Volume 2008, Article ID 745010,7pages doi:10.1155/2008/745010
Research Article
Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals
Satit Saejung
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Correspondence should be addressed to Satit Saejung,[email protected] Received 28 November 2007; Revised 15 January 2008; Accepted 30 January 2008 Recommended by William A. Kirk
We prove a convergence theorem by the new iterative method introduced by Takahashi et al.2007.
Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also cor- rect the strong convergence theorem recently proved by He and Chen2007.
Copyrightq2008 Satit Saejung. 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
LetHbe a real Hilbert space with the inner product·,·and the norm · . Let{Tt:t≥0}
be a family of mappings from a subsetCofH into itself. We call it a nonexpansive semigroup onCif the following conditions are satisfied:
1T0xxfor allx∈C;
2Tst TsTtfor alls, t≥0;
3for eachx∈Cthe mappingt→Ttxis continuous;
4Ttx−Tty ≤ x−yfor allx, y∈Candt≥0.
Motivated by Suzuki’s result1and Nakajo-Takahashi’s results2, He and Chen3recently proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hy- brid method in the mathematical programming. However, their proof of the main result3, Theorem 2.3is very questionable. Indeed, the existence of the subsequence {sj}such that 2.16of3are satisfied, that is,
sj−→0, xj−T sj
xj
sj −→0, 1.1
needs to be proved precisely. So, the aim of this short paper is to correct He-Chen’s result and also to give a new result by using the method recently introduced by Takahashi et al.
We need the following lemma proved by Suzuki4, Lemma 1.
Lemma 1.1. Let {tn} be a real sequence and let τ be a real number such that lim infntn ≤ τ ≤ lim supntn. Suppose that either of the following holds:
ilim supntn1−tn≤0, or iilim infntn1−tn≥0.
Thenτis a cluster point of{tn}. Moreover, forε >0,k, m∈N, there existsm0≥msuch that|tj−τ|< ε for every integerjwithm0≤j≤m0k.
2. Results
2.1. The shrinking projection method
The following method is introduced by Takahashi et al. in5. We use this method to approx- imate a common fixed point of a nonexpansive semigroup without Bochner integrals as was the case in5, Theorem 4.4.
Theorem 2.1. LetC be a closed convex subset of a real Hilbert spaceH. Let {Tt : t ≥ 0}be a nonexpansive semigroup onCwith a nonempty common fixed pointF, that is,F ∩t≥0FTt/∅.
Suppose that{xn}is a sequence iteratively generated by the following scheme:
x0∈H taken arbitrary, C1C, x1PC1
x0
, ynαnxn
1−αnT tn
xn, Cn1
z∈Cn:yn−z≤xn−z, xn1PCn1
x0
.
2.1
where{αn} ⊂0, a ⊂0,1, lim infntn 0, lim supntn >0, and limntn1−tn 0. Thenxn → PFx0.
Proof. It is well known thatFis closed and convex. We first show that the iterative scheme is well defined. To see that eachCnis nonempty, it suffices to show thatF ⊂Cn. The proof is by induction. Clearly,F⊂C1. Suppose thatF⊂Ck. Then, forz∈F⊂Ck,
yk−z≤αkxk−z
1−αkT tk
xk−z
≤αkxk−z
1−αkxk−z xk−z.
2.2
That is,z∈Ck1as required.
Notice that
Cn:
z∈H :yn−z≤xn−z 2.3
is convex since
yn−z≤xn−z⇐⇒2
xn−yn, z ≤xn2−yn2. 2.4 This implies that each subsetCnC∩C1∩ · · · ∩Cn−1is convex. It is also clear thatCnis closed.
Hence the first claim is proved.
Next, we prove that{xn}is bounded. AsxnPCnx0,
xn−x0≤z−x0 ∀z∈Cn. 2.5 In particular, forz∈F ⊂Cnfor alln∈N, the sequence{xn−x0}is bounded and hence so is {xn}.
Next, we show that{xn}is a Cauchy sequence. Asxn1∈Cn1⊂CnandxnPCnx0, xn−x0≤xn1−x0 ∀n. 2.6
Moreover, since the sequence{xn}is bounded,
n→∞limxn−x0exists. 2.7
Note that
x0−xn, xn−v ≥0 ∀v∈Cn. 2.8
In particular, sincexnk∈Cnk⊂Cnfor allk∈N,
xnk−xn2xnk−x02−xn−x02−2
xnk−xn, xn−x0
≤xnk−x02−xn−x02. 2.9 It then follows from the existence of limnxn−x02that{xn}is a Cauchy sequence. In fact, for ε >0, there exists a natural numberNsuch that, for alln≥N,
xn−x02−a< ε
2, 2.10
wherealimnxn−x02. In particular, ifn≥Nandk∈N, then xnk−xn2≤xnk−x02−xn−x02
≤aε 2 −
a−ε 2
ε. 2.11 Moreover,
xn1−xn−→0. 2.12
We now assume thatxn → pfor somep ∈C. Now sinceαn ≤ a < 1 for alln∈ Nand xn1∈Cn,
xn−T tn
xn 1
1−αnyn−xn
≤ 1
1−ayn−xn1xn1−xn
≤ 2
1−axn1−xn−→0.
2.13
The last convergence follows from2.12. We choose a sequence{tnk}of positive real number such that
tnk−→0, 1
tnkxnk−T tnk
xnk−→0. 2.14
We now show that how such a special subsequence can be constructed. First we fixδ >0 such that
lim inf
n tn0< δ <lim sup
n tn. 2.15
From2.13, there existsm1 ∈Nsuch thatTtnxn−xn<1/32for alln≥m1. ByLemma 1.1, δ/2 is a cluster point of{tn}. In particular, there existsn1 > m1such thatδ/3< tn1 < δ. Next, we choosem2 > n1 such thatTtnxn−xn<1/42for alln≥ m2. Again, byLemma 1.1,δ/3 is a cluster point of{tn}and this implies that there existsn2 > m2such thatδ/4 < tn2 < δ/2.
Continuing in this way, we obtain a subsequence{nk}of{n}satisfying T
tnk
xnk−xnk< 1
k22, δ
k2 < tnk <δ
k ∀k∈N. 2.16
Consequently,2.14is satisfied.
We next show thatp∈F. To see this, we fixt >0, xnk−Ttp
≤
t/tnk−1 j0
T jtnk
xnk−T
j1tnk xnk
T t
tnk
tnk
xnk−T
t tnk
tnk
p
T
t tnk
tnk
p−Ttp
≤ t
tnk
xnk−Ttnkxnkxnk−p T
t−
t tnk
tnk
p−p
≤ t tnk
xnk−T tnk
xnkxnk−psupTsp−p: 0≤s≤tnk
.
2.17
Asxnk →pand2.14, we havexnk →Ttpand soTtpp.
Finally, we show thatpPFx0. SinceF⊂Cn1andxn1PCn1x0,
xn1−x0≤q−x0 ∀n∈N, q∈F. 2.18
Butxn→p; we have
p−x0≤q−x0 ∀q∈F. 2.19 HencepPFx0as required. This completes the proof.
2.2. The hybrid method
We consider the iterative scheme computing by the hybrid methodsome authors call the CQ- method. The following result is proved by He and Chen 3. However, the important part of the proof seems to be overlooked. Here we present the correction under some additional restriction on the parameter{tn}.
Theorem 2.2. LetC be a closed convex subset of a real Hilbert spaceH. Let {Tt : t ≥ 0}be a nonexpansive semigroup onCwith a nonempty common fixed pointF, that is,F ∩t≥0FTt/∅.
Suppose that{xn}is a sequence iteratively generated by the following scheme:
x0∈Ctaken arbitrary, ynαnxn
1−αn T
tn xn, Cn
z∈C:yn−z≤xn−z, Qn
z∈C:
xn−x0, z−xn ≥0 , xn1PCn∩Qn
x0
,
2.20
where{αn} ⊂0, a ⊂0,1, lim infntn 0, lim supntn >0, and limntn1−tn 0. Thenxn → PFx0.
Proof. For the sake of clarity, we give the whole sketch proof even though some parts of the proof are the same as3. To see that the scheme is well defined, it suffices to show that both Cn andQn are closed and convex, and Cn∩Qn/∅ for alln ∈ N. It follows easily from the definition thatCnandQnare just the intersection ofCand the half-spaces, respectively,
Cn:
z∈H : 2
xn−yn, z ≤xn2−yn2 , Qn:
z∈H:
xn−x0, z−xn ≥0
. 2.21
As in the proof of the preceding theorem, we haveF ⊂Cnfor alln∈N. Clearly,F ⊂CQ1. Suppose thatF⊂Qkfor somek∈N, we havep∈Ck∩Qk. In particular,xk1−x0, p−xk1 ≥0, that is,p∈Qk1. It follows from the induction thatF⊂Qnfor alln∈N. This proves the claim.
We next show thatxn−Ttnxn→0. To see this, we first prove that
xn1−xn−→0. 2.22
Asxn1∈QnandxnPQnx0,
xn−x0≤xn1−x0 ∀n∈N. 2.23
For fixedz∈F. It follows fromF⊂Qnfor alln∈Nthat
xn−x0 ≤ z−x0 ∀n∈N. 2.24 This implies that sequence{xn}is bounded and
n→∞limxn−x0exists. 2.25
Notice that
xn1−xn, xn−x0 ≥0. 2.26 This implies that
xn1−xn2xn1−x02−xn−x02−2
xn1−xn, xn−x0
≤xn1−x02−xn−x02−→0. 2.27 It then follows fromxn1∈Cnthatyn−xn1 ≤ xn−xn1and hence
T tn
xn−xn 1
αnyn−xn
≤ 1 αn
yn−xn1xn1−xn−→0.
2.28
As inTheorem 2.1, we can choose a subsequence{nk}of{n}such that xnk
−−−→w p∈C, tnk−→0, 1 tnk
xnk−T tnk
xnk−→0. 2.29 Consequently, for anyt >0,
xnk−Ttp≤ t tnk
xnk−T tnk
xnkxnk−psupTsp−p: 0≤s≤tnk .
2.30 This implies that
lim sup
k→∞
xnk−Ttp≤lim sup
k→∞
xnk−p. 2.31 In virtue of Opial’s condition ofH, we havep Ttpfor all t > 0, that is,p ∈ F. Next, we observe that
x0−PF
x0≤x0−p≤lim inf
k→∞ x0−xnk≤lim sup
k→∞
x0−xnk≤x0−PF x0.
2.32 This implies that
k→∞limx0−xnkx0−PF
x0x0−p. 2.33
Consequently,
xnk −→PF x0
p. 2.34 Hence the whole sequence must converge toPFx0 p, as required.
Acknowledgments
The author would like to thank the referees for his comments and suggestions on the manuscript. This work is supported by the Commission on Higher Education and the Thai- land Research FundGrant MRG4980022.
References
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3H. He and R. Chen, “Strong convergence theorems of the CQ method for nonexpansive semigroups,”
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