andμ>
w∗=J(Ui,Ki)
μ
I–γG∗Gw∗. (.)
Proof Indeed, ifw∗= (x∗,y∗)∈H×H is a solution of (GSEVIP) (.), then by Lem-ma .(ii), for eachi≥, and for anyγ> andμ> we have
x∗∈Ui–() =FJUi
μ
, y∗∈Ki–() =FJKi
μ
and Ax∗=By∗
⇔ x∗=JUi
μ x∗, y∗=JμKiy∗ and Ax∗=By∗.
Hence we haveG(w∗) =Ax∗–By∗= , and so
J(Ui,Ki)
μi
I–γG∗Gw∗=J(Ui,Ki)
μ
w∗=JUi
μ x∗,JμKiy∗
=w∗.
This implies that (.) is true.
Conversely, ifw∗= (x∗,y∗)∈H×Hsatisfies (.), then we have
x∗=JUi
μ [x∗–γA∗(Ax∗–By∗)],
y∗=JKi
μ[y∗+γB∗(Ax∗–By∗)].
(.)
We make the assumption that the solution set of (GSEVIP) (.) is nonempty. Hence
the setsU–
i () andKi–() both are nonempty. By Lemma .(v) and (.), we have
x∗–x∗–γA∗Ax∗–By∗,x–x∗≥, ∀x∈Ui–(), and so
Similarly, by Lemma .(v) and (.) again, one gets
Ax∗–By∗,By∗–By≥, ∀y∈Ki–(). (.)
Adding up (.) and (.), we have
Ax∗–By∗,Ax–Ax∗+By∗–By≥, ∀x∈Ui–() andy∈Ki–(). Simplifying it, we have
Ax∗–By∗
≤Ax∗–By∗,Ax–By, ∀x∈Ui–() andy∈Ki–(). (.)
Since =∅, takingw¯ = (¯x,y¯)∈ , for eachi≥, we havex¯∈U–
i () andy¯∈Ki–() and Ax¯=By¯. In (.), takingx=x¯andy=y¯, we have
Ax∗–By∗= , i.e., Ax∗=By∗. (.)
Hence from (.) and (.)
x∗=JUi
μ (x∗),
y∗=JKi
μ(y∗),
⇔ ∈Ui
x∗, ∈Ki
y∗, ∀i≥. (.)
It follows from (.) and (.) thatw∗is a solution of (GSEVIP) (.).
This completes the proof of Lemma ..
Lemma . Ifλ∈(,L),where L=G,then(I–λG∗G) :H×H→H×His 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+λG∗G(u–w)– λu–w,G∗G(u–w) ≤ u–w+λLG(u–w)– λG(u–w),G(u–w) =u–w+λLG(u–w)– λG(u–w) =u–w–λ( –λL)G(u–w)
≤ u–w.
This completes the proof.
Theorem . Let H,H,H,A,B,A∗,B∗,{Ui},{Ki},J(Ui,Ki)
μi ,G,G∗,f be the same as above.
Let{wn}be the sequence defined by(.).If the solution set of(GSEVIP) (.)is nonempty and the following conditions are satisfied:
(i) αn+βn+
∞
i=γn,i= ,for eachn≥;
(ii) limn→∞βn= ,and
∞
(iii) lim infn→∞αnγn,i> for eachi≥;
(iv) {λn,i} ⊂(,L)for eachi≥,whereL=G,
then the sequence{wn}converges strongly to w∗=P f(w∗),which is a solution of(GSEVIP) (.).
Proof (I) First we prove that the sequence{wn}is bounded.
In fact, for any givenz∈ , it follows from Lemma ., Lemma ., and condition (iv) that
z=J(Ui,Ki)
μ
I–λn,iG∗Gz, for eachi≥,
and (I–λn,iG∗G) :H×H→H×His a nonexpansive mapping. Also by Lemma .(i), for eachi≥,J(Ui,Ki)
μi is a firmly nonexpansive mapping. Hence we have
wn+–z=
αnwn+βnf(wn) +
∞
i=
γn,iJ(Ui,Ki)
μi
I–λn,iG∗Gwn
–z
≤αnwn–z+βnf(wn) –z+ ∞
i=
γn,iJμ(Uii,Ki)
I–λn,iG∗Gwn–z
≤αnwn–z+βnf(wn) –z+ ∞
i=
γn,iI–λn,iG∗G
wn–z
≤αnwn–z+βnf(wn) –z
+
∞
i=
γn,iI–λn,iG∗G
wn–
I–λn,iG∗G
z
≤αnwn–z+βnf(wn) –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 , ∀n≥.
This shows that{wn}is bounded, and so is{f(wn)}.
(II) Now we prove that the following inequality holds:
αnγn,iwn–Jμ(Uii,Ki)
I–λn,iG∗Gwn
≤ wn–z–wn+–z+βnf(wn) –z
Indeed, it follows from (.) and Lemma . that for eachi≥
wn+–z=
αn(wn–z) +βn
f(wn) –z+
∞
j=
γn,j
Jμ(Ujj,Kj)
I–λn,jG∗Gwn–z
≤αnwn–z+βnf(wn) –z
+
∞
j=
γn,jJ (Uj,Kj)
μj
I–λn,jG∗Gwn–z
–αnγn,iwn–Jμ(Uii,Ki)
I–λn,iG∗Gwn
≤αnwn–z+βnf(wn) –z
+
∞
j=
γn,jwn–z
–αnγn,iwn–Jμ(Uii,Ki)
I–λn,iG∗G
wn
= ( –βn)wn–z+βnf(wn) –z
–αnγn,iwn–Jμ(Uii,Ki)
I–λn,iG∗Gwn.
This implies that for eachi≥
αnγn,iwn–Jμ(Uii,Ki)
I–λn,iG∗Gwn≤ wn–z–wn+–z+βnf(wn) –z
.
Inequality (.) is proved.
It is easy to see that the solution set of (GSEVIP) (.) is a closed and convex subset inH×H. By the assumption that is nonempty, so it is a nonempty closed and convex subset inH×H. Hence the metric projectionP is well defined. In addition, sinceP f :
H×H→ is a contractive mapping, there exists a uniquew∗∈ such that
w∗=P fw∗. (.)
(III) Now we prove that{wn}converges strongly tow∗. For the 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 (.) takingz=w∗ and letting n→ ∞, in view of conditions (ii) and (iii), we have
lim n→∞wn–J
(Ui,Ki)
μi
I–λn,iG∗G
wn= , for eachi≥. (.)
On the other hand, by Lemma . and (.), we have
wn+–w∗=
αnwn+βnf(wn) +
∞
i=
γn,iJ(Ui,Ki)
μi
I–λn,iG∗Gwn
–w∗
= αn
wn–w∗+βn
f(wn) –w∗
+ ∞ i= γn,i
J(Ui,Ki)
μi
I–λn,iG∗Gwn–w∗
≤αn
wn–w∗+
∞
i=
γn,i
J(Ui,Ki)
μi
I–λn,iG∗Gwn–w∗
+ βn
f(wn) –w∗,wn+–w∗ (by Lemma .) ≤
αnwn–w∗+ ∞
i=
γn,iwn–w∗
+ βn
f(wn) –f
w∗,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 it 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, (.)
where
ηn=
( –k)βn
–βnk , δn=
βnM
( –k)+ –k
fw∗–w∗,wn+–w∗
,
M=sup n≥
wn–w∗.
By condition (ii),limn→∞βn= and
∞
n=βn=∞, and so is
∞
n=ηn=∞.
Next we prove that
lim sup
n→∞ δn≤. (.)
In fact, since{wn}is bounded inH×H, there exists a subsequence{wnk} ⊂ {wn}with
wnkv∗(some point inC×Q), andλnk,i→λi∈(,
L) such that
lim n→∞
fw∗–w∗,wnk–w
∗=lim sup n→∞
fw∗–w∗,wn–w∗
Since
wnk–J
(Ui,Ki)
μi
I–λnk,iG
∗Gwn
k→, for eachi≥
andJ(Ui,Ki)
μi (I–λnk,iG∗G) is a nonexpansive mapping, by Remark .,I–J
(Bi,Ki)
μi (I–λn,iG∗G) is demiclosed at zero, hence we have
v∗=J(Ui,Ki)
μi
I–λn,iG∗Gv∗, ∀i≥. (.)
By Lemma ., this implies thatv∗∈ . In addition, sincew∗=P f(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, andcn=δnηnin Lemma .,
therefore 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(wherenlarge enough) such that
τ(n) =maxk≤n:wk–w∗≤wk+–w∗. (.)
Clearly{τ(n)}is a 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∗=P f(w∗) is a solution of (GSEVIP) (.).
This completes the proof of Theorem ..
Remark . Theorem . extends and improves the main results in Moudafiet al.[, , ], Eslamian and Latif [], Chenet al.[], Chuang [], Naraghirad [] and Ansari and Rehan [].
4 General split equality optimization problem and strong convergence theorems
LetH,H, andHbe three real Hilbert spaces. LetA:H→HandB:H→Hbe two linear and bounded operators. The so-calledgeneral split equality optimization problem
(GSEOP) is to findx∗∈H, andy∗∈Hsuch that for eachi≥
hi
x∗=min x∈H
hi(x), gi
y∗=min z∈H
wherehi:H→Randgi:H→Rare two families of proper, lower semicontinuous, and convex functions.
For eachi≥ denote by∂hi=Uiand∂gi=Ki. Then the mappingsUi:H→Hand
Ki:H→H,i= , , . . . both are set-valued maximal monotone mappings, and
hi
x∗=min x∈H
hi(x) ⇔ ∈∂hi
x∗=Ui
x∗,
giy∗=min z∈H
gi(z) ⇔ ∈∂giy∗=Kiy∗.
Therefore (GSEOP) (.) is equivalent to the following general split equality variational inclusion problem (GSEVIP): to findx∗∈Handy∗∈Hsuch that
∈
∞
i=
Uix∗, ∈
∞
i=
Kiy∗ and Ax∗=By∗. (.)
Therefore, the following theorem can be obtained from Theorem . immediately.
Theorem . Let H,H,H,A,B,A∗,B∗,{Ui},{Ki}be the same as above.Let J(Ui,Ki)
μi ,G,
G∗,f be the same as in Theorem..Let{wn}be the sequence defined by(.).If the solution set of(GSEVIP) (.)is nonempty and the following conditions are satisfied:
(i) αn+βn+
∞
i=γn,i= ,for eachn≥;
(ii) limn→∞βn= ,and
∞
n=βn=∞;
(iii) lim infn→∞αnγn,i> for eachi≥;
(iv) {λn,i} ⊂(,L)for eachi≥,whereL=G,
then the sequence{wn}converges strongly to w∗=P f(w∗),which is a solution of(GSEOP) (.).
By using Theorem . and Theorem ., now we give some corollaries for the split equal-ity feasibilequal-ity problem, the split equalequal-ity equilibrium problem, and the split optimization problem.
LetH,H,H,C,Q,A,Bbe the same as in the split equality feasibility problem (.). LetiCandiQbe the indicator function ofCandQ, respectively, defined by (.). In The-orem ., take{U}={∂iC},{K}={∂iQ}, andJ(U,K)
μ =PC×Q:=
PC
PQ
, therefore we have the following.
Corollary . Let H,H,H,A,B,A∗,B∗,PC×Qbe the same as above.Let G,G∗,f be the same as in Theorem..Let{wn}be the sequence generated by w∈H×H
wn+=αnwn+βnf(wn) +γn
PC×Q
I–λnG∗G
wn
, n≥, (.)
or its equivalent form
xn+=αnxn+βnf(xn) +γn(PC(xn–λn(A∗(Axn–Byn)))), yn+=αnyn+βnf(yn) +γn(PQ(yn+λn(B∗(Axn–Byn)))).
(.)
If the solution setof(SEFP) (.)is nonempty and the following conditions are satisfied:
(ii) limn→∞βn= ,and
∞
n=βn=∞;
(iii) lim infn→∞αnγn> ;
(iv) {λn} ⊂(,L)for eachi≥,whereL=G,
then the sequence{wn}converges strongly to w∗=Pf(w∗),which is a solution of(SEFP) (.).
Remark . Since the simultaneous iterative sequence{(xn,yn)}(.) converges strongly
to a solution of (SEFP) (.). Therefore it provides anaffirmative answer to Moudafi’s open question .[].
Leth,g:D×D→(–∞, +∞) be two equilibrium functions. For givenλ> , letRλ,hand Rλ,gbe the resolvents ofhandg(defined by (.)), respectively.
The so-calledsplit equality equilibrium problem with respective to h, g, and D(SEEP(h,
g,D)) is to findx∗∈D,y∗∈Dsuch that
hx∗,u≥, ∀u∈D, gy∗,v≥, ∀v∈D and Ax∗=By∗, (.)
whereA,B:D→Dare two linear and bounded operators.
By Proposition ., the (SEEP(h,g,D)) (.) is equivalent to findx∗∈D,y∗∈Dsuch that for eachλ>
x∗∈EP(h,D), y∗∈EP(g,D) and Ax∗=By∗
⇔ x∗∈F(Rλh), y∗∈F(Rλg) and Ax∗=By∗.
(.)
LettingC=F(Rλh),Q=F(Rλg), by Proposition .,CandQboth are nonempty closed and
convex subset ofD. Hence the problem (.) (and so the problem (.)) is equivalent to the following split equality feasibility problem:
to findx∗∈C,y∗∈Qsuch thatAx∗=By∗. (.)
In Corollary . takingH=H=H=D, from Corollary . we have the following.
Corollary . Let D,C,Q be the same as above.Let A,B,A∗,B∗,PC×Q,G,G∗,f be the same as in Corollary..For any given w∈D×D,let{wn}be the sequence generated by
wn+=αnwn+βnf(wn) +γn
PC×Q
I–λnG∗Gwn, n≥. (.)
If the solution setof(SEEP(h,g,D)) (.)is nonempty and the following conditions are satisfied:
(i) αn+βn+γn= ,for eachn≥;
(ii) limn→∞βn= ,and
∞
n=βn=∞;
(iii) lim infn→∞αnγn> ;
(iv) {λn} ⊂(,L)for eachi≥,whereL=G,
then the sequence{wn}converges strongly to w∗=Pf(w∗),which is a solution of(SEEP(h,
LetHandHbe two real Hilbert spaces,A:H→Hbe a linear and bounded
opera-tors,h:H→Randg:H→Rbe two proper convex and lower semicontinuous
func-tions. Thesplit optimization problem(SOP) is to findx∗∈H,Ax∗∈Hsuch that
hx∗=min x∈H
hi(x) and g
Ax∗=min z∈H
g(z). (.)
DenoteU=∂handK=∂g, then the (SOP) (.) is equivalent to the followingsplit vari-ational inclusion problem (SVIP): to findx∗∈Hsuch that
∈Ux∗, ∈KAx∗. (.)
In Theorem . takingH=H,B=I(the identity mapping onH) and
˜
G= [A–I], G˜∗=
A∗
–I
, G˜∗G˜ =
A∗A –A∗
–A I
,
then from Theorem . we have the following.
Corollary . Let H,H,A,I,G˜,G˜∗,U,K,be the same as above.Let Jμ(U,K),f be the same
as in Theorem..For any given w= (x,y)∈H×H,let{wn= (xn,yn)}be the sequence defined by
xn+=αnxn+βnf(xn) +γnJμU(xn–λnA∗(Axn–yn)), yn+=αnyn+βnf(yn) +γnJμK(yn+λn(Axn–yn)),
(.)
or its equivalent form:
wn+=αnwn+βnf(wn) +γn
Jμ(U,K)
I–λnG˜∗G˜
wn, n≥, (.)
If:={x∗∈U–()∩A–K–()},the solution set of (SOP) (.)is nonempty,and the following conditions are satisfied:
(i) αn+βn+γn= ,for eachn≥;
(ii) limn→∞βn= ,and
∞
n=βn=∞;
(iii) lim infn→∞αnγn> ;
(iv) {λn} ⊂(,L),whereL= ˜G,
then the sequence{wn}converges strongly to w∗=Pf(w∗),which is a solution of(SOP) (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors would like to express their thanks to the editors and the referees for their helpful suggestion and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
References
1. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms8, 221-239 (1994)
2. Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl.18, 441-453 (2002)
3. Censor, Y, Bortfeld, T, Martin, N, Trofimov, A: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys. Med. Biol.51, 2353-2365 (2006)
4. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications. Inverse Probl.
21, 2071-2084 (2005)
5. Censor, Y, Motova, A, Segal, A: Perturbed projections ans subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl.327, 1244-1256 (2007)
6. Xu, HK: A variable Krasnosel’skii-Mann algorithm and the multiple-sets split feasibility problem. Inverse Probl.22, 2021-2034 (2006)
7. Yang, Q: The relaxedCQalgorithm for solving the split feasibility problem. Inverse Probl.20, 1261-1266 (2004) 8. Zhao, J, Yang, Q: Several solution methods for the split feasibility problem. Inverse Probl.21, 1791-1799 (2005) 9. Chang, SS, Cho, YJ, Kim, JK, Zhang, WB, Yang, L: Multiple-set split feasibility problems for asymptotically strict
pseudocontractions. Abstr. Appl. Anal.2012, Article ID 491760 (2012). doi:10:1155/2012/491760
10. Chang, SS, Wang, L, Tang, YK, Yang, L: The split common fixed point problem for total asymptotically strictly pseudocontractive mappings. J. Appl. Math.2012, Article ID 385638 (2012). doi:10.1155/2012.385638
11. Moudafi, A: A relaxed alternatingCQalgorithm for convex feasibility problems. Nonlinear Anal.79, 117-121 (2013) 12. Moudafi, A, Al-Shemas, E: Simultaneous iterative methods for split equality problem. Trans. Math. Program. Appl.1,
1-11 (2013)
13. Eslamian, M, Latif, A: General split feasibility problems in Hilbert spaces. Abstr. Appl. Anal.2013, Article ID 805104 (2013)
14. Chen, RD, Wang, J, Zhang, HW: General split equality problems in Hilbert spaces. Fixed Point Theory Appl.2014, Article ID 35 (2014)
15. Chuang, C-S: Strong convergence theorems for the split variational inclusion problem in Hilbert spaces. Fixed Point Theory Appl.2013, Article ID 350 (2013)
16. Chang, S-s, Wang, L: Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces. Fixed Point Theory Appl.2014, Article ID 171 (2014)
17. Ansari, QH, Rehan, A: Split feasibility and fixed point problems. In: Ansari, QH (ed.) Nonlinear Analysis: Approximation Theory, Optimization and Applications, pp. 282-322. Springer, New York (2014)
18. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63(1/4), 123-145 (1994)
19. Moudafi, A: Split monotone variational inclusions. J. Optim. Theory Appl.150, 275-283 (2011)
20. Censor, Y, Segal, A: The split common fixed point problem for directed operators. J. Convex Anal.16, 587-600 (2009) 21. Naraghirad, E: On an open question of Moudafi for convex feasibility problem in Hilbert spaces. Taiwan. J. Math.18(2),
371-408 (2014)
22. Chang, SS, Joseph Lee, HW Chan, CK, Zhang, WB: A modified Halpern-type iterative algorithm for totally quasi-asymptotically nonexpansive mappings with applications. Appl. Math. Comput.218, 6489-6497 (2012) 23. Chang, SS: On Chidume’s open questions and approximate solutions for multi-valued strongly accretive mapping
equations in Banach spaces. J. Math. Anal. Appl.216, 94-111 (1997)
24. Maingé, P-E: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal.16(7-8), 899-912 (2008)
25. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory, vol. 28. Cambridge University Press, Cambridge (1990)
10.1186/1687-1812-2014-215
