Volume 2010, Article ID 732872, 16 pagesdoi:10.1155/2010/732872 Research Article Some Characterizations for a Family of Nonexpansive Mappings and Convergence of a Generated Sequence to T
Trang 1Volume 2010, Article ID 732872, 16 pages
doi:10.1155/2010/732872
Research Article
Some Characterizations for
a Family of Nonexpansive Mappings and
Convergence of a Generated Sequence to
Their Common Fixed Point
1 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology,
O-okayama, Meguro-ku, Tokyo 152-8552, Japan
2 Faculty of Engineering, Tamagawa University, Tamagawa-Gakuen, Machida-shi, Tokyo 194-8610, Japan
Correspondence should be addressed to Yasunori Kimura,yasunori@is.titech.ac.jp
Received 7 October 2009; Accepted 19 October 2009
Academic Editor: Anthony To Ming Lau
Copyrightq 2010 Y Kimura and K Nakajo 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
Motivated by the method of Xu2006 and Matsushita and Takahashi 2008 , we characterize the set of all common fixed points of a family of nonexpansive mappings by the notion of Mosco convergence and prove strong convergence theorems for nonexpansive mappings and semigroups
in a uniformly convex Banach space
1 Introduction
Let C be a nonempty bounded closed convex subset of a Banach space and T : C → C
a nonexpansive mapping; that is, T satisfies Tx − Ty ≤ x − y for any x, y ∈ C, and consider approximating a fixed point of T This problem has been investigated by many
researchers and various types of strong convergent algorithm have been established For implicit algorithms, see Browder 1, Reich 2, Takahashi and Ueda 3, and others For explicit iterative schemes, see Halpern4, Wittmann 5, Shioji and Takahashi 6, and others Nakajo and Takahashi 7 introduced a hybrid type iterative scheme by using the metric projection, and recently Takahashi et al 8 established a modified type of this projection method, also known as the shrinking projection method
Let us focus on the following methods generating an approximating sequence to a
fixed point of a nonexpansive mapping Let C be a nonempty bounded closed convex subset
of a uniformly convex and smooth Banach space E and let T be a nonexpansive mapping of
Trang 2C into itself Xu9 considered a sequence {x n} generated by
x1 x ∈ C,
C n clco {z ∈ C : z − Tz ≤ t n x n − Tx n },
D n {z ∈ C : x n − z, Jx − Jx n ≥ 0},
x n1 ΠC n ∩D n x
1.1
for each n ∈ N, where clco D is the closure of the convex hull of D, Π C n ∩D nis the generalized
projection onto C n ∩ D n, and{t n } is a sequence in 0, 1 with t n → 0 as n → ∞ Then, he
proved that{x n} converges strongly to ΠF T x Matsushita and Takahashi10 considered a sequence{y n} generated by
y1 x ∈ C,
C n clcoz ∈ C : z − Tz ≤ t ny n − Ty n,
D nz ∈ C :y n − z, Jx − y n
≥ 0,
y n1 P C n ∩D n x
1.2
for each n ∈ N, where P C n ∩D n is the metric projection onto C n ∩ D nand{t n} is a sequence in
0, 1 with t n → 0 as n → ∞ They proved that {y n } converges strongly to P F T x.
In this paper, motivated by these results, we characterize the set of all common fixed points of a family of nonexpansive mappings by the notion of Mosco convergence and prove strong convergence theorems for nonexpansive mappings and semigroups in a uniformly convex Banach space
2 Preliminaries
Throughout this paper, we denote by E a real Banach space with norm · We write x n x
to indicate that a sequence{x n } converges weakly to x Similarly, x n → x will symbolize strong convergence Let G be the family of all strictly increasing continuous convex functions
g : 0, ∞ → 0, ∞ satisfying that g0 0 We have the following theorem 11, Theorem 2 for a uniformly convex Banach space
Theorem 2.1 Xu 11 E is a uniformly convex Banach space if and only if, for every bounded
λx 1 − λy2≤ λx2 1 − λy2− λ1 − λg Bx − y 2.1
Bruck12 proved the following result for nonexpansive mappings
Trang 3Theorem 2.2 Bruck 12 Let C be a bounded closed convex subset of a uniformly convex Banach
T
n
i1
λ i x i
− n
i1
λ i Tx i
≤ max
1≤j<k≤nx j − x k − Tx j − Tx k 2.2
for all n ∈ N, {x1, x2, , x n } ⊂ C, {λ1, λ2, , λ n } ⊂ 0, 1 with n
i1λ i 1 and nonexpansive
mapping T of C into E.
Let{C n} be a sequence of nonempty closed convex subsets of a reflexive Banach space
only if there exists{x n } ⊂ E such that {x n } converges strongly to x and that x n ∈ C nfor all
n ∈ N Similarly the set of all weak subsequential limit points by w-Lsn C n ; y ∈ w-Lsn C nif and only if there exist a subsequence{C n i } of {C n } and a sequence {y i } ⊂ E such that {y i}
converges weakly to y and that y i ∈ C n i for all i ∈ N If C0 satisfies that C0 s-Lin C n w-Lsn C n, then we say that{C n } converges to C0 in the sense of Mosco and we write C0 M-limn C n By definition, it always holds that s-Lin C n ⊂ w-Lsn C n Therefore, to prove C0 M-limn C n, it suffices to show that
w-Ls
n C n ⊂ C0 ⊂ s-Li
One of the simplest examples of Mosco convergence is a decreasing sequence {C n} with respect to inclusion The Mosco limit of such a sequence is ∞n1C n For more details, see
13
Suppose that E is smooth, strictly convex, and reflexive The normalized duality mapping of E is denoted by J, that is,
Jxx∗∈ E∗:x2 x, x∗ x∗2
2.4
for x ∈ E In this setting, we may show that J is a single-valued one-to-one mapping onto E∗ For more details, see14–16
Let C be a nonempty closed convex subset of a strictly convex and reflexive Banach
minimizer y x ∈ C Using such a point, we define the metric projection P C : E → C by P C x
y x for every x ∈ E The metric projection has the following important property: x0 P C x if
and only if x0∈ C and x0− z, Jx − x0 ≥ 0 for all z ∈ C.
In the same manner, we define the generalized projection17 ΠC : E → C for a nonempty closed convex subset C of a strictly convex, smooth, and reflexive Banach space E
minimizer and we defineΠC x by this point We know that the following characterization
holds for the generalized projection17,18: x0 ΠC x if and only if x0 ∈ C and x0− z, Jx −
Jx0 ≥ 0 for all z ∈ C.
Tsukada19 proved the following theorem for a sequence of metric projections in a Banach space
Theorem 2.3 Tsukada 19 Let E be a reflexive and strictly convex Banach space and let {C n}
Trang 4for each x ∈ E, {P C n x } converges weakly to P C0x, where P K is the metric projection onto a nonempty closed convex subset K of E Moreover, if E has the Kadec-Klee property, the convergence is in the strong topology.
On the other hand, Ibaraki et al.20 proved the following theorem for a sequence of generalized projections in a Banach space
Theorem 2.4 Ibaraki et al 20 Let E be a strictly convex, smooth, and reflexive Banach space
projection onto a nonempty closed convex subset K of E Moreover, if E has the Kadec-Klee property, the convergence is in the strong topology.
Kimura21 obtained the further generalization of this theorem by using the Bregman projection; see also22
Theorem 2.5 Kimura 21 Let C be a nonempty closed convex subset of a reflexive Banach space
{C n } be a sequence of nonempty closed convex subsets of C such that C0 M-lim n C n exists and is
{y n } of the domain of f, lim n→ ∞D f y n , x n 0 implies lim n→ ∞y n − x n 0 Then, the sequence
{Πf
C n x } of Bregman projections converges strongly to Π f
C0x for all x ∈ C.
We note that the generalized duality mapping J coincides with ∇f if the function f is defined by fx x2/2 for x ∈ E In this case, the Bregman projection Π f
K with respect to f
becomes the generalized projectionΠK for any nonempty closed convex subset K of E.
3 Main Results
Let C be a nonempty closed convex subset of E and let {T n } be a sequence of mappings of C into itself such that F ∞n1F T n / ∅ We consider the following conditions.
I For every bounded sequence {z n } in C, lim n→ ∞z n − T n z n 0 implies ω w z n ⊂ F, where ω w z n is the set of all weak cluster points of {z n}; see 23–25
II for every sequence {z n } in C and z ∈ C, z n → z and T n z n → z imply z ∈ F.
We know that conditionI implies condition II Then, we have the following results
Theorem 3.1 Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space
n1F T n / ∅ Let
C n t n clco {z ∈ C : z − T n z ≤ t n } for each n ∈ N, where {t n } ⊂ 0, ∞ Then, the following are
equivalent:
i {T n } satisfies condition (I);
ii for every {t n } ⊂ 0, ∞ with t n → 0 as n → ∞, M-lim n C n t n F.
Trang 5Proof First, let us prove that i implies ii Let {t n } ⊂ 0, ∞ with t n → 0 as n → ∞ It is obvious that F ⊂ C n t n and C n t n is closed and convex for all n ∈ N Thus we have
F⊂ s-Li
Let z ∈ w-Lsn C n t n Then, there exists a sequence {z i } such that z i ∈ C n i t n i for all i ∈ N and z i z as i → ∞ Let {u n } be a sequence in C such that u n ∈ C n t n for every n ∈ N and that u n i z i for all i ∈ N Fix n ∈ N From the definition of C n t n , there exist m ∈ N, {λ1, λ2, , λ m } ⊂ 0, 1, and {y1, y2, , y m } ⊂ C such that
m
i1
u n− m
i1
λ i y i
for each i 1, 2, , m On the other hand, byTheorem 2.2, there exists a strictly increasing
continuous convex function γ : 0, ∞ → 0, ∞ with γ0 0 such that
T
n
i1
λ i x i
− n
i1
λ i Tx i
≤ max
1≤j<k≤nx j − x k − Tx j − Tx k 3.3
for all n ∈ N, {x1, x2, , x n } ⊂ C, {λ1, λ2, , λ n } ⊂ 0, 1 with n
i1λ i 1 and nonexpansive
mapping T of C into E Thus we get
u n − T n u n ≤
u n− m
i1
λ i y i
m
i1
λ i y i− m
i1
λ i T n y i
m
i1
λ i T n y i − T n
i1
λ i y i
T n
i1
λ i y i
− T n u n
≤ 3t n γ−1
max
1≤j<k≤my j − y k − T n y j − T n y k
≤ 3t n γ−1
max
1≤j<k≤my j − T n y j y k − T n y k
≤ 3t n γ−12t n
3.4
for every n ∈ N, which implies u n − T n u n → 0 as n → ∞ From condition I, we get
z ∈ ω w z i ⊂ ω w u n ⊂ F, that is,
w-Ls
By3.1 and 3.5, we have
M-lim
Trang 6Next we show thatii implies i Let {z n } be a sequence in C such that
lim
and define{t n } by t n z n − T n z n for each n ∈ N Suppose that a subsequence {z n k } of {z n}
converges weakly to z Then since z n ∈ C n t n for all n ∈ N and M-lim n C n t n F, we have
z ∈ F; that is, condition I holds.
For a sequence of mappings satisfying condition II, we have the following characterization
Theorem 3.2 Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space
D0t0 C and D n t n clco {z ∈ D n−1t n−1 : z−T n z ≤ t n } for each n ∈ N, where {t n } ⊂ 0, ∞.
Then, the following are equivalent:
i {T n } satisfies condition (II);
ii for every {t n } ⊂ 0, ∞ with t n → 0 as n → ∞, M-lim n D n t n F.
F ⊂ D n t n ⊂ D n−1t n−1 for all n ∈ N Thus we get
n0
D n t n M-lim
n0D n t n We have z ∈ D n t n for all n ∈ N As in the proof ofTheorem 3.1, we get limn→ ∞z − T n z 0 By condition II, we obtain z ∈ F, which implies ∞n0D n t n ⊂ F.
Hence we have M-limn D n t n F.
Suppose that conditionii holds Let {z n } be a sequence in C and z ∈ C such that
z n → z and that T n z n → z Since
z − T n z ≤ z − z n z n − T n z n T n z n − T n z
for each n∈ N, we have limn→ ∞z − T n z 0 Letting t n z − T n z for each n ∈ N, we have
z ∈ D n t n for every n ∈ N and t n → 0 as n → ∞, which implies z ∈ M-lim n D n t n F.
Hencei holds, which is the desired result
Remark 3.3 In Theorem 3.2 , it is obvious by definition that {D n t n } is a decreasing sequence with
ii for every {t n } ⊂ 0, ∞ with t n → 0 as n → ∞, PK-lim n D n t n F,
where PK-lim n D n t n is the Painlev´e-Kuratowski limit of {D n t n }; see, for example, [ 13 ] for more details.
Trang 7In the next section, we will see various types of sequences of nonexpansive mappings which satisfy conditionsI and II
4 The Sequences of Mappings Satisfying Conditions (I) and (II)
First let us show some known results which play important roles for our results
Theorem 4.1 Browder 1 Let C be a nonempty closed convex subset of a uniformly convex
z ∈ C and {x n − Tx n } converges strongly to 0, then z is a fixed point of T.
Theorem 4.2 Bruck 26 Let C be a nonempty closed convex subset of a strictly convex Banach
k1β k 1 If ∞
k1F T k is nonempty, then the mapping T ∞
k1β k T k is well defined and
F T ∞
k1
Theorems4.3,4.5i,4.6–4.9show the examples of a family of nonexpansive mappings satisfying conditionI Theorems4.5ii,4.11, and4.12show those satisfying conditionII
Theorem 4.3 Let C be a nonempty closed convex subset of a uniformly convex Banach space E and
(I).
Remark 4.4 In the previous theorem, if C is bounded, then FT is guaranteed to be nonempty by
Kirk’s fixed point theorem [ 27 ].
Let E be a Banach space and A a set-valued operator on E A is called an accretive
operator ifx1− x2 ≤ x1− x2 λy1 − y2 for every λ > 0 and x1, x2, y1, y2 ∈ E with
y1∈ Ax1and y2∈ Ax2
Let A be an accretive operator and r > 0 We know that the operator I rA has a single-valued inverse, where I is the identity operator on E We call I rA−1the resolvent
of A and denote it by J r We also know that J r is a nonexpansive mapping with F J r A−10
for any r > 0, where A−10 {z ∈ E : 0 ∈ Az} For more details, see, for example, 15
We have the following result for the resolvents of an accretive operator by25; see also15, Theorem 4.6.3, and 16, Theorem 3.4.3
Theorem 4.5 Let C be a nonempty closed convex subset of a uniformly convex Banach space E and
Trang 8into itself with ∞n1F T n A−10 and the following hold:
i if inf n∈Nr n > 0, then {T n } satisfies condition (I),
ii if there exists a subsequence {r n i } of {r n } such that inf i∈Nr n i > 0, then {T n } satisfies
condition (II).
For i, suppose infn∈Nr n > 0 and let {z n } be a bounded sequence in C such that
limn→ ∞z n − T n z n 0 By 25, Lemma 3.5, we have lim n→ ∞z n − J1z n 0 Using Theorem 4.1we obtain ω w z n ⊂ FJ1 A−10
Let us showii Let {r n i } be a subsequence of {r n} with infi∈Nr n i > 0 and let {z n} be
a sequence in C and z ∈ C such that z n → z and T n z n → z As in the proof of i, we get
limi→ ∞z n i − J1z n i 0 and z ∈ A−10
Let C be a nonempty closed convex subset of E Let {S n} be a family of mappings of
0≤ β i,j ≤ 1 for every i, j ∈ N with i ≥ j Takahashi 16,28 introduced a mapping W n of C into itself for each n∈ N as follows:
U n,n β n,n S n1− β n,n
I,
U n,n−1 β n,n−1S n−1U n,n1− β n,n−1
I,
U n,k β n,k S k U n,k11− β n,k
I,
U n,2 β n,2 S2U n,31− β n,2
I,
W n U n,1 β n,1 S1U n,21− β n,1
I.
4.2
Such a mapping W n is called the W-mapping generated by S n , S n−1, , S1 and β n,n , β n,n−1,
3.6.
Theorem 4.6 Let C be a nonempty closed convex subset of a uniformly convex Banach space E and
let {S n } be a family of nonexpansive mappings of C into itself with F ∞n1F S n / ∅ Let {β n,k :
n, k ∈ N, 1 ≤ k ≤ n} be a sequence of real numbers such that 0 < a ≤ β i,j ≤ b < 1 for every i, j ∈ N
with i ≥ j and let W n be the W-mapping generated by S n , S n−1, , S1 and β n,n , β n,n−1, , β n,1 Let
∞
n1F T n F and satisfies condition (I).
Lemma 3.1, FT n n
i1F S i for all n ∈ N, which implies ∞n1F T n F Let {z n} be a
bounded sequence in C such that lim n→ ∞z n −T n z n 0 We have limn→ ∞z n −S1U n,2 z n 0
Trang 9Let z ∈ F FromTheorem 2.1, for a bounded subset B of C containing {z n } and z, there exists
g B0∈ G, where B0 {y ∈ E : y ≤ 2 sup x ∈B x}, such that
z n − z2≤ z n − S1U n,2 z n S1U n,2 z n − z2
z n − S1U n,2 z n z n − S1U n,2 z n 2S1U n,2 z n − z
S1U n,2 z n − z2
≤ Mz n − S1U n,2 z n U n,2 z n − z2
≤ Mz n − S1U n,2 z n β n,2 S2U n,3 z n − z21− β n,2
z n − z2
− β n,2
1− β n,2
g B0S2U n,3 z n − z n
≤ Mz n − S1U n,2 z n z n − z2− β n,2
1− β n,2
g B0S2U n,3 z n − z n
4.3
for every n ∈ N, where M sup n∈Nz n − S1U n,2 z n 2S1U n,2 z n − z Thus we obtain
limn→ ∞S2U n,3 z n − z n 0 Let m ∈ N Similarly, we have
lim
n→ ∞S m U n,m1z n − z n lim
n→ ∞S m1U n,m2z n − z n 0. 4.4
As in the proof of30, Theorem 3.1, we get lim n→ ∞z n − S k z n 0 for each k ∈ N Using
Theorem 4.1we obtain ω w z n ⊂ F.
We have the following result for a convex combination of nonexpansive mappings which Aoyama et al.31 proposed
Theorem 4.7 Let C be a nonempty closed convex subset of a uniformly convex Banach space E and
let {S n } be a family of nonexpansive mappings of C into itself such that F ∞n1F S n / ∅ Let {β k}
i n
k1β k 1 for every n ∈ N,
ii limn→ ∞β k > 0 for each k ∈ N,
and let T n α n I 1 − α n n
k1β k S k for all n ∈ N, where {α n } ⊂ a, b for some a, b ∈ 0, 1 with
a ≤ b Then, {T n } is a family of nonexpansive mappings of C into itself with ∞
n1F T n F and
satisfies condition (I).
Theorem 4.2, we have F n
k1β k S k n
k1F S k and thus FT n n
k1F S k It follows that
n1
F S n ∞
n1
n
k1
F S k ∞
n1
Trang 10Let{z n } be a bounded sequence in C such that lim n→ ∞z n − T n z n 0 Let z ∈ F, m ∈ N, and
γ m
n α n 1 − α n β m
n for n∈ N ByTheorem 2.1, for a bounded subset B of C containing {z n}
and z, there exists g B0 ∈ G with B0 {y ∈ E : y ≤ 2 sup x ∈B x} which satisfies that
z n − z2≤ z n − T n z n T n z n − z2≤ Mz n − T n z n T n z n − z2
Mz n − T n z n
α n z n − z 1 − α n n
k1
β k n S k z n − z
2
≤ Mz n − T n z n γ m
n
α n z n − z 1 − α n β m
n S m z n − z
γ n m
2
1− γ m n
1 − α n m−1
k1β k S k z n − z n
k m1 β k S k z n − z
1− γ m n
2
≤ Mz n − T n z n α n z n − z2 1 − α n β m
n S m z n − z2
−α n 1 − α n β m
n
γ m n
g B0z n − S m z n 1− γ m
n
z n − z2
Mz n − T n z n z n − z2− α n 1 − α n β m n
α n 1 − α n β m
n
g B0zn − S m z n
4.6
for n ∈ N, where M sup n∈N{z n − T n z n 2T n z n − z} Since a ≤ α n ≤ b for all n ∈ N and
limn→ ∞β m
n > 0, we get lim n→ ∞gB
0z n − S m z n 0 and hence limn→ ∞z n − S m z n 0 for
each m∈ N Therefore, usingTheorem 4.1we obtain ω w z n ⊂ F.
Let C be a nonempty closed convex subset of a Banach space E and let S be a
semigroup A familyS {Tt : t ∈ S} is said to be a nonexpansive semigroup on C if
i for each t ∈ S, Tt is a nonexpansive mapping of C into itself;
ii Tst TsTt for every s, t ∈ S.
We denote by FS the set of all common fixed points of S, that is, FS t ∈S F Tt We
have the following result for nonexpansive semigroups by25, Lemma 3.9; see also 32,33
Theorem 4.8 Let C be a nonempty closed convex subset of a uniformly convex Banach space E and
s μ n → 0 as n → ∞ for all s ∈ S and let T n T μ n for
n1F T n FS
and satisfies condition (I).
Lemma 3.9, we have FS ∞n1F T n Let {z n } be a bounded sequence in C such that
limn→ ∞z n − T n z n 0 Then we get limn→ ∞z n − Ttz n 0 for every t ∈ S Using
Theorem 4.1we have ω w z n ⊂ FS.
...y1∈ Ax1and y2∈ Ax2
Let A be an accretive operator and r > We know that the operator I rA has a single-valued... k for all n ∈ N, where {α n } ⊂ a, b for some a, b ∈ 0, 1 with
a ≤ b Then, {T n } is a family of nonexpansive mappings of C into itself...
Trang 4for each x ∈ E, {P C n x } converges weakly to P C0x,