Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 914702, 12 pages
doi:10.1155/2010/914702
Research Article
Convergence Theorems of
Modified Ishikawa Iterative Scheme for
Two Nonexpansive Semigroups
Kriengsak Wattanawitoon and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th
Received 26 September 2009; Accepted 24 November 2009
Academic Editor: Tomonari Suzuki
Copyright q 2010 K. Wattanawitoon and P. Kumam. 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.
We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive
semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the
corresponding results announced by Saejung 2008 and some others.
1. Introduction
Let C be a subset of real Hilbert spaces H with the inner product ·, · and the norm · .
T : C → C is called a nonexpansive mapping if
T x − T y ≤ x − y ∀x, y ∈ C.
1.1
We denote by FT the set of fixed points of T , that is, FT {x ∈ C : x T x}.
Let {T t : t ≥ 0} be a family of mappings from a subset C of H into itself. We call it a
nonexpansive semigroup on C if the following conditions are satisfied:
i T 0x x for all x ∈ C;
ii T s t T sT t for all s, t ≥ 0;
iii for each x ∈ C the mapping t → T tx is continuous;
iv T tx − T ty ≤ x − y for all x, y ∈ C and t ≥ 0.
2
Fixed Point Theory and Applications
The Mann’s iterative algorithm was introduced by Mann 1 in 1953. This iterative process is
now known as Mann’s iterative process, which is defined as
xn1 αn xn 1 − αn T xn ,
n ≥ 0,
1.2
where the initial guess x0 is taken in C arbitrarily and the sequence {αn }∞
n0 is in the interval
0, 1.
In 1967, Halpern 2 first introduced the following iterative scheme:
x0 u ∈ C chosen arbitrarily,
xn1 αn u 1 − αn T xn ,
1.3
see also Browder 3. He pointed out that the conditions limn → ∞ αn 0 and ∞
n1 αn ∞ are
necessary in the sence that, if the iteration 1.3 converges to a fixed point of T , then these
conditions must be satisfied.
On the other hand, in 2002, Suzuki 4 was the first to introduce the following implicit
iteration process in Hilbert spaces:
xn αn u 1 − αn T tn xn ,
n ≥ 1,
1.4
for the nonexpansive semigroup. In 2005, Xu 5 established a Banach space version of the
sequence 1.4 of Suzuki 4.
In 2007, Chen and He 6 studied the viscosity approximation process for a
nonexpansive semigroup and prove another strong convergence theorem for a nonexpansive
semigroup in Banach spaces, which is defined by
xn1 αn fxn 1 − αn T tn xn ,
∀n ∈ N,
1.5
where f : C → C is a fixed contractive mapping.
Recently He and Chen 7 is proved a strong convergence theorem for nonexpansive
semigroups in Hilbert spaces by hybrid method in the mathematical programming. Very
recently, Saejung 8 proved a convergence theorem by the new iterative method introduced
by Takahashi et al. 9 without Bochner integrals for a nonexpansive semigroup {T t : t ≥ 0}
∅ in Hilbert spaces:
with F : ∞
t0 FT t /
x0 ∈ H taken arbitrary,
C1 C,
x1 PC1 x0 ,
yn αn xn 1 − αn T tn xn ,
Cn1 z ∈ Cn : yn − z ≤ xn − z ,
xn1 PCn1 x0 ,
where PC denotes the metric projection from H onto a closed convex subset C of H.
1.6
Fixed Point Theory and Applications
3
In 1974, Ishikawa 10 introduced a new iterative scheme, which is defined recursively
by
yn βn xn 1 − βn T xn ,
xn1 αn xn 1 − αn T yn ,
1.7
where the initial guess x0 is taken in C arbitrarily and the sequences {αn } and {βn } are in the
interval 0, 1.
In this paper, motivated by the iterative sequences 1.6 given by Saejung in 8 and
Ishikawa 10, we introduce the modified Ishikawa iterative scheme for two nonexpansive
semigroups in Hilbert spaces. Further, we obtain strong convergence theorems by using the
hybrid methods. This result extends and improves the result of Saejung 8 and some others.
2. Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the
next section.
It is known that every Hilbert space H satisfies the Opial’s condition 11, that is,
lim infxn − x < lim infxn − y,
n→∞
n→∞
∀y ∈ X, y /
x.
2.1
Recall that the metric nearest point projection PC from a Hilbert space H to a closed
convex subset C of H is defined as follows. Given x ∈ H, PC x is the only point in C with the
property
x − PC x inf x − y : y ∈ C .
2.2
PC x is characterized as follows.
Lemma 2.1. Let H be a real Hilbert space, C a closed convex subset of H. Given x ∈ H and y ∈ C.
Then y PC x if and only if there holds the inequality
x − y, y − z ≥ 0,
∀z ∈ C.
2.3
Lemma 2.2. There holds the identity in a Hilbert space H
λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2
for all x, y ∈ H and λ ∈ 0, 1.
2.4
4
Fixed Point Theory and Applications
Lemma 2.3 see 12, Lemma 1. Let {tn } be a real sequence and let τ be a real number such that
lim infn tn ≤ τ ≤ lim supn tn . Suppose that either of the following holds:
i lim supn tn1 − tn ≤ 0 or
ii lim infn tn1 − tn ≥ 0,
then τ is a cluster point of {tn }. Moreover, for ε > 0, k, m ∈ N, there exists m0 ≥ m such that
|tj − τ| < ε for every integer j with m0 ≤ j ≤ m0 k.
3. Main Results
3.1. The Shrinking Projection Method
In this section, we prove strong convergence of an iterative sequence generated by the
shrinking hybrid projection method in mathematical programming.
Theorem 3.1. Let C be a closed convex subset of a real Hilbert space H. Let {T t : t ≥ 0} and
{St : t ≥ 0} be nonexpansive semigroups on C with a nonempty common fixed point set F, that
∞
is, F : ∞
/ ∅. Let {αn } ⊂ 0, a ⊂ 0, 1, {βn } ⊂ b, c ⊂ 0, 1 and
t0 FT t ∩ t0 FSt {tn } be the sequences such that lim infn → ∞ tn 0, lim supn → ∞ tn > 0, and limn → ∞ tn1 − tn 0.
Suppose that {xn } is a sequence generated by the following iterative scheme:
x0 ∈ H taken arbitrary,
C1 C,
x1 PC1 x0 ,
zn βn xn 1 − βn T tn xn ,
Cn1
3.1
yn αn xn 1 − αn Stn zn ,
u ∈ Cn : yn − u ≤ xn − u ,
xn1 PCn1 x0 ,
then {xn } converges strongly to PF x0 .
Proof. We first show that Cn1 is closed and convex for each n ≥ 0. From the definition of Cn1
it is obvious that Cn1 is closed for each n ≥ 0. We show that Cn1 is convex for any n ≥ 0.
Since
yn − u ≤ xn − u ⇐⇒ 2xn − yn , u ≤ xn 2 − yn 2 ,
3.2
Fixed Point Theory and Applications
5
and hence Cn1 is convex. Next we show that F ⊂ Cn1 for all n ≥ 0. Let p ∈ F, then we have
zn − p βn xn 1 − βn T tn xn − p
≤ βn xn − p 1 − βn T tn xn − p
≤ βn xn − p 1 − βn xn − p
≤ xn − p,
3.3
yn − p αn xn 1 − αn Stn zn − p
≤ αn xn − p 1 − αn Stn zn − p
≤ αn xn − p 1 − αn zn − p.
3.4
Substituting 3.3 into 3.4, we have
yn − p ≤ xn − p.
3.5
This means that p ∈ Cn1 for all n ≥ 0. Thus, {xn } is well defined. Since xn PCn x0 and
xn1 ∈ Cn1 ⊂ Cn , we get
x0 − xn , xn − xn1 ≥ 0 ∀n ∈ N.
3.6
Consequently,
0 ≤ x0 − xn , xn − xn1 x0 − xn , xn − x0 x0 − xn1 −xn − x0 , xn − x0 x0 − xn , x0 − xn1 3.7
≤ −xn − x0 2 x0 − xn x0 − xn1 ,
for n ∈ N. This implies that
x0 − xn ≤ x0 − xn1 ∀n ∈ N.
3.8
Therefore, {x0 − xn } is nondecreasing. From xn PCn x0 , we also have x0 − xn , xn − p ≥ 0,
for all p ∈ Cn .
Since F ⊆ Cn , we get
x0 − xn , xn − p ≥ 0 ∀p ∈ F.
3.9
6
Fixed Point Theory and Applications
Thus, for p ∈ F, we obtain
0 ≤ x0 − xn , xn − p
−xn − x0 , xn − x0 x0 − xn , x0 − p
≤ −xn − x0 2 x0 − xn x0 − p.
3.10
Thus, xn − x0 ≤ x0 − p, for all p ∈ F and n ∈ N. Then limn → ∞ xn − x0 exists and {xn } is
bounded.
Next, we show that xn1 − xn → 0 as n → ∞. From 3.6 we have
xn − xn1 2 xn − x0 x0 − xn1 2
xn − x0 2 2xn − x0 , x0 − xn1 x0 − xn1 2
xn − x0 2 2xn − x0 , x0 − xn xn − xn1 x0 − xn1 2
xn − x0 2 − 2x0 − xn , x0 − xn − 2x0 − xn , xn − xn1 x0 − xn1 2
3.11
≤ xn − x0 2 − 2xn − x0 2 x0 − xn1 2
−xn − x0 2 x0 − xn1 2 .
Since limn → ∞ xn − x0 exists, then
lim xn − xn1 0.
3.12
n→∞
Further, as in the proof of 8, page 3, we have {xn } which is a Cauchy sequence. So, we have
xn z. By definition of yn , we have
yn − xn 1 − αn Stn zn − xn .
3.13
Since xn1 ∈ Cn1 and 3.12, we obtain
Stn zn − xn ≤
1 yn − xn 1 − αn
1 yn − xn1 xn1 − xn 1 − αn
1
≤
xn − xn1 xn1 − xn 1 − αn
≤
2
xn − xn1 −→ 0
1 − αn
We now show that T tn xn − xn → 0.
as n −→ ∞.
3.14
Fixed Point Theory and Applications
7
For p ∈ F, we have xn − p ≤ xn − Stn zn ≤ Stn zn − p. This implies that
0 ≤ xn − p − zn − p ≤ xn − Stn zn → 0 and hence xn − p2 − zn − p2 → 0. Moreover,
since
zn − p2 βn xn − p2 1 − βn T tn xn − p2 − βn 1 − βn xn − T tn xn 2 ,
3.15
we have
bcxn − T tn xn 2 ≤ βn 1 − βn xn − T tn xn 2
2 2 2
≤ βn xn − p 1 − βn T tn xn − p − zn − p
2 2
≤ xn − p − zn − p −→ 0.
3.16
And since Stn is a nonexpansive mapping, we obtain
xn − Stn xn ≤ xn − Stn zn Stn zn − Stn xn ,
≤ xn − Stn zn zn − xn .
3.17
Since zn − xn 1 − βn T tn xn − xn → 0 and xn − Stn zn → 0, we obtain
lim xn − Stn xn 0.
3.18
n→∞
As in the proof of 12, Theorem 4, by Lemma 2.3, we can choose a sequence {tnk } of positive
real numbers such that
tnk −→ 0,
1
xnk − T tnk xnk −→ 0,
tnk
as k −→ ∞.
3.19
1
xnk − Stnk xnk −→ 0,
tnk
as k −→ ∞.
3.20
In similar way, we also have
tnk −→ 0,
8
Fixed Point Theory and Applications
Next, we show that z ∈ F. To see this, we fix t > 0,
xnk − T tz ≤
t/tnk −1
T jtn xn − T j 1 tn xn k
k
k
k
j0
t
t
t
T
tnk xnk − T
tnk z T
tnk z − T tz
t
t
t
≤
≤
t
tnk
t
tnk
nk
nk
xnk − T tnk xnk xnk
nk
t
− z T t −
tnk z − z
tnk
3.21
xnk − T tnk xnk xnk − z sup{T sz − z : 0 ≤ s ≤ tnk }.
As xnk → z and 3.19, we obtain xnk → T tz and so T tz z. Similarly, we have Stz z.
Thus z ∈ F.
Finally, we show that z PF x0 . Since F ⊂ Cn1 and xn1 PCn1 x0 ,
xn1 − x0 ≤ q − x0 ∀n ∈ N, q ∈ F.
3.22
But xn → z as n → ∞, we have
z − x0 ≤ q − x0 ∀q ∈ F.
3.23
Hence z PF x0 as required. This completes the proof.
Corollary 3.2. Let C be a closed convex subset of a real Hilbert space H. Let {T t : t ≥
0} be nonexpansive semigroups on C with a nonempty common fixed point set F, that is, F :
∞
/ ∅. Let {αn } ⊂ 0, a ⊂ 0, 1, {βn } ⊂ b, c ⊂ 0, 1 and {tn } be the sequences such
t0 FT t that lim infn → ∞ tn 0, lim supn → ∞ tn > 0, and limn → ∞ tn1 − tn 0. Suppose that {xn } is a
sequence iteratively generated by the following iterative scheme:
x0 ∈ H taken arbitrary,
C1 C,
x1 PC1 x0 ,
Cn1
yn αn xn 1 − αn T tn zn ,
zn βn xn 1 − βn T tn xn ,
u ∈ Cn : yn − u ≤ xn − u ,
xn1 PCn1 x0 ,
then {xn } converges strongly to PF x0 .
Proof. Putting Stn T tn , in Theorem 3.1, we obtain the conclusion immediately.
3.24
Fixed Point Theory and Applications
9
Corollary 3.3 see 8, Theorem 2.1. Let C be a closed convex subset of a real Hilbert space H.
Let {T t : t ≥ 0} be a nonexpansive semigroups on C with a nonempty common fixed point set
∅. Let {αn } ⊂ 0, a ⊂ 0, 1 and {tn } be the sequences such that
F, that is, F : ∞
t0 FT t /
lim infn → ∞ tn 0, lim supn → ∞ tn > 0, and limn → ∞ tn1 − tn 0. Suppose that {xn } is a sequence
iteratively generated by the following iterative scheme:
x0 ∈ H taken arbitrary,
C1 C,
x1 PC1 x0 ,
Cn1
zn αn xn 1 − αn T tn xn ,
u ∈ Cn : yn − u ≤ xn − u ,
3.25
xn1 PCn1 x0 ,
then xn → PF x0 .
Proof. If Stn T tn for all n ∈ N and T t I for every t > 0 in Theorem 3.1 then 3.1
reduced to 3.25. By using Theorem 3.1, we get the following conclusion.
3.2. The CQ Hybrid Method
In this section, we consider the modified Ishikawa iterative scheme computing by the CQ
hybrid method 13–15. We use the same idea as Saejung’s Theorem 2.2 in 8 and our
Theorem 3.1 to obtain the following result and the proof is omitted.
Theorem 3.4. Let C be a closed convex subset of a real Hilbert space H. Let {T t : t ≥ 0} and
{St : t ≥ 0} be nonexpansive semigroups on C with a nonempty common fixed point set F, that
∞
is, F : ∞
/ ∅. Let {αn } ⊂ 0, a ⊂ 0, 1, {βn } ⊂ b, c ⊂ 0, 1 and
t0 FT t ∩ t0 FSt {tn } be the sequences such that lim infn → ∞ tn 0, lim supn → ∞ tn > 0, and limn → ∞ tn1 − tn 0.
Suppose that {xn } is a sequence generated by the following iterative scheme:
x0 ∈ H taken arbitrary,
yn αn xn 1 − αn Stn zn ,
zn βn xn 1 − βn T tn xn ,
Cn u ∈ C : yn − u ≤ xn − u ,
3.26
Qn {u ∈ C : xn − x0 , u − xn ≥ 0},
xn1 PCn ∩Qn x0 ,
then {xn } converges strongly to PF x0 .
Proof. First, we show that both Cn and Qn are closed and convex, and Cn ∩ Qn /
∅ for all
n ∈ N ∪ {0}. It follows easily from the definition that Cn and Qn are just intersection of C
and the half-spaces see also 9. As in the proof of the preceding theorem, we have F ⊂
Cn for all n ∈ N ∪ {0}. Clearly, F ⊂ C Q0 . Suppose that F ⊂ Qk for some k ∈ N ∪ {0},
10
Fixed Point Theory and Applications
we have p ∈ Ck ∩ Qk . In particular, xk1 − x0 , p − xk1 ≥ 0, that is, p ∈ Qk1 . It follows from
the induction that F ⊂ Qn for all n ∈ N ∪ {0}. This proves the claim.
Next, we show that xn − T tn xn → 0, and xn − Stn xn → 0.
We first claim that xn1 − xn → 0. Indeed, as xn1 ∈ Qn and xn PQn x0 ,
xn − x0 ≤ xn1 − x0 ∀n ∈ N.
3.27
For fixed z ∈ F. It follows from F ⊂ Qn for all n ∈ N that
xn − x0 ≤ z − x0 ∀n ∈ N.
3.28
This implies that sequence {xn } is bounded and
lim xn − x0 exists.
3.29
xn1 − xn , xn − x0 ≥ 0.
3.30
n→∞
Notice that
This implies that
xn1 − xn 2 xn1 − x0 2 − 2xn1 − xn , xn − x0 − x0 − xn 2
≤ xn1 − x0 2 − xn − x0 2 −→ 0.
3.31
By using the same argument of Saejung 8, Theorem 2.2, page 6 and in the proof of
Theorem 3.1, we have T tn xn − xn → 0 and Stn xn − xn → 0. And we can choose a
subsequence {nk } of {n} such that xnk z ∈ C, tnk → 0, 1/tnk xnk − T tnk xnk → 0 and
1/tnk xnk − Stnk xnk → 0 as k → ∞.
From 3.21, we obtain
lim supxnk − T tz ≤ lim supxnk − z,
k→∞
k→∞
lim supxnk − Stz ≤ lim supxnk − z.
k→∞
3.32
k→∞
By the Opial’s condition of H, we have z T tz and z Stz for all t > 0, that is, z ∈ F.
We note that
x0 − PF x0 ≤ x0 − z ≤ lim infx0 − xnk ≤ lim supx0 − xnk ≤ x0 − PF x0 .
k→∞
k→∞
3.33
This implies that
lim x0 − xnk x0 − PF x0 x0 − z.
k→∞
3.34
Fixed Point Theory and Applications
11
Therefore,
xnk −→ PF x0 z,
as k −→ ∞.
3.35
Hence the whole sequence must converge to PF x0 z, as required. This completes the
proof.
Corollary 3.5. Let C be a closed convex subset of a real Hilbert space H. Let {T t : t ≥ 0}
be nonexpansive semigroups on C with a nonempty common fixed point set F, that is, F :
∞
∅. Let {αn } ⊂ 0, a ⊂ 0, 1, {βn } ⊂ b, c ⊂ 0, 1 and {tn } be the sequences such that
t0 FT t /
lim infn → ∞ tn 0, lim supn → ∞ tn > 0, and limn → ∞ tn1 − tn 0. Suppose that {xn } is a sequence
iteratively generated by the following iterative scheme:
x0 ∈ H taken arbitrary,
yn αn xn 1 − αn T tn zn ,
zn βn xn 1 − βn T tn xn ,
Cn u ∈ C : yn − u ≤ xn − u ,
3.36
Qn {u ∈ C : xn − x0 , u − xn ≥ 0},
xn1 PCn ∩Qn x0 ,
then {xn } converges strongly to PF x0 .
Proof. If Stn T tn for all n ∈ N ∪ {0}, in Theorem 3.4 then 3.26 reduced to 3.36. So, we
obtain the result immediately.
We also deduce the following corollary.
Corollary 3.6 see 8, Theorem 2.2. Let C be a closed convex subset of a real Hilbert space H.
Let {T t : t ≥ 0} be a nonexpansive semigroups on C with a nonempty common fixed point set
∅. Let {αn } ⊂ 0, a ⊂ 0, 1 and {tn } be the sequences such that
F, that is, F : ∞
t0 FT t /
lim infn → ∞ tn 0, lim supn → ∞ tn > 0 and limn → ∞ tn1 − tn 0. Suppose that {xn } is a sequence
iteratively generated by the following iterative scheme:
x0 ∈ H taken arbitrary,
zn αn xn 1 − αn T tn xn ,
Cn u ∈ C : yn − u ≤ xn − u ,
Qn {u ∈ C : xn − x0 , u − xn ≥ 0},
xn1 PCn ∩Qn x0 ,
then xn → PF x0 .
3.37
12
Fixed Point Theory and Applications
Acknowledgments
The authors would like to thank the editors and the anonymous referees for their valuable
suggestions which help to improve this paper. This research was supported by the
Computational Science and Engineering Research Cluster, King Mongkut’s University of
Technology Thonburi KMUTT National Research Universities under CSEC Project no.
E01008.
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