Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 739561, 17 pages
doi:10.1155/2010/739561
Research Article
Strong Convergence Theorems for a Countable
Family of Lipschitzian Mappings
Weerayuth Nilsrakoo1 and Satit Saejung2
1
Department of Mathematics, Statistics, and Computer, Ubon Ratchathani University,
Ubon Ratchathani 34190, Thailand
2
Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
Correspondence should be addressed to Weerayuth Nilsrakoo, nilsrakoo@hotmail.com
Received 22 June 2010; Revised 14 September 2010; Accepted 12 October 2010
Academic Editor: W. A. Kirk
Copyright q 2010 W. Nilsrakoo and S. 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.
We modify the iterative method introduced by Kim and Xu 2006 for a countable family of
Lipschitzian mappings by the hybrid method of Takahashi et al. 2008. Our results include recent
ones concerning asymptotically nonexpansive mappings due to Plubtieng and Ungchittrakool
2007 and Zegeye and Shahzad 2008, 2010 as special cases.
1. Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping T : C → C
is said to be Lipschitzian if there exists a positive constant L such that
Tx − Ty ≤ Lx − y
∀x, y ∈ C.
1.1
In this case, T is also said to be L-Lipschitzian. Clearly, if T is L1 -Lipschitzian and L1 < L2 ,
then T is L2 -Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping
is L-Lipschitzian with L ≥ 1. If L 1, then T is known as a nonexpansive mapping. We
denote by FT the set of fixed points of T. If C is nonempty bounded closed convex and
T is a nonexpansive of C into itself, then FT /
see 1. There are many methods for
approximating fixed points of a nonexpansive mapping. In 1953, Mann 2 introduced the
iteration as follows: a sequence {xn } defined by
xn1 αn xn 1 − αn Txn ,
1.2
2
Abstract and Applied Analysis
where the initial guess element x1 ∈ C is arbitrary and {αn } is a real sequence in 0, 1.
Mann iteration has been extensively investigated for nonexpansive mappings. One of the
fundamental convergence results is proved by Reich 3. In an infinite-dimensional Hilbert
space, Mann iteration can conclude only weak convergence 4. Attempts to modify the Mann
iteration method 1.2 so that strong convergence is guaranteed have recently been made.
Nakajo and Takahashi 5 proposed the following modification of Mann iteration method
1.2:
x1 x ∈ C is arbitrary,
yn αn xn 1 − αn Txn ,
Cn z ∈ C : yn − z ≤ xn − z ,
1.3
Qn {z ∈ C : xn − z, x − xn ≥ 0},
xn1 PCn ∩Qn x,
n 1, 2, 3, . . . ,
where PK denotes the metric projection from H onto a closed convex subset K of H. They
prove that if the sequence {αn } bounded above from one, then {xn } defined by 1.3 converges
strongly to PFT x. Takahashi et al. 6 modified 1.3 so-called the shrinking projection method
for a countable family of nonexpansive mappings {Tn }∞
n1 as follows:
x1 x ∈ H,
C1 C,
Cn1
yn αn xn 1 − αn Tn xn ,
z ∈ Cn : yn − z ≤ xn − z ,
xn1 PCn1 x,
1.4
n 1, 2, 3, . . . ,
and prove that if the sequence {αn } bounded above from one, then {xn } defined by 1.4
converges strongly to P∩∞n1 FTn x.
Recently, the present authors 7 extended 1.3 to obtain a strong convergence
theorem for common fixed points of a countable family of Ln -Lipschitzian mappings {Tn }∞
n1
by
x1 x ∈ C is arbitrary,
yn αn xn 1 − αn Tn xn ,
Cn z ∈ C : yn − z2 ≤ xn − z2 θn ,
Qn {z ∈ C : xn − z, x − xn ≥ 0},
xn1 PCn ∩Qn x,
n 1, 2, 3, . . . ,
1.5
Abstract and Applied Analysis
3
where θn 1 − αn L2n − 1diam C2 → 0 as n → ∞ and prove that {xn } defined by 1.5
converges strongly to P∩∞n1 FTn x.
In this paper, we establish strong convergence theorems for finding common fixed
points of a countable family of Lipschitzian mappings in a real Hilbert space. Moreover, we
also apply our results for asymptotically nonexpansive mappings.
2. Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · . Then,
x − y2 x2 − y2 − 2 x − y, y ,
2.1
λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2 ,
2.2
for all x, y ∈ H and λ ∈ 0, 1. For any n points x1 , x2 , . . . , xn in H, the following generalized
identity holds:
2
n
n
λi xi 2 − λi λj xi − xj 2 ,
λi xi i1
i1
i<j
2.3
where λi ∈ 0, 1 and ni1 λi 1.
We write xn → x xn x, resp. if {xn } converges strongly weakly, resp. to x. It is
also known that H satisfies:
1 the Opial’s condition 8 that is, for any sequence {xn } with xn x,
lim infxn − x < lim infxn − y
n→∞
n→∞
2.4
holds for every y ∈ H with y /x
2 the Kadec-Klee property 9, 10; that is, for any sequence {xn } with xn x and
xn → x together imply xn → x.
Let C be a nonempty closed convex subset of H. Then, for any x ∈ H, there exists a unique
nearest point in C, denoted by PC x, such that
x − PC x ≤ x − y
∀y ∈ C.
2.5
Such a mapping PC is called the metric projection of H onto C. We know that PC is
nonexpansive. Furthermore, for x ∈ H and z ∈ C,
z PC x
iff x − z, z − y ≥ 0, ∀y ∈ C.
2.6
To deal with a family of mappings, the following conditions are introduced: let C be a subset
,
of a Banach space, let {Tn } and T be families of mappings of C with ∞
n1 FTn FT /
where FT is the set of all common fixed points of all mappings in T. {Tn } is said to satisfy
4
Abstract and Applied Analysis
a the AKTT-condition 11 if for each bounded subset B of C,
∞
sup{Tn1 z − Tn z : z ∈ B} < ∞,
2.7
n1
b the NST-condition (I) with T 12 if for each bounded sequence {zn } in C,
lim zn − Tn zn 0 implies lim zn − Tzn 0
n→∞
n→∞
∀T ∈ T,
2.8
c the NST-condition (II) 12 if for each bounded sequence {zn } in C,
lim zn1 − Tn zn 0 implies lim zn − Tm zn 0
n→∞
n→∞
∀m ∈ ,
2.9
d NST∗ -condition with T 13 if for each bounded sequence {zn } in C,
lim zn − Tn zn 0,
n→∞
lim zn − zn1 0,
n→∞
2.10
imply limn → ∞ zn − Tzn 0 for all T ∈ T.
In particular, if T {T}, then we simply say that {Tn } satisfies the NST-condition I with T
NST∗ -condition with T, resp. rather than NST-condition I with {T} NST∗ -condition with
{T}, resp..
Remark 2.1. It follows directly from the definitions above that
i if {Tn } satisfies the NST-condition I with T, then {Tn } satisfies the NST∗ -condition
with T
ii if {Tn } satisfies the NST-condition II, then {Tn } satisfies the NST∗ -condition with
{Tn }.
Lemma 2.2 see 11, Lemma 3.2. Let C be a nonempty closed subset of a Banach space, and let
{Tn } be a family of mappings of C into itself which satisfies the AKTT-condition, then the mapping
T : C → C defined by
Tx lim Tn x
n→∞
∀x ∈ C
2.11
satisfies
lim sup{Tz − Tn z : z ∈ B} 0,
n→∞
2.12
for each bounded subset B of C.
From now on, we will write {Tn }, T satisfies AKTT-condition if {Tn } satisfies AKTTcondition and T is defined by 2.11.
Abstract and Applied Analysis
5
Lemma 2.3 see 13, Lemma 2.6. Let C be a nonempty closed subset of a Banach space. Suppose
. Then, {Tn } satisfies the NSTthat {Tn }, T satisfies AKTT-condition and FT ∞
n1 FTn /
condition (I) with T. Consequently, {Tn } satisfies the NST∗ -condition with T.
Remark 2.4. There are families of mappings {Tn } and T such that
1 {Tn } satisfies the NST∗ -condition with T, and
2 {Tn } fails the NST-condition I with T and the NST-condition II.
The following example shows that the NST∗ -condition with T is strictly weaker than
NST-condition I with T and the NST-condition II.
Example 2.5 see 13, Example 2.9. Let H : 2 and C : 0, 1 × 0, 1. Define T1 , T2 : C → C
as follows:
T1 x, y x, 1 − y ,
T2 x, y 1 − x, y ,
2.13
for all x, y ∈ C. Hence, T1 and T2 are nonexpansive mappings with
FT1 ∩ FT2 0, 1 ×
1
1 1
1
∩
× 0, 1 ,
.
/
2
2
2 2
2.14
Let Tn Tn−1 mod 21 . Then, {Tn } satisfies NST∗ -condition but it fails NST-condition I with
T and the NST-condition II.
Lemma 2.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn } and {Sn }
be two families of tn -Lipschitzian and sn -Lipschitzian mappings of C into itself, respectively. Let {Un }
be a family of mappings of C into itself defined by
∀n ∈ ,
Un Tn βn I 1 − βn Sn
2.15
where {βn } is a sequence in a, b for some a, b ∈ 0, 1 and I is an identity mapping. Assume that
{tn } and {sn } are two sequences such that tn → 1 and sn → 1. Then, the following statements hold.
i {Un } is a family of Ln -Lipschitzian mappings of C into itself, where Ln βn t2n 1 −
βn t2n s2n 1/2 and Ln → 1.
ii Suppose that T1 and T2 are families of mappings of C into itself such that FT1 ∞
∞
. If {Tn } and {Sn } satisfy the
n1 FTn , FT2 n1 FSn and FT1 ∩ FT2 /
NST∗ -condition with T1 and T2 , respectively, then {Un } satisfies the NST∗ -condition with
T1 ∪ T2 and
∞
n1
FUn FT1 ∩ FT2 .
2.16
6
Abstract and Applied Analysis
Proof. i We first observe that
Un x − Un y2 ≤ t2n βn x − y 1 − βn Sn x − Sn y 2
≤ t2n βn x − y2 1 − βn Sn x − Sn y2
≤
βn t2n x
− y 1 − βn t2n s2n x − y2
2.17
2
L2n x − y2 ,
for all x, y ∈ C. That is, Un is Ln -Lipschitzian. Since tn → 1 and sn → 1, it follows that
Ln → 1.
ii Let {zn } be a bounded sequence in C such that limn → ∞ zn −Un zn limn → ∞ zn1 −
zn 0. Let p ∈ FT1 ∩ FT2 , and let M sup{zn − Un zn , zn − p : n ∈ }. Then
2
zn − p2 ≤ zn − Un zn Un zn − p
zn − Un zn 2 2zn − Un zn Un zn − p Un zn − p2
≤ 3Mzn − Un zn Tn βn zn 1 − βn Sn zn − Tn p2
≤ 3Mzn − Un zn t2n βn zn − p 1 − βn Sn zn − p 2
3Mzn − Un zn βn t2n zn − p2 1 − βn t2n Sn zn − p2
− βn 1 − βn t2n zn − Sn zn 2
≤ 3Mzn − Un zn βn t2n zn − p2 1 − βn t2n s2n zn − p2
2.18
− a1 − bzn − Sn zn 2
3Mzn − Un zn L2n zn − p2 − a1 − bzn − Sn zn 2 ,
for all n ∈ . In particular,
a1 − bzn − Sn zn 2 ≤ 3Mzn − Un zn L2n − 1 zn − p2 .
2.19
lim zn − Sn zn 0.
2.20
So, we get
n→∞
Since {Sn } satisfies the NST∗ -condition with T2 , we have
lim zn − Szn 0
n→∞
∀S ∈ T2 .
2.21
Abstract and Applied Analysis
7
Since
zn − Tn zn ≤ zn − Un zn Un zn − Tn zn ≤ zn − Un zn 1 − βn tn zn − Sn zn ,
2.22
lim zn − Tn zn 0.
2.23
it follows that
n→∞
Since {Tn } satisfies the NST∗ -condition with T1 , we have
lim zn − Tzn 0
n→∞
∀T ∈ T1 .
2.24
It is easy to see that FT1 ∩ FT2 ⊂ ∞
n1 FUn . To see the reverse inclusion, let z ∈
∞
n1 FUn follow the first part of the proof above but now let zn ≡ z. Then, z ∈ FT1 ∩
FT2 FT1 ∪ T2 . Hence, {Un } satisfies the NST∗ -condition with T1 ∪ T2 .
Lemma 2.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tni }∞
n1 be
families of Lni -Lipschitzian mappings of C into itself for i 1, 2, . . . , r, respectively. Let {Tn } be a
family of mappings of C into itself defined by
r
βni Tni
Tn ∀n ∈ ,
2.25
i1
r
where {βni }∞
n1 are sequences in a, b for some a, b ∈ 0, 1 satisfying
i1 βni 1 for all n ∈ .
Assume that {Lni }∞
n1 are sequences such that Lni → 1 as n → ∞ for all i 1, 2, . . . , r. Then, the
following statements hold.
i {Tn } is a family of Ln -Lipschitzian mappings of C into itself, where Ln ri1 βni L2ni 1/2
and Ln → 1.
ii Suppose that Ti are families of mappings of C into itself such that FTi ∞
n1 FTni ∗
.
If
{T
}
satisfies
the
NST
-condition
with
Ti for all
for i 1, 2, . . . , r and ri1 FTi /
ni
i 1, 2, . . . , r, then {Tn } satisfies the NST∗ -condition with ri1 Ti and
∞
FTn n1
r
i1
FTi .
2.26
8
Abstract and Applied Analysis
Proof. i From 2.3, we have
2
r
Tn x − Tn y2 βni Tni x − Tni y
i1
≤
r
βni Tni x − Tni y2
i1
≤
2.27
r
βni L2ni x − y2
i1
L2n x
− y2 ,
for all x, y ∈ C. That is, Tn is Ln -Lipschitzian. Since Lni → 1 for i 1, 2, . . . , r and ri1 βni 1,
it follows that Ln → 1.
ii Let {zn } be a bounded sequence in C such that limn → ∞ zn −Tn zn limn → ∞ zn1 −
zn 0. Let p ∈ ri1 FTi , and let M sup{zn − Tn zn , zn − p : n ∈ }; it follows from
2.3 that
2
zn − p2 ≤ zn − Tn zn Tn zn − p
zn − Tn zn 2 2zn − Tn zn Tn zn − p Tn zn − p2
2
r
≤ 3Mzn − Tn zn βni Tni zn − p
i1
3Mzn − Tn zn r
βni Tni zn − p2 − βni βnj Tni zn − Tnj zn 2
i1
≤ 3Mzn − Tn zn 2.28
i<j
r
βni L2ni zn − p2 − a2 Tni zn − Tnj zn 2
i1
i<j
3Mzn − Tn zn L2n zn − p2 − a2 Tni zn − Tnj zn 2 .
i<j
So, by i, we get
lim Tni zn − Tnj zn 0 ∀i, j ∈ {1, 2, . . . , r}.
n→∞
2.29
Abstract and Applied Analysis
9
For each k 1, 2, . . . , r, we have
zn − Tnk zn ≤ zn − Tn zn Tn zn − Tnk zn r
zn − Tn zn βni Tni zn − Tnk zn i1
≤ zn − Tn zn 2.30
r
βni Tni zn − Tnk zn −→ 0.
i1
∗
Since each family {Tnk }∞
n1 satisfies the NST -condition with Tk ,
lim zn − Tzn 0 ∀T ∈
n→∞
r
Ti .
2.31
i1
let z ∈ ∞
It is easy to see that ri1 FTi ⊂ ∞
n1 FTn . To see the reverse inclusion,
n1 FTn .
r
Follow the first part of the proof above but now let zn ≡ z. Then, z ∈ i1 FTi F∪ri1 Ti .
Hence, {Tn } satisfies the NST∗ -condition with ri1 Ti .
Lemma 2.8. Let C be a nonempty closed convex subset of a real Hilbert space H, and let {Tn } be
. If {Tn }
a family of Ln -Lipschitzian mappings of C into itself with Ln → 1 and ∞
n1 FTn /
satisfies the NST∗ -condition with T, where T is a family of mappings of C into itself such that FT ∞
n1 FTn , then FT is closed and convex.
Proof. It follows from the continuity of Tn that FTn is closed and so is FT ∞
n1 FTn .
Now, we prove that FT is convex. To this end, let x, y ∈ FT. Put z tx 1 − ty, where
t ∈ 0, 1. From 2.2, we have
z − Tn z2 tx − Tn z2 1 − ty − Tn z2 − t1 − tx − y2
≤ tL2n x − z2 1 − tL2n y − z2 − t1 − tx − y2
t1 − t L2n − 1 x − y2 .
2.32
So, we get
lim z − Tn z 0.
n→∞
2.33
Since {Tn } satisfies the NST∗ -condition with T, we have z − Tz 0 for all T ∈ T. Then,
z ∈ FT and so FT is convex.
Remark 2.9. The conclusions of Lemmas 2.6, 2.7, and 2.8 remain true if we replace a Hilbert
space with a uniformly convex Banach space. Recall a Banach space X is uniformly convex if for
any ε > 0, there exists δ > 0 such that x y 1 and x − y ≥ ε imply x y/2 ≤ 1 − δ.
10
Abstract and Applied Analysis
3. Main Results
In this section, using the method introduced by Takahashi et al. 6, we obtain a strong
convergence theorem for a countable family of Lipschitzian mappings.
Recall that a mapping T : C → C is closed demiclosed, resp. at y if whenever {xn } is a
sequence in C satisfying xn → x xn x, resp. and Txn → y, then x ∈ C and Tx y.
Theorem 3.1. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let {Tn }
be a family of Ln -Lipschitzian mappings of C into itself with a common fixed point. Assume that {αn }
is a sequence in 0, b for some b ∈ 0, 1. For x1 x ∈ H and C1 C, one defines a sequence {xn }
of C as follows:
yn αn xn 1 − αn Tn xn ,
Cn1 z ∈ Cn : yn − z2 ≤ xn − z2 θn ,
xn1 PCn1 x,
3.1
n 1, 2, 3, . . . ,
where
θn 1 − αn L2n − 1 diam C2 −→ 0
as n −→ ∞.
3.2
Suppose that T is a family of mappings of C into itself such that FT ∞
n1 FTn and I − T is
closed at 0 for all T ∈ T. If {Tn } satisfies the NST∗ -condition with T, then {xn } converges strongly to
PFT x.
Proof. By Lemma 2.8, we have FT is closed and convex. We now prove that Cn is closed and
convex for each n ∈ by induction. It is obvious that C1 C is closed and convex. Assume
that Ck is closed and convex for some k ∈ . For z ∈ Ck , we know that
yk − z2 ≤ xk − z2 θk
3.3
2xk − yk , z ≤ xk 2 − yk 2 θk .
3.4
is equivalent to
It follows that Ck1 is closed and convex. Next, we show that
FT ⊂ Cn
∀n ∈ .
3.5
Abstract and Applied Analysis
11
It is clear that FT ⊂ C1 C. Suppose that FT ⊂ Ck for some k ∈ . Then, for p ∈ FT,
yk − p2 αk xk 1 − αn Tk xk − p2
≤ αk xk − p2 1 − αk Tk xk − p2
≤ αk xk − p2 1 − αk L2k xk − p2
xk − p2 1 − αk L2k − 1 xk − p2
3.6
≤ xk − p2 θk ,
we have p ∈ Ck1 . Therefore, we obtain 3.5. Now, the sequence {xn } is well defined. As
xn PCn x,
xn − x ≤ z − x
∀z ∈ Cn , ∀ n ∈ .
3.7
∀p ∈ FT, ∀n ∈ .
3.8
In particular, since FT ⊂ Cn ,
xn − x ≤ p − x
On the other hand, from xn PCn x and xn1 ∈ Cn1 ⊂ Cn , we have
xn − x ≤ xn1 − x
∀n ∈ .
3.9
Therefore, {xn − x} is nondecreasing and bounded. So,
lim xn − x exists.
n→∞
3.10
Noticing again that xn PCn x and for any positive integer k, xnk ∈ Cnk−1 ⊂ Cn , we have
xn − xnk , x − xn ≥ 0.
3.11
It follows from 2.1 that
xnk − xn 2 xnk − x − xn − x2
xnk − x2 − xn − x2 − 2xnk − xn , xn − x
3.12
≤ xnk − x2 − xn − x2 .
It then follows from the existence of limn → ∞ xn − x2 that {xn } is a Cauchy sequence.
Moreover,
lim xn1 − xn 0.
n→∞
3.13
12
Abstract and Applied Analysis
We now assume that xn → w for some w ∈ C. Now, since xn1 ∈ Cn1 and Cn1 ⊂ Cn ,
yn − xn1 2 ≤ xn − xn1 2 θn which implies that
yn − xn1 ≤ xn − xn1 θn −→ 0.
3.14
From αn ≤ b < 1, we get
xn − Tn xn 1
yn − xn 1 − αn
1 ≤
yn − xn1 xn − xn1 −→ 0.
1−b
3.15
Since {Tn } satisfies the NST∗ -condition with T, we have
lim xn − Txn 0 ∀T ∈ T.
n→∞
3.16
Since I − T is closed at 0 for all T ∈ T, we have I − Tw 0. This implies that w ∈ FT.
Furthermore, by 3.8,
w − x lim xn − x ≤ p − x
n→∞
∀p ∈ FT.
3.17
Hence, w PFT x. This completes the proof.
Lemma 3.2 see 9, Theorem 10.4. Let C be a nonempty closed convex subset of a real Hilbert
space, and let T : C → C be a nonexpansive mapping. Then, I − T is demiclosed at 0.
It is not difficult to see from the proof of Theorem 3.1 that the boundedness of C can
be discarded if {Tn } is a family of nonexpansive mappings. So, we immediately obtain the
following theorem.
Theorem 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn } and
and
T be two families of nonexpansive mappings of C into itself such that ∞
n1 FTn FT /
suppose that {Tn } satisfies the NST∗ -condition with T. Assume that {αn } is a sequence in 0, b for
some b ∈ 0, 1. Then, the sequence {xn } in C defined by 1.4 converges strongly to PFT x.
Remark 3.4. Theorem 3.3 includes 6, Theorem 3.3 as a special case since the NST-condition
I with T implies the NST∗ -condition with T.
Theorem 3.5. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let {Tn }
be a family of Ln -Lipschitzian mappings of C into itself with a common fixed point. Suppose that T
is a family of mappings from C into itself such that FT ∞
n1 FTn and I − T is demiclosed at
0 for all T ∈ T. Assume that {αn } is a sequence in 0, b for some b ∈ 0, 1. If {Tn } satisfies the
NST∗ -condition with T, then the sequence {xn } in C defined by 1.5 converges strongly to PFT x.
Proof. The proof is analogous to the proof of 7, Theorem 10 and Theorem 3.1, so it is omitted.
Abstract and Applied Analysis
13
4. Deduced Results
Let C be a subset of a real Hilbert space H. A mapping T : C → C is said to be an
asymptotically nonexpansive if there exists a sequence {kn } of real numbers such that kn ∈
1, ∞, kn → 1, and
T n x − T n y ≤ kn x − y
∀x, y ∈ C, n ∈ .
4.1
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk 14
as an important generalization of the class of nonexpansive mappings. They proved that if C
is nonempty bounded closed convex and T is an asymptotically nonexpansive self-mapping
of C, then T has a fixed point.
In this section, we use the NST∗ -condition to obtain recent results proved by Kim and
Xu 15, Plubtieng and Ungchittrakool 16, and Zegeye and Shahzad 17, 18. We start with
the following auxiliary result.
Lemma 4.1. Let C be a nonempty closed convex subset of a Hilbert space H, and let T be an
asymptotically nonexpansive mappings of C into itself with a sequence {kn } in 1, ∞ satisfying
kn → 1 and FT / . Then, {T n } is a family of kn -Lipschitzian mappings of C into itself and
∗
satisfies the NST -condition with T.
Proof. We note that {T n } is a family of kn -Lipschitzian mappings of C into itself. Let {zn } be a
bounded sequence in C such that
lim zn − T n zn 0,
n→∞
lim zn1 − zn 0.
n→∞
4.2
Since
zn1 − Tzn1 ≤ zn1 − T n1 zn1 T n1 zn1 − Tzn1 ≤ zn1 − T n1 zn1 k1 T n zn1 − zn1 ≤ zn1 − T n1 zn1 k1 T n zn1 − T n zn T n zn − zn zn1 − zn 4.3
≤ zn1 − T n1 zn1 k1 kn 1zn1 − zn k1 T n zn − zn ,
it follows that
lim zn − Tzn 0.
n→∞
4.4
n
n
It is easy to see that FT ⊂ ∞
∈ ∞
n1 FT . To see the reverse inclusion, let z
n1 FT ∞
n
following from the first part of the proof above, but now let zn ≡ z. Then, z ∈ n1 FT , and
n
n
∗
hence ∞
n1 FT ⊂ FT. This implies that {T } satisfies the NST -condition with T.
Lemma 4.2 see 19. Let C be a nonempty bounded closed convex subset of a Hilbert space H, and
let T be an asymptotically nonexpansive mappings of C into itself. Then, I − T is demiclosed at 0.
14
Abstract and Applied Analysis
Using Theorem 3.1 and Lemmas 2.6 and 4.1, we have the following result.
Theorem 4.3. Let C be a nonempty bounded closed convex subset of a real Hilbert space H, and let
S, T be two asymptotically nonexpansive mappings of C into itself with sequences {sn } and {tn },
respectively, and FS ∩ FT /
. Assume that {αn } is a sequence in 0, b and {βn } is a sequence in
a, b for some a, b ∈ 0, 1. For x1 x ∈ H and C1 C, one defines a sequence {xn } of C as follows:
zn βn xn 1 − βn Sn xn ,
yn αn xn 1 − αn T n zn ,
Cn1 z ∈ Cn : yn − z2 ≤ xn − z2 θn ,
xn1 PCn1 x,
4.5
n 1, 2, 3, . . . ,
where
θn 1 − αn t2n − 1 1 − βn t2n s2n − 1 diam C2 −→ 0
as n −→ ∞.
4.6
Then, {xn } converges strongly to PFS∩FT x.
Using Theorem 3.5 and Lemmas 2.6 and 4.1, we have the following result.
Theorem 4.4 see 16, Theorem 3.1. Let C be a nonempty bounded closed convex subset of a real
Hilbert space H, and let S, T be two asymptotically nonexpansive mappings of C into itself with
sequences {sn } and {tn }, respectively, and FS ∩ FT /
. Assume that {αn } is a sequence in 0, b
and {βn } is a sequence in a, b for some a, b ∈ 0, 1. For x1 x ∈ C, one defines a sequence {xn } of
C as follows:
zn βn xn 1 − βn Sn xn ,
yn αn xn 1 − αn T n zn ,
Cn z ∈ C : yn − z2 ≤ xn − z2 θn ,
4.7
Qn {z ∈ C : xn − z, x − xn ≥ 0},
xn1 PCn ∩Qn x,
n 1, 2, 3, . . . ,
where
θn 1 − αn t2n − 1 1 − βn t2n s2n − 1 diam C2 −→ 0
Then, {xn } converges strongly to PFS∩FT x.
as n −→ ∞.
4.8
Abstract and Applied Analysis
15
Using Theorem 3.1 and Lemmas 2.7 and 4.1, we have the following result.
Theorem 4.5. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let {Ti }ri1
be a finite family of asymptotically nonexpansive mappings of C into itself with sequences {kni } for
. Assume that {αni }∞
i 1, 2, . . . , r, respectively, and suppose that ri1 FTi /
n1 are sequences in
0, 1 such that αn0 ≤ b < 1, αni ≥ a > 0 for some a, b ∈ 0, 1 and ri0 αni 1 for all n ∈ . For
x1 x ∈ H and C1 C, one defines a sequence {xn } of C as follows:
yn αn0 xn αn1 T1n xn αn2 T2n xn · · · αnr Trn xn ,
Cn1 z ∈ Cn : yn − z2 ≤ xn − z2 θn ,
xn1 PCn1 x,
4.9
n 1, 2, 3, . . . ,
where
2
2
2
θn αn1 kn1
− 1 αn2 kn2
− 1 · · · αnr knr
− 1 diam C2 −→ 0
as n −→ ∞.
4.10
Then, {xn } converges strongly to P∩ri1 FTi x.
Using Theorem 3.5 and Lemmas 2.7 and 4.1, we have the following two results which
were proved by Zegeye and Shahzad 17, 18.
Theorem 4.6. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let
{Ti }ri1 be a finite family of asymptotically nonexpansive mappings of C with sequences {kni } for
∞
i 1, 2, . . . , r, respectively, and suppose that ri1 FTi / . Assume
that {αni }n1 are sequences in
r
0, 1 such that αn0 ≤ b < 1, αni ≥ a > 0 for some a, b ∈ 0, 1 and i0 αni 1 for all n ∈ . For
x1 x ∈ C, one defines a sequence {xn } of C as follows:
yn αn0 xn αn1 T1n xn αn2 T2n xn · · · αnr Trn xn ,
Cn z ∈ C : yn − z2 ≤ xn − z2 θn ,
4.11
Qn {z ∈ C : xn − z, x − xn ≥ 0},
xn1 PCn ∩Qn x,
n 1, 2, 3, . . . ,
where
2
2
2
− 1 αn2 kn2
− 1 · · · αnr knr
− 1 diam C2 −→ 0
θn αn1 kn1
as n −→ ∞.
4.12
Then, {xn } converges strongly to P∩ri1 FTi x.
16
Abstract and Applied Analysis
Theorem 4.7. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let
Ìi {Ti t : t ∈ , i 1, 2, . . . , r} be a finite family of asymptotically nonexpansive semigroups
. Assume that {αni }∞
such that F ri1 FÌi /
n1 are sequences in 0, 1 such that
αn0 ≤ b < 1,
αni ≥ a > 0,
4.13
for some a, b ∈ 0, 1 and ri0 αni 1 for all n ∈ . Let {tni , i 1, 2, . . . , r} be finite positive and
divergent real sequences. For x1 x ∈ C, one defines a sequence {xn } of C as follows:
1 tn1
1 tn2
T1 uxn du αn2
T2 uxn du
tn1 0
tn2 0
1 tnr
· · · αnr
Tr uxn du,
tnr 0
Cn z ∈ C : yn − z2 ≤ xn − z2 θn ,
yn αn0 xn αn1
4.14
Qn {z ∈ C : xn − z, x − xn ≥ 0},
xn1 PCn ∩Qn x,
n 1, 2, 3, . . . ,
where
θ n αn1 L2u1 − 1 αn2 L2u2 − 1 · · · αnr L2ur − 1 diam C2 −→ 0
with Lui 1/tni tni
0
as n −→ ∞, 4.15
LTi i du. Then {xn } converges strongly to PF x.
Acknowledgments
The authors would like to thank the referee for the insightful comments and suggestions.
The second author was supported by the Commission on Higher Education, the Thailand
Research Fund, and Khon Kaen University under Grant no. RMU5380039.
References
1 F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National
Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.
2 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.
4, pp. 506–510, 1953.
3 S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of
Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.
4 A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics,
vol. 22, no. 1, pp. 81–86, 1975.
5 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and
nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–
379, 2003.
6 W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods
for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and
Applications, vol. 341, no. 1, pp. 276–286, 2008.
Abstract and Applied Analysis
17
7 W. Nilsrakoo and S. Saejung, “Weak and strong convergence theorems for countable Lipschitzian
mappings and its applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp.
2695–2708, 2008.
8 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive
mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
9 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced
Mathematics, Cambridge University Press, Cambridge, UK, 1990.
10 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000, Fixed
point theory and Its application.
11 K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a
countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods
& Applications, vol. 67, no. 8, pp. 2350–2360, 2007.
12 K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence to common fixed points of families of
nonexpansive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp.
11–34, 2007.
13 K. Nakprasit, W. Nilsrakoo, and S. Saejung, “Weak and strong convergence theorems of an
implicit iteration process for a countable family of nonexpansive mappings,” Fixed Point Theory and
Applications, vol. 2008, Article ID 732193, 18 pages, 2008.
14 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”
Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
15 T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically
nonexpansive mappings and semigroups,” Nonlinear Analysis. Theory, Methods & Applications, vol.
64, no. 5, pp. 1140–1152, 2006.
16 S. Plubtieng and K. Ungchittrakool, “Strong convergence of modified Ishikawa iteration for two
asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis. Theory, Methods &
Applications, vol. 67, no. 7, pp. 2306–2315, 2007.
17 H. Zegeye and N. Shahzad, “Strong convergence theorems for a finite family of asymptotically
nonexpansive mappings and semigroups,” Nonlinear Analysis. Theory, Methods & Applications, vol.
69, no. 12, pp. 4496–4503, 2008.
18 H. Zegeye and N. Shahzad, “Corrigendum to: Strong convergence theorems for a finite family
of asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis. Theory, Methods
& Applications, vol. 69, no. 12, pp. 4496–4503, 2008, Nonlinear Analysis, Theory, Methods and
Applications, vol. 73, pp. 1905–1907, 2010.
19 P.-K. Lin, K.-K. Tan, and H. K. Xu, “Demiclosedness principle and asymptotic behavior for
asymptotically nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 24,
no. 6, pp. 929–946, 1995.