STRONG CONVERGENCE THEOREMS FOR
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS AND
ASYMPTOTICALLY NONEXPANSIVE SEMIGROUPS
YONGFU SU AND XIAOLONG QIN
Received 22 April 2006; Accepted 14 July 2006
Strong convergence theorems are obtained from modified Halpern iterative scheme for
asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.
Copyright © 2006 Y. Su and X. Qin. 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 preliminary
Let H be a real Hilbert space, C a nonempty closed convex subset of H, and T : C → C a
mapping. Recall that T is nonexpansive if
Tx − T y ≤ x − y ∀x, y ∈ C,
(1.1)
and T is asymptotically nonexpansive if there exists a sequence {kn } of positive real numbers with limn→∞ kn = 1 and such that
n
T x − T n y ≤ kn x − y ∀n ≥ 1, x, y ∈ C.
(1.2)
A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points
of T; that is, F(T) = {x ∈ C : Tx = x}. Also, recall that a family S = {T(s) | 0 ≤ s < ∞} of
mappings from C into itself is called an asymptotically nonexpansive semigroup on C if
it satisfies the following conditions:
(i) T(0)x = x for all x ∈ C;
(ii) T(s + t) = T(s)T(t) for all s,t ≥ 0;
(iii) there exists a positive valued function L : [0, ∞) → [1, ∞) such that lims→∞ Ls = 1
and T(s)x − T(s)y ≤ Ls x − y for all x, y ∈ C and s ≥ 0;
(iv) for all x ∈ C, s → T(s)x is continuous.
Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2006, Article ID 96215, Pages 1–11
DOI 10.1155/FPTA/2006/96215
2
Nonexpansive mapping
We denote by F(S) the set of all common fixed points of S, that is, F(S) = 0≤s<∞
F(T(s)). It is known that F(S) is closed and convex. Construction of fixed point of nonexpansive mapping is an important subject in the theory of nonexpansive mappings and
finds applications in a number of applied areas, in particular, in image recovery and signal
processing (see, e.g., [14, 15]). However, the sequence {T n x}∞
n=0 of iterates of the mapping T at a point x ∈ C may not converge in the weak topology. Thus averaged iterations
prevail. In fact, Mann’s iterations do have weak convergence. More precisely, Mann’s iteration procedure is a sequence {xn } defined by
xn+1 = αn xn + 1 − αn Txn ,
(1.3)
where the initial guess x0 ∈ C is chosen arbitrarily.
Reich [9] proved that if E is a uniformly convex Banach space with a Fréchet differentiable norm and if {αn } is chosen such that ∞
n=1 αn (1 − αn ) = ∞, then the sequence {xn }
defined by (1.3) converges weakly to a fixed point of T. However we note that Mann’s
iterations have only weak convergence even in a Hilbert space [1].
Recently many authors want to modify the Mann iteration method (1.3) so that strong
convergence is guaranteed have recently been made. Nakajo and Takahashi [8] proposed
the following modification of the Mann iteration (1.3) for a single nonexpansive mapping
T in a Hilbert space:
x0 ∈ C
arbitrarily,
yn = αn xn + 1 − αn Txn ,
Cn = z ∈ C : yn − z ≤ xn − z ,
(1.4)
Qn = z ∈ C : x0 − xn ,xn − z ≥ 0 ,
xn+1 = PCn ∩Qn x0 ,
where PK denotes the metric projection from H onto a closed convex subset K of H and
proved that sequence {xn } converges strongly to PF(T) x0 .
They also proposed the following iteration process for a nonexpansive semigroup S =
{T(s)|0 ≤ s < ∞} in a Hilbert space H:
x0 ∈ C
arbitrarily,
yn = αn xn + 1 − αn
1
tn
tn
0
T(s)xn ds,
Cn = z ∈ C : yn − z ≤ xn − z ,
(1.5)
Qn = z ∈ C : x0 − xn ,xn − z ≥ 0 ,
xn+1 = PCn ∩Qn x0 .
They proved that if the sequence {αn } is bounded from one and if {tn } is a positive
real divergent sequence, then the sequence {xn } generated by (1.5) converges strongly to
PF(S) x0 .
Y. Su and X. Qin 3
Halpern [3] firstly studied iteration scheme as follows:
xn+1 = αn u + 1 − αn Txn ,
n ≥ 0,
(1.6)
where u,x0 ∈ C are arbitrary(but fixed) and {αn } ⊂ (0,1). He pointed out that the conditions limn→∞ αn = 0 and ∞
n=1 αn = ∞ are necessary in the sense that if the iteration
scheme (1.6) converges to a fixed point of T, then these conditions must be satisfied. Ten
years later, Lions [6] investigated the general case in Hilbert space under the conditions
∞
lim αn = 0,
n→∞
αn = ∞,
n =1
lim
n→∞
αn − αn+1
=0
α2n+1
(1.7)
on the parameters. However, Lions’ conditions on the parameters were more restrictive and did not include the natural candidate {αn = 1/n}. Reich [10] gave the iteration
scheme (1.6) in the case when E is uniformly smooth and αn = n−δ with 0 < δ < 1.
Wittmann [13] studied the iteration scheme (1.6) in the case when E is a Hilbert space
and {αn } satisfies
∞
lim αn = 0,
n→∞
αn = ∞,
n =1
∞
αn+1 − αn < ∞.
(1.8)
n =1
Reich [11] obtained a strong convergence of the iterates (1.6) with two necessary and
decreasing conditions on parameters for convergence in the case when E is uniformly
smooth with a weakly continuous duality mapping.
Recently, Martinez-Yanes and Xu [7] adapted the iteration (1.6) in Hilbert space as
follows:
x0 ∈ C
arbitrarily,
yn = αn x0 + 1 − αn Txn ,
2 2
2
Cn = z ∈ C : yn − z ≤ xn − z + αn x0 + 2 xn − x0 ,z ,
(1.9)
Qn = z ∈ C : x0 − xn ,xn − z ≥ 0 ,
xn+1 = PCn ∩Qn x0 .
More precisely, they prove the following theorem.
Theorem 1.1 (Martinez-Yanes and Xu [7]). Let H be a real Hilbert space, C a closed convex
subset of H, and T : C → C a nonexpansive mapping such that F(T) = ∅. Assume that
{αn } ⊂ (0,1) is such that limn→∞ αn = 0. Then the sequence {xn } defined by (1.9) converges
strongly to PF(T) x0 .
The purpose of this paper is to employ Nakajo and Takahashi’s [8] idea to modify process (1.6) for asymptotically nonexpansive mappings and asymptotically nonexpansive
semigroup to have strong convergence theorem in Hilbert space.
In the sequel, we need the following lemmas for the proof of our main results.
4
Nonexpansive mapping
Lemma 1.2. Let K be a closed convex subset of real Hilbert space H and let PK be the metric
projection from H onto K (i.e., for x ∈ H, Pk is the only point in K such that x − Pk x =
inf {x − z : z ∈ K }). Given x ∈ H and z ∈ K. Then z = PK x if and only if there holds the
relations
x − z, y − z ≤ 0
∀ y ∈ K.
(1.10)
Lemma 1.3 (Lin et al. [5]). Let T be an asymptotically nonexpansive mapping defined on a
bounded closed convex subset C of a Hilbert space H. Assume that {xn } is a sequence in C
with the properties (i) xn p and (ii) Txn − xn → 0. Then p ∈ F(T).
Lemma 1.4 (Kim and Xu [4]). Let C be a nonexpansive bounded closed convex subset of H
and let S = {T(t) : 0 ≤ t < ∞} be an asymptotically nonexpansive semigroup on C. Then it
holds that
t
1
1 t
T(u)xn du −
T(u)xn du
limsup limsup sup T(s)
= 0.
t
t
s→∞
n→∞
0
x ∈C
(1.11)
0
Lemma 1.5. Let C be a nonexpansive bounded closed convex subset of H and let S = {T(s) :
0 ≤ s < ∞} be an asymptotically nonexpansive semigroup on C. If {xn } is a sequence in
C satisfying the properties (i) xn z; (ii) limsups→∞ limsupn→∞ T(s)xn − xn = 0, then
z ∈ F(S).
Proof. This lemma is the continuous version of [12, Lemma 2.3]. The proof given in [12]
is easily extended to the continuous case.
2. Main results
In this section we propose a modification of the Halpern iteration method to have strong
convergence for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroup in Hilbert space.
Theorem 2.1. Let C be a bounded closed convex subset of a Hilbert space H and let T : C →
C be an asymptotically nonexpansive mapping with sequence {kn }. Assume that {αn }∞
n =0
and {βn }∞
n=0 are sequences in (0,1) such that limn→∞ αn = 0, limn→∞ βn = 1, and M is an
appropriate constant such that M ≥ x0 − v2 , for all v ∈ C. Define a sequence {xn } in C by
the following algorithm:
x0 ∈ C
chosen arbitrarily,
zn = βn xn + 1 − βn T n xn ,
2
yn = αn x0 + 1 − αn T n zn ,
2
2
2
Cn = v ∈ C : yn − v ≤ xn − v + zn − xn + 2 xn − zn ,v + αn M ,
Qn = v ∈ C : x0 − xn ,xn − v ≥ 0 ,
xn+1 = PCn ∩Qn x0 .
Then {xn } converges to PF(T) x0 , provided kn2 (1 − αn ) − 1 ≤ 0.
(2.1)
Y. Su and X. Qin 5
Proof. From [2] we know that T has a fixed point in C. That is, F(T) = ∅. It is obviously
that Cn is closed and Qn is closed and convex for each n ≥ 0. Next observe that C is
convex. For v1 ,v2 ∈ Cn and t ∈ (0,1), putting v = tv1 + (1 − t)v2 . It is sufficient to show
that v ∈ Cn . Indeed, the defining inequality in Cn is equivalent to the inequality
2
2
2 zn − yn , v ≤ zn − yn + αn M.
(2.2)
Therefore, we have
2 zn − yn ,v = 2 zn − yn ,tv1 + (1 − t)v2
= 2t zn − yn ,v1 + 2(1 − t) zn − yn ,v2
2 2
≤ zn − yn + αn M,
(2.3)
which implies that C is convex. Next, we show that F(T) ⊂ Cn for all n. Indeed, for each
p ∈ F(T),
yn − p2 = αn x0 − p + 1 − αn T n zn − p 2
2 2
≤ αn x0 − p + 1 − αn kn2 zn − p
2 2
2 2
≤ xn − p − xn − p + αn x0 − p + 1 − αn kn2 zn − p
2 2 2 2
≤ xn − p + zn − p − xn − p + αn x0 − p
2 2 2
≤ xn − p + zn − xn + 2 xn − zn , p + αn M.
(2.4)
Therefore, p ∈ Cn for each n ≥ 1, which implies that F(T) ⊂ Cn . Next we show that
F(T) ⊂ Qn
∀n ≥ 0.
(2.5)
We prove this by induction. For n = 0, we have F(T) ⊂ C = Q0 . Assume that F(T) ⊂ Qn .
Since xn+1 is the projection of x0 onto Cn ∩ Qn , by Lemma 1.2 we have
x0 − xn+1 ,xn+1 − z ≥ 0 ∀z ∈ Cn ∩ Qn .
(2.6)
As F(T) ⊂ Cn ∩ Qn by the induction assumptions, the last inequality holds, in particular, for all z ∈ F(T). This together with the definition of Qn+1 implies that F(T) ⊂ Qn+1 .
Hence (2.5) holds for all n ≥ 0. In order to prove limn→∞ xn+1 − xn = 0, from the definition of Qn we have xn = PQn x0 which together with the fact that xn+1 ∈ Cn ∩ Qn ⊂ Qn
implies that
x0 − xn ≤ x0 − xn+1 .
(2.7)
This shows that the sequence {xn − x0 } is nondecreasing. Since C is bounded. We obtain that limn→∞ xn − x0 exists. Notice again that xn = PQn x0 and xn+1 ∈ Qn which give
6
Nonexpansive mapping
xn+1 − xn ,xn − x0 ≥ 0. Therefore, we have
xn+1 − xn 2 = xn+1 − x0 − xn − x0 2
2 2
≤ xn+1 − x0 − xn − x0 − 2 xn+1 − xn ,xn − x0
(2.8)
2 2
≤ xn+1 − x0 − xn − x0 .
It follows that
lim xn − xn+1 = 0.
n→∞
(2.9)
On the other hand, It follows from xn+1 ∈ Cn that
yn − xn+1 2 ≤ xn − xn+1 2 + zn 2 − xn 2 + 2 xn − zn ,xn+1 + αn M.
(2.10)
It follows from (2.1) and limn→∞ βn = 1 that
zn − xn = 1 − βn xn − T n xn −→ 0.
(2.11)
Next, we consider
2 2
zn − xn + 2 xn − zn ,xn+1
2
2 2
= zn + xn − 2 zn ,xn + 2 xn − zn ,xn+1 − 2xn + 2 zn ,xn
2
2
= zn − xn + 2 xn − zn ,xn+1 − 2xn + 2 zn ,xn
(2.12)
2
2
= zn − xn + 2 zn ,xn − xn+1 − 2xn + 2 xn ,xn+1 .
Therefore, it follows from (2.9) and (2.11) that
2 2
zn − xn + 2 xn − zn ,xn+1 −→ 0.
(2.13)
Furthermore, from (2.9), (2.13), and limn→∞ αn = 0, we obtain
lim yn − xn+1 = 0.
n→∞
(2.14)
On the other hand, we consider
yn − T n xn ≤ yn − T n zn + T n zn − T n xn ≤ αn x0 − T n zn + kn zn − xn = αn x0 − T n zn + kn 1 − βn xn − T n xn .
(2.15)
Y. Su and X. Qin 7
Therefore, it follows that
xn − T n xn ≤ xn − xn+1 + xn+1 − yn + yn − T n xn ≤ xn − xn+1 + xn+1 − yn + αn x0 − T n zn (2.16)
+ kn 1 − βn xn − T n xn .
That is,
1 − kn 1 − βn xn − T n xn ≤ xn − xn+1 + xn+1 − yn + αn x0 − T n zn .
(2.17)
It follows from limn→∞ βn = 1, limn→∞ αn = 0, (2.9), and (2.14) that
lim xn − T n xn −→ 0.
(2.18)
n→∞
Putting k = sup{kn : n ≥ 1} < ∞, we obtain
Txn − xn ≤ Txn − T n+1 xn + T n+1 xn − T n+1 xn+1 + T n+1 xn+1 − xn+1 + xn+1 − xn ≤ k xn − T n xn + 1 + k xn − xn+1 (2.19)
+ T n+1 xn+1 − xn+1 ,
which implies that
Txn − xn −→ 0.
(2.20)
Assume that {xni } is a subsequence of {xn } such that xni x. By Lemma 1.3 we have
x ∈ F(T). Next we show that x = PF(T) x0 and the convergence is strong. Put x = PF(T) x0
and consider the sequence {x0 − xni }. Then we have x0 − xni x0 − x and by the weak
lower semicontinuity of the norm and by the fact that x0 − xn+1 ≤ x0 − x for all n ≥ 0
which is implied by the fact that xn+1 = PCn ∩Qn x0 , we have
x0 − x ≤ x0 − x ≤ liminf x0 − xn ≤ limsup x0 − xn ≤ x0 − x.
i→∞
i
i
i→∞
(2.21)
This gives
x0 − x = x0 − x,
x0 − xn −→ x0 − x.
i
(2.22)
It follows that x0 − xni → x0 − x; hence, xni → x. Since {xni } is an arbitrary subsequence of
{xn }, we conclude that xn → x. The proof is completed.
8
Nonexpansive mapping
Theorem 2.2. Let C be a nonempty bounded closed convex subset of H and let S = {T(s) :
0 ≤ s < ∞} be an asymptotically nonexpansive semigroup on C. Assume that {αn }∞
n=0 and
{βn }∞
are
sequences
in
(0,1)
such
that
lim
α
=
0
and
lim
β
=
1.
{
t
}
is
a
positive
n→∞ n
n→∞ n
n
n =0
real divergent sequence and M is an appropriate constant such that M ≥ x0 − v for all
v ∈ C. Define a sequence {xn } in C by the following algorithm:
x0 ∈ C
chosen arbitrarily,
zn = βn xn + 1 − βn
yn = αn x0 + 1 − αn
2
2
tn
1
tn
T(s)xn ds,
0
tn
1
tn
0
T(s)zn ds,
2
2
Cn = v ∈ C : yn − v ≤ αn xn − v + zn − xn + 2 xn − zn ,v + αn M ,
Qn = v ∈ C : x0 − xn ,xn − z ≥ 0 ,
xn+1 = PCn ∩Qn x0 .
(2.23)
Then {xn } converges to PF(S) x0 , provided ((1/tn )
tn
0
Ls dt)2 (1 − αn ) − 1 ≤ 0.
Proof. We only conclude the difference. First we show F(S) ⊂ Cn . It follows from C is
bounded, we obtain that F(S) = ∅ (see [12]). Taking p ∈ F(S), we have
2
tn
yn − p2 ≤ αn x0 − p2 + 1 − αn 1
T(s)zn ds − p
t
n
0
2
2 1 tn T(s)zn − pds
≤ αn x0 − p + 1 − αn
tn
2 1
≤ αn x0 − p + 1 − αn
tn
0
tn
0
2
2
Ls ds zn − p
(2.24)
2 2 2 2
= xn − p + zn − p − xn − p + αn x0 − p
2 2 2
≤ xn − p + zn − xn + 2 xn − zn , p + αn M.
It follows that F(S) ⊂ Cn for each n ≥ 0. From the proof of Theorem 2.1 we have the sequence {xn } is well defined and F(S) ⊂ Cn ∩ Qn for each n ≥ 0. Similarly to the argument
of Theorem 2.1 and noticing q = PF(S) x0 , we have xn+1 − x0 ≤ q − x0 for each n ≥ 0
and xn+1 − xn → 0. Next, we assume that a subsequence {xni } of {xn } converges weakly
Y. Su and X. Qin 9
to q. It follows that
tn
T(s)xn − xn ≤ T(s)xn − T(s) 1
T(s)xn ds tn 0
tn
tn
1
1
+
T(s)
T(s)x
ds
T(s)x
ds
−
n
n tn 0
tn 0
tn
1
+
T(s)xn ds − xn tn
(2.25)
0
tn
1
T(s)xn ds − xn ≤ 2
tn 0
tn
1
1 tn
+
T(s)
T(s)x
ds
T(s)x
ds
−
n
n ,
t
t
0
n
n
0
for each n ≥ 0. It follows from (2.23) that
tn
1 tn
yn − 1
≤
T(s)z
ds
α
x
−
T(s)z
ds
n n 0
n .
t
t
n
0
n
(2.26)
0
Therefore, we obtain
tn
yn − 1
T(s)z
ds
n −→ 0.
t
n
(2.27)
0
Next, we consider the first term on the right-hand side of (2.25)
tn
1
1 tn
≤
T(s)x
ds
−
x
x
−
x
+
x
−
y
+
y
−
T(s)x
ds
n
n
n
n+1
n+1
n
n .
t
n t
n
0
n
(2.28)
0
Since xn+1 ∈ Cn , we have
yn − xn+1 2 ≤ αn xn − xn+1 2 + zn 2 − xn 2 + 2 xn − zn ,xn+1 + αn M.
(2.29)
Similar to the proof of Theorem 2.1, we have
lim yn − xn+1 = 0,
(2.30)
n→∞
and hence
tn
yn − 1
T(s)x
ds
n tn 0
tn
1
1 tn
1 tn
T(s)zn ds
T(s)zn ds −
T(s)xn ds
≤
+
yn −
tn
0
tn
0
tn
0
tn
0
tn
0
1 tn 1 tn
T(s)zn − T(s)xn ds
≤ yn −
T(s)zn ds +
tn 0
tn 0
tn
1 tn
1
zn − xn 2 .
≤
y
−
T(s)z
ds
+
L
ds
n s
n
(2.31)
10
Nonexpansive mapping
Since limn→∞ βn = 1, we have
tn
zn − xn = 1 − βn xn − 1
T(s)x
ds
n −→ 0.
t
n
(2.32)
0
It follows from (2.27) and (2.32) that
tn
yn − 1
−→ 0.
T(s)x
ds
n
t
n
(2.33)
0
It follows from (2.30) and (2.33) that
tn
x n − 1
T(s)xn ds
−→ 0.
t
n
(2.34)
0
On the other hand, by using Lemma 1.4 we obtain
limsup limsup T(s)
s→∞
n→∞
1
tn
tn
0
T(s)xn ds −
1
tn
tn
0
T(s)xn ds
= 0.
(2.35)
It follows from (2.34) and (2.35) that
limsup limsup T(s)xn − xn = 0.
s→∞
(2.36)
n→∞
Assume that a {xni } is a subsequence of {xn } such that {xni } q ∈ C, then q ∈ F(S)
(by Lemma 1.5). Next we show that q = ΠF(S) x0 and the convergence is strong. Put q =
ΠF(S) x0 , from xn+1 = ΠCn ∩Qn x0 and q ∈ F(S) ⊂ Cn ∩ Qn , we have xn+1 − x0 ≤ q − x0 .
On the other hand, from weakly lower semicontinuity of the norm, we obtain
q − x0 ≤ x0 − q ≤ liminf x0 − xn i→∞
≤ limsup x0 − xn i→∞
i
i
(2.37)
≤ q − x 0 .
It follows from definition of ΠF(S) x0 that we obtain q = ΠF(S) x0 and hence
q − x 0 = q − x 0 .
(2.38)
It follows that xni → q . Since {xni } is an arbitrarily weakly convergent sequence of {xn },
we can conclude that {xn } converges strongly to one point of ΠF(S) x0 . This completes the
proof.
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Yongfu Su: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
E-mail address: suyongfu@tjpu.edu.cn
Xiaolong Qin: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
E-mail address: qxlxajh@163.com