Theoretical Mathematics & Applications, vol.2, no.2, 2012, 99-114
ISSN: 1792-9687 (print), 1792-9709 (online)
International Scientific Press, 2012
Iteration methods for two nonexpansive
mappings and semigroups on two sets
Nguyen Buong1 and Nguyen Duc Lang2
Abstract
In this paper, we introduce some new iteration methods based on the
hybrid method in mathematical programming, the descent-like iterative
method and the Halpern’s method for finding a common fixed point of
two nonexpansive mappings and nonexpansive semigroups on two closed
and convex subsets in Hilbert spaces.
Mathematics Subject Classification: 47J05, 47H09, 49J30
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, e-mail: nbuong@ioit.ac.vn
Viet Nam, Thainguyen University, University of Sciences,
e-mail: nguyenduclang2002@yahoo.com
Article Info: Received : March 7, 2012. Revised : May 2, 2012
Published online : June 30, 2012
100
Iteration methods for two nonexpansive mappings and semigroups
and convex subset of H. Denote by PC x the metric projection from x ∈ H
onto C. Let T be a nonexpansive mapping on C, i.e., 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 (t1 + t2 ) = T (t1 ) ◦ T (t2 ) for all t1 , t2 > 0; and
(4) for each x ∈ C, the mapping T (.)x from (0, ∞) into C is continuous.
Denote by F = ∩t>0 F (T (t)) the set of common fixed points for the semigroup {T (t) : t > 0}. We know that F is a closed convex subset in H and
F 6= ∅ if C is compact (see, [2]).
Let Ci , i = 1, 2, be two closed and convex subsets in H. Let Ti and
{Ti (t) : t > 0}, i = 1, 2, be two nonexpansive mappings and semigroups
on Ci , respectively. The problems studied in this paper is to find two elements
p ∈ F := F (T1 ) ∩ F (T2 )
(1.1)
q ∈ F1,2 := F1 ∩ F2 ,
(1.2)
and
where Fi = ∩t>0 F (Ti (t)). Assume that F and F1,2 are not empty. Some
particular cases of (1.1) and (1.2) are the following:
(i) when T1 = T2 = I, the indentity mapping in H, (1.1) is the convex feasibility
problem studied in [3].
(ii) when C1 = C2 = C, problems (1.1) and (1.2) are considered in [4]-[6].
For finding a fixed point of a nonexpansive mapping T on C, in 1953, Mann
[7] proposed the following method:
x0 ∈ C
any element,
xn+1 = αn xn + (1 − αn )T xn ,
n ≥ 0,
(1.3)
that converges only weakly, in general (see [8] for an example). In 1967,
Halpern [9] firstly proposed the following iteration process:
xn+1 = βn u + (1 − βn )T xn ,
n ≥ 0,
(1.4)
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N. Buong and N.D. Lang
where u, x0 are two fixed elements in C and {βn } ⊂ (0, 1). He pointed out
P
that the conditions limn→∞ βn = 0 and ∞
n=0 βn = ∞ are necessary in the
sense that, if the iteration (1.4) converges to a fixed point of T , then these
conditions must be satisfied. Further, the iteration method was investigated
by Lions [10], Reich [11], Wittmann [12] and Song [13]. Recently, Alber [14]
proposed the following descent-like method
xn+1 = PC (xn − µn [xn − T xn ]), n ≥ 0,
(1.5)
and proved that if {µn } : µn > 0, 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 ); and
(iii) if F (T ) is a singleton, i.e., F (T ) = {x̃}, then {xn } converges weakly to x̃.
To obtain strong convergence for (1.3), Nakajo and Takahashi [15] introduced the hybrid Mann’s iteration method:
x0 ∈ C,
yn = αn xn + (1 − αn )T xn ,
Cn = {z ∈ C : kyn − zk ≤ kxn − zk},
(1.6)
Qn = {z ∈ C : hxn − z, x0 − xn i ≥ 0},
xn+1 = PCn ∩Qn x0 ,
where {αn } ⊂ [0, a] for some a ∈ [0, 1). They showed that {xn } defined by
(1.6) converges strongly to PF (T ) x0 as n → ∞. Recently, Yanes and Xu [16]
adapted the iteration process (1.4) as follows:
x0 ∈ C
any element,
yn = βn x0 + (1 − βn )T xn ,
Cn = {z ∈ C : kyn − zk2 ≤ kxn − zk2
+ βn (kx0 k2 + 2hxn − x0 , zi)},
(1.7)
Qn = {z ∈ C : hxn − z, x0 − xn i ≥ 0},
xn+1 = PCn ∩Qn x0 .
They proved that if T is a nonexpansive mapping on a closed convex subset C
with F (T ) 6= ∅ and the sequence {βn } ⊂ (0, 1) is chosen such that
lim βn = 0,
n→∞
102
Iteration methods for two nonexpansive mappings and semigroups
then the sequence {xn } defined by (1.7) converges strongly to PF (T ) x0 as n →
∞.
For finding an element p ∈ F, Nakajo and Takahashi [15] also introduced
an iteration procedure as follows:
x0 ∈ C
any element,
1
yn = αn xn + (1 − αn )
tn
Z
tn
T (s)xn ds,
0
Cn = {z ∈ C : kyn − zk ≤ kxn − zk},
(1.8)
Qn = {z ∈ C : hxn − z, x0 − xn i ≥ 0},
xn+1 = PCn ∩Qn x0 , n ≥ 0,
where αn ∈ [0,a] for some a ∈ [0,1) and {tn } is a positive real number divergent
sequence. Under the conditions on {αn } and {tn }, the sequence {xn } defined
by (1.8) converges strongly to PF x0 .
If C ≡ H, then Cn and Qn in (1.6)-(1.8) are two halfspaces. So, the
projection xn+1 onto Cn ∩ Qn in these methods can be found by an explicit
formula [17]. Clearly, if C is a proper subset of H, then Cn and Qn in (1.6)(1.8) 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
(1.6)-(1.8) in a similar form as in [17]? This problem is solved very recently
in [18]-[20]. In this works, Cn and Qn are replaced by two halfspaces and yn is
the right hand side of (1.5) with a modification. In this paper, motivated by
(1.5), (1.7) and [14], [15], to solve problems (1.1) and (1.2) we introduce the
following new iteration processes:
x0 ∈ H
any element,
zn = xn − µn (xn − T1 PC1 xn ),
yn = βn x0 + (1 − βn )T2 PC2 zn ,
Hn = {z ∈ H : kyn − zk2 ≤ kxn − zk2
+ βn (kx0 k2 + 2hxn − x0 , zi)},
Wn = {z ∈ H : hxn − z, x0 − xn i ≥ 0},
xn+1 = PHn ∩Wn x0 , n ≥ 0;
(1.9)
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N. Buong and N.D. Lang
and
x0 ∈ H
any element,
Z
1 tn
z n = xn − µ n xn −
T1 (s)PC1 xn ds ,
tn 0
Z
1 tn
T2 (s)PC2 zn ds,
yn = βn x0 + (1 − βn )
tn 0
Hn = {z ∈ H : kyn − zk2 ≤ kxn − zk2
(1.10)
+ βn (kx0 k2 + 2hxn − x0 , zi)},
Wn = {z ∈ H : hxn − z, x0 − xn i ≥ 0},
xn+1 = PHn ∩Wn x0 ,
n ≥ 0.
We shall prove the strong convergence of the sequences {xn }, {yn } and
{zn } defined by (1.9) and (1.10) to some elements p and q in Sections 2 and 3,
respectively.
Below, the symbols * and → denote weak and strong convergences, respectively.
2
Strong convergence to a common fixed point
of two nonexpansive mappings
We formulate the following facts needed in the proof of our results.
Lemma 2.1. [21] Let H be a real Hilbert. There holds the following identity:
kλx + (1 − λ)yk2 = λkxk2 + (1 − λ)kyk2 − λ(1 − λ)kx − yk2 .
Lemma 2.2. [16] Let C be a nonempty, closed and 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.
Lemma 2.3. (Demiclosedness principle) [21] If C is a nonempty, closed and
convex subset of a real Hilbert space H, T is a nonexpansive mapping on C,
{xn } is a sequence in C such that xn * x and xn − T xn → 0, then x − T x = 0.
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Iteration methods for two nonexpansive mappings and semigroups
Lemma 2.4. [22] Every Hilbert space H has Randon-Riesz property or KadecKlee property, that is, for a sequence {xn } ⊂ H with xn * x and kxn k → kxk,
then there hodls xn → x.
Now, we are in a position to prove the following result.
Theorem 2.5. Let C1 and C2 be two nonempty, closed and convex subsets
in a real Hilbert space H and let T1 and T2 be two nonexpansive mappings on
C1 and C2 , respectively, such that F := F (T1 ) ∩ F (T2 ) 6= ∅. Assume that {µn }
and {βn } are sequences in [0, 1] such that µn ∈ (a, b) for some a, b ∈ (0, 1) and
βn → 0. Then, the sequences {xn }, {zn } and {yn }, defined by (1.9), converge
strongly to the same point u0 = PF x0 , as n → ∞.
Proof. First, note that
kyn − zk2 ≤ kxn − zk2 + βn (kx0 k2 + 2hxn − x0 , zi)
is equivalent to
βn
1
h(1 − βn )xn + βn x0 − yn , zi ≤ hxn − yn , xn i − kyn − xn k2 + kx0 k2 .
2
2
Thus, Hn is a halfspace. 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 that F = F (T̃1 ) ∩ F (T̃2 ) where
T̃i = Ti PCi , i = 1, 2, and T̃i , i = 1, 2, are also two nonexpansive mappings on
H. Hence, by (1.9) and Lemma 2.1, we obtain for any p ∈ F that
kzn − pk2 = k(1 − µn )(xn − p) + µn (T̃1 xn − p)k2
= (1 − µn )kxn − pk2 + µn kT̃1 xn − pk2
− (1 − µn )µn kxn − T̃1 xn k2
≤ (1 − µn )kxn − pk2 + µn kxn − pk2
− (1 − µn )µn kxn − T̃1 xn k2
≤ kxn − pk2 − (1 − µn )µn kxn − T̃1 xn k2 ≤ kxn − pk2 .
(2.1)
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N. Buong and N.D. Lang
By the similar argument and the convexity of k.k2 , we also obtain
kyn − pk2 = kβn x0 + (1 − βn )T̃2 zn − pk2
≤ βn kx0 − pk2 + (1 − βn )kT̃2 zn − T̃2 pk2
≤ βn kx0 − pk2 + (1 − βn )kzn − pk2
≤ βn kx0 − pk2 + (1 − βn )kxn − pk2
= kxn − pk2 + βn (kx0 − pk2 − kxn − pk2 )
= kxn − pk2 + βn (kx0 k2 + 2hxn − x0 , pi).
Therefore, p ∈ Hn for all n ≥ 0. It means that F (T ) ⊂ Hn for all n ≥ 0.
Next, we show by mathematical induction that F (T ) ⊂ Hn ∩ Wn for each
n ≥ 0. For n = 0, we have W0 = H, and hence F (T ) ⊂ H0 ∩ W0 . Suppose
that xi is given and F (T ) ⊂ Hi ∩ Wi for some i > 0. There exists a unique
element xi+1 ∈ Hi ∩ Wi such that xi+1 = PHi ∩Wi x0 . Therefore, by Lemma 2.2,
hxi+1 − x0 , p − xi+1 i ≥ 0
for each p ∈ Hi ∩ Wi . Since F (T ) ⊂ Hi ∩ Wi , we get F (T ) ⊂ Wi+1 . So, we
have F (T ) ⊂ Hi+1 ∩ Wi+1 .
Further, since F (T ) is a nonempty, closed and convex subset of H, there
exists a unique element u0 ∈ F (T ) such that u0 = PF (T ) x0 . From xn+1 =
PHn ∩Wn (x0 ), we obtain
kxn+1 − x0 k ≤ kz − x0 k
for every z ∈ Hn ∩ Wn . As u0 ∈ F (T ) ⊂ Wn , we get
kxn+1 − x0 k ≤ ku0 − x0 k ∀ n ≥ 0.
(2.2)
This implies that {xn } is bounded. Now, we show that
lim kxn+1 − xn k = 0.
n→∞
(2.3)
From the definition of Wn and Lemma 2.2, we have xn = PWn x0 . As xn+1 ∈
Hn ∩ Wn , we obtain
kxn+1 − x0 k ≥ kxn − x0 k ∀ n ≥ 0.
Therefore, {kxn − x0 k} is a nondecreasing and bounded sequence. So, there
exists limn→∞ kxn − x0 k = c. On the other hand, from xn+1 ∈ Wn , it follows
that
hxn − x0 , xn+1 − xn i ≥ 0,
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Iteration methods for two nonexpansive mappings and semigroups
and hence
kxn − xn+1 k2 = kxn − x0 − (xn+1 − x0 )k2
= kxn − x0 k2 − 2hxn − x0 , xn+1 − x0 i + kxn+1 − x0 k2
≤ kxn+1 − x0 k2 − kxn − x0 k2
∀n ≥ 0.
Thus, (2.3) is proved by using the last inequality and limn→∞ kxn − x0 k = c.
Next, since xn+1 ∈ Hn we have that
kyn − xn+1 k2 ≤ kxn − xn+1 k2 + βn (kx0 k + 2hxn − x0 , zi)}.
Therefore, from (2.3), the boundedness of {xn }, βn → 0 and the last inequality,
it follows that
lim kyn − xn+1 k = 0.
(2.4)
n→∞
This together with (2.3) implies that
lim kyn − xn k = 0.
n→∞
(2.5)
Noticing that T̃2 zn = yn − βn (xn − T̃2 zn ) + βn (xn − x0 ), we have
kxn − T̃2 zn k ≤ kxn − yn k + βn kxn − T̃2 zn k + βn kxn − x0 k.
From (2.2) and the last inequality, it follows that
1
kxn − yn k + βn ku0 − x0 k .
kxn − T̃2 zn k ≤
1 − βn
By βn → 0 (βn ≤ 1 − β for some β ∈ (0, 1)), (2.5) and the last inequality, we
obtain
lim kxn − T̃2 zn k = 0.
(2.6)
n→∞
Now, we shall prove that kxn − T̃1 xn k → 0 and kxn − T̃2 xn k → 0, as n → ∞.
Indeed, since {xn } is bounded, for any p ∈ F and any subsequence {T̃1 xnk −
xnk } of {T̃1 xn − xn } there exists a subsequence {xnj } ⊂ {xnk } such that
lim kxnj − pk = lim sup kxnk − pk = a.
j→∞
k→∞
By (2.6), (2.1) and the following inequalities
kxnj − pk ≤ kxnj − T̃2 znj k + kT̃2 znj − pk
≤ kxnj − T̃2 znj k + kznj − pk
≤ kxnj − T̃2 znj k + kxnj − pk,
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N. Buong and N.D. Lang
we get that
lim kxnj − pk = lim kznj − pk = a.
j→∞
j→∞
Again from (2.1) and the condition on µn , it implies that
a(1 − b)kT̃1 xnj − xnj k ≤ kxnj − pk − kznj − pk.
So, kT̃1 xnj − xnj k → 0 and hence kT̃1 xn − xn k → 0, as n → ∞. Further, since
kT̃2 xn − xn k ≤ kT̃2 xn − T̃2 zn k + kT̃2 zn − xn k
≤ kxn − zn k + kT̃2 zn − xn k,
lim kzn − xn k = lim µn kT̃1 xn − xn k = 0,
n→∞
n→∞
(2.7)
by (2.6) and kT̃1 xn −xn k → 0, we also obtain that kT̃2 xn −xn k → 0. Since {xn }
is bounded, there exists a subsequence {xni } of {xn } that convegers weakly to
some element p ∈ H as i → ∞. By Lemmas 2.3 and kT̃1 xn −xn k, kT̃2 xn −xn k →
0, we have that p ∈ F .
Now, from (2.2) and the weak lower semicontinuity of the norm it implies
that
kx0 − u0 k ≤ kx0 − pk ≤ lim inf kx0 − xnj k ≤ lim sup kx0 − xnj k ≤ kx0 − u0 k.
j→∞
j→∞
Thus, we obtain limj→∞ kx0 − xnj k = kx0 − u0 k = kx0 − pk. This implies
xkj → p = u0 by Lemma 2.4. By the uniqueness of the projection u0 = PF x0 ,
we have that xn → u0 . Consequently, from (2.7) it follows that zn → u0 . From
(2.5), we also get that yn → u0 . This completes the proof.
We have the following corollaries.
Corollary 2.6. Let Ci , i = 1, 2, be two nonempty, closed and convex subsets
in a real Hilbert space H. Let Ti , i = 1, 2, be two nonexpansive mappings on
Ci such that F (T1 ) ∩ F (T2 ) 6= ∅. Assume that {µn } is a sequence such that
0 < a ≤ µn ≤ b < 1. Then, the sequences {xn } and {yn }, defined by
x0 ∈ H
any element,
yn = T2 PC2 (xn − µn (xn − T1 PC1 xn )),
Hn = {z ∈ H : kyn − zk ≤ kxn − zk},
Wn = {z ∈ H : hxn − z, x0 − xn i ≥ 0},
xn+1 = PHn ∩Wn (x0 ), n ≥ 0,
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Iteration methods for two nonexpansive mappings and semigroups
converge strongly to the same point u0 = PF (T ) x0 , as n → ∞.
Proof. By putting βn ≡ 0 in Theorem 2.5, we obtain the conclusion.
Corollary 2.7. Let Ci , i = 1, 2, be two nonempty, closed and convex subsets
in a real Hilbert space H such that C := C1 ∩ C2 6= ∅. Assume that {µn } and
{βn } are sequences in [0, 1] such that µn ∈ (a, b) for some a, b ∈ (0, 1) and
βn → 0. Then, the sequences {xn }, {zn } and {yn }, defined by
x0 ∈ H
any element,
zn = xn − µn (xn − PC1 xn ),
yn = βn x0 + (1 − βn )PC2 zn ,
Hn = {z ∈ H : kyn − zk2 ≤ kxn − zk2
+ βn (kx0 k + 2hxn − x0 , zi)},
Wn = {z ∈ H : hxn − z, x0 − xn i ≥ 0},
xn+1 = PHn ∩Wn x0 , n ≥ 0,
converge strongly to the same point u0 = PC x0 , as n → ∞.
Proof. By putting T1 = T2 = I in Theorem 2.5, we obtain the conclusion.
3
Strong convergence to a common fixed point
of two nonexpansive semigroups
We need the following Lemma in the proof of our result.
Lemma 3.1. [23] 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
Z t
Z
1 t
1
T (s)yds −
T (s)yds
lim sup supT (h)
= 0.
t 0
t 0
t→∞ y∈C
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N. Buong and N.D. Lang
Now, we prove the following result.
Theorem 3.2. Let C1 and C2 be two nonempty closed convex subsets in a real
Hilbert space H and let {T1 (t) : t > 0} and {T2 (t) : t > 0} be two nonexpansive
semigroups on C1 and C2 , respectively, such that F = F1 ∩ F2 6= ∅ where
Fi = ∩t>0 F (Ti (t)), i = 1, 2. Assume that {µn } and {βn } are sequences in
[0, 1] such that µn ∈ (a, b) for some a, b ∈ (0, 1) and βn → 0 and {tn } is
a positive real divergent sequence. Then, the sequences {xn }, {zn } and {yn },
defined by (1.10), converge strongly to the same point u0 = PF x0 , as n → ∞.
Proof. For each p ∈ F, we have for each s > 0 that
p = PCi p = T̃i (s)p,
i = 1, 2,
where T̃i (s) = Ti (s)PCi , and hence from (1.10) and Lemma 2.1, we obtain that
Z tn
2
1
2
T̃1 (s)xn )ds − p kzn − pk = (1 − µn )(xn − p) + µn
tn 0
Z tn
2
1
= (1 − µn )(xn − p) + µn
[T̃1 (s)xn − T̃1 (s)p]ds tn 0
2
Z tn
1
T̃
(s)x
−
T̃
(s)pds
= (1 − µn )kxn − pk2 + µn 1
n
1
tn
0
2
Z tn
1
(3.1)
T̃
(s)x
ds
− (1 − µn )µn x
−
1
n
n
tn 0
2
Z
1 tn
2
T̃1 (s)xn ds
≤ kxn − pk − (1 − µn )µn xn −
tn
0
2
≤ kxn − pk .
By the similar argument and the convexity of k.k2 , we also obtain
2
Z tn
1
2
T̃2 (s)zn ds − p kyn − pk = βn (x0 − p) + (1 − βn ) tn
0
2
Z tn
1
2
[T̃2 (s)zn − T̃2 (s)p]ds
≤ βn kx0 − pk + (1 − βn )
tn 0
≤ βn kx0 − pk2 + (1 − βn )kzn − pk2
≤ βn kx0 − pk2 + (1 − βn )kxn − pk2
= kxn − pk2 + βn (kx0 − pk2 − kxn − pk2 )
= kxn − pk2 + βn (kx0 k2 + 2hxn − x0 , pi).
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Iteration methods for two nonexpansive mappings and semigroups
Therefore, p ∈ Hn for n ≥ 0. It means that F ⊂ Hn for n ≥ 0. As in the proof
of Theorem 2.5, we can obtain the following properties:
(i) F ⊂ Hn ∩ Wn ,
kxn+1 − x0 k ≤ ku0 − x0 k, u0 = PF x0
(3.2)
for n ≥ 0. This implies that {xn } is bounded.
(ii)
lim kxn+1 − xn k = 0.
(3.3)
lim kyn − xn+1 k = 0.
(3.4)
lim kyn − xn k = 0.
(3.5)
n→∞
n→∞
n→∞
Noticing that
Z
Z
1 tn
1 tn
T̃2 (s)zn ds = yn − βn xn −
T̃2 (s)zn ds +βn (xn − x0 ),
tn 0
tn 0
we have
1
kxn −
tn
Z
tn
0
T̃2 (s)zn dsk ≤ kxn − yn k
Z
1 tn
T̃2 (s)zn ds
+βn xn −
+βn kxn − x0 k.
tn 0
From (3.2) and the last inequality, it follows that
Z tn
1
xn − 1
kxn − yn k + βn ku0 − x0 k .
T̃2 (s)zn ds≤
tn 0
1 − βn
By βn → 0 (βn ≤ 1 − β for some β ∈ (0, 1)), (3.5) and the last inequality, we
obtain
Z
1 tn
lim xn −
T̃2 (s)zn ds
(3.6)
= 0.
n→∞
tn
0
As in the proof of Theorem 2.5, by using (3.6) we can obtain that
Z tn
1
= 0, i = 1, 2,
T̃
(s)x
ds
lim x
−
i
n
n
n→∞
tn 0
(3.7)
and
lim kxn − zn k = 0.
n→∞
(3.8)
111
N. Buong and N.D. Lang
Since
1
tn
Z
tn
T̃i (s)xn ds ∈ Ci , i = 1, 2,
0
we have that
Z tn
Z tn
1
1
PC xn −
= PC xn − PC
T̃
(s)x
ds
T̃
(s)x
ds
i
n
i
n
i
i
i
tn 0
tn 0
Z tn
1
,
T̃
(s)x
ds
x
−
≤
i
n
n
tn
0
and hence from (3.7) it implies that
Z tn
1
= 0, i = 1, 2.
T̃
(s)x
ds
lim P
x
−
i
n
C
n
i
n→∞
tn 0
(3.9)
Since {xn } is bounded, there exists a subsequence {xnj } of {xn } that converges
weakly to some element q ∈ H as j → ∞. From (3.7) and (3.9), we also obtain
that uinj := PCi xnj → q as j → ∞. It means that q ∈ C1 ∩ C2 . Then, for each
h > 0, we have that
Z tn
1
i
i
i
i
T
(h)u
−
T
(h)
kTi (h)un − un k ≤ T
(s)u
ds
i
i
i
n
n
tn 0
Z tn
Z tn
1
1
i
i
+
T
(s)u
ds
−
T
(s)u
ds
T
(h)
i
i
i
n
n
tn 0
tn 0
Z tn
1
(3.10)
+
Ti (s)uin ds − uin tn
0
Z tn
1
i
i
T
(s)u
ds
−
u
≤ 2
i
n
n
tn
0
Z tn
Z
1
1 tn
i
i
+ T (h)
Ti (s)un ds −
Ti (s)un ds
.
tn 0
tn 0
Let C0i = {z ∈ Ci : kz − u0 k ≤ 2kx0 − u0 k}. Since u0 = PF x0 ∈ Ci , we have
that
kuinj − u0 k = kPCi xnj − PCi u0 k ≤ kxnj − u0 k ≤ 2kx0 − uo k.
So, C0i is a nonempty bounded closed convex subset. It is easy to verify that
{Ti (t) : t > 0} is a nonexpansive semigroup on C0i . By Lemma 3.1, we get
Z tn
Z tn
1
1
i
i
T (s)un ds −
T (s)un ds
lim Ti (h)
= 0
n→∞
tn 0
tn 0
for every fixed h > 0 and hence by (3.9)-(3.10) we obtain that
lim kTi (h)uinj − uinj k = 0
j→∞
112
Iteration methods for two nonexpansive mappings and semigroups
for each h > 0. By Lemma 2.3, q ∈ F (Ti (h)) for all h > 0. It means that
q ∈ F. As in the proof of Theorem 2.5, by using (3.2), (3.5) and (3.8), we
also obtain that the sequences {xn }, {yn } and {zn }, defined by (1.10), converge
strongly to u0 as n → ∞. This completes the proof.
Corollary 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 {βn } is a sequence in [0, 1] such that
βn → 0. Then, the sequences {xn } and {yn }, defined by
x0 ∈ H
any element,
1
yn = βn x0 + (1 − βn )
tn
Z
tn
T (s)PC (xn )ds,
0
Hn = {z ∈ H : kyn − zk2 ≤ kxn − zk2
+ βn (kx0 k + 2hxn − x0 , zi)},
Wn = {z ∈ H : hxn − z, x0 − xn i ≥ 0},
xn+1 = PHn ∩Wn (x0 ), n ≥ 0,
converge strongly to the same point u0 = PF x0 , as n → ∞.
Proof. By putting T1 (s) = I for all s > 0, C1 = H, C2 = C and T2 (s) = T (s)
in Theorem 3.2, we obtain the conclusion.
Corollary 3.4. 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 {αn } is a sequence in [0, 1] such that
αn → 1. Then, the sequences {xn } and {yn }, defined by
x0 ∈ H
any element,
Z
1 tn
1 tn
T (s)PC xn − µn xn −
T (s)PC xn ds ds ,
yn =
tn 0
tn 0
Z
Hn = {z ∈ H : kyn − zk ≤ kxn − zk},
Wn = {z ∈ H : hxn − z, x0 − xn i ≥ 0},
xn+1 = PHn ∩Wn (x0 ), n ≥ 0,
converge strongly to the same point u0 = PF x0 , as n → ∞.
Proof. By putting βn ≡ 0, C2 = H, C1 = C, T2 (s) = I and T1 (s) = T (s) for
all s > 0 in Theorem 3.2, we obtain the conclusion.
N. Buong and N.D. Lang
113
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