Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 962303, 7 pages
doi:10.1155/2009/962303
Research Article
Strong Convergence Theorems for Infinitely
Nonexpansive Mappings in Hilbert Space
Yi-An Chen
College of Mathematics and Statistics, Chongqing Technology and Business University,
Chongqing 400067, China
Correspondence should be addressed to Yi-An Chen, chenyian1969@sohu.com
Received 23 June 2009; Accepted 12 October 2009
Recommended by Anthony To Ming Lau
We introduce a modified Ishikawa iterative process for approximating a fixed point of two
infinitely nonexpansive self-mappings by using the hybrid method in a Hilbert space and prove
that the modified Ishikawa iterative sequence converges strongly to a common fixed point of two
infinitely nonexpansive self-mappings.
Copyright q 2009 Yi-An Chen. 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
Let C be a nonempty closed convex subset of a Hilbert space H, T a self-mapping of C. Recall
that T is said to be nonexpansive if T x − T y ≤ x − y, for all x, y ∈ C.
Construction of fixed points of nonexpansive mappings via Mann’s iteration 1 has
extensively been investigated in literature see, e.g., 2–5 and reference therein. But the
convergence about Mann’s iteration and Ishikawa’s iteration is in general not strong see
the counterexample in 6. In order to get strong convergence, one must modify them. In
2003, Nakajo and Takahashi 7 proposed such a modification for a nonexpansive mapping
T.
Consider the algorithm,
x0 ∈ C chosen arbitrarity,
yn αn xn 1 − αn T xn ,
Cn v ∈ C : yn − v ≤ xn − v ,
Qn {v ∈ C : xn − v, xn − x0 ≤ 0},
xn1 PCn ∩Qn x0 ,
1.1
2
Fixed Point Theory and Applications
where PC denotes the metric projection from H onto a closed convex subset C of H. They
prove the sequence {xn } generated by that algorithm 1.1 converges strongly to a fixed point
of T provided that the control sequence {αn } is chosen so that supn≥0 αn < 1.
∞
Let {Tn }∞
n1 be a sequence of nonexpansive self-mappings of C, {λn }n1 a sequence of
nonnegative numbers in 0, 1. For each n ≥ 1, defined a mapping Wn of C into itself as
follows:
Un,n1 I,
Un,n λn Tn Un,n1 1 − λn I,
Un,n−1 λn−1 Tn−1 Un,n 1 − λn−1 I,
..
.
Un,k λk Tk Un,k1 1 − λk I,
1.2
Un,k−1 λk−1 Tk−1 Un,k 1 − λk−1 I,
..
.
Un,2 λ2 T2 Un,3 1 − λ2 I,
Wn Un,1 λ1 T1 Un,2 1 − λ1 I.
Such a mapping Wn is called the W-mapping generated by Tn , Tn−1 , . . . , T1 and λn , λn−1 , . . . , λ1 ;
see 8.
In this paper, motivated by 9, for any given xi ∈ C i 0, 1, . . . , q, q ∈ N is a fixed
number, we will propose the following iterative progress for two infinitely nonexpansive
1
2
mappings {Tn } and {Tn } in a Hilbert space H:
x0 , x1 , . . . , xq ∈ C chosen arbitrarity,
1
yn αn xn 1 − αn Wn zn−q ,
2
zn αn xn 1 − αn Wn xn ,
2
2
Cn v ∈ K : yn − v ≤ xn − v2 1 − αn xn−q − x∗ − xn − x∗ 2 ,
Qn v ∈ K : xn − v, xn − xq ≤ 0 ,
xn1 PCn ∩Qn xq , n ≥ q
1
2
and prove, {xn } converges strongly to a fixed point of {Tn } and {Tn }.
We will use the notation:
for weak convergence and → for strong convergence.
ωw xn {x : ∃xnj x} denotes the weak ω-limit set of xn .
1.3
Fixed Point Theory and Applications
3
2. Preliminaries
In this paper, we need some facts and tools which are listed as lemmas below.
Lemma 2.1 see 10. Let H be a Hilbert space, C a nonempty closed convex subset of H, and T
a nonexpansive mapping with FixT /
∅. If {xn } is a sequence in C weakly converging to x and if
{I − T xn } converges strongly to y, then I − T x y.
Lemma 2.2 see 11. Let C be a nonempty bounded closed convex subset of a Hilbert space H.
Given also a real number a ∈ R and x, y, z ∈ H. Then the set D : {v ∈ C : y − v2 ≤ x − v2 z, v a} is closed and convex.
Let {Tn }∞
n1 be a sequence of nonexpansive self-mappings on C, where C is a nonempty closed
convex subset of a strictly convex Banach space E. Given a sequence {λn }∞
n1 in 0, 1, one defines a
of
self-mappings
on
C
by
1.2.
Then
one
has
the
following
results.
sequence {Wn }∞
n1
Lemma 2.3 see 8. Let C be a nonempty closed convex subset of a strictly convex Banach space
∞
∅ and let {λn } be a
E, {Tn }∞
n1 a sequence of nonexpansive self-mappings on C such that n1 FTn /
sequence in 0, b for some b ∈ 0, 1. Then, for every x ∈ C and k ≥ 1 the limit limn → ∞ Un,k x exists.
Remark 2.4. It can be known from Lemma 2.3 that if D is a nonempty bounded subset of C,
then for ε > 0 there exists n0 ≥ k such that supx∈D Un,k x − Uk x ≤ ε for all n > n0 .
Remark 2.5. Using Lemma 2.3, we can define a mapping W : C → C as follows:
Wx lim Wn x lim Un,1 x
n→∞
n→∞
2.1
for all x ∈ C. Such a W is called the W-mapping generated by T1 , T2 , . . . and λ1 , λ2 , . . . . Since
Wn is nonexpansive mapping, W : C → C is also nonexpansive. Indeed, observe that for
each x, y ∈ C,
Wx − Wy lim Wn x − Wn y ≤ x − y.
n→∞
2.2
If {xn } is a bounded sequence in C, then we put D {xn : n ≥ 0}. Hence, it is clear
from Remark 2.4 that for ε > 0 there exists N0 ≥ 1 such that for all n > N0 , Wn xn − Wxn Un,1 xn − U1 xn ≤ supx∈D Un,1 x − U1 x ≤ ε. This implies that
lim Wn xn − Wxn 0.
n→∞
2.3
Lemma 2.6 see 8. Let C be a nonempty closed convex subset of a strictly convex Banach space E.
C such that ∞
∅ and let {λn }
Let {Tn }∞
n1 be a sequence of nonexpansive self-mappings on
n1 FTn /
∞
be a sequence in 0, b for some b ∈ 0, 1. Then, FW n1 FTn .
4
Fixed Point Theory and Applications
3. Strong Convergence Theorem
1
2
Theorem 3.1. Let C be a closed convex subset of a Hilbert space H and let {Wn } and {Wn } be
defined as 1.2. Assume that αn ≤ a for all n and for some 0 < a < 1, and {αn } ∈ b, c for all n and
1 2
FTn /
∅, then {xn } generated by 1.3 converges strongly
0 < b < c < 1. If F ∞
n1 FTn to PF xq .
Proof. Firstly, we observe that Cn is convex by Lemma 2.2. Next, we show that F ⊂ Cn for all
n.
Indeed, for all x∗ ∈ F,
yn − x∗ 2 ≤ αn xn − x∗ 2 1 − αn zn−q − x∗ 2
2
xn − x∗ 2 1 − αn zn−q − x∗ − xn − x∗ 2 ,
2
zn−q − x∗ 2 αn−q xn−q 1 − αn−q Wn−q xn−q − x∗ 2
2 2
αn−q xn−q − x∗ 1 − αn−q Wn−q xn−q − x∗ 2
2
− αn−q 1 − αn−q Wn−q xn−q − xn−q 3.1
2 2
≤ αn−q xn−q − x∗ 1 − αn−q xn−q − x∗ 2
2
− αn−q 1 − αn−q Wn−q xn−q − xn−q 2
2
2
xn−q − x∗ − αn−q 1 − αn−q Wn−q xn−q − xn−q 2
≤ xn−q − x∗ .
Therefore,
yn − x∗ 2 ≤ xn − x∗ 2 1 − αn xn−q − x∗ 2 − xn − x∗ 2 .
3.2
That is x∗ ∈ Cn for all n ≥ q. Next we show that F ⊂ Qn for all n ≥ q.
We prove this by induction. For n q, we have F ⊂ C Qq . Assume that F ⊂ Qn for
all n ≥ q 1, since xn1 is the projection of xq onto Cn Qn , so
xn1 − z, xq − xn1 ≥ 0,
∀z ∈ Cn
Qn .
3.3
As F ⊂ Cn Qn by the induction assumption, the last inequality holds, in particular,
for all x∗ ∈ F. This together with definition of Qn1 implies that F ⊂ Qn1 . Hence F ⊂ Cn Qn
for all n ≥ q.
Notice that the definition of Qn implies xn PQn xq . This together with the fact F ⊂ Qn
further implies xn − xq ≤ x∗ − xq for all x∗ ∈ F.
Fixed Point Theory and Applications
5
The fact xn1 ∈ Qn asserts that xn1 − xn , xn − xq ≥ 0 implies
2
xn1 − xn 2 xn1 − xq − xn − xq 2 2
xn1 − xq − xn − xq − 2xn1 − xn , xn − xq 2 2
≤ xn1 − xq − xn − xq −→ 0 n −→ ∞.
3.4
We now claim that W 1 xn − xn → 0 and W 2 xn − xn → 0. Indeed,
xn − yn 1
xn − Wn zn−q 1 − αn
xn − xn1 xn1 − yn ≤
,
1 − αn
3.5
since xn1 ∈ Cn , we have
yn − xn1 2 ≤ xn − xn1 2 1 − αn xn−q − x∗ 2 − xn − x∗ 2 −→ 0.
3.6
1
xn − Wn zn−q −→ 0.
3.7
Thus
2
2
We now show limn → ∞ Wn xn − xn 0. Let {Wnk xnk − xnk } be any subsequence of
2
{Wn xn − xn }. Since C is a bounded subset of H, there exists a subsequence {xnkj } of {xnk }
such that
lim xnkj − x∗ lim supxnk − x∗ : r.
3.8
j →∞
k→∞
Since
1
1
xnkj − x∗ ≤ xnkj − Wnkj znkj −q Wnkj znkj −q − x∗ 1
≤ xnkj − Wnkj znkj −q znkj −q − x∗ ,
3.9
it follows that r limj → ∞ xnkj − x∗ ≤ lim infj → ∞ znkj − x∗ . By 3.1, we have
2
znkj − x∗ ≤ xnkj − x∗ .
3.10
lim supznkj − x∗ ≤ lim xnkj − x∗ r.
3.11
Hence
j →∞
j →∞
6
Fixed Point Theory and Applications
Thus,
lim znkj − x∗ r lim xnkj − x∗ .
j →∞
j →∞
3.12
Using 3.1 again, we obtain that
2 2 2
2
αnkj −q 1 − αnkj −q Wnkj −q xnkj −q − xnkj −q ≤ xnkj −q − x∗ − znkj −q − x∗ −→ 0.
3.13
2
This imply that limj → ∞ Wnkj xnkj − xnkj 0. For the arbitrariness of {xnk } ⊂ {xn }, we have
2
limn → ∞ Wn xn − xn 0 and
2
zn − xn 1 − αn Wn xn − xn −→ 0.
3.14
Thus, by 3.4, 3.7 and 3.14, we have
1
1
1
1
Wn xn − xn ≤ Wn xn − Wn zn−q Wn zn−q − xn 1
≤ zn−q − xn Wn zn−q − xn 1
≤ Wn zn−q − xn zn−q − xn−q xn−q − xn−q1 3.15
xn−q1 − xn−q2 · · · xn−1 − xn −→ 0.
1
2
Since limn → ∞ Wn xn − W 1 xn 0 and limn → ∞ Wn xn − W 2 xn 0, we have
lim W 1 xn − xn 0,
n→∞
lim W 2 xn − xn 0.
3.16
n→∞
Thus, using 3.16, Lemma 2.1, and the boundedness of {xn }, we get that ∅ /
ωw xn ⊂ F.
Since xn PQn xq and F ⊂ Qn , we have xn − xq ≤ x∗ − xq where x∗ : PF xq . By the weak
lower semicontinuity of the norm, we have w − xq ≤ x∗ − xq for all w ∈ ωw xn . However,
since ωw xn ⊂ F, we must have w x∗ for all w ∈ ωw xn . Hence xn x∗ PF xq and
2
2
xn − x∗ 2 xn − xq 2xn − xq , xq − x∗ xq − x∗ 2 −→ 0.
≤ 2 x∗ − xq xn − xq , xq − x∗
That is, {xn } converges to PF xq .
This completes the proof.
3.17
Fixed Point Theory and Applications
7
Acknowledgment
This work is supported by Grant KJ080725 of the Chongqing Municipal Education
Commission.
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