Iterative techniques for approximat-ing fixed points of nonexpansive mappapproximat-ings have been studied by various authors see, e.g., [1,4–10], etc., using famous Mann iteration metho
Trang 1Volume 2007, Article ID 28619, 8 pages
doi:10.1155/2007/28619
Research Article
An Iteration Method for Nonexpansive Mappings
in Hilbert Spaces
Lin Wang
Received 22 August 2006; Revised 2 November 2006; Accepted 2 November 2006 Recommended by Nan-Jing Huang
In real Hilbert spaceH, from an arbitrary initial point x0∈ H, an explicit iteration scheme
is defined as follows: x n+1 = α n x n+ (1− α n)T λ n+1 x n,n ≥0, where T λ n+1 x n = Tx n −
λ n+1 μF(Tx n),T : H → H is a nonexpansive mapping such that F(T) = { x ∈ K : Tx = x }
is nonempty,F : H → H is a η-strongly monotone and k-Lipschitzian mapping, { α n } ⊂
(0, 1), and{ λ n } ⊂[0, 1) Under some suitable conditions, the sequence{ x n }is shown to converge strongly to a fixed point ofT and the necessary and sufficient conditions that { x n }converges strongly to a fixed point ofT are obtained.
Copyright © 2007 Lin Wang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited
1 Introduction
LetH be a Hilbert space with inner product ·,·and norm · A mappingT : H → H is
said to be nonexpansive if Tx − T y ≤ x − y for anyx, y ∈ H A mapping F : H → H is
said to beη-strongly monotone if there exists constant η > 0 such that Fx − F y,x − y ≥
η x − y 2for anyx, y ∈ H F : H → H is said to be k-Lipschitzian if there exists constant
k > 0 such that Fx − F y ≤ k x − y for anyx, y ∈ H.
The interest and importance of construction of fixed points of nonexpansive map-pings stem mainly from the fact that it may be applied in many areas, such as imagine recovery and signal processing (see, e.g., [1–3]) Iterative techniques for approximat-ing fixed points of nonexpansive mappapproximat-ings have been studied by various authors (see, e.g., [1,4–10], etc.), using famous Mann iteration method, Ishikawa iteration method, and many other iteration methods such as, viscosity approximation method [6] and CQ method [7]
Let F : H → H be a nonlinear mapping and K nonempty closed convex subset of
H The variational inequality problem is formulated as finding a point u ∗ ∈ K such
Trang 2
VI(F,K)
F
u ∗
,v − u ∗
The variational inequalities were initially studied by Kinderlehrer and Stampacchia [11], and ever since have been widely studied It is well known that the VI(F,K) is equivalent
to the fixed point equation
u ∗ = P K
u ∗ − μF
u ∗
where P K is the projection from H onto K and μ is an arbitrarily fixed constant In
fact, whenF is an η-strongly monotone and Lipschitzian mapping on K and μ > 0 small
enough, then the mapping defined by the right-hand side of (1.2) is a contraction For reducing the complexity of computation caused by the projectionP K, Yamada [12] proposed an iteration method to solve the variational inequalities VI(F,K) For arbitrary
u0∈ H,
u n+1 = Tu n − λ n+1 μF
T
u n
whereT is a nonexpansive mapping from H into itself, K is the fixed point set of T, F
is anη-strongly monotone and k-Lipschitzian mapping on K, { λ n }is a real sequence in [0, 1), and 0< μ < 2η/k2 Then Yamada [12] proved that{ u n }converges strongly to the unique solution of the VI(F,K) as { λ n }satisfies the following conditions:
(1) limn →∞ λ n =0,
(2)∞
n =0λ n = ∞,
(3) limn →∞(λ n − λ n+1)/λ2
n+1 =0
Motivated by the above work, we propose a new explicit iteration scheme with map-pingF to approximate the fixed point of nonexpansive mapping T in Hilbert space The
strong and weak convergence theorems to a fixed point ofT are obtained The necessary
and sufficient conditions for strong convergence of this iteration scheme are obtained, too
2 Preliminaries
LetT be a nonexpansive mapping from H into itself, F : H → H an η-strongly
mono-tone andk-Lipschitzian mapping, { λ n } ⊂(0, 1),{ λ n } ⊂[0, 1), andμ a fixed constant in
(0, 2η/k2) Starting with an initial pointx0∈ H, the explicit iteration scheme with
map-pingF is defined as follows:
x n+1 = α n x n+
1− α n
Tx n − λ n+1 μF
Tx n
, n ≥0. (2.1) For simplicity, we define a mappingT λ:H → H by
Then (2.1) may be written as follows:
x n+1 = α n x n+
1− α n
T λ n+1 x n, n ≥0. (2.3)
Trang 3In fact, asλ n =0,n ≥1, then the iteration scheme (2.3) reduces to the famous Mann iteration scheme
A Banach spaceE is said to satisfy Opial’s condition if for any sequence { x n }inE,
x n x implies that limsup n →∞ x n − x < limsup n →∞ x n − y for ally ∈ E with y x,
wherex n x denotes that { x n }converges weakly tox It is well known that every Hilbert
space satisfies Opial’s condition
A mappingT : K → E is said to be semicompact if, for any sequence { x n }inK such that
x n − Tx n →0 (n → ∞), there exists subsequence{ x n j }of{ x n }such that{ x n j }converges strongly tox ∗ ∈ K.
A mappingT with domain D(T) and range R(T) in E is said to be demiclosed at p; if
whenever{ x n }is a sequence inD(T) such that { x n }converges weakly tox ∗ ∈ D(T) and { Tx n }converges strongly top, then Tx ∗ = p.
Lemma 2.1 [13] Let { α n } and { t n } be two nonnegative sequences satisfying
α n+1 ≤1 +a n
α n+b n, ∀ n ≥1. (2.4)
If∞
n =1a n < ∞ and∞
n =1b n < ∞ , then lim n →∞ α n exists.
Lemma 2.2 [12] Let T λ x = Tx − λμF(Tx), where T : H → H is a nonexpansive mapping from H into itself and F is an η-strongly monotone and k-Lipschitzian mapping from H into itself If 0 ≤ λ < 1 and 0 < μ < 2η/k2, then T λ is a contraction and satisfies
T λ x − T λ y ≤(1− λτ) x − y , ∀ x, y ∈ H, (2.5)
where τ =1−1− μ(2η − μk2).
Lemma 2.3 [14] Let K be a nonempty closed convex subset of a real Hilbert space H and T
a nonexpansive mapping from K into itself If T has a fixed point, then I − T is demiclosed
at zero, where I is the identity mapping of H, that is, whenever { x n } is a sequence in K weakly converging to some x ∈ K and the sequence {(I − T)x n } strongly converges to some
y, it follows that (I − T)x = y.
3 Main results
Lemma 3.1 Let H be a Hilbert space, T : H → H a nonexpansive mapping with F(T) φ, and F : H → H an η-strongly monotone and k-Lipschitzian mapping For any given x0∈ H, { x n } is defined by
x n+1 = α n x n+
1− α n
T λ n+1 x n, n ≥0, (3.1)
where { α n } and { λ n } ⊂ [0, 1) satisfy the following conditions:
(1)α ≤ α n ≤ β for some α,β ∈ (0, 1);
(2)∞
n =1λ n < ∞ ;
(3) 0< μ < 2η/k2.
Then,
(1) limn →∞ x n − q exists for each q ∈ F(T);
(2) limn →∞ x n − Tx n = 0.
Trang 4Proof (1) For any q ∈ F(T), we have
x
n+1 − q 2
=α
n
x n − q
+
1− α n
T λ n+1 x n − q 2
= α nx n − q 2
+
1− α nT λ n+1 x n − q 2− α n
1− α nx n − T λ n+1 x n 2
, (3.2) where (byLemma 2.2)
T λ n+1 x n − q = T λ n+1 x n − T λ n+1 q + T λ n+1 q − q
≤T λ n+1 x n − T λ n+1 q+T λ n+1 q − q
≤1− λ n+1 τx n − q+λ n+1 μF(q). (3.3)
Furthermore,
T λ n+1 x n − q 2
≤1− λ n+1 τx n − q 2
+λ n+1 μ2
τ F(q) 2
Thus,
x n+1 − q 2
≤ α nx n − q 2
+
1− α n
1− λ n+1 τx n − q 2
+
1− α nλ n+1 μ2
τ F(q) 2
− α n
1− α nx n − T λ n+1 x n 2
≤ α nx n − q 2
+
1− α n
1− λ n+1 τx n − q 2
+
1− α nλ n+1 μ2
τ F(q) 2
− α nx
n+1 − x n 2
≤x n − q 2
+λ n+1 μ2
τ F(q) 2
− α nx n+1 − x n 2
.
(3.5)
Since∞
n =1λ n < ∞, it follows fromLemma 2.1that limn →∞ x n − q exists for eachq ∈ F(T) It also implies that { x n }is bounded
(2) From (3.5), we have
αx n+1 − x n 2
≤ α nx n+1 − x n 2
≤x n − q 2
−x n+1 − q 2
+λ n+1 μ2
τ F(q) 2
. (3.6) Therefore, limn →∞ x n+1 − x n =0 In addition,
(1− β)x
n − T λ n+1 x n ≤ 1− α nx n − T λ n+1 x n = x n+1 − x n. (3.7)
Hence, limn →∞ x n − T λ n+1 =0 Thus,
x n − Tx n = x n − T λ n+1 x n+T λ n+1 x n − Tx n
≤x n − T λ n+1 x n+λ n+1 μF
Tx n. (3.8) Since { x n } is bounded, then { Tx n } and { F(Tx n)} are bounded, as well Therefore, limn →∞ x n − Tx n =0 The proof is completed
Trang 5Theorem 3.2 Let H be a Hilbert space, T : H → H a nonexpansive mapping with F(T)
φ, and F : H → H an η-strongly monotone and k-Lipschitzian mapping For any given x0∈
H, { x n } is defined by
x n+1 = α n x n+
1− α n
T λ n+1 x n, n ≥0, (3.9)
where { α n } and { λ n } ⊂ [0, 1) satisfy the following conditions:
(1)α ≤ α n ≤ β for some α,β ∈ (0, 1);
(2)∞
n =1λ n < ∞ ;
(3) 0< μ < 2η/k2.
Then,
(1){ x n } converges weakly to a fixed point of T;
(2){ x n } converges strongly to a fixed point of T if and only if liminf n →∞ d(x n,F(T)) = 0.
Proof (1) It follows fromLemma 3.1that{ x n }is bounded Thus, letq1andq2be weak limits of subsequences{ x n k }and{ x n j }of{ x n }, respectively It follows from Lemmas2.3 and3.1thatq1,q2∈ F(T) Assume q1 q2, then by Opial’s condition, we obtain
lim
n →∞x
n − q1 =lim
k →∞
x
n k − q1 < lim
k →∞
x
n k − q2
=lim
j →∞x n
j − q2 < lim
k →∞
x n
k − q1 =lim
n →∞x n − q1, (3.10) which is a contradiction; hence,q1= q2 Then,{ x n }converges weakly to a common fixed point ofT.
(2) Suppose that{ x n }converges strongly to a fixed pointq of T, then lim n →∞ x n −
q =0 Since 0≤ d(x n,F(T)) ≤ x n − q , we have lim infn →∞ d(x n,F(T)) =0
Conversely, suppose that lim infn →∞ d(x n,F(T)) =0 For any p ∈ F(T), F(p) ≤
F(p) − F(x n)+ F(x n) ≤ k x n − p + F(x n) Since{ x n }and{ F(x n)}are bounded,
F(p) is bounded for any p ∈ F(T), that is, there exists constant M > 0
such that F(p) ≤ M for all p ∈ F(T) In addition, it follows from (3.5) that
x n+1 − p 2
≤x n − p 2
+λ n+1 μ2
τ F(p) 2
So,
x
n+1 − p 2
≤x
n − p 2
+λ n+1 μ2
τ
2k2 x
n − p 2
+ 2F
x n 2
=
1 + 2k2λ n+1 μ
2
τ
x n − p 2
+ 2λ n+1 μ2
τ F
x n 2
.
(3.12)
Thus,
d
x n+1,F(T) 2
≤ 1 + 2k2λ n+1 μ2
x n,F(T) 2
+ 2λ n+1 μ2
τ F
x n 2
. (3.13)
In addition, we obtain that∞
n =12k2(λ n+1 μ2/τ) < ∞and∞
n =12(λ n+1 μ2/τ) F(x n)2<
∞ since ∞
n =1λ n < ∞ and { F(x n)} is bounded It follows from Lemma 2.1 that
Trang 6limn →∞ d(x n,F(T)) exists Furthermore, since liminf n →∞ d(x n,F(T)) =0, we have limn →∞ d(x n,F(T)) =0 We now prove that{ x n }is a Cauchy sequence
TakingM1=max{2e(2μ2
k2
/τ)∞
i =1λ i, 4(μ2M2/τ)e(2μ2
k2
/τ)∞
i =1λ i }, for any > 0, there exists
positive integerN such that d(x n,F(T)) <
/4M1and∞
i = n λ i < /4M1asn ≥ N Taking
q ∈ F(T), for any n,m ≥ N, it follows from (3.12) that
x n − x m 2
2 ≤x n − q 2
+x m − q 2
≤
1 + 2k2λ n μ2
τ
x n −1− q 2
+ 2λ n μ2
τ F
x n −1 2
+
1 + 2k2λ m μ2
τ
x m −1− q 2
+ 2λ m μ2
τ F
x m −1 2
≤
1 + 2k2λ n μ2
τ
x n −1− q 2
+ 2λ n μ2
2
+
1 + 2k2λ m μ2
τ
x m −1− q 2
+ 2λ m μ2
2
≤ n
i = N+1
1 + 2k2λ i μ2
τ
x N − q 2
+
n−1
i = N+1
2λ i μ2
2 n
j = i+1
1 + 2k2λ j μ2
τ
+ 2λ n μ2
2+
m
i = N+1
1 + 2k2λ i μ2
τ
x N − q 2
+
m−1
i = N+1
2λ i μ2
2 m
j = i+1
1 + 2k2λ j μ2
τ
+ 2λ m μ2
2
≤2e(2μ2k2/τ)∞
i = N+1 λ ix
N − q 2
+ 4μ2M2
(2μ2k2/τ)∞
i = N+1 λ i
∞
i = N+1
λ i
(3.14) Thus,
x n − x m 2
≤2M1 x N − q 2
+ 2M1
∞
i = N+1
Taking the infimum for allq ∈ F(T), we have
x n − x m 2
≤2M1 d
x N,F(T) 2
+ 2M1
∞
i = N+1
λ i < (3.16)
This implies that{ x n }is a Cauchy sequence Therefore, there existsp ∈ H such that { x n }
converges strongly top It follows fromLemma 3.1that
Trang 7Corollary 3.3 Under the conditions of Lemma 3.1 , if T is completely continuous, then { x n } converges strongly to a fixed point of T.
Proof ByLemma 3.1,{ x n }is bounded and limn →∞ x n − Tx n =0, then{ Tx n }is also bounded SinceT is completely continuous, there exists subsequence { Tx n j }of { Tx n }
such thatTx n j → p as j → ∞ It follows fromLemma 3.1that limj →∞ x n j − Tx n j =0
So by the continuity ofT andLemma 2.3, we have limj →∞ x n j − p =0 andp ∈ F(T).
Furthermore, byLemma 3.1, we get that limn →∞ x n − p exists Thus, limn →∞ x n − p =
Corollary 3.4 Under the conditions of Lemma 3.1 , if T is demicompact, then { x n } con-verges strongly to a fixed point of T.
Proof Since T is demicompact, { x n }is bounded and limn →∞ x n − Tx n =0, then there exists subsequence{ x n j }of{ x n }such that{ x n j }converges strongly toq ∈ H It follows
fromLemma 2.3 thatq ∈ F(T) Thus, lim n →∞ x n − q exists byLemma 3.1 Since the subsequence{ x n j }of{ x n }such that{ x n j }converges strongly toq, then { x n }converges strongly to the common fixed pointq ∈ F(T) The proof is completed. For studying the strong convergence of fixed points of a nonexpansive mapping, Sen-ter and Dotson [9] introduced Condition (A) Later on, Maiti and Ghosh [5] well as Tan and Xu [10] studied Condition (A) and pointed out that Condition (A) is weaker than
the requirement of demicompactness for nonexpansive mappings A mappingT : K → K
withF(T) = { x ∈ K : Tx = x φ is said to satisfy condition (A) if there exists a
non-decreasing function f : [0, ∞)→[0,∞) with f (0) =0 and f (t) > 0 for all t ∈(0,∞) such that x − Tx ≥ f (d(x,F(T))) for all x ∈ K, where d(x,F(T)) =inf{ x − q :q ∈ F(T) }
Theorem 3.5 Under the conditions of Lemma 3.1 , if T satisfies condition (A), then { x n } converges strongly to a fixed point of T.
Proof Since T satisfies condition (A), then f (d(x n,F(T))) ≤ x n − Tx n It follows from Lemma 3.1that lim infn →∞ d(x n,F(T)) =0 Thus, it follows fromTheorem 3.2that{ x n }
converges strongly to a fixed point ofT The proof is completed.
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Lin Wang: Department of Mathematics, Kunming Teachers College, Kunming,
Yunnan 650031, China
Email address:wl64mail@yahoo.com.cn
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