Theoretical Mathematics & Applications, vol.3, no.1, 2013, 35-45
ISSN: 1792-9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
Strong convergence to a common fixed point
of nonexpansive mappings semigroups
Nguyen Buong1 and Nguyen Duc Lang2
Abstract
In this paper, we introduce some new iteration methods based on
the hybrid method in mathematical programming and the descent-like
method for finding a fixed point of a nonexpansive mapping and a common fixed point of a nonexpansive semigroup in Hilbert spaces. The
main results in this paper modify and improve some well-known results
in the literature.
Mathematics Subject Classification : 41A65, 47H17, 47H20
Keywords: Metric projection; Fixed point; Nonexpansive Mappings and
Semigroups
1
Introduction
Let H be a real Hilbert space with the scalar product and the norm denoted
by the symbols h., .i and k.k, respectively, and let C be a nonempty closed
1
2
Vietnamese Academy of Science and Technology Institute of Information Technology
18, Hoang Quoc Viet, Cau Giay, Ha Noi, Viet Nam
Viet Nam, Thainguyen University, University of Sciences,
e-mail: nguyenduclang2002@yahoo.com
Article Info: Received : May 6, 2012. Revised : August 9, 2012
Published online : April 15, 2013
36
Strong convergence to a common fixed point...
and convex subset of H. Denote by PC x the metric projection of an element
x ∈ H onto C. It is well-known that PC is a nonexpansive mapping on H
for any closed convex subset C in H. Recall that a mapping T is said to be
nonexpansive on C, if T : C → C and kT x−T yk ≤ kx−yk for all x, y ∈ C. We
use F (T ) to denote the set of fixed points of T , i.e., F (T ) = {x ∈ C : x = T x}.
We know that F (T ) is nonempty, if C is bounded, for more details see [1].
Let {T (t) : t > 0} be a nonexpansive semigroup on C, that is,
(1) for each t > 0, T (t) is a nonexpansive mapping on C;
(2) T (0)x = x for all x ∈ C;
(3) T (s + t) = T (s) ◦ T (t) for all s, t > 0; and
(4) for each x ∈ C, the mapping T (.)x from (0, ∞) into C is continuous.
Assume that F = ∩t>0 F (T (t)) 6= ∅. We know that F is a closed convex
subset [2] and that F 6= ∅, if C is bounded [3].
For finding a fixed point of a nonexpansive mapping T on C, Alber [4]
proposed the following descent-like method:
xn+1 = PC (xn − µn (I − T )xn ), n ≥ 0, x0 ∈ C,
(1)
where I denotes the identity mapping in H, and proved that if the sequence
of positive real numbers {µn } is chosen such that µn → 0 as n → ∞ and {xn }
is bounded, then:
(i) there exists a weak accumulation point x̃ ∈ C of {xn };
(ii) all weak accumulation points of {xn } belong to F (T );
(iii) if F (T ) is a singleton, i.e., F (T ) = {x̃}, then {xn } converges weakly to x̃.
Motivated by Solodov and Svaiter’s algorithm [5] , Nakajo and Takahashi
[2] introduced the following strongly convergence iteration procedures:
x0 ∈ C
any element,
yn = αn xn + (1 − αn )T xn ,
Cn = {z ∈ C : kyn − zk ≤ kxn − zk},
(2)
Qn = {z ∈ C : hxn − x0 , z − xn i ≥ 0},
xn+1 = PCn ∩Qn (x0 ), n ≥ 0,
where {αn } ⊂ [0, a] for some a ∈ [0, 1), for finding a fixed point of a nonex-
37
Nguyen Buong and Nguyen Duc Lang
pansive mapping T on C, and
x0 ∈ C
any element,
yn = αn xn + (1 − αn )Tn xn ,
Cn = {z ∈ C : kyn − zk ≤ kxn − zk},
(3)
Qn = {z ∈ C : hxn − x0 , z − xn i ≥ 0},
xn+1 = PCn ∩Qn (x0 ), n ≥ 0,
where where Tn is defined by
1
Tn y =
λn
Z
λn
T (s)yds,
0
for each y ∈ C, αn ∈ [0,a] for some a ∈ [0,1) and {λn } is a positive real
number divergent sequence, for finding a common fixed point of a nonexpansive
semigroup {T (t) : t > 0}.
Further, in 2008, Takahashi, Takeuchi and Kubota [6] proposed a simple
variant of (3) that has the following form:
x0 ∈ H, C1 = C, x1 = PC1 x0 ,
yn = αn xn + (1 − αn )Tn xn ,
Cn+1 = {z ∈ Cn : kyn − zk ≤ kxn − zk},
(4)
xn+1 = PCn+1 x0 , n ≥ 1.
They showed (Theorem 4.4 in [6]) that if 0 ≤ αn ≤ a < 1, 0 < λn < ∞ for all
n ≥ 1 and λn → ∞, then {xn } converges strongly to u0 = PF x0 . At the time,
Saejung [7] considered the following analogue without Bochner integral:
x0 ∈ H, C1 = C, x1 = PC1 x0 ,
yn = αn xn + (1 − αn )T (tn )xn ,
Cn+1 = {z ∈ Cn : kyn − zk ≤ kxn − zk},
(5)
xn+1 = PCn+1 x0 , n ≥ 1,
where 0 ≤ αn ≤ a < 1, lim inf n tn = 0, lim supn tn > 0, and limn (tn+1 − tn ) = 0.
Then {xn } converges strongly to u0 = PF x0 .
If C ≡ H, then Cn and Qn in (2)-(5) are two halfspaces. So, the projection
xn+1 onto Cn ∩Qn or Cn+1 in these methods can be found by an explicit formula
38
Strong convergence to a common fixed point...
[5]. Clearly, if C is a proper subset of H, then Cn and Qn in these algorithms
are not two halfspaces. Then, the following problem is posed: how to construct
the closed convex subsets Cn and Qn and if we can express xn+1 of the above
algoritms in a similar form as in [5]? This problem is solved very recently in [8],
[9] and [10]. In the works, Cn and Qn in (2)-(3) are replaced by two halfspaces
and yn is the right hand side of (1) with a modification. In this paper, using
the idea, we present a new variant for (4)-(5) where Cn+1 becomes a halfspace
Hn+1 defined below. More precisely, we consider the following algorithms:
x0 ∈ H = H0 , yn = xn − µn (I − T PC )xn ,
Hn+1 = {z ∈ Hn : kyn − zk ≤ kxn − zk},
(6)
xn+1 = PHn+1 x0 , n ≥ 0,
for finding an element in F (T );
x0 ∈ H = H0 , yn = xn − µn (I − Tn PC )xn ),
Hn+1 = {z ∈ Hn : kyn − zk ≤ kxn − zk},
(7)
xn+1 = PHn+1 x0 , n ≥ 0;
and
x0 ∈ H = H0 , yn = xn − µn (I − T (tn )PC )xn ,
Hn+1 = {z ∈ Hn : kyn − zk ≤ kxn − zk},
(8)
xn+1 = PHn+1 x0 , n ≥ 0,
for finding an element in F.
We shall prove that iteration processes (6) and (7), (8) converge strongly
to a fixed point of T and a common fixed point of {T (t) : t > 0} in sections 2
and 3, respectively.
The symbols * and → denote weak and strong convergences, respectively.
2
Strong convergence to a fixed point of nonexpansive mappings
We formulate the following facts needed in the proof of our results.
39
Nguyen Buong and Nguyen Duc Lang
Lemma 2.1 [11]. Let C be a nonempty closed convex subset of a real Hilbert
space H. For any x ∈ H, there exists a unique z ∈ C such that
kz − xk ≤ ky − xk
for all y ∈ C, and z = PC x if and only if hz − x, y − zi ≥ 0 for all y ∈ C.
Theorem 2.2. Let C be a nonempty closed convex subset in a real Hilbert space
H and let T be a nonexpansive mapping on C such that F (T ) 6= ∅. Assume
that {µn } is a sequence in (a, 1) for some a ∈ (0, 1]. Then, the sequences {xn }
and {yn }, defined by (6), converge strongly to the same point u0 = PF (T ) x0 .
Proof. First, note that kyn − zk ≤ kxn − zk is equivalent to
1
hyn − xn , xn − zi ≤ − kyn − xn k2 .
2
Thus, Hn is a halfspace. Next, we show that F (T ) ⊂ Hn for all n ≥ 0. It is
clear that F (T ) = F (T PC ) := {p ∈ H : T PC p = p} for any mapping T from
C into C. So, we have for each p ∈ F (T ) that
kyn − pk = k(1 − µn )xn + µn T PC xn − pk
= k(1 − µn )(xn − p) + µn (T PC xn − T PC p)k
≤ (1 − µn )kxn − pk + µn kxn − pk
= kxn − pk.
Therefore, p ∈ Hn for all n ≥ 0.
Further, since F (T ) is a nonempty closed convex subset of H, by Lemma
2.1, there exists a unique element u0 ∈ F (T ) such that u0 = PF (T ) x0 . From
xn+1 = PHn+1 x0 , we obtain that
kxn+1 − x0 k ≤ kz − x0 k
for every z ∈ Hn+1 . As u0 ∈ F (T ) ⊂ Hn+1 , we get
kxn+1 − x0 k ≤ ku0 − x0 k ∀n ≥ 0.
(9)
lim kxn+m − xn k = 0,
(10)
Now, we show that
n→∞
40
Strong convergence to a common fixed point...
for each fixed integer m > 0. Indeed, from the definition of Hn+1 , it implies
that Hn+1 ⊆ Hn and hence we have that
kxn − x0 k ≤ kxn+1 − x0 k ∀n ≥ 0.
Therefore, there exists limn kxn − x0 k = c. Next, by Lemma 2.1, xn+m ∈ Hn
and xn = PHn x0 , we get that
hxn − x0 , xn+m − xn i ≥ 0.
Thus,
kxn+m − xn k2 = kxn+m − x0 k2 − kxn − x0 k2 − 2hxn − x0 , xn+m − xn i
≤ kxn+m − x0 k2 − kxn − x0 k2
from that and limn kxn − x0 k = c, (10) is implied. So, {xn } is a Cauchy
sequence. We assume that xn → p ∈ H. On the other hand, from (10) and
the following inequalities
1
kyn − xn k
µn
1
≤ (kyn − xn+m k + kxn+m − xn k)
a
2
≤ kxn+m − xn k,
a
kxn − T PC xn k =
we get
lim kxn − T PC xn k = 0.
n→∞
So, p = T PC p. It means that p ∈ F (T ). Now, from (9) and Lemma 2.1, it
implies that p = u0 . The strong convergence of the sequence {yn } to u0 is
followed from
lim kyn − xn k = lim µn kxn − T PC xn k = 0
n→∞
n→∞
and xn → u0 . This completes the proof.
41
Nguyen Buong and Nguyen Duc Lang
3
Strong convergence to a common fixed point
of nonexpansive semigroups
Lemma 3.1 [12]. Let C be a nonempty bounded closed convex subset in a real
Hilbert space H and let {T (t) : t > 0} be a nonexpansive semigroup on C.
Then, for any h > 0
°
°
µ Z t
¶
Z t
°
°
1
1
°= 0.
T
(h)
T
(s)yds
−
T
(s)yds
lim sup sup°
°
°
t 0
t 0
t→∞ y∈C
Theorem 3.2. Let C be a nonempty closed convex subset in a real Hilbert
space H and let {T (t) : t > 0} be a nonexpansive semigroup on C such that
F = ∩t>0 F (T (t)) 6= ∅. Assume that {µn } is a sequence in (a, 1] for some
a ∈ (0, 1] and {λn } is a positive real number divergent sequence. Then, the
sequences {xn } and {yn } defined by (7), converge strongly to the same point
u0 = P F x 0 .
Proof. For each p ∈ F ⊆ C, we have from (7) and p = PC p that
°
¶°
µ Z λn
°
°
1
°
kyn − pk = °(1 − µn )(xn − p) + µn
T (s)PC xn ds − p °
°
λn
°
° Z λ0n
°
°1
°
(T
(s)P
x
−
T
(s)P
p)ds
≤ (1 − µn )kxn − pk + µn °
C
n
C
°
° λn
0
Z λn
1
≤ (1 − µn )kxn − pk + µn
kxn − pkds
λn 0
= kxn − pk.
Therefore, p ∈ Hn . It means that F ⊂ Hn for all n ≥ 0. As in the proof
of Theorem 2.2, we get that {xn } is well defined, it converges strongly to an
element p ∈ H, and
°
°
Z λn
°
°
1
xn −
T (s)PC xn ds°
lim °
°= 0,
°
n→∞
λn 0
kxn+1 − x0 k ≤ ku0 − x0 k,
where u0 = PF x0 . Since
1
λn
Z
λn
T (s)PC xn ds ∈ C
0
(11)
42
Strong convergence to a common fixed point...
and PC is a nonexpansive mapping, we have that
°
° °
°
Z λn
Z λn
°
° °
°
1
1
°PC xn −
° = °PC xn − PC
°
T
(s)P
x
ds
T
(s)P
x
ds
C
n
C
n
°
° °
°
λn 0
λn 0
°
°
Z λn
°
°
1
≤°
T (s)PC xn ds°
°x n − λ n
°.
0
So, we obtain from (11) that
°
°
Z λn
°
°
1
°= 0.
P
x
−
lim °
T
(s)P
x
ds
C
n
C
n
°
n→∞°
λn 0
(12)
This together with (11) and xn → p implies that the sequence {PC xn } also
converges strongly to p. Since C is closed, we get p ∈ C.
On the other hand, we have for each h > 0 that
°
µ Z λn
¶°
°
°
1
°
kT (h)PC xn − PC xn k ≤ °T (h)PC xn − T (h)
T (s)PC xn ds °
°
λn 0
°
°
¶
µ Z λn
Z λn
°
°
1
1
T (s)PC xn ds −
T (s)PC xn ds°
+°
°
°T (h) λn
λn 0
0
° Z λn
°
°1
°
°
(13)
+°
T
(s)P
x
ds
−
P
x
C
n
C
n
° λn
°
0
°
° Z λn
°
°1
°
T (s)PC xn ds − PC xn °
≤ 2°
°
λn 0
°
°
¶
µ Z λn
Z λn
°
°
1
1
°.
T
(s)P
x
ds
−
T
(s)P
x
ds
+°
T
(h)
C
n
C
n
°
°
λn 0
λn 0
Let C0 = {z ∈ C : kz − u0 k ≤ 2kx0 − u0 k}. Since u0 = PF x0 ∈ C, we have
from (11) and
kPC xn − u0 k = kPC xn − PC u0 k
≤ kxn − u0 k
≤ kxn − x0 k + kx0 − u0 k
≤ 2kx0 − u0 k.
So, C0 is a nonempty bounded closed convex subset. It is easy to verify that
{T (t) : t > 0} also is a nonexpansive semigroup on C0 . By Lemma 3.1, (13)
and PC xn → p, we get p = T (h)p for each h > 0. So, p ∈ F. Again, from (11)
and p ∈ F, it implies that p = u0 and yn → u0 as n → ∞. This completes the
proof.
43
Nguyen Buong and Nguyen Duc Lang
Theorem 3.3. Let C be a nonempty closed convex subset in a real Hilbert
space H and let {T (t) : t > 0} be a nonexpansive semigroup on C such that
F = ∩t>0 F (T (t)) 6= ∅. Assume that {µn } is a sequence in (a, 1] for some a ∈
(0, 1] and {tn } is a sequence of positive real numbers satisfying the condition
lim inf n tn = 0, lim supn tn > 0, and limn (tn+1 − tn ) = 0. Then, the sequences
{xn } and {yn } defined by (8), converge strongly to the same point u0 = PF x0 .
Proof. As in the proof of Theorems 2.2 and 3.2, we get
kxn+1 − x0 k ≤ ku0 − x0 k,
lim kxn − T (tn )PC xn k = 0,
n→∞
lim kPC xn − T (tn )PC xn k = 0,
n→∞
(14)
(15)
and the sequence {xn } and {PC xn } also converge strongly to p ∈ C.
Without loss of generality, as in [7], let
kPC xnj − T (tnj )PC xnj k
= 0.
j→∞
t nj
lim tnj = lim
j→∞
(16)
Now, we prove that p = T (t)p for a fixed t > 0. It is easy to see that
[t−tnj ]−1
X
kPC xnj − T (t)pk ≤
kT (ltnj )PC xnj − T ((l + 1)tnj )PC xkj k
°l=0µ· ¸¶
µ· ¸¶ °
°
°
t
t
°
P
z
p
−
T
+°
T
C
n
j
°
°
tnj
t nj
°
° µ· ¸¶
°
°
t
°
p
−
T
(t)p
+°
T
°
°
tkj
t
kPC xnj − T (tnj )PC xnj k + kPC xnj − pk
≤
tnj
°
° µ
· ¸ ¶
°
°
t
°
+°
°T t − tn tnj p − p°.
j
Therefore,
kPC xnj − T (t)pk ≤
t
tnj
kPC xnj − T (tnj )PC xnj k
+ kPC xnj − pk + sup{kT (s)p − pk : 0 ≤ s ≤ tnj }.
This fact, together with (16) and property (4) for the semigroup, implies that
lim kPC xnj − T (t)pk = 0.
j→∞
44
Strong convergence to a common fixed point...
Therefore, p ∈ F . So, from (14), we have that the sequence {xn } converges
strongly to u0 as n → ∞. The strong convergence of the sequence {yn } to u0
is followed from (8), (14), µn ∈ (a, 1] and xn → u0 as n → ∞. The theorem is
proved.
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