Volume 2009, Article ID 962303, 7 pagesdoi:10.1155/2009/962303 Research Article Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space Yi-An Chen College of Ma
Trang 1Volume 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
Copyrightq 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
LetC be a nonempty closed convex subset of a Hilbert space H, T a self-mapping of C Recall
thatT is said to be nonexpansive if Tx − Ty ≤ x − y, for all x, y ∈ C.
Construction of fixed points of nonexpansive mappings via Mann’s iteration1 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 in6 In order to get strong convergence, one must modify them In
2003, Nakajo and Takahashi7 proposed such a modification for a nonexpansive mapping
T.
Consider the algorithm,
x0 ∈ C chosen arbitrarity,
y n α n x n 1 − α n Tx n ,
C nv ∈ C : y n − v ≤ x n − v,
Q n {v ∈ C : x n − v, x n − x0 ≤ 0},
x n1 P C n ∩Q n x0,
1.1
Trang 2whereP C denotes the metric projection fromH onto a closed convex subset C of H They
prove the sequence{x n} generated by that algorithm 1.1 converges strongly to a fixed point
ofT provided that the control sequence {α n} is chosen so that supn≥0 α n < 1.
Let{T n}∞n1be a sequence of nonexpansive self-mappings ofC, {λ n}∞n1a sequence of nonnegative numbers in 0, 1 For each n ≥ 1, defined a mapping W n of C into itself as
follows:
U n,n1 I,
U n,n λ n T n U n,n1 1 − λ n I,
U n,n−1 λ n−1 T n−1 U n,n 1 − λ n−1 I,
U n,k λ k T k U n,k1 1 − λ k I,
U n,k−1 λ k−1 T k−1 U n,k 1 − λ k−1 I,
U n,2 λ2T2U n,3 1 − λ2I,
W n U n,1 λ1T1U n,2 1 − λ1I.
1.2
Such a mappingW nis called theW-mapping generated by T n , T n−1 , , T1andλ n , λ n−1 , , λ1; see8
In this paper, motivated by9, for any given x i ∈ C i 0, 1, , q, q ∈ N is a fixed
number, we will propose the following iterative progress for two infinitely nonexpansive mappings{T n1} and {T n2} in a Hilbert space H:
x0, x1, , x q ∈ C chosen arbitrarity,
y n α n x n 1 − α n W n1z n−q ,
z n α n x n 1 − α n W n2x n ,
C nv ∈ K : y n − v2
≤ x n − v2 1 − α nx
n−q − x∗2
− x n − x∗2
,
Q nv ∈ K : x n − v, x n − x q ≤ 0,
x n1 P C n ∩Q n
x q
, n ≥ q
1.3
and prove,{x n } converges strongly to a fixed point of {T n1} and {T n2}
We will use the notation:
for weak convergence and → for strong convergence.
ω w x n {x : ∃x n x} denotes the weak ω-limit set of x n
Trang 32 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 Fix T / ∅ If {x n } is a sequence in C weakly converging to x and if {I − Tx n } converges strongly to y, then I − Tx 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 {T n}∞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 {λ n}∞n1 in 0, 1, one defines a
sequence {W n}∞
n1 of self-mappings on C by 1.2 Then one has the following results.
Lemma 2.3 see 8 Let C be a nonempty closed convex subset of a strictly convex Banach space
E, {T n}∞n1 a sequence of nonexpansive self-mappings on C such that ∞n1 FT n / ∅ and let {λ n } be a
sequence in 0, b for some b ∈ 0, 1 Then, for every x ∈ C and k ≥ 1 the limit lim n → ∞ U n,k x exists Remark 2.4 It can be known fromLemma 2.3that ifD is a nonempty bounded subset of C,
then forε > 0 there exists n0≥ k such that sup x∈D U n,k x − U k x ≤ ε for all n > n0
Remark 2.5 UsingLemma 2.3, we can define a mappingW : C → C as follows:
Wx lim
n → ∞ W n x lim
for allx ∈ C Such a W is called the W-mapping generated by T1, T2, and λ1, λ2, Since
W n is nonexpansive mapping,W : C → C is also nonexpansive Indeed, observe that for
eachx, y ∈ C,
Wx − Wy lim n → ∞ W n x − W n y ≤ x − y. 2.2
If{x n } is a bounded sequence in C, then we put D {x n : n ≥ 0} Hence, it is clear
fromRemark 2.4that forε > 0 there exists N0≥ 1 such that for all n > N0, W n x n − Wx n
U n,1 x n − U1x n ≤ supx∈D U n,1 x − U1x ≤ ε This implies that
lim
Lemma 2.6 see 8 Let C be a nonempty closed convex subset of a strictly convex Banach space E.
Let {T n}∞n1 be a sequence of nonexpansive self-mappings on C such that ∞n1 FT n / ∅ and let {λ n}
be a sequence in 0, b for some b ∈ 0, 1 Then, FW ∞
n1 FT n .
Trang 43 Strong Convergence Theorem
Theorem 3.1 Let C be a closed convex subset of a Hilbert space H and let {W n1} and {W n2} be
defined as1.2 Assume that α n ≤ a for all n and for some 0 < a < 1, and {α n } ∈ b, c for all n and
0< b < c < 1 If F ∞
n1 FT n1 FT n2 / ∅, then {x n } generated by 1.3 converges strongly
to P F x q .
Proof Firstly, we observe that C nis convex byLemma 2.2 Next, we show thatF ⊂ C nfor all
n.
Indeed, for allx∗∈ F,
y n − x∗2≤ α n x n − x∗2 1 − α nzn−q − x∗2
x n − x∗2 1 − α nz
n−q − x∗2
− x n − x∗2
,
z n−q − x∗2α
n−q x n−q 1− α n−q W n−q2x n−q − x∗
α n−q x n−q − x∗2 1− α n−q W2
n−q x n−q − x∗2
− α n−q 1− α n−q W2
n−q x n−q − x n−q2
≤ α n−q x n−q − x∗2 1− α n−q x n−q − x∗2
− α n−q
1− α n−q W2
n−q x n−q − x n−q2
x n−q − x∗2− α n−q
1− α n−q W2
n−q x n−q − x n−q2
≤x n−q − x∗2.
3.1
Therefore,
y n − x∗2
≤ x n − x∗2 1 − α nx
n−q − x∗2
− x n − x∗2
That isx∗∈ C nfor alln ≥ q Next we show that F ⊂ Q nfor alln ≥ q.
We prove this by induction Forn q, we have F ⊂ C Q q Assume that F ⊂ Q n for alln ≥ q 1, since x n1is the projection ofx qontoC n Q n , so
x n1 − z, x q − x n1 ≥ 0, ∀z ∈ C nQ n 3.3
AsF ⊂ C n Q nby the induction assumption, the last inequality holds, in particular, for allx∗∈ F This together with definition of Q n1implies thatF ⊂ Q n1 HenceF ⊂ C n Q n
for alln ≥ q.
Notice that the definition ofQ nimpliesx n P Q n x q This together with the factF ⊂ Q n
further impliesx n − x q ≤ x∗− x q for all x∗∈ F.
Trang 5The factx n1 ∈ Q nasserts thatx n1 − x n , x n − x q ≥ 0 implies
x n1 − x n2x n1 − x q − x n − x q2
x n1 − x q2−x n − x q2− 2x n1 − x n , x n − x q
≤x n1 − x q2−x n − x q2
−→ 0 n −→ ∞.
3.4
We now claim thatW1x n − x n → 0 and W2x n − x n → 0 Indeed,
x n − W n1z n−q x n − y n
1− α n
≤ x n − x n1 x n1 − y n
1− α n ,
3.5
sincex n1 ∈ C n, we have
y n − x n12
≤ x n − x n12 1 − α nx
n−q − x∗2
− x n − x∗2
−→ 0. 3.6
Thus
x n − W n1z n−q −→ 0. 3.7
We now show limn → ∞ W n2x n − x n 0 Let {W n2k x n k − x n k} be any subsequence of
{W n2x n − x n } Since C is a bounded subset of H, there exists a subsequence {x n kj } of {x n k} such that
lim
j → ∞
x n kj − x∗ lim sup
k → ∞ x n k − x∗ : r. 3.8
Since
x n kj − x∗ ≤x
n kj − W n1kj z n kj −q W1
n kj z n kj −q − x∗
≤x
n kj − W n1kj z n kj −q z
n kj −q − x∗, 3.9
it follows thatr lim j → ∞ x n kj − x∗ ≤ lim infj → ∞ z n kj − x∗ By 3.1, we have
z n kj − x∗ ≤x
n kj − x∗2
Hence
lim sup
j → ∞
z n kj − x∗ ≤ lim
j → ∞
x n kj − x∗ r.
3.11
Trang 6lim
j → ∞
z n kj − x∗ r lim
j → ∞
x n kj − x∗. 3.12 Using3.1 again, we obtain that
α n kj −q1− α n kj −qW2
n kj −q x n kj −q − x n kj −q2
≤x
n kj −q − x∗2
−z
n kj −q − x∗2
−→ 0. 3.13
This imply that limj → ∞ W n2kj x n kj − x n kj 0 For the arbitrariness of {x n k } ⊂ {x n}, we have limn → ∞ W n2x n − x n 0 and
z n − x n 1 − α nW2
n x n − x n −→ 0. 3.14
Thus, by3.4, 3.7 and 3.14, we have
W n1x n − x n ≤W1
n x n − W n1z n−q W1
n z n−q − x n
≤z n−q − x n W n1z n−q − x n
≤W1
n z n−q − x n z n−q − x n−q x n−q − x n−q1
x n−q1 − x n−q2 ··· x n−1 − x n
−→ 0.
3.15
Since limn → ∞ W n1x n − W1x n 0 and limn → ∞ W n2x n − W2x n 0, we have
lim
n → ∞
W1x n − x n 0, lim
n → ∞
W2x n − x n 0. 3.16
Thus, using3.16,Lemma 2.1, and the boundedness of {x n }, we get that ∅ / ω w x n ⊂ F.
Sincex n P Q n x q and F ⊂ Q n, we havex n −x q ≤ x∗−x q where x∗: PF x q By the weak lower semicontinuity of the norm, we havew −x q ≤ x∗−x q for all w ∈ ω w x n However, sinceω w x n ⊂ F, we must have w x∗for allw ∈ ω w x n Hence x n x∗ P F x q and
x n − x∗2x n − x q2 2x n − x q , x q − x∗ x q − x∗2
≤ 2x∗
− x q2 x n − x q , x q − x∗ −→ 0. 3.17
That is,{x n } converges to P F x q
This completes the proof
Trang 7This work is supported by Grant KJ080725 of the Chongqing Municipal Education Commission
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We will use the notation:
for weak convergence and → for strong convergence.