Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 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 space H, from an arbitrary initial point x0 ∈ H, an explicit iteration scheme
is defined as follows: xn+1 = αn xn + (1 − αn )T λn+1 xn ,n ≥ 0, where T λn+1 xn = Txn −
λn+1 μF(Txn ), 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, {αn } ⊂
(0,1), and {λn } ⊂ [0,1). Under some suitable conditions, the sequence {xn } is shown to
converge strongly to a fixed point of T and the necessary and sufficient conditions that
{xn } converges strongly to a fixed point of T 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 reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let H be a Hilbert space with inner product ·, · and norm · . A mapping T : H → H is
said to be nonexpansive if Tx − T y ≤ x − y for any x, y ∈ H. A mapping F : H → H is
said to be η-strongly monotone if there exists constant η > 0 such that Fx − F y,x − y ≥
ηx − y 2 for any x, y ∈ H. F : H → H is said to be k-Lipschitzian if there exists constant
k > 0 such that Fx − F y ≤ kx − y for any x, y ∈ H.
The interest and importance of construction of fixed points of nonexpansive mappings 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 approximating fixed points of nonexpansive mappings 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
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Fixed Point Theory and Applications
that
VI(F,K) F u∗ ,v − u∗ ≥ 0,
∀v ∈ K.
(1.1)
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∗ = PK u∗ − μF u∗ ,
(1.2)
where PK is the projection from H onto K and μ is an arbitrarily fixed constant. In
fact, when F 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 projection PK , Yamada [12]
proposed an iteration method to solve the variational inequalities VI(F,K). For arbitrary
u0 ∈ H,
un+1 = Tun − λn+1 μF T un ,
n ≥ 0,
(1.3)
where T is a nonexpansive mapping from H into itself, K is the fixed point set of T, F
is an η-strongly monotone and k-Lipschitzian mapping on K, {λn } is a real sequence in
[0,1), and 0 < μ < 2η/k2 . Then Yamada [12] proved that {un } converges strongly to the
unique solution of the VI(F,K) as {λn } satisfies the following conditions:
(1) limn→∞ λn = 0,
(2) ∞
n =0 λ n = ∞ ,
(3) limn→∞ (λn − λn+1 )/λ2n+1 = 0.
Motivated by the above work, we propose a new explicit iteration scheme with mapping F to approximate the fixed point of nonexpansive mapping T in Hilbert space. The
strong and weak convergence theorems to a fixed point of T are obtained. The necessary
and sufficient conditions for strong convergence of this iteration scheme are obtained,
too.
2. Preliminaries
Let T be a nonexpansive mapping from H into itself, F : H → H an η-strongly monotone and k-Lipschitzian mapping, {λn } ⊂ (0,1), {λn } ⊂ [0,1), and μ a fixed constant in
(0,2η/k2 ). Starting with an initial point x0 ∈ H, the explicit iteration scheme with mapping F is defined as follows:
xn+1 = αn xn + 1 − αn Txn − λn+1 μF Txn ,
n ≥ 0.
(2.1)
For simplicity, we define a mapping T λ : H → H by
T λ x = Tx − λμF(Tx),
∀x ∈ H.
(2.2)
Then (2.1) may be written as follows:
xn+1 = αn xn + 1 − αn T λn+1 xn ,
n ≥ 0.
(2.3)
Lin Wang 3
In fact, as λn = 0, n ≥ 1, then the iteration scheme (2.3) reduces to the famous Mann
iteration scheme.
A Banach space E is said to satisfy Opial’s condition if for any sequence {xn } in E,
xn x implies that limsupn→∞ xn − x < limsupn→∞ xn − y for all y ∈ E with y = x,
where xn x denotes that {xn } converges weakly to x. It is well known that every Hilbert
space satisfies Opial’s condition.
A mapping T : K → E is said to be semicompact if, for any sequence {xn } in K such that
xn − Txn → 0 (n → ∞), there exists subsequence {xn j } of {xn } such that {xn j } converges
strongly to x∗ ∈ K.
A mapping T with domain D(T) and range R(T) in E is said to be demiclosed at p; if
whenever {xn } is a sequence in D(T) such that {xn } converges weakly to x∗ ∈ D(T) and
{Txn } converges strongly to p, then Tx∗ = p.
Lemma 2.1 [13]. Let {αn } and {tn } be two nonnegative sequences satisfying
αn+1 ≤ 1 + an αn + bn ,
If
∞
n =1 a n
< ∞ and
∞
n=1 bn
∀n ≥ 1.
(2.4)
< ∞, then limn→∞ αn 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
λ
T 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 {xn } is a sequence in K
weakly converging to some x ∈ K and the sequence {(I − T)xn } 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,
{xn } is defined by
xn+1 = αn xn + 1 − αn T λn+1 xn ,
n ≥ 0,
where {αn } and {λn } ⊂ [0,1) satisfy the following conditions:
(1) α ≤ αn ≤ β for some α,β ∈ (0,1);
(2) ∞
n =1 λ n < ∞ ;
(3) 0 < μ < 2η/k2 .
Then,
(1) limn→∞ xn − q exists for each q ∈ F(T);
(2) limn→∞ xn − Txn = 0.
(3.1)
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Fixed Point Theory and Applications
Proof. (1) For any q ∈ F(T), we have
xn+1 − q2 = αn xn − q + 1 − αn T λn+1 xn − q 2
2 2
= αn xn − q + 1 − αn T λn+1 xn − q2 − αn 1 − αn xn − T λn+1 xn ,
(3.2)
where (by Lemma 2.2)
λ
T n+1 xn − q = T λn+1 xn − T λn+1 q + T λn+1 q − q
≤ T λn+1 xn − T λn+1 q + T λn+1 q − q
≤ 1 − λn+1 τ xn − q + λn+1 μF(q).
(3.3)
Furthermore,
2
λ
T n+1 xn − q2 ≤ 1 − λn+1 τ xn − q2 + λn+1 μ F(q)2 .
τ
(3.4)
Thus,
xn+1 − q2 ≤ αn xn − q2 + 1 − αn 1 − λn+1 τ xn − q2
+ 1 − αn
λn+1 μ2 F(q)2 − αn 1 − αn xn − T λn+1 xn 2
τ
2 2
≤ αn xn − q + 1 − αn 1 − λn+1 τ xn − q
(3.5)
λn+1 μ2 F(q)2 − αn xn+1 − xn 2
+ 1 − αn
τ
2 λn+1 μ2 F(q)2 − αn xn+1 − xn 2 .
≤ x n − q +
τ
∞
Since n=1 λn < ∞, it follows from Lemma 2.1 that limn→∞ xn − q exists for each q ∈
F(T). It also implies that {xn } is bounded.
(2) From (3.5), we have
2
2
2
2
αxn+1 − xn ≤ αn xn+1 − xn ≤ xn − q − xn+1 − q +
λn+1 μ2 F(q)2 .
τ
(3.6)
Therefore, limn→∞ xn+1 − xn = 0. In addition,
(1 − β)xn − T λn+1 xn ≤ 1 − αn xn − T λn+1 xn = xn+1 − xn .
(3.7)
Hence, limn→∞ xn − T λn+1 = 0. Thus,
xn − Txn = xn − T λn+1 xn + T λn+1 xn − Txn ≤ xn − T λn+1 xn + λn+1 μF Txn .
(3.8)
Since {xn } is bounded, then {Txn } and {F(Txn )} are bounded, as well. Therefore,
limn→∞ xn − Txn = 0. The proof is completed.
Lin Wang 5
Theorem 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, {xn } is defined by
xn+1 = αn xn + 1 − αn T λn+1 xn ,
n ≥ 0,
(3.9)
where {αn } and {λn } ⊂ [0,1) satisfy the following conditions:
(1) α ≤ αn ≤ β for some α,β ∈ (0,1);
(2) ∞
n =1 λ n < ∞ ;
(3) 0 < μ < 2η/k2 .
Then,
(1) {xn } converges weakly to a fixed point of T;
(2) {xn } converges strongly to a fixed point of T if and only if liminf n→∞ d(xn ,F(T)) = 0.
Proof. (1) It follows from Lemma 3.1 that {xn } is bounded. Thus, let q1 and q2 be weak
limits of subsequences {xnk } and {xn j } of {xn }, respectively. It follows from Lemmas 2.3
and 3.1 that q1 , q2 ∈ F(T). Assume q1 = q2 , then by Opial’s condition, we obtain
lim xn − q1 = lim xnk − q1 < lim xnk − q2 n→∞
k→∞
k→∞
j →∞
k→∞
= lim xn j − q2 < lim xnk − q1 = lim xn − q1 ,
(3.10)
n→∞
which is a contradiction; hence, q1 = q2 . Then, {xn } converges weakly to a common fixed
point of T.
(2) Suppose that {xn } converges strongly to a fixed point q of T, then limn→∞ xn −
q = 0. Since 0 ≤ d(xn ,F(T)) ≤ xn − q, we have liminf n→∞ d(xn ,F(T)) = 0.
Conversely, suppose that liminf n→∞ d(xn ,F(T)) = 0. For any p ∈ F(T), F(p) ≤
F(p) − F(xn ) + F(xn ) ≤ k xn − p + F(xn ). Since {xn } and {F(xn )} 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
2
xn+1 − p2 ≤ xn − p2 + λn+1 μ F(p)2 .
τ
(3.11)
So,
2
xn+1 − p2 ≤ xn − p2 + λn+1 μ 2k 2 xn − p2 + 2F xn 2
τ
2 2 2 λn+1 μ
xn − p2 + 2 λn+1 μ F xn 2 .
= 1 + 2k
τ
τ
(3.12)
Thus,
d xn+1 ,F(T)
2
≤ 1 + 2k
2 λn+1 μ
τ
2 d xn ,F(T)
2
+2
λn+1 μ2 F x n 2 .
τ
(3.13)
∞
2
2
2
2
In addition,
we obtain that ∞
n=1 2k (λn+1 μ /τ) < ∞ and
n=1 2(λn+1 μ /τ)F(xn ) <
∞
∞ since
n=1 λn < ∞ and {F(xn )} is bounded. It follows from Lemma 2.1 that
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Fixed Point Theory and Applications
limn→∞ d(xn ,F(T)) exists. Furthermore, since liminf n→∞ d(xn ,F(T)) = 0, we have
sequence.
limn→∞ d(xn ,F(T)) = 0. We now prove that {xn } is a Cauchy
∞
2 2
2 /τ)e(2μ2 k2 /τ) ∞
i=1 λi }, for any > 0, there exists
Taking M1 = max{2e(2μ k /τ) i=1 λi ,4(μ2 M
positive integer N such that d(xn ,F(T)) < /4M1 and ∞
i=n λi < /4M1 as n ≥ N. Taking
q ∈ F(T), for any n,m ≥ N, it follows from (3.12) that
xn − xm 2
2
2 2
≤ x n − q + x m − q 2 λ n μ2 xn−1 − q2 + 2 λn μ F xn−1 2
≤ 1 + 2k 2
τ
τ
2 λ m μ2 xm−1 − q2 + 2 λm μ F xm−1 2
+ 1 + 2k2
τ
τ
2
2
λn μ λn μ 2
2
≤ 1 + 2k 2
x n −1 − q + 2
M
τ
τ
2 2
2 λm μ
xm−1 − q2 + 2 λm μ M 2
+ 1 + 2k
τ
τ
≤
n 1 + 2k2
i=N+1
+2
+
n n
−1
λ j μ2
λ i μ2 2 λ i μ2 xN − q2 +
2
1 + 2k2
M
τ
τ
τ
i=N+1
j =i+1
m λ i μ2 λ n μ2 2 xN − q2
1 + 2k2
M +
τ
τ
i=N+1
m
−1
i=N+1
≤ 2e(2μ
2
m
λ j μ2
λ i μ2 2 λ m μ2 2
1 + 2k2
+2
M
M
τ
τ
τ
j =i+1
2 k 2 /τ)
∞
i=N+1 λi
∞
2 2
xN − q2 + 4 μ M e(2μ2 k2 /τ) ∞i=N+1 λi
λi .
τ
i=N+1
(3.14)
Thus,
∞
xn − xm 2 ≤ 2M1 xN − q2 + 2M1
λi .
(3.15)
i=N+1
Taking the infimum for all q ∈ F(T), we have
∞
xn − xm 2 ≤ 2M1 d xN ,F(T) 2 + 2M1
λi < .
(3.16)
i=N+1
This implies that {xn } is a Cauchy sequence. Therefore, there exists p ∈ H such that {xn }
converges strongly to p. It follows from Lemma 3.1 that
p − T p ≤ p − xn + xn − Txn −→ 0,
Hence, p ∈ F(T). The proof is completed.
as n −→ ∞.
(3.17)
Lin Wang 7
Corollary 3.3. Under the conditions of Lemma 3.1, if T is completely continuous, then
{xn } converges strongly to a fixed point of T.
Proof. By Lemma 3.1, {xn } is bounded and limn→∞ xn − Txn = 0, then {Txn } is also
bounded. Since T is completely continuous, there exists subsequence {Txn j } of {Txn }
such that Txn j → p as j → ∞. It follows from Lemma 3.1 that lim j →∞ xn j − Txn j = 0.
So by the continuity of T and Lemma 2.3, we have lim j →∞ xn j − p = 0 and p ∈ F(T).
Furthermore, by Lemma 3.1, we get that limn→∞ xn − p exists. Thus, limn→∞ xn − p =
0. The proof is completed.
Corollary 3.4. Under the conditions of Lemma 3.1, if T is demicompact, then {xn } converges strongly to a fixed point of T.
Proof. Since T is demicompact, {xn } is bounded and limn→∞ xn − Txn = 0, then there
exists subsequence {xn j } of {xn } such that {xn j } converges strongly to q ∈ H. It follows
from Lemma 2.3 that q ∈ F(T). Thus, limn→∞ xn − q exists by Lemma 3.1. Since the
subsequence {xn j } of {xn } such that {xn j } converges strongly to q, then {xn } converges
strongly to the common fixed point q ∈ F(T). The proof is completed.
For studying the strong convergence of fixed points of a nonexpansive mapping, Senter 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 mapping T : K → K
with F(T) = {x ∈ K : Tx = x} = φ is said to satisfy condition (A) if there exists a nondecreasing 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 {xn }
converges strongly to a fixed point of T.
Proof. Since T satisfies condition (A), then f (d(xn ,F(T))) ≤ xn − Txn . It follows from
Lemma 3.1 that liminf n→∞ d(xn ,F(T)) = 0. Thus, it follows from Theorem 3.2 that {xn }
converges strongly to a fixed point of T. The proof is completed.
References
[1] F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in
Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, no. 2, pp. 197–228,
1967.
[2] C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
[3] C. I. Podilchuk and R. J. Mammone, “Image recovery by convex projections using a least-squares
constraint,” Journal of the Optical Society of America. A, vol. 7, no. 3, pp. 517–521, 1990.
[4] G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert
spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
[5] M. Maiti and M. K. Ghosh, “Approximating fixed points by Ishikawa iterates,” Bulletin of the
Australian Mathematical Society, vol. 40, no. 1, pp. 113–117, 1989.
[6] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
8
Fixed Point Theory and Applications
[7] K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and
nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2,
pp. 372–379, 2003.
[8] K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method
for families of nonexpansive mappings in Hilbert spaces,” Taiwanese Journal of Mathematics,
vol. 10, no. 2, pp. 339–360, 2006.
[9] H. F. Senter and W. G. Dotson Jr., “Approximating fixed points of nonexpansive mappings,”
Proceedings of the American Mathematical Society, vol. 44, no. 2, pp. 375–380, 1974.
[10] K.-K. Tan and H.-K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa
iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–
308, 1993.
[11] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980.
[12] I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the
intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in
Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and
S. Reich, Eds., vol. 8 of Stud. Comput. Math., pp. 473–504, North-Holland, Amsterdam, The
Netherlands, 2001.
[13] M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp.
77–88, 2002.
[14] K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in
Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
Lin Wang: Department of Mathematics, Kunming Teachers College, Kunming,
Yunnan 650031, China
Email address: wl64mail@yahoo.com.cn