Volume 2007, Article ID 59735,8pages doi:10.1155/2007/59735
Research Article
Strong Convergence Theorems of the CQ Method for
Nonexpansive Semigroups
Huimin He and Rudong Chen
Received 25 January 2007; Accepted 19 March 2007
Recommended by Jerzy Jezierski
Motivated by T. Suzuki, we show strong convergence theorems of the CQ method for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. The results presented extend and improve the corresponding results of Kazuhide Nakajo and Wataru Takahashi (2003).
Copyright © 2007 H. He and R. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
Throughout this paper, letHbe a real Hilbert space with inner product·,·and · . We usexnxto indicate that the sequence{xn}converges weakly tox. Similarly,xn→xwill symbolize strong convergence. we denote byNandR+the sets of nonnegative integers and nonnegative real numbers, respectively. letCbe a closed convex subset of a Hilbert spaceH, and LetT:C→Cbe a nonexpansive mapping (i.e.,Tx−T y ≤ x−yfor allx,y∈C). We use Fix(T) to denote the set of fixed points ofT; that is, Fix(T)= {x∈
C:x=Tx}. We know that Fix(T) is nonempty ifCis bounded, for more details see [1]. In [2], Shioji and Takahashi introduce in a Hilbert space the implicit iteration
xn=αnu+
1−αn 1
tn tn
0 T(s)xnds, n∈N, (1.1)
Where{αn}is a sequence in (0,1),{tn}is a sequence of positive real numbers divergent to ∞, for eacht≥0 andu∈C. In 2003, Suzuki [3] is the first to introduce again in a Hilbert space the following implicit iteration process:
xn=αnu+
1−αn
Ttn
xn
for the nonexpansive semigroup case. In 2005, Xu [4] established a Banach space version of the sequence (1.2) of Suzuki [3], he proved that ifEis a uniformly convex Banach space with a weakly continuous duality map (e.g.,lpfor 1< p <∞), ifCis a closed convex sub-set ofE, and if{T(t) :t∈R+}is a nonexpansive semigroup on a closed convex subset Csuch that Fix(T)= ∅, then under certain appropriate assumptions made and the se-quencesαnandtnof the parameters, he showed that the sequencexnimplicitly defined by (1.2) for alln≥1 converges strongly to a member ofF=t≥0Fix(T(t)).
Recently, Chen and He [5] extend and improve the corresponding results of Suzuki [3], ifEis a reflexive Banach space which admits a weakly sequentially continuous duality mappingJfromEtoE∗, supposeCis a nonempty closed convex subset ofE. Let{T(t) : t∈R+}be a nonexpansive semigroup onCsuch thatF(T)= ∅, and f :C→Cis a fixed contraction onC. Let{αn}and{tn}be sequences of real numbers satisfying 0< αn<1, tn>0 and limn→∞tn=limn→∞αn/tn=0. Define a sequence{xn}inCby
xn=αnf
xn
+1−αn
Ttn
xn
, n≥1. (1.3)
Then{xn}converges strongly toq, asn→ ∞.qis the element ofF, such thatq is the unique solution inFto the following variational inequality:
(f −I)q,j(x−q)≤0 ∀x∈F(T). (1.4)
Some other results can be seen in [6–8].
Nakajo and Takahashi [9] introduced an iteration procedure for nonexpansive self-mappingsTonCas follows:
x0=x∈C,
yn=αnxn+1−αnTxn,
Cn=z∈C; yn−z ≤ xn−z , Qn=
z∈C;xn−z,x0−xn
≥0, xn+1=PCn∩Qn
x0
(1.5)
for eachn∈N∪ {0}, whereαn∈[0,a] for somea∈[0, 1), and{xn}converges strongly toPFix(T)x0
Let{T(t) :t∈R+}be a nonexpansive semigroup on a closed convex subsetC of a Hilbert spaceH, that is,
(1) for eacht∈R+,T(t) is a nonexpansive mapping onC; (2)T(0)x=xfor allx∈C;
(3)T(s+t)=T(s)◦T(t) for alls,t∈R+;
(4) for each x∈X, the mappingT(·)xfromR+intoCis continuous. We putF=
t≥0Fix(T(t)). We know thatFis nonempty ifCis bounded, see [10].
Nakajo and Takahashi [9] also introduced an iteration procedure for nonexpansive semi-group{T(t) :t∈R+}onCas follows:
x0=x∈C,
yn=αnxn+
1−αn 1
tn tn
0 T(s)xnds, Cn=
z∈C; yn−z ≤ xn−z ,
Qn=
z∈C;xn−z,x0−xn
≥0,
xn+1=PCn∩Qn
x0
(1.6)
for eachn∈N∪ {0}, whereαn∈[0,a] for some a∈[0, 1) and {tn} is a positive real number divergent sequence, and the sequence{xn}converges strongly toPFx0.
In 2006, Martinez-Yanes and Xu [11] employ Nakajo-Takahashi [9] idea and prove some strong convergence theorems for nonexpansive mappings and maximal monotone operators.
In this paper, we consider an iteration procedure for nonexpansive semigroups{T(t) : t∈R+}onCas follows:
x0=x∈C,
yn=αnxn+1−αnTtnxn,
Cn=z∈C; yn−z ≤ xn−z ,
Qn=
z∈C;xn−z,x0−xn
≥0,
xn+1=PCn∩Qn
x0
(1.7)
for eachn∈N∪ {0}, whereαn∈[0,a] for somea∈[0, 1) andtn≥0 limn→∞tn=0. then the sequence{xn}converges strongly toPFx0.
In the sequel, we will need the following definitions and results.
Definition 1.1. A Banach spaceEis said to satisfy Opial’s condition [12] if whenever{xn}
is a sequence inEwhich converges weakly tox, asn→ ∞, then
lim sup n→∞
xn−x <lim sup n→∞
xn−y , ∀y∈Ewithx=y. (1.8)
It is well known that Hilbert space andlp(1< l <∞) space satisfy Opial’s condition [13].
Lemma1.2 [14]. LetC be a nonempty closed convex subset of a Hilbert spaceH. Given
x∈Handy∈C, theny=PCxif and only ifx−y,y−z ≥0,is satisfied for allz∈C.
Lemma1.3 [14,15]. Every Hilbert spaceHhas Radon-Riesz property or Kadets-Klee
2. Main results
Lemma2.1. LetCbe a closed convex subset of a Hilbert spaceH. Let{T(t) :t∈R+}be a
nonexpansive semigroup onCsuch thatF= ∅, and the sequence{xn}generated by (1.7),
whereαn∈[0,a]for somea∈[0, 1), Then{xn}is well defined andF⊂Cn∩Qnfor every
n∈N∪ {0}.
Proof. It is obvious thatCnis closed andQnis closed and convex for everyn∈N∪ {0}.
It follows from thatCnis convex for everyn∈N∪ {0}becauseyn−z ≤ xn−zis equivalent to
y n−xn
2
+ 2yn−xn,xn−z
≤0. (2.1)
So,Cn∩Qnis closed and convex for everyn∈N∪ {0}. Letu∈F. Then from y
n−u = αnxn+
1−αn
Ttn
xn−u ≤αn xn−u +
1−αn T
tn
xn−u ≤ xn−u .
(2.2)
we haveu∈Cnfor eachn∈N∪ {0}. So, we haveF⊂Cnfor alln∈N∪ {0}.
Next, we show by mathematical induction that{xn}is well defined andF⊂Cn∩Qn for everyn∈N∪ {0}. Forn=0, we havex0=x∈CandQ0=C, and henceF⊂C0∩Q0. Suppose thatxk is given and F⊂Ck∩Qk for some k∈N∪ {0}.There exists a unique elementxk+1∈Ck∩Qk such thatxk+1=PCk∩Qk(x0). Fromxk+1=PCk∩Qk(x0), it holds that
xk+1−z,x0−xk+1
≥0 (2.3)
for eachz∈Ck∩Qk. SinceF⊂Ck∩Qk, we getF⊂Qk+1, therefore we haveF⊂Ck+1∩ Qk+1.
The proof is completed.
Lemma2.2. LetCbe a closed convex subset of a Hilbert spaceH. Let{T(t) :t∈R+}be a
nonexpansive semigroup onCsuch thatF= ∅, and the sequence{xn}generated by (1.7),
whereαn∈[0,a]for somea∈[0, 1), Thenlimn→∞xn+1−xn =0.
Proof. At first, we show thatFis a closed convex subset ofC. SinceT(t) :C→C,t >0 is
nonexpansive, we claim thatFis closed. In fact, if pn⊂F=
t≥0Fix(T(t)),n≥1, such that limn→∞pn=p, then we have
T(t)p=lim
n→∞T(t)pn=nlim→∞pn=p ∀t∈R+. (2.4)
Thusp∈F.
Next, we show thatFis convex, we will use the following identity in Hilbert space:
tx+ (1−t)y 2
which holds for allx,y∈Hand for allt∈[0, 1] indeed,
tx+ (1−t)y 2
=t2x2+ (1−t)2y2+ 2t(1−t)x,y
=tx2+ (1−t)y2+ 2t(1−t)x,y
−t(1−t)x2−t(1−t)y2
=tx2+ (1−t)y2−t(1−t)x2+y2−2x,y
=tx2+ (1−t)y2−t(1−t)x−y2.
(2.6)
Letp1,p2∈Fand for allt∈[0, 1],p=tp1+ (1−t)p2, then
p−p1=(1−t)p2−p1, p−p2=(1−t)p1−p2. (2.7)
From (2.5) and (2.7), we have
p−T(t)p 2
= tp1−T(t)p
+ (1−t)p2−T(t)p 2
=t p1−T(t)p 2
+ (1−t) p2−T(t)p 2
−t(1−t) p1−p2 2
≤t p1−p 2+ (1−t) p2−p 2−t(1−t) p1−p2 2
=t(1−t)2 p1−p2 2+t2(1−t) p1−p2 2−t(1−t) p1−p2 2
=t(1−t)(1−t+t−1) p1−p2 2
=0.
(2.8)
Thusp=T(t)p, for allt >0, that is,p∈F.
Secondly, we show that{xn}is bounded. SinceFis a nonempty closed convex subset ofC, there exists a unique elementz0∈Fsuch thatz0=PF(x0).Fromxn+1=PCn∩Qn(x0), we have
x
n+1−x0 ≤ z−x0 ∀z∈Cn∩Qn. (2.9)
It follows fromLemma 2.1thatF⊂Cn∩Qnfor everyn∈N∪ {0}, together withz0∈ F(T), we have
xn+1−x0 ≤ z0−x0 ∀n∈N∪ {0}. (2.10)
This implies that{xn}is bounded, so{T(t)xn}is also bounded, and moreover so is {yn}sinceyn ≤αnxn+ (1−αn)T(t)xn.
Thirdly, we show thatxn+1−xn →0 asn→ ∞. SinceQn= {z∈C;xn−z,x0−xn ≥ 0},xn=PQn(x0). Asxn+1∈Cn∩Qn⊂Qn, we obtain
x
Therefore the sequence{xn−x0}is bounded and nondecreasing. So
lim
n→∞ xn−x0 exists. (2.12)
On the other hand, fromxn+1∈Qn, we getxn−xn+1,x0−xn ≥0, and hence
xn−xn+1 2
= xn−x0−xn+1−x0 2
= xn−x0 2
−2xn−x0,xn+1−x0
+ xn+1−x0 2
= xn−x0 2
+ xn+1−x0 2
−2xn−x0,xn+1−xn+xn−x0
= xn+1−x0 2
− xn−x0 2
−2xn−xn+1,x0−xn
≤ xn+1−x0 2
− xn−x0 2
−→0 (n−→ ∞).
(2.13)
So
lim
n→∞ xn+1−xn =0. (2.14)
This proof is completed.
Theorem2.3. LetCbe a closed convex subset of a Hilbert space H. Let{T(t) :t∈R+}
be a nonexpansive semigroup onCsuch thatF= ∅, and the sequence{xn}generated by
(1.7), whereαn∈[0,a]for some a∈[0, 1), andtn≥0 limn→∞tn=0. then the sequence
{xn}converges strongly toPFx0.
Proof. It follows fromxn+1∈Cnthat
T
tn
xn−xn =1−1 αn
yn−xn
≤ 1
1−αn
yn−xn+1 + x
n+1−xn
≤ 2
1−αn
xn+1−xn
(2.15)
for everyn∈N∪ {0}. ByLemma 2.2, we getT(tn)xn−xn →0.
We claim that{xn} is relatively sequentially compact. Indeed, there exists a weakly convergence subsequence{xnj} ⊆ {xn} by reflexivity ofH and boundedness of the se-quence{xn}, now we supposexnjx∈C(j→ ∞). Now we show thatx∈F. Putxj=xnj, βj=αnj, andsj=tnj forj∈N, letsj≥0 be such that
sj−→0, T
sjxj−xj
Fixt >0, from
x
j−T(t)x ≤
[t/sj]−1
k=0
T
(k+ 1)sj
xj−T
ksj
xj
+ T
t sj sj
xj−T t sj sj
x + Tt/sj
sj
x−T(t)x
≤
t sj T
sjxj−xj + xj−x + T t− t sj sj
x−x
≤t T
sj
xj−xj
sj +
x
j−x + max T(s)x−x : 0≤s≤sj
.
(2.17)
For allj∈N∪ {0}, as every Hilbert space satisfies Opial’s condition, then we have
lim sup j→∞
xj−T(t)x ≤lim sup j→∞
xj−x .
(2.18)
This implies thatT(t)x=x. Therefore,
x∈F. (2.19)
Ifz0=PF(x0), it follows from (2.10), (2.19), and the lower semicontinuity of the norm that
x0−z0 ≤ x0−x ≤lim inf
j→∞ x0−xnj ≤lim sup j→∞
x0−xn
j ≤ x0−z0 . (2.20)
Thus, we obtain
lim
j→∞ xnj−x0 = x0−x = x0−z0 . (2.21)
This implies that
xnj−→x=z0. (2.22)
This shows that{xn}is relatively sequentially compact. Therefore, we havexn→z0.
The proof is completed.
Acknowledgment
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Huimin He: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address:[email protected]
