R E S E A R C H
Open Access
An algorithm with strong convergence for
the split common fixed point problem of total
asymptotically strict pseudocontraction
mappings
Zhaoli Ma
1and Lin Wang
2**Correspondence:
2College of Statistics and
Mathematics, Yunnan University of Finance and Economics, Longquan Road, Kunming, 650221, China Full list of author information is available at the end of the article
Abstract
The purpose of this paper is to propose an algorithm to solve the split common fixed point problems for total asymptotically strict pseudocontraction mappings in Hilbert spaces. Without the assumption of semi-compactness on the mappings, the iterative scheme is shown to converge strongly to a split common fixed point of such mappings. The results presented in the paper improve and extend some recent corresponding results.
MSC: 47H09; 47J25
Keywords: split common fixed point problem; split feasibility problem; total asymptotically strict pseudocontraction mapping; Hilbert space
1 Introduction
LetHandH be two real Hilbert spaces,CandQbe nonempty closed convex subsets ofHandH, respectively. The split feasibility problem is formulated as finding a point
q∈Hwith the properties
q∈C and Aq∈Q, (.)
whereA:H→His a bounded linear operator. Assuming that SFP (.) is consistent (i.e., (.) has a solution), it is not hard to see thatx∈Csolve (.) if and only if it solves the following fixed point equation:
x=PC
I–γA∗(I–PQ)A
x, x∈C, (.)
wherePC andPQare the (orthogonal) projections ontoC andQ, respectively,γ > is any positive constant, andA∗ denotes the adjoint ofA. The split feasibility problem in finite dimensional Hilbert spaces was introduced by Censor and Elfving [] in for modeling inverse problems which arise from phase retrievals and in medical image re-construction []. Recently, it has been found that split feasibility problems can be used in various disciplines, such as image restoration, computer tomography and radiation ther-apy treatment planning [–]. The split feasibility problem in Hilbert space can be found
in [, , , , ]. Also the convex feasibility formalism is at the core of the modeling of many inverse problems and has been used to model significant real-world problems. The split common fixed point problem is a generalization of the split feasibility problem and the convex feasibility problem. If CandQare the sets of fixed points of two nonlinear mappings, respectively, andCandQare nonempty closed convex subsets, thenqis said to be a split common fixed point for the two nonlinear mappings. The split common fixed point problem (SCFP) for mappingsSandTis to find a pointq∈Hwith the properties
q∈F(S) and Aq∈F(T), (.)
we useto denote the set of solutions of SCFP (.), that is,={q∈F(S) :Aq∈F(T)}. LetHbe a real Hilbert space,Cbe a nonempty and closed convex subset ofH. A mapping
T :C→Cis said to be ak-strictly pseudocontractive mapping, if there existsk∈[, ) such that
Tx–Ty≤ x–y+k(I–T)x– (I–T)y, ∀x,y∈C. (.)
A mappingT:C→Cis said to be (k,{kn})-asymptotically strictly pseudocontractive, if there exist a constantk∈[, ) and a sequence{kn} ⊂[,∞) withkn→ such that
Tnx–Tny≤knx–y+kI–Tn
x–I–Tny, ∀x,y∈C. (.) A mappingT:C→Cis said to be (k,{μn},{ξn},φ)-totally asymptotically strictly pseu-docontractive, if there exist a constantk∈[, ) and sequences{μn} ⊂[,∞) and{ξn} ⊂ [,∞) withμn→ andξn→ such that for allx,y∈C,
Tnx–Tny≤ x–y+kI–Tnx–I–Tny
+μnφ
x–y+ξn, ∀n≥, (.)
whereφ: [,∞)→[,∞) is a continuous and strictly increasing function withφ() = . Now, we give two examples of (k,{μn},{ξn},φ)-total asymptotically strict pseudocon-traction mappings.
Example . LetCbe a unit ball in a real Hilbert spaceland letS:C→Cbe a mapping defined by
S: (x,x, . . .)→
,x,ax,ax, . . .
,
where{ai}is a sequence in (, ) such that ∞
i=ai=. It is proved by Goebel and Kirk [] that
(i) Sx–Sy ≤x–y,∀x,y∈C; (ii) Snx–Sny ≤n
j=ajx–y,∀x,y∈C,∀n≥. Denote byk
= ,k
n = nj=aj,n≥, then
lim
n→∞kn=n→∞lim
n
j=
aj
Lettingμn= (kn– ),∀n≥,φ(t) =t,∀t≥,k= , and{ξn}be a nonnegative real se-quence withξn→, then∀x,y∈C,n≥, we have
Snx–Sny≤ x–y+μnφ
x–y+kx–y–Snx–Sny+ξn.
Especially, asan+= – n,n
j=aj=.
Example . Let D be an orthogonal subspace of Rn with the norm x= n i=xi and the inner product x,y=ni=xiyi forx= (x, . . . ,xn) and y= (y, . . . ,yn). For each
x= (x,x, . . . ,xn)∈D, we define a mappingT:D→Dby
Tx=
(x,x, . . . ,xn), if n
i=xi< , (–x, –x, . . . , –xn), ifni=xi≥.
(.)
Next we prove thatT is a (k,{μn},{ξn},φ)-total asymptotically strict pseudocontraction mapping.
In fact, for anyx,y∈D, letμn= (kn– ),∀n≥,φ(t) =t,∀t≥,k= , and letting{ξn} be a nonnegative real sequence withξn→, we have the following:
Case . Ifni=xi< and n
i=yi< , then we haveTnx=x,Tny=y, and so inequality (.) holds.
Case . Ifni=xi< , and n
i=yi≥, then we haveTnx=x,Tny= (–)ny. This implies that
⎧ ⎪ ⎨ ⎪ ⎩
Tnx–Tny=x– (–)ny=x+y,
knx–y=kn(x+y),
x–Tnx– (y–Tny)= [ – (–)n]y. Therefore the inequality (.) holds.
Case . Ifni=xi≥ andni=yi< , then we haveTnx= (–)nx,Tny=y. Therefore we obtain
⎧ ⎪ ⎨ ⎪ ⎩
Tnx–Tny=(–)nx–y=x+y,
knx–y=kn(x+y),
x–Tnx– (y–Tny)= [ – (–)n]x. So the inequality (.) holds.
Case . Ifni=xi≥ and n
i=yi≥, then we haveTnx= (–)nx,Tny= (–)ny. Hence we have
⎧ ⎪ ⎨ ⎪ ⎩
Tnx–Tny=(–)nx– (–)ny=x–y=x+y,
knx–y=kn(x+y),
x–Tnx– (y–Tny)= [ – (–)n]x–y= [ – (–)n](x+y).
Thus the inequality (.) still holds. Therefore the mapping T defined by (.) is a (k,{μn},{ξn},φ)-total asymptotically strict pseudocontraction mapping.
A mappingT:C→Cis said to beL-Lipschitzian, if there exists a constantL> , such that
Tx–Ty ≤Lx–y, ∀x,y∈Candn≥. (.)
A mappingT :C→Cis said to be uniformlyL-Lipschitzian, if there exists a constant
L> , such that
Tnx–Tny≤Lx–y, ∀x,y∈Candn≥. (.)
Recently, Changet al.[] proposed the following iterative algorithm for solving a split common fixed point problem for total asymptotically strict pseudocontraction mappings in the framework of infinite-dimensional Hilbert spaces:
⎧ ⎪ ⎨ ⎪ ⎩
x∈H chosen arbitrary,
un=xn+γA∗(Tn–I)Axn,
xn+= ( –αn)un+αnSn(un), n∈N,
they proved that{xn}converges weakly to a split common fixed pointx∗of the mappingsS andT, whereS:H→HandT:H→Hare two total asymptotically strict pseudocon-traction mappings,A:H→His a bounded linear operator. In addition, they also show that{xn}converges strongly to a split common fixed pointx∗for mappingsSandT when
Sis semi-compact.
Inspired and motivated by the recent work of Changet al.[], Moudafi [, ],etc., the purpose of this paper is to propose an algorithm to solve the split common fixed point problems for total asymptotically strict pseudocontraction mappings in Hilbert spaces. Under suitable conditions on the control parameters and without the assumption of semi-compactness on the mappings, a strong convergence theorem is established. The results presented in the paper improve and extend some recent corresponding results in [, , , –].
2 Preliminaries
Throughout this paper, letHbe a Hilbert space with inner product·,·and norm · . We denote the strong convergence and weak convergence of a sequence{xn}to a point
x∈Hbyxn→x,xnx, respectively.
LetH be a Hilbert space. A mappingT:H→His said to be demi-closed at origin, if for any sequence{xn} ⊂Hwithxnx∗and(I–T)xn →, thenx∗=Tx∗.
A mappingT:C→Cis said to be semi-compact, if for any bounded sequence{xn} ⊂C withlimn→∞xn–Txn= , then there exists a subsequence{xni} ⊂ {xn}such that{xni}
converges strongly to some pointx∗∈C.
For every pointx∈H, there exists a unique nearest point ofC, denoted byPCx, such thatx–PCx ≤ x–yfor ally∈C. Such aPCis called the metric projection fromH ontoC. It is well known thatPCis a firmly nonexpansive mapping fromHtoC,i.e.,
Further, for anyx∈Handz∈C,z=PCxif and only if
x–z,z–y ≥, ∀y∈C. (.)
Lemma .([]) Let C be a nonempty closed convex subset of a real Hilbert space H and PC:H→C be the metric projection from H onto C.Then the following inequality holds:
y–PC(x)
+x–PC(x)
≤ x–y, ∀y∈C,∀x∈H. (.)
Lemma .([]) Let H be a real Hilbert space,then the following equalities hold:
(i) λx+ ( –λ)y=λx+ ( –λ)y–λ( –λ)x–y,∀x,y∈H,∀λ∈R;
(ii) x,y=x+y–x–y,∀x,y∈H.
Lemma .([]) Let H be a real Hilbert space.If{xn}is a sequence in H,weakly
conver-gent to z,then
lim sup
n→∞ xn–y
=lim sup
n→∞ xn–z
+z–y, ∀y∈H.
Lemma .([]) Let T:C→C be a(ρ,{μn},{ξn},φ)-total asymptotically strictly
pseu-docontractive mapping.If F(T)=∅,then for each q∈F(T)and for each x∈C,the following equivalent inequalities hold:
Tnx–q≤ x–q+ρx–Tnx+μnφ
x–q+ξn, (.)
x–Tnx,x–q≥ –ρ
x–T
nx–μn φ
x–q–ξn
, (.)
x–Tnx,q–Tnx≤ρ+
T
nx–x+μn φ
x–q+ξn
. (.)
Lemma . ([]) Let H be a real Hilbert space and let T : H→H be a uniformly L-Lipschitzian and(k,{μn},{ξn},φ)-total asymptotically strictly pseudocontractive
map-ping.Then the demi-closedness principle holds for T in the sense that if{xn}is a sequence
in H such that xnx∗,andlim supm→∞lim supn→∞xn–Tmxn= ,then(I–T)x∗= .
In particular,if xnx∗,and(I–T)xn →,then(I–T)x∗= ,i.e.,T is demi-closed at
the origin.
3 Main results
Theorem . Let H and Hbe two real Hilbert spaces,and A:H→Hbe a bounded
linear operator, S : H → H be a uniformly L-Lipschitzian and (ρ,{μ()n },{ξn()},φ)
-total asymptotically strict pseudocontraction mapping and T :H→Hbe a uniformly
˜
L-Lipschitzian and(k,{μ()n },{ξn()},φ)-total asymptotically strict pseudocontraction
map-ping satisfying F(S)=∅ and F(T)=∅,respectively. Let{xn}be a sequence generated by
x∈C=H, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
yn= ( –αn)zn+αnSn(zn),
zn=xn+γA∗(Tn–I)Axn, ∀n≥,
Cn+={ν∈Cn:yn–ν≤( +μnM∗)zn–ν+μnφ(M) +ξn,
zn–ν≤( +γ μnM∗A)xn–ν+γ μnφ(M) +γ ξn},
xn+=PCn+(x),
where{αn} is a sequence in(, ),γ is a positive constant, {μn}, {ξn}, andφ satisfy the
following conditions:
(i) αn∈(δ, –ρ),∀n≥andγ ∈(,A–k),whereδis a constant in(, –ρ);
(ii) μn=max{μn(),μ()n },ξn=max{ξn(),ξn()},n≥,and ∞
n=μn<∞, ∞
n=ξn<∞;
(iii) φ=max{φ,φ}and there exist two positive constantsMandM∗such that
φ(λ)≤M∗λfor allλ≥M.
If ={p∈F(S) :Ap∈F(T)} =∅, then the sequence{xn} converges strongly to a split
common fixed point x∗∈.
Proof We shall divide the proof into five steps.
Step . We first show thatCnis closed and convex for eachn≥.
SinceC=H,Cis closed and convex. Suppose thatCnis closed and convex for some
n> . Since for anyν∈Cn, we have
yn–ν≤
+μnM∗
zn–ν+μnφ(M) +ξn
⇔ +μnM∗
zn–yn–μnM∗ν,ν
≤ +μnM∗
zn–yn+μnφ(M) +ξn (.)
and
zn–ν≤
+γ μnM∗A
xn–ν+γ μnφ(M) +γ ξn
⇔ +γ μnM∗A
xn–zn–γ μnM∗Aν,ν
≤ +γ μnM∗A
xn–zn+γ μnφ(M) +γ ξn, (.)
hence the setCn+is closed and convex. ThereforeCnis closed and convex for eachn≥, andPCn+xis well defined.
Step . We show that⊂Cnfor alln≥.
In fact, since φ is a continuous and increasing function,φ(λ)≤φ(M), ifλ≤M, and
φ(λ)≤M∗λ, ifλ≥M. In either case, we can obtain
φ(λ)≤φ(M) +M∗λ, ∀λ≥. (.) For any givenp∈, thenp∈F(S) andAp∈F(T). It follows from (.) that
zn–p=xn–p+γA∗
Tn–IAxn
=xn–p+γA∗
Tn–IAxn
+ γxn–p,A∗
Tn–IAxn
, (.)
where
γxn–p,A∗
Tn–IAxn
= γAxn–Ap,
Tn–IAxn
= γAxn–Ap+
Tn–IAxn–
Tn–IAxn,Tn–IAxn
= γTnAxn–Ap,TnAxn–Axn
≤γ
+k
T
n–IAx n+
μn φ
Axn–Ap
+ξn –T
n n–I
Axn
≤γ(k– )yTnn–IAxn
+γ μnM∗Axn–p+γ μnφ(M) +γ ξn. (.)
Substituting (.) into (.), we have
zn–p=xn–p+γA∗
Tn–IAxn
≤ xn–p+γA∗
Tn–IAxn+γ(k– )Tn–I
Axn
+γ μnM∗Axn–p+γ μnφ(M) +γ ξn
≤ xn–p+γATnAxn–Axn
+γ(k– )Tn–IAxn
+γ μnM∗Axn–p+γ μnφ(M) +γ ξn = +γ μnM∗A
xn–p–γ
–k–γATnAxn–Axn
+γ μnφ(M) +γ ξn. (.)
On the other hand, since
yn–p =zn–p–αn
zn–Snzn
=zn–p– αn
zn–p,zn–Snzn
+αnzn–Snzn
≤ zn–p–αn( –ρ)zn–Snzn
+αnμnφ
zn–p
+αnξn+αnzn–Snzn
≤ zn–p–αn( –ρ–αn)zn–Snzn +αnμn
φ(M) +M∗zn–p
+αnξn
= +αnμnM∗
zn–p–αn( –ρ–αn)zn–Snzn
+αnμnφ(M) +αnξn, (.)
so, it follows from (.) and (.) thatp∈Cn, and then⊂Cnfor anyn≥. Step . We prove that{xn}is a Cauchy sequence.
From the definition ofCn+, we know thatxn=PCnx. Since⊂Cn+⊂Cn, andxn+∈
Cn+⊂Cn,∀n> , we have
xn–x ≤ xn+–x (.)
and
xn–x ≤ p–x, ∀n∈Nandp∈. (.)
It means that{xn}is nondecreasing and bounded. So,limn→∞xn–xexists. Form>n, by the definition ofCn, we havexm=PCmx∈Cm⊂Cn, it from Lemma . that
xm–xn+x–xn=xm–PCnx
+x
–PCnx
≤ x
Sincelimn→∞xn–xexists, from (.), we obtainlimn→∞xn–xm= . Therefore{xn} is a Cauchy sequence.
Step . We prove thatlimn→∞zn–Szn=limn→∞Axn–TAxn= . Sincexn+=PCn+x∈Cn+⊂Cn, we obtain
zn–xn≤ zn–xn++xn+–xn+ zn–xn+ · xn+–xn
≤ +γ μnM∗A
+ xn+–xn+γ μnφ(M) +γ ξn + +γ μnM∗A
xn+–xn+γ μnφ(M) +γ ξn
× xn+–xn, (.)
since∞n=μn<∞,∞n=ξn<∞, andlimn→∞xn–xm= , therefore lim
n→∞zn–xn= . (.)
And
yn–xn≤ yn–xn++xn+–xn+ yn–xn+xn+–xn
≤ +γ μnM∗
zn–xn++μnφ(M) +ξn+xn+–xn + yn–xn+xn+–xn
≤ +γ μnM∗
+γ μnM∗A
+ xn+–xn + +γ μnM∗
γ + μnφ(M) +
+γ μnM∗
γ+ ξn
+ +γ μnM∗
zn–xn++μnφ(M) +ξnxn+–xn,
by∞n=μn<∞, ∞
n=ξn<∞, andlimn→∞xn–xm= , we have lim
n→∞yn–xn= . (.)
Further,
zn–yn ≤ zn–xn+xn–yn →. (.)
It follows from (.) that
γ –k–γATnAxn–Axn
≤ +γ μnM∗A
xn–p–zn–p+γ μnφ(M) +γ ξn
≤ xn–zn
xn–p+zn–p
+γ μnM∗Axn–p
+γ μnφ(M) +γ ξn. (.)
Since∞n=μn<∞, ∞
n=ξn<∞,γ( –k–γA) > , and{xn}is bounded, by (.) and (.), we get
lim n→∞T
nAx
On the other hand, from (.), we have
αn( –ρ–αn)zn–Snzn
≤ +αnμnM∗
zn–p–yn–p+αnμnφ(M) +αnξn
≤ zn–yn
zn–p+yn–p
+αnμnM∗zn–p+αnμnφ(M) +αnξn. (.)
This together with the conditions (i), (ii), and{zn}being bounded, from (.) and (.), we have
lim n→∞zn–S
nz
n= . (.)
In addition, sincezn+–zn ≤ zn+–xn++xn+–xn+xn–zn, this means that
lim
n→∞zn+–zn= . (.)
SinceSis uniformlyL-Lipschitzian continuous,
zn–Szn ≤zn–Snzn+Snzn–Szn
≤zn–Snzn+LSn–zn–zn
≤zn–Snzn+LSn–zn–Sn–zn–+Sn–zn––zn
≤zn–Snzn+Lzn–zn–+LSn–zn––zn–+zn––zn
≤zn–Snzn+L( +L)zn–zn–+Lzn––Sn–zn–. (.)
It follows from (.), (.), and (.) that
lim
n→∞zn–Szn= . (.)
Similarly, in the same way as above, from (.), we can also obtain
lim
n→∞Axn–TAxn= . (.)
Step . We prove that{xn}converges strongly tox∗∈.
Since{xn}is a Cauchy sequence, we may assume thatxn→x∗. Thus we havezn→x∗ from (.), which implies thatznx∗. So it follows from (.) and Lemma . that
x∗∈F(S).
On the other hand, sinceAis a bounded linear operator, we know thatlimn→∞Axn–
Ax∗= . Therefore, it follows from Lemma . and (.) thatAx∗∈F(T). This means thatx∗∈and{xn}converges strongly tox∗∈. The proof is completed.
The following result can be obtained from Theorem . immediately.
Corollary . Let Hand Hbe two real Hilbert spaces,A:H→Hbe a bounded
pseudocontraction mapping and T:H→Hbe a uniformlyL-Lipschitzian and˜ (k,{kn()})
-asymptotically strict pseudocontraction mapping satisfying F(S)=∅and F(T)=∅, respec-tively.Let{xn}be a sequence defined as follows:x∈C=H,
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
yn= ( –αn)zn+αnSn(zn),
zn=xn+γA∗(Tn–I)Axn, ∀n≥,
Cn+={ν∈Cn:yn–ν≤( + (kn– )αn)zn–ν,
zn–ν≤( + (kn– )γA)xn–ν},
xn+=PCn+(x),
(.)
where{αn}is a sequence in (, ),γ is a positive constant and {kn} satisfy the following
conditions:
(i) kn=max{kn(),k()n },and ∞
n=(kn– ) <∞;
(ii) αn∈(δ, –ρ),∀n≥andγ ∈(,A–k),whereδis a constant in(, –ρ).
If=∅,then the sequence{xn}converges strongly to a split common fixed point x∗∈. Remark . Theorem . extends and improves the result of Changet al.[, ] from weak convergence to strong convergence by using the modified iterative scheme that we propose.
Remark . In Theorem ., asSandTare two nonexpansive mappings, demi-contrac-tive mappings or asymptotically strict pseudocontraction mappings, we can also obtain similar results.
Example . LetCandSbe the same as in Example ., andDandT be the same as in Example .. It is obvious thatF(T) ={(, , . . . , )} ∪ {(x,x, . . . ,xn) :
n
i=xi< },F(S) =
{(, , . . . , , . . .)},CandDare nonempty closed convex subsets oflandRn, respectively. LetA:C→Dbe defined byAx= (x,x, . . . ,xn) forx= (x,x, . . .)∈C. ThenAis a bounded linear operator with adjoint operatorA∗z= (x,x, . . . ,xn, , , . . .) forz= (x,x, . . . ,xn)∈D. Clearly,A=A∗= . By using algorithm (.) with <αn< andγ ∈(, ). We can verifyxn→(, , . . .)∈F(S) andA(, , . . .) = (, , . . . , )∈F(T).
4 Applications
4.1 Application to hierarchical variational inequality problem
LetHbe a real Hilbert space,TandSbe two nonexpansive mappings fromHtoHsuch thatF(T)=∅andF(S)=∅.
The so-called hierarchical variational inequality problem for nonexpansive mappingT
with respect to nonexpansive mappingS:H→His to find a pointx∗∈F(S) such that
x∗–Tx∗,x∗–x≤, ∀x∈F(S). (.) It is easy to see that (.) is equivalent to the following fixed point problem:
the problem (.) is equivalent to the following split feasibility problem:
findx∗∈Csuch thatAx∗∈Q. (.)
Hence from Theorem . we have the following theorem.
Theorem . Let H,S,T,C,and Q be the same as above.Let{xn}be a sequence generated
by x∈C=H, ⎧
⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
yn= ( –αn)zn+αnSzn,
zn=xn+γ(T–I)xn, ∀n≥,
Cn+={ν∈Cn:yn–ν ≤ zn–ν ≤ xn–ν},
xn+=PCn+(x),
(.)
whereγ ∈(, )and{αn}is a sequence in(, )satisfyinglim infn→∞αn( –αn) > .If C∩
Q=∅,then the sequence{xn}converges strongly to a solution of the hierarchical variational
inequality problem(.).
Proof SinceSis nonexpansive, it is uniformlyL-Lipschitzian continuous and (ρ,{μ()n },
{ξn()},φ)-total asymptotically strict pseudocontractive withL= ,μ()n = ,ξn()= ,φ= . Again sinceT is nonexpansive, it is uniformlyL˜-Lipschitzian continuous and (k,{μ()n },
{ξn()},φ)-total asymptotically strict pseudocontractive withL˜= ,μ()n = ,ξn()= ,φ= . Therefore, all conditions in Theorem . are satisfied. The conclusions of Theorem . can
be obtained from Theorem ..
4.2 Application to quadratic minimization problem over a fixed point set
LetK:H→Hbe a linear boundedη-strongly positive operator withη> ,i.e.,
Kx,x ≥ηx.
Letf :H→Hbe a-contraction with∈(, ) andγ∈(–(η–η–),η),S:H→Hbe a nonexpansive mapping withF(S)=∅andT:=I–η(K–γf) be a mapping fromHtoH. Lemma .([]) Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficientγ¯> and <ρ≤ A–.ThenI–ρA ≤ –ργ¯.
Now we prove thatT:H→His a nonexpansive mapping. In fact, for∀x,y∈H,γ ∈
(–(η–η–),η
), from Lemma ., we have
Tx–Ty ≤(I–ηK)x– (I–ηK)y+ηγf(x) –f(y)
≤ I–ηKx–y+ηγ x–y
≤ –η+ηγ x–y
≤ x–y. (.)
Then the hierarchical variational inequality problem (.) reduces to finding x∗∈F(S) such that
It is easy to see that (.) is exactly the optimality condition of the following quadratic minimization problem:
min x∈F(S)
Kx,x–h(x), (.)
wherehis the potential forγf,i.e.,h(x) =γf. LetC=F(S),Q=F(PF(S)(I–η(K–γf))) and
A=I, then the quadratic minimization problem (.) is equivalent to the following split feasibility problem:
findx∗∈Cand such thatAx∗∈Q.
Hence from Theorem . we have the following result.
Theorem . Let H,K,f,S,T,C,and Q be the same as above.Let{xn}be a sequence
generated by x∈C=H, ⎧
⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
yn= ( –αn)zn+αnSzn,
zn=xn+γ(T–I)xn, ∀n≥,
Cn+={ν∈Cn:yn–ν ≤ zn–ν ≤ xn–ν},
xn+=PCn+(x),
(.)
whereγ ∈(, )and{αn}is a sequence in(, )satisfyinglim infn→∞αn( –αn) > .If C∩
Q=∅,then the sequence{xn}converges strongly to a solution of problem(.).
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
Both authors contributed equally to this work. Both authors read and approved the final manuscript. Author details
1School of Information Engineering, College of Arts and Sciences, Yunnan Normal University, Longquan Road, Kunming,
650222, China.2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Longquan Road,
Kunming, 650221, China. Acknowledgements
The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
Received: 1 May 2014 Accepted: 14 January 2015 References
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