R E S E A R C H
Open Access
A general iterative algorithm for an infinite
family of nonexpansive operators in Hilbert
spaces
Cuijie Zhang
*and Songnian He
*Correspondence:
[email protected] College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China
Abstract
In this paper, we introduce a new general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to a common point of the sets of fixed points, which solves a variational inequality. Our results improve and extend the corresponding results announced by many others. As applications, at the end of the paper, we apply our results to the split common fixed point problem.
Keywords: an infinite family of nonexpansive operators; strong convergence;
k-Lipschitizian;
η-strongly monotone; split common fixed point problem
1 Introduction
LetHbe a real Hilbert space with the inner product·,·and the norm · . LetT be a nonexpansive operator. The set of fixed points ofTis denoted byFix(T). In , Moudafi [] introduced the viscosity approximation method for a nonexpansive operator and con-sidered the sequence{xn}by
xn+=αnfxn+ ( –αn)Txn, (.)
wheref is a contraction onHand{αn}is a sequence in (, ). In , Xu [] proved that
under some conditions on{αn}, the sequence{xn}generated by (.) strongly converges to
x∗inFix(T) which is the unique solution of the variational inequality
(I–f)x∗,x–x∗≥, ∀x∈Fix(T).
It is well known that iterative methods for nonexpansive operators have been used to solve convex minimization problems; see,e.g., [, ]. A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive operatorT on a real Hilbert spaceH:
min
x∈Fix(T)
Ax,x–x,b, (.)
wherebis a given point inHandAis a strongly positive bounded linear operator. In [], Xu proved that the sequence{xn}defined by the following iterative method:
xn+= (I–αnA)Txn+αnb, (.)
converges strongly to the unique solution of the minimization problem (.). In [], Marino and Xu combined the iterative method (.) and the viscosity method (.) and considered the following general iterative method:
xn+=αnγfxn+ (I–αnA)Txn. (.)
They proved that the sequence{xn}generated by (.) converges strongly to the unique
solution of the variational inequality
(A–γf)x∗,x–x∗≥, ∀x∈Fix(T),
which is the optimality condition for the minimization problem
min
x∈Fix(T)
Ax,x–h(x),
wherehis a potential function forγf (i.e.,h(x) =γf(x) forx∈H).
On the other hand, Yamada [] in introduced the following hybrid iterative method:
xn+=Txn–μλnFTxn, n≥, (.)
where F is a k-Lipschitzian and η-strongly monotone operator withk> , η> and <μ< η/k. Under some appropriate conditions, he proved that the sequence{x
n}
gen-erated by (.) converges strongly to the unique solution of the variational inequality
Fx˜,x–x˜ ≥, ∀x∈Fix(T).
Recently, combining (.) and (.), Tian [] considered the following general iterative method:
xn+=αnγfxn+ (I–μαnF)Txn. (.)
Improving and extending the corresponding results given by Marino, Xu and Yamada, he proved that the sequence{xn}generated by (.) converges strongly to the unique solution
x∗∈Fix(T) of the variational inequality
(γf–μF)x˜,x–x˜≤, ∀x∈Fix(T).
operators has attracted much attention because of its extraordinary utility and broad ap-plicability in many branches of mathematical science and engineering. For example, if the nonexpansive operators are projection onto some closed convex setsCi(i∈N) in a real
Hilbert spaceH, then such a fixed point problem becomes the convex feasibility problem of finding a point ini∈NCi. Many previous results [–] and many results not cited
here considered the common fixed point about an infinite family of nonexpansive opera-tors byWn-mappings.
Motivated and inspired by the above results, we consider the following iterative algo-rithm withoutWn-mappings:
⎧ ⎨ ⎩
yn=βnxn+
n
i=(βi––βi)Tixn,
xn+=αnγVxn+ (I–μαnF)yn,
(.)
where{αn}is a sequence in (, ] and{βn}is a strictly decreasing sequence in (, ]. Under
some appropriate conditions, we proved the sequence{xn}generated by (.) converges
strongly to the unique solution of the variational inequality:
(μF–γV)x˜,z–x˜≥, ∀z∈
∞
i= Fix(Ti).
Our results improve and extend the corresponding results announced by many others. As applications, at the end of the paper, we apply our results to the split common fixed point problem.
2 Preliminaries
Throughout this paper, we writexnxandxn→xto indicate that{xn}converges weakly
toxand converges strongly tox, respectively.
An operatorT:H→His said to be nonexpansive ifTx–Ty ≤ x–yfor allx,y∈H. It is well known thatFix(T) is closed and convex. It is known thatAis called strongly positive if there exists a constantγ > such thatAx,x ≥γxfor allx∈H. The operatorFis calledη-strongly monotone if there exists a constantη> such that
x–y,Fx–Fy ≥ηx–y
for allx,y∈H.
In order to prove our main results, we collect the following lemmas in this section.
Lemma .(Demiclosedness principle []) Let H be a Hilbert space,C be a closed convex subset of H,and T:C→C be a nonexpansive operator withFix(T)=∅.If{xn}is a sequence
in C weakly converging to x∈C and{(I–T)xn}converges strongly to y∈C,then(I–T)x=y.
In particular,if y= ,then x∈Fix(T).
Lemma .[] Assume that{an}is a sequence of nonnegative real numbers such that
an+≤( –γn)an+δn, n≥,
(i) ∞n=γn=∞,
(ii) lim supn→∞δn γn ≤or
∞
n=|δn|<∞.
Thenlimn→∞an= .
Lemma .[] Let H be a real Hilbert space,let V:H→H be an L-Lipschitzian oper-ator with L> ,and let F:H→H be a k-Lipschitzian continuous operator andη-strongly monotone operator with k> ,η> .Then,for <γ <μηL,μF–γV is strongly monotone with coefficientμη–γL.
Lemma .[] Let C be a closed convex subset of a real Hilbert space H,given x∈H and y∈C.Then y=PCx if and only if the following inequality holds:
x–y,z–y ≤
for every z∈C.
3 Main results
Lemma . Let{Tn}:H→H be an infinite family of nonexpansive operators,let F:H→
H be a k-Lipschitzian andη-strongly monotone operator with k> andη> ,and let V:H→H be an L-Lipschitzian operator.Let <μ<kη and <γ<
μ(η–μk)
L =
τ
L.Assume
that Sn=βnI+
n
i=(βi––βi)Ti,where{βn}is a strictly decreasing sequence withβ=
andβn∈(, ].Consider the following mapping Gnon H defined by
Gnx=αnγVx+ (I–μαnF)Snx,
where{αn}is a sequence in(, ].Then Gnis a contraction.
Proof Observe that
Gnx–Gny ≤αnγVx–Vy+ ( –αnτ)Snx–Sny
≤αnγLx–y+ ( –αnτ)
βn(x–y) + n
i=
(βi––βi)(Tix–Tiy)
≤αnγLx–y+ ( –αnτ)
βnx–y+ n
i=
(βi––βi)x–y
=αnγLx–y+ ( –αnτ)x–y
= –αn(τ–γL)
x–y.
Since < –αn(τ–γL) < ,Gnis a contraction. This completes the proof.
SinceGnis a contraction, using the Banach contraction principle,Gnhas a unique fixed
pointxV
n ∈Hsuch that
xVn =αnγVxVn+ (I–μαnF)SnxVn.
For simplicity, we denotexnforxVn without confusion.
Theorem . Let{Tn}be an infinite family of nonexpansive self-mappings of a real Hilbert
space H,let F be a k-Lipschitzian andη-strongly monotone operator on H with k> and
η> ,and let V be an L-Lipschitzian operator.Suppose that=i∞=Fix(Ti)is nonempty.
Suppose that{xn}is generated by the following algorithm:
⎧ ⎨ ⎩
yn=βnxn+
n
i=(βi––βi)Tixn,
xn=αnγVxn+ (I–μαnF)yn,
(.)
where <μ<kη and <γ <τL withτ=μ(η–μk).If the following conditions are
satis-fied:
(i) {αn}is a sequence in(, ]andlimn→∞αn= ;
(ii) {βn}is a strictly decreasing sequence in(, ]andβ= .
Then{xn}converges strongly tox˜∈,which solves the variational inequality:
(μF–γV)x˜,z–x˜≥, ∀z∈. (.)
Equivalently,we have P(I–μF+γV)x˜=x˜.
Proof We proceed with the following steps: Step : First we show that{xn}is bounded.
In fact, letp∈, then for everyi∈N,Tip=p. Observe that
yn–p ≤βnxn–p+ n
i=
(βi––βi)Tixn–Tip ≤ xn–p.
Thus it follows that
xn–p=αnγVxn+ (I–μαnF)yn–p
=αn(γVxn–μFp) + (I–μαnF)yn– (I–μαnF)p
≤( –αnτ)yn–p+αn
γVxn–γVp+γVp–μFp
≤( –αnτ)yn–p+αnγLxn–p+αnγVp–μFp
≤ –αn(τ–γL)
xn–p+αnγVp–μFp.
Then we have
xn–p ≤
τ–γLγVp–μFp,
which implies that{xn}is bounded. Hence we can obtain{yn},{Tixn},{Fyn}and{Vxn}are
bounded. Note that
xn–yn=αnγVxn+ (I–μαnF)yn–yn=αnγVxn–μFyn,
we immediately obtain that
lim
Step : We showlimn→∞xn–Tixn= .
Sincep∈, we note that
xn–p≥ Tixn–p=Tixn–xn+xn–p
=Tixn–xn+xn–p+ Tixn–xn,xn–p,
which implies that
Tixn–xn ≤ x
n–Tixn,xn–p.
Thus
n
i=
(βi––βi)Tixn–xn≤ n
i=
(βi––βi)xn–Tixn,xn–p
=
( –βn)xn– n
i=
(βi––βi)Tixn,xn–p
=( –βn)xn–yn+βnxn,xn–p
=xn–yn,xn–p
≤ xn–ynxn–p.
Then we immediately obtainlimn→∞ni=(βi––βi)Tixn–xn= . Since{βn}is strictly
decreasing, it follows that
lim
n→∞Tixn–xn= (.)
for everyi∈N. SinceSn=βnI+
n
i=(βi––βi)Ti, thus
xn–Snxn≤ n
i=
(βi––βi)Tixn–xn.
It shows that
lim
n→∞xn–Snxn= . (.)
Step : We show that there exists a subsequence{xnk}of{xn}such thatxnk→ ˜x.
Since{xn}is bounded, there exist a pointx˜∈H and a subsequence{xnk}of{xn}such
thatxnkx˜. By Lemma . and (.), we obtainx˜∈Fix(Ti) for anyi∈N. This shows that
˜
x∈. On the other hand, we note that
xn–x˜=αnγVxn+ (I–μαnF)yn–x˜
=I–αn(μF–γV)
yn–
I–αn(μF–γV)
˜
x
Hence we obtain
xn–x˜ =
I–αn(μF–γV)
yn–
I–αn(μF–γV)
˜
x,xn–x˜
–αn
(μF–γV)x˜,xn–x˜
+αnγVxn–γVyn,xn–x˜
≤I–αn(μF–γV)
yn–
I–αn(μF–γV)
˜
xxn–x˜
–αn
(μF–γV)x˜,xn–x˜
+αnγLxn–ynxn–x˜
≤ –αn(τ–γL)
xn–x˜–αn
(μF–γV)x˜,xn–x˜
+αnγLxn–ynxn–x˜.
Then it follows that
xn–x˜≤ γL
τ–γLxn–ynxn–x˜–
τ–γL
(μF–γV)x˜,xn–x˜
.
In particular,
xnk–x˜
≤ γL
τ–γLxnk–ynkxnk–x˜–
τ–γL
(μF–γV)x˜,xnk–x˜
.
Fromxnkx˜and (.), it follows thatxnk→ ˜x.
Step : We show thatx˜solves the variational inequality (.). Observe that
xn=αnγVxn+ (I–μαnF)Snxn.
Hence, we conclude that
(μF–γV)xn= (μF–γV)(xn–Snxn) +μFSnxn–γVSnxn
= (μF–γV)(xn–Snxn) + (γVxn–γVSnxn) –γVxn+μFSnxn
= (μF–γV)(xn–Snxn) + (γVxn–γVSnxn) –
αn
(I–Sn)xn.
SinceSnis nonexpansive, we have thatI–Snis monotone. Note that for anyz∈,Snz=z.
Then we deduce
(μF–γV)xn,xn–z
= –
αn
(I–Sn)xn– (I–Sn)z,xn–z
+(μF–γV)(I–Sn)xn,xn–z
+γVxn–γVSnxn,xn–z
≤(μF–γV)(I–Sn)xn,xn–z
+γLxn–Snxnxn–z.
Now, replacingnwithnkin the above inequality, and lettingk→ ∞, by (.) we have
(μF–γV)x˜,x˜–z=lim
k→
(μF–γV)xnk,xnk–z
≤lim
k→
+γLxnk–Snkxnkxnk–z
= .
That is,(μF–γV)x˜,z–x˜ ≥ for everyz∈. It follows thatx˜is a solution of the vari-ational inequality (.). SinceμF–γV is (μη–γL)-strongly monotone and (μk–γL )-Lipschitzian, the variational inequality (.) has a unique solution. So, we conclude that
xn→ ˜xasn→ ∞. The variational inequality (.) can be written as
(I–μF+γV)x˜–x˜,z–x˜≤, ∀z∈.
By Lemma ., we haveP(I–μF+γV)x˜=x˜.
Theorem . Let{Tn}be an infinite family of nonexpansive self-mappings of a real Hilbert
space H,let F be a k-Lipschitzian andη-strongly monotone operator on H with k> and
η> ,and let V be an L-Lipschitzian operator.Suppose that=i∞=Fix(Ti)is nonempty.
Suppose that x∈H, <μ<kη and <γ < τLwithτ=μ(η–μk).Letβ= ,{αn}be
a sequence in(, ],and let{βn}be a strictly decreasing sequence in(, ].If the following
conditions are satisfied: (i) limn→∞αn= ;
(ii) ∞n=αn=∞;
(iii) ∞n=|αn+–αn|<∞;
(iv) ∞n=|βn+–βn|<∞.
Then{xn}generated by(.)converges strongly tox˜∈,which solves the variational
in-equality
(μF–γV)x˜,z–x˜≥, ∀z∈. (.)
Equivalently,we have P(I–μF+γV)x˜=x˜.
Proof We proceed with the following steps:
Step : First show that there existsx˜∈such thatx˜=P(I–μF+γV)x˜.
In fact, by Lemma .,μF–γVis strongly monotone. So, the variational inequality (.) has only one solution. We setx˜∈to indicate the unique solution of (.). The variational inequality (.) can be written as
(I–μF+γV)x˜–x˜,z–x˜≤, ∀z∈.
So, by Lemma ., it is equivalent to the fixed point equation
P(I–μF+γV)x˜=x˜.
Step : Now we show that{xn}is bounded.
Letp∈, then for everyi∈N,Tip=p. Observe that
yn–p ≤βnxn–p+ n
i=
(βi––βi)Tixn–p
≤βnxn–p+ n
i=
Thus it follows that
xn+–p=αnγVxn+ (I–μαnF)yn–p
=αn(γVxn–μFp) + (I–μαnF)yn– (I–μαnF)p
≤( –αnτ)yn–p+αn
γVxn–γVp+γVp–μFp
≤( –αnτ)yn–p+αnγLxn–p+αnγVp–μFp
≤ –αn(τ–γL)
xn–p+αn(τ–γL)
γVp–μFp
τ–γL
≤max
xn–p,
γVp–μFp
τ–γL
≤ · · · ≤max
x–p,
γVp–μFp
τ–γL
.
Therefore,{xn} is bounded. Hence we can obtain that{yn},{Tixn}, {Fyn}and{Vxn}are
bounded.
Step : we showlimn→∞xn+–xn= .
We observe that
xn+–xn=αnγVxn+ (I–μαnF)yn–αn–γVxn–– (I–μαn–F)yn– =αnγ(Vxn–Vxn–) + (αn–αn–)γVxn–
+(I–μαnF)yn– (I–μαnF)yn–
+ (αn––αn)μFyn–. It follows that
xn+–xn ≤αnγVxn–Vxn–+|αn–αn–|γVxn–
+ ( –αnτ)yn–yn–+|αn––αn|μFyn–. (.)
We have
yn–yn–=
βnxn+
n
i=
(βi––βi)Tixn–βn–xn––
n–
i=
(βi––βi)Tixn–
≤βnxn–xn–+|βn–βn–|xn–
+
n
i=
(βi––βi)Tixn–Tixn–+|βn–βn–|Tnxn–
≤ xn–xn–+|βn–βn–|
xn–+Tnxn–
. (.)
Combining (.) and (.), we obtain that
xn+–xn ≤αnγLxn–xn–+|αn–αn–|
γVxn–+μFyn–
+ ( –αnτ)xn–xn–+|βn–βn–|( –αnτ)
xn–+Tnxn–
= –αn(τ–γL)
xn–xn–+|αn–αn–|
γVxn–+μFyn–
+|βn–βn–|( –αnτ)
xn–+Tnxn–
.
Step : We showlimn→∞Tixn–xn= for alli∈N.
Sincep∈, we note that
xn–p≥ Tixn–Tip=Tixn–xn+xn–p
=Tixn–xn+xn–p+ Tixn–xn,xn–p,
which implies that
Tixn–xn ≤ x
n–Tixn,xn–p. (.)
From (.) and (.), we deduce
n
i=
(βi––βi)Tixn–xn≤ n
i=
(βi––βi)xn–Tixn,xn–p
=
( –βn)xn– n
i=
(βi––βi)Tixn,xn–p
=( –βn)xn–yn+βnxn,xn–p
=xn–yn,xn–p
=xn–xn+,xn–p+xn+–yn,xn–p.
Using (.), we can have
n
i=
(βi––βi)Tixn–xn≤ xn–xn+,xn–p+αnγVxn–μFyn,xn–p
≤ xn–xn+xn–p+αnγVxn–μFynxn–p.
Noting thatlimn→∞xn–xn+= andlimn→∞αn= , we immediately obtain
lim
n→∞
n
i=
(βi––βi)Tixn–xn= .
Since{βn}is strictly decreasing, it follows that for everyi∈N,
lim
n→∞Tixn–xn= . (.)
Step : Showlim supn→∞(γV–μF)x˜,xn–x˜ ≤, wherex˜=P(I–μF+γV)x˜.
Since{xn}is bounded, there exist a pointv∈Hand a subsequence{xnk}of{xn}such that
lim sup
n→∞
(γV–μF)x˜,xn–x˜
= lim
k→∞
(γV–μF)x˜,xnk–x˜
andxnkv. Now, applying (.) and Lemma ., we conclude thatv∈Fix(Ti) for every
i∈N. Hence,v∈. Sinceis closed and convex, by Lemma ., we get lim sup
n→∞
(γV–μF)x˜,xn–x˜
= lim
k→∞
(γV–μF)x˜,xnk–x˜
Step : Showxn→ ˜x=P(I–μF+γV)(x˜).
Sincex˜∈, we haveTix˜=x˜for everyi∈N. Using (.), we have
xn+–x˜ =αnγVxn+ (I–μαnF)yn–x˜
=(I–μαnF)yn– (I–μαnF)x˜+αn(γVxn–μFx˜)
≤(I–μαnF)yn– (I–μαnF)x˜+ αnγVxn–μFx˜,xn+–x˜ ≤( –αnτ)yn–x˜+ αnγVxn–Vx˜,xn+–x˜
+ αnγVx˜–μFx˜,xn+–x˜ ≤( –αnτ)xn–x˜+αnLγ
xn–x˜+xn+–x˜
+ αnγVx˜–μFx˜,xn+–x˜,
which implies that
xn+–x˜≤
( –αnτ)+αnγL
–αnγL
xn–x˜+
αn
–αnγL
γVx˜–μFx˜,xn+–x˜
≤
–αn(τ–γL) –αnγL
xn–x˜
+αn(τ–γL) –αnγL
τ–γLγVx˜–μFx˜,xn+–x˜
.
Consequently, according to (.) and Lemma ., we deduce that{xn}converges strongly
tox˜=P(I–μF+γV)x˜. This completes the proof.
Corollary . Let T be a nonexpansive self-mapping of a real Hilbert space H,let F be a k-Lipschitzian andη-strongly monotone operator on H with k> andη> ,and let V be an L-Lipschitzian operator.Suppose that=Fix(T)is nonempty.Suppose that x∈H and
that{xn}is generated by the following algorithm:
xn+=αnγV(xn) + (I–μαnF)Txn,
where <μ<kη and <γ <τLwithτ =μ(η–μk).Let{αn}be in(, ].If the following
conditions are satisfied: (i) limn→∞αn= ;
(ii) ∞n=αn=∞;
(iii) ∞n=|αn+–αn|<∞.
Then{xn}converges strongly tox˜∈,which solves the variational inequality
(μF–γV)x˜,z–x˜≥, ∀z∈.
Equivalently,we have P(I–μF+γV)x˜=x˜.
Proof Set{Tn}to be the sequences of operators defined byTn=Tfor alln∈Nin
4 Application in the split common fixed point problem
LetHandHbe Hilbert spaces, letA:H→Hbe a bounded linear operator. The split common fixed point problem (SCFPP) is to find a pointx∗∈Hsatisfied with
x∗∈
p
i=
Fix(Ui), Ax∗∈ r
j= Fix(Tj),
whereUi:H→H(i= , , . . . ,p) andTj:H→H(j= , , . . . ,r) are nonlinear operators. The concept of the SCFPP in finite-dimensional Hilbert spaces was firstly introduced by Censor and Segal in []. Now we consider a generalized split common fixed point prob-lem (GSCFPP) which is to find a point
x∗∈
∞
i=
Fix(Ui), Ax∗∈
∞
j=
Fix(Tj). (.)
We know that if for alliandj,UiandTjare nonexpansive operators, the GSCFPP is
equiv-alent to the following common fixed point problem:
x∗∈
∞
i=
Fix(Ui), x∗∈
∞
j= Fix(Vj),
whereVj=I–γA∗(I–Tj)Awith <γ ≤A for everyj∈N(see []). The solution set
of GSCFPP (.) is denoted byS.
Theorem . Let{Un}and{Tn}be sequences of nonexpansive operators on real Hilbert
spaces Hand H,respectively.Let F be a k-Lipschitzian andη-strongly monotone operator
on Hwith k> andη> .Let V be an L-Lipschitzian operator.Suppose that S is nonempty.
Suppose that x∈H and that{xn}is generated by the following algorithm:
⎧ ⎨ ⎩
yn=βnxn+
n
i=(βi––βi)Ui(I–γA∗(I–Ti)A)xn,
xn+=αnγV(xn) + (I–μαnF)yn,
where <μ<kη and <γ <τLwithτ=μ(η–μk).Let{αn}be in(, ],{βn}be a strictly
decreasing sequence in(, ]andβ= .If the following conditions are satisfied: (i) limn→∞αn= ;
(ii) ∞n=αn=∞;
(iii) ∞n=|αn+–αn|<∞;
(iv) ∞n=|βn+–βn|<∞.
Then{xn}converges strongly tox˜∈S,which solves the variational inequality
(μF–γV)x˜,z–x˜≥, ∀z∈S.
Equivalently,we have P(I–μF+γV)x˜=x˜.
Proof Set{Tn}to be the sequences of operators defined byTn:=Un(I–γA∗(I–Tn)A) for
alln∈Nin Theorem .. By Theorem ., we can obtain lim
n→∞Ui
I–γA∗(I–Ti)A
in Step . But it does not imply that the set of cluster points of the weak topologyωw(xn)
is a subset of S. In order to prove this, we only show limn→∞TiAxn–xn= and
limn→∞Uixn–xn= .
Sincep∈,TiAp=Ap. Hence, for everyi∈N,
Axn–Ap≥ TiAxn–TiAp=TiAxn–Ap
=TiAxn–Axn+Axn–Ap
=TiAxn–Axn+Axn–Ap+ TiAxn–Axn,Axn–Ap,
which yields that
TiAxn–Axn,Axn–Ap ≤–
TiAxn–Axn
(.)
for everyi∈N. Using (.), we note that
yn–p =
βn(xn–p) + n
i=
(βi––βi)
Ui
I–γA∗(I–Ti)A
xn–p
≤βnxn–p+ n
i=
(βi––βi)xn+γA∗(Ti–I)Axn–p
≤βnxn–p+ n
i=
(βi––βi)
xn–p+γATiAxn–Axn
+ γAxn–Ap,TiAxn–Axn
≤ xn–p+ n
i=
(βi––βi)
γATiAxn–Axn
–γTiAxn–Axn
=xn–p+γ
γA–
n
i=
(βi––βi)TiAxn–Axn.
Thus
γ –γA
n
i=
(βi––βi)TiAxn–Axn
=xn–p–yn–p
=xn–p–yn–p
xn–p+yn–p
≤ xn–yn
xn–p+yn–p
.
It follows thatlimn→∞TiAxn–Axn= . Now we show thatlimn→∞Uixn–xn= . Note
that
Uixn–xn ≤Uixn–Ui
I–γA∗(I–Ti)A
xn+Ui
I–γA∗(I–Ti)A
xn–xn
≤xn–
I–γA∗(I–Ti)A
xn+Ui
I–γA∗(I–Ti)A
xn–xn
≤γA(Ti–I)Axn+Ui
I–γA∗(I–Ti)A
Then we havelimn→∞Uixn–xn= for every i∈N. Then we can have ωw(xn)⊂S.
Hence, by Theorem ., we obtain the desired result.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Acknowledgements
This research is supported by the Fundamental Science Research Funds for the Central Universities (Program No. ZXH2012K001).
Received: 5 December 2012 Accepted: 9 May 2013 Published: 29 May 2013 References
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