Volume 2010, Article ID 852030, 12 pagesdoi:10.1155/2010/852030 Research Article Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces 1 School of M
Trang 1Volume 2010, Article ID 852030, 12 pages
doi:10.1155/2010/852030
Research Article
Two New Iterative Methods for a Countable Family
of Nonexpansive Mappings in Hilbert Spaces
1 School of Mathematical Sciences, Yancheng Teachers University, Yancheng, Jiangsu 224051, China
2 Department of Mathematics, Hubei Normal University, Huangshi 435002, China
Correspondence should be addressed to Shuang Wang,wangshuang19841119@163.com
Received 6 August 2010; Accepted 5 October 2010
Academic Editor: Tomonari Suzuki
Copyrightq 2010 S Wang and C Hu 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
We consider two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces We proved that the proposed algorithms strongly converge to a common fixed point of a countable family of nonexpansive mappings which solves the corresponding variational inequality Our results improve and extend the corresponding ones announced by many others
1 Introduction
LetH be a real Hilbert space and let C be a nonempty closed convex subset of H Recall that a
mappingT : C → C is said to be nonexpansive if Tx−Ty ≤ x−y, for all x, y ∈ C We use FT to denote the set of fixed points of T A mapping F : H → H is called k-Lipschitzian if
there exists a positive constantk such that
Fx − Fy ≤ kx − y, ∀x,y ∈ H. 1.1
F is said to be η-strongly monotone if there exists a positive constant η such that
Fx − Fy, x − y≥ ηx − y2, ∀x, y ∈ H. 1.2
Let A be a strongly positive bounded linear operator on H, that is, there exists a
constantγ > 0 such that
Ax, x ≥ γx2, ∀x ∈ H. 1.3
Trang 2A typical problem is that of minimizing a quadratic function over the set of the fixed points
of a nonexpansive mapping on a real Hilbert spaceH:
min
x∈FT
1
2Ax, x − x, b , 1.4 whereb is a given point in H.
Remark 1.1 From the definition of A, we note that a strongly positive bounded linear operator
A is a A-Lipschitzian and γ-strongly monotone operator.
Construction of fixed points of nonlinear mappings is an important and active research area In particular, iterative algorithms for finding fixed points of nonexpansive mappings have received vast investigationcf 1,2 since these algorithms find applications in variety
of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see3 8 One classical way to find the fixed point of a nonexpansive mapping T
is to use a contraction to approximate it More precisely, taket ∈ 0, 1 and define a contraction
T t : C → C by T t x tu 1 − tTx, where u ∈ C is a fixed point Banach’s Contraction
Mapping Principle guarantees thatT thas a unique fixed pointx tinC, that is,
x t tu 1 − tTxt , u ∈ C. 1.5
The strong convergence of the pathx thas been studied by Browder9 and Halpern 10 in
a Hilbert space
Recently, Yao et al.11 considered the following algorithms:
x t TPC1 − txt, 1.6 and forx0∈ C arbitrarily,
y n PC 1 − αnxn,
x n11− βnx n βn Ty n , n ≥ 0. 1.7
They proved that if{αn} and {βn} satisfying appropriate conditions, then the {xt} defined by
1.6 and {x n} defined by 1.7 converge strongly to a fixed point of T.
On the other hand, Yamada12 introduced the following hybrid iterative method for solving the variational inequality:
x n1 Tx n − μλ n FTx n , n ≥ 0, 1.8
whereF is a k-Lipschitzian and η-strongly monotone operator with k > 0, η > 0, 0 < μ <
2η/k2 Then he proved that{xn} generated by 1.8 converges strongly to the unique solution
of variational inequalityF x, x − x ≥ 0, x ∈ FT.
In this paper, motivated and inspired by the above results, we introduce two new algorithms 3.3 and 3.13 for a countable family of nonexpansive mappings in Hilbert spaces We prove that the proposed algorithms strongly converge tox∗ ∈∞
n1 FT n which solves the variational inequality:Fx∗, x∗− u ≤ 0, u ∈∞
n1 FT n
Trang 32 Preliminaries
LetH be a real Hilbert space with inner product ·, · and norm · For the sequence {x n} in
H, we write x n x to indicate that the sequence {x n } converges weakly to x x n → x implies
that{xn} converges strongly to x For every point x ∈ H, there exists a unique nearest point
inC, denoted by P C x such that
x − PC x ≤ x − y, ∀y ∈ C. 2.1
The mappingP C is called the metric projection of H onto C It is well know that P C is a nonexpansive mapping In a real Hilbert spaceH, we have
x − y2
x2y2− 2x, y, ∀x, y ∈ H. 2.2
In order to prove our main results, we need the following lemmas
Lemma 2.1 see 13 Let H be a Hilbert space, C a closed convex subset of H, and T : C → C
a nonexpansive mapping with FT / ∅, if {x n} is a sequence in C weakly converging to x and if {I − Txn} converges strongly to y, then I − Tx y.
Lemma 2.2 see 14 Let {x n } and {z n } be bounded sequences in Banach space E and {γ n } a
sequence in 0, 1 which satisfies the following condition:
0< lim inf
n → ∞ γ n≤ lim sup
n → ∞ γ n < 1. 2.3
Suppose that x n1 γn x n 1 − γnzn , n ≥ 0 and lim sup n → ∞ zn1 − zn − xn1 − xn ≤ 0.
Then lim n → ∞zn − xn 0.
Lemma 2.3 see 15,16 Let {s n } be a sequence of nonnegative real numbers satisfying
s n1 ≤ 1 − λnsn λn δ n γn , n ≥ 0, 2.4
where {λn}, {δn}, and {γn} satisfy the following conditions: i {λn} ⊂ 0, 1 and ∞n0 λ n ∞, ii
lim supn → ∞ δ n ≤ 0 or ∞n0 λ n δ n < ∞, iii γ n ≥ 0 n ≥ 0, ∞n0 γ n < ∞ Then lim n → ∞ s n 0.
Lemma 2.4 see 17, Lemma 3.2 Let C be a nonempty closed convex subset of a Banach space E Suppose that
∞
n1
sup{Tn1z − T n z : z ∈ C} < ∞. 2.5
Then, for each y ∈ C, {T n y} converges strongly to some point of C Moreover, let T be a mapping of C into itself defined by Ty lim n → ∞ T n y, for all y ∈ C Then lim n → ∞sup{Tz − Tn z : z ∈ C} 0.
Trang 4Lemma 2.5 Let F be a k-Lipschitzian and η-strongly monotone operator on a Hilbert space H with
0 < η ≤ k and 0 < t < η/k2 Then S I − tF : H → H is a contraction with contraction coefficient τ t 1− t2η − tk2.
Proof From1.1, 1.2, and 2.2, we have
Sx − Sy2x − y − tFx − Fy2
x − y2 t2Fx − Fy2− 2tFx − Fy, x − y
≤x − y2 t2k2x − y2− 2tηx − y2
1− t 2η − tk2x − y2,
2.6
for allx, y ∈ H From 0 < η ≤ k and 0 < t < η/k2, we have
Sx − Sy ≤ τtx − y, 2.7 whereτ t 1− t2η − tk2 Hence S is a contraction with contraction coefficient τ t
3 Main Results
LetF be a k-Lipschitzian and η-strongly monotone operator on H with 0 < η ≤ k and T : C →
C a nonexpansive mapping Let t ∈ 0, η/k2 and τ t 1− t2η − tk2; consider a mapping
S tonC defined by
It is easy to see thatS tis a contraction Indeed, fromLemma 2.5, we have
St x − S t y ≤ TPCI − tFx − TPC I − tFy
≤I − tFx − I − tFy
≤ τtx − y, 3.2
for allx, y ∈ C Hence it has a unique fixed point, denoted x t, which uniquely solves the fixed point equation
x t TPC I − tFxt, xt ∈ C. 3.3
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → C
be a nonexpansive mapping such that FT / ∅ Let F be a k-Lipschitzian and η-strongly monotone
Trang 5operator on H with 0 < η ≤ k For each t ∈ 0, η/k2, let the net {xt} be generated by 3.3 Then, as
t → 0, the net {x t } converges strongly to a fixed point x∗of T which solves the variational inequality:
Fx∗, x∗− u ≤ 0, u ∈ FT. 3.4
Proof We first show the uniqueness of a solution of the variational inequality3.4, which is indeed a consequence of the strong monotonicity ofF Suppose x∗∈ FT and x ∈ FT both
are solutions to3.4; then
Fx∗, x∗− x ≤ 0,
F x, x − x∗ ≤ 0. 3.5
Adding up3.5 gets
Fx∗− F x, x∗− x ≤ 0. 3.6
The strong monotonicity ofF implies that x∗ x and the uniqueness is proved Below we
usex∗∈ FT to denote the unique solution of 3.4
Next, we prove that{xt} is bounded Take u ∈ FT; from 3.3 and usingLemma 2.5,
we have
xt − u TPC I − tFxt − TPC u
≤ I − tFxt − u
≤ I − tFxt − I − tFu − tFu
≤ I − tFxt − I − tFu tFu
≤ τ t x t − u tFu,
3.7
that is,
xt − u ≤ 1− τ t
Observe that
lim
t → 0
t
1− τ t
1
Fromt → 0, we may assume, without loss of generality, that t ≤ η/k2
t/1 − τ t is continuous, for all t ∈ 0, η/k2
sup
t
1− τt : t ∈
0, η
k2
From3.8 and 3.10, we have that {x t } is bounded and so is {Fx t}
Trang 6On the other hand, from3.3, we obtain
xt − Txt TPC I − tFxt − TPC x t ≤ I − tFxt − xt tFxt −→ 0 t −→ 0.
3.11
To prove thatx t → x∗ For a givenu ∈ FT, by 2.2 and usingLemma 2.5, we have
x t − u2 TPC I − tFx t − TP C u2
≤ I − tFxt − I − tFu − tFu2
≤ τ t2x t − u2 t2Fu2 2tI − tFu − I − tFxt , Fu
≤ τ t x t − u2 t2Fu2 2tu − xt , Fu 2t2Fx t − Fu, Fu
≤ τ t x t − u2 t2Fu2 2tu − xt , Fu 2t2kx t − uFu.
3.12
Therefore,
xt − u2≤ t2
1− τt Fu2
2t
1− τt u − xt , Fu 12− τt t2k xt − uFu. 3.13
Fromτ t 1− t2η − tk2, we have limt → 0 t2/1 − τ t 0 and limt → 0 2t2k/1 − τ t 0 Observe that, ifx t u, we have lim t → 0 2t/1 − τ t u − x t , Fu 0.
Since{xt} is bounded, we see that if {tn} is a sequence in 0, η/k2
n → 0 andx t n x, then by 3.13, we see xt n → x Moreover, by 3.11 and usingLemma 2.1, we have x ∈ FT We next prove that x solves the variational inequality 3.4 From 3.3 and
u ∈ FT, we have
xt − u2≤ I − tFxt − u2
xt − u2 t2Fxt2− 2tFxt , x t − u , 3.14
that is,
Fxt , x t − u ≤ 2t Fxt2. 3.15 Now replacingt in 3.15 with t nand lettingn → ∞, we have
F x, x − u ≤ 0. 3.16 That is x ∈ FT is a solution of 3.4; hence x x∗by uniqueness In a summary, we have shown that each cluster point of{x t } as t → 0 equals x∗ Therefore,x t → x∗ast → 0.
Trang 7SettingF A inTheorem 3.1, we can obtain the following result.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → C
be a nonexpansive mapping such that FT / ∅ Let A be a strongly positive bounded linear operator with coefficient 0 < γ ≤ A For each t ∈ 0, γ/A2, let the net {xt} be generated by xt
TP C I −tAx t Then, as t → 0, the net {x t } converges strongly to a fixed point x∗of T which solves the variational inequality:
Ax∗, x∗− u ≤ 0, u ∈ FT. 3.17 SettingF I, the identity mapping, inTheorem 3.1, we can obtain the following result
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → C
be a nonexpansive mapping such that FT / ∅ For each t ∈ 0, 1, let the net {x t } be generated
by1.6 Then, as t → 0, the net {x t } converges strongly to a fixed point x∗of T which solves the variational inequality:
x∗, x∗− u ≤ 0, u ∈ FT. 3.18
Remark 3.4 TheCorollary 3.3complements the results of Theorem 3.1 in Yao et al 11, that
is,x∗is the solution of the variational inequality:x∗, x∗− u ≤ 0, u ∈ FT.
Theorem 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H Let {T n } be a
sequence of nonexpansive mappings of C into itself such that∞n1 FT n / ∅ Let F be a k-Lipschitzian
and η-strongly monotone operator on H with 0 < η ≤ k Let {α n} and {βn} be two real sequences in
0, 1 and satisfy the conditions:
A1 lim n → ∞ α n 0 and ∞
n1 α n ∞;
A2 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1.
Suppose that ∞
n1sup{Tn1z − T n z : z ∈ B} < ∞ for any bounded subset B of C Let T be a mapping of C into itself defined by Tz lim n → ∞ T n z for all z ∈ C and suppose that FT ∞
n1 FT n For given x1∈ C arbitrarily, let the sequence {x n } be generated by
y n PCI − αn Fx n,
x n11− β nx n β n T n y n , n ≥ 1. 3.19
Then the sequence {xn} strongly converges to a x∗ ∈ ∞n1 FT n which solves the variational
inequality:
Fx∗, x∗− u ≤ 0, u ∈ ∞
n1 FT n. 3.20
Trang 8Proof We proceed with the following steps.
Step 1 We claim that {x n} is bounded From limn → ∞ α n 0, we may assume, without loss of generality, that 0< α n ≤ η/k2 ∞
n1 FT n, from 3.19 and using
Lemma 2.5, we have
yn − u PCI − αn Fx n − PC u
≤ I − αn Fx n − I − αn Fu − α n Fu
≤ τ α n x n − u αn Fu,
3.21
whereτ α n 1− αn2η − αn k2 Then from 3.19 and 3.21, we obtain
x n1 − u 1− β n
x n − u βnT n y n − u
≤1− βnxn − u βnyn − u
≤1− β nxn − u βnτα n xn − u αnFu
≤1− β n 1 − τ α nx n − u βn α n Fu
≤ max
x n − u, α nFu
1− τ α n
.
3.22
By induction, we have
xn − u ≤ max{x1− u, M1Fu}, 3.23 whereM1 sup{α n /1 − τ α n : 0 < α n ≤ η/k2
n} is bounded We also obtain that{yn}, {Tn y n}, and {Fxn} are bounded Without loss of generality, we may assume
that{xn}, {yn}, {Tn y n}, and {Fxn} ⊂ B, where B is a bounded set of C.
Step 2 We claim that lim n → ∞ x n1 −x n 0 To this end, define a sequence {z n } by z n T n y n
It follows that
z n1 − z n Tn1 y n1 − T n y n
≤Tn1 y n1 − Tn1 y n Tn1 y n − Tn y n
≤yn1 − y n Tn1 y n − T n y n
≤ I − αn1 Fx n1 − I − αn Fx n Tn1 y n − T n y n
≤ xn1 − xn αn1Fxn1 αnFxn sup{Tn1 z − T n z : z ∈ B}.
3.24
Thus, we have
zn1 − zn − xn1 − xn ≤ αn1Fxn1 αnFxn sup{Tn1 z − T n z : z ∈ B}. 3.25
Trang 9From limn → ∞ α n 0 and 3.25, we have
lim sup
n → ∞ zn1 − zn − xn1 − xn ≤ 0. 3.26
By3.26, A2, and usingLemma 2.2, we have limn → ∞zn − xn 0 Therefore,
lim
n → ∞ xn1 − x n lim
n → ∞ β nzn − x n 0. 3.27
Step 3 We claim that lim n → ∞ x n − T n x n 0 Observe that
x n − T n x n ≤ x n − x n1 x n1 − T n x n
≤ xn − xn1 1− βnxn − Tn x n βnTn y n − Tn x n
≤ xn − xn1 1− βnxn − Tn x n βnyn − xn
≤ xn − x n1 1− β nx n − T n x n α n Fx n ,
3.28
that is,
x n − T n x n ≤ β1
n x n1 − x n α n Fx n −→ 0 n −→ ∞. 3.29
Step 4 We claim that lim n → ∞xn − Txn 0 Observe that
xn − Txn ≤ xn − Tn x n Tn x n − Txn
≤ xn − T n x n sup{T n z − Tz : z ∈ B}. 3.30
Hence, fromStep 3and usingLemma 2.4, we have
lim
n → ∞ xn − Txn 0. 3.31
Step 5 We claim that lim sup n → ∞ Fx∗, x∗−xn ≤ 0, where x∗ limt → 0 x tandx tis defined by
3.3 Since xnis bounded, there exists a subsequence{xn k } of {xn} which converges weakly
toω From Step 4, we obtain Tx n k ω From Lemma 2.1, we haveω ∈ FT Hence, by
Theorem 3.1, we have
lim sup
n → ∞ Fx∗, x∗− xn lim
k → ∞ Fx∗, x∗− xn k Fx∗, x∗− ω ≤ 0. 3.32
Trang 10Step 6 We claim that {xn} converges strongly to x∗∈ ∞n1 FT n From 3.19, we have
x n1 − x∗2≤1− β nx n − x∗2 β nTn y n − x∗2
≤1− β nx n − x∗2 β nyn − x∗2
≤1− βnxn − x∗2 βnI − αn Fx n − I − αn Fx∗− αn Fx∗2
≤1− βnxn − x∗2
βn τ2
α n xn − x∗2 α2
n Fx∗ 2αnI − αn Fx∗− I − αn Fx n , Fx∗
≤1− βnxn − x∗2 βn τ α n xn − x∗2 βn α2
n Fx∗2 2αn β nx∗− xn , Fx∗
2βn α2
n Fxn − Fx∗, Fx∗
≤1− βn1 − τα nxn − x∗2 βn α2
n Fx∗2 2αn β nx∗− xn , Fx∗
2β n α2
n kx n − x∗Fx∗
≤1− β n 1 − τ α nx n − x∗2 β n α2
n M2 2α n β n x∗− x n , Fx∗ 2β n α2
n M2
≤1− β n 1 − τ α nx n − x∗2 β n 1 − τ α n
3α2
n M2
1− τα n
2M1x∗− x n , Fx∗
1 − λnxn − x∗2 λn δ n ,
3.33
whereM2 sup{Fx∗2, kx n − x∗Fx∗}, λ n β n 1 − τ α n , and δ n 3α2
n M2/1 − τ α n
2M1x∗ − xn , Fx∗ It is easy to see that λn → 0, ∞n1 λ n ∞, and lim supn → ∞ δ n ≤ 0 Hence, byLemma 2.3, the sequence{xn} converges strongly to x∗ ∈ ∞n1 FT n From x∗ limt → 0 x tandTheorem 3.1, we have thatx∗is the unique solution of the variational inequality:
Fx∗, x∗− u ≤ 0, u ∈∞n1 FT n.
Remark 3.6 From Remark 3.1 of Peng and Yao 18, we obtain that {Wn} is a sequence of
nonexpansive mappings satisfying condition ∞
n1sup{Wn1 z − W n z : z ∈ B} < ∞ for
any bounded subset B of H Moreover, let W be the W-mapping; we know that Wy
limn → ∞ W n y for all y ∈ C and that FW ∞n1 FW n If we replace {Tn} by {Wn}
in the recursion formula 3.19, we can obtain the corresponding results of the so-called
W-mapping.
SettingF A and T n T inTheorem 3.5, we can obtain the following result
Corollary 3.7 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → C
be a nonexpansive mapping such that FT / ∅ Let A be a strongly positive bounded linear operator with coefficient 0 < γ ≤ A Let {α n} and {βn} be two real sequences in 0, 1 and satisfy the
conditions (A1) and (A2) For given x1∈ C arbitrarily, let the sequence {x n } be generated by
y n PCI − αn Ax n,
x n11− βnx n βn Ty n , n ≥ 1. 3.34
... x∗ast → 0. Trang 7SettingF A in< /i>Theorem 3.1, we can obtain the following...
be a nonexpansive mapping such that FT / ∅ Let A be a strongly positive bounded linear operator with coefficient < γ ≤ A Let {α n} and {βn} be two real sequences in 0, 1 and satisfy... η/k2
n} is bounded We also obtain that{yn}, {Tn y n}, and {Fxn} are bounded Without loss of generality, we may assume
that{xn},