The viscosity approximation methods are employed to establish strong convergence theorems of the modified Mann iteration scheme toλ-strict pseudocontractions in p-uniformly convex Banach
Trang 1Volume 2010, Article ID 150539, 9 pages
doi:10.1155/2010/150539
Research Article
Strong Convergence Theorems for Strict
Pseudocontractions in Uniformly Convex
Banach Spaces
1 Department of Mathematics, Ningbo University, Zhejiang 315211, China
2 School of Computer Science and Engineering, South China University of Technology,
Guangzhou 510640, China
Correspondence should be addressed to Liang-Gen Hu,hulianggen@yahoo.cn
Received 20 April 2010; Accepted 26 August 2010
Academic Editor: W Takahashi
Copyrightq 2010 Liang-Gen Hu et al 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
The viscosity approximation methods are employed to establish strong convergence theorems of the modified Mann iteration scheme toλ-strict pseudocontractions in p-uniformly convex Banach
spaces with a uniformly Gˆateaux differentiable norm The main result improves and extends many nice results existing in the current literature
1 Introduction
LetE be a real Banach space, and let C be a nonempty closed convex subset E We denote by
J the normalized duality map from E to 2 E∗
defined by
Jx x∗∈ E∗: x, x∗ x2 x∗2, ∀x ∈ E. 1.1
A mappingT : C → C is said to be a λ-strictly pseudocontractive mapping see, e.g.,
1 if there exists a constant 0 ≤ λ < 1 such that
Tx − Ty2≤x − y2 λI − Tx − I − Ty2, 1.2
Trang 2for allx, y ∈ C We note that the class of λ-strict pseudocontractions strictly includes the class
of nonexpansive mappings which are mappingT on C such that
for allx, y ∈ C Obviously, T is nonexpansive if and only if T is a 0-strict pseudocontraction.
A mappingT : C → C is said to be a λ-strictly pseudocontractive mapping with respect to
p if, for all x, y ∈ C, there exists a constant 0 ≤ λ < 1 such that
Tx − Ty p≤x − y p λI − Tx − I − Ty p 1.4
A mappingf : C → C is called k-contraction if there exists a constant k ∈ 0, 1 such that
We denote by FixT the set of fixed point of T, that is, FixT {x ∈ C : Tx x}
Recall the definition of Mann’s iteration; let C be a nonempty convex subset E, and let T
be a self-mapping of C For any x1∈ C, the sequence {x n } is defined by
x n1 1 − α n x n α n Tx n , n ≥ 1, 1.6
where{α n } is a real sequence in 0, 1.
In the last ten years or so, there have been many nice papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudocontractive mappings by utilizing the Mann iteration process Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and more general class of mappings; see, for example,1 6 and the references therein for more information about this problem
In 2007, Marino and Xu 2 showed that the Mann iterative sequence converges weakly to a fixed point of λ-strict pseudocontractions in Hilbert spaces Meanwhile, they
have proposed an open question; that is, is the result of 2, Theorem 3.1 true in uniformly convex Banach spaces with Fr´echet differentiable norm? In other words, can Reich’s theorem 7, Theorem 2, with respect to nonexpansive mappings, be extended to λ-strict pseudocontractions in uniformly convex Banach spaces?
In 2008, using the Mann iteration and the modified Ishikawa iteration, Zhou 3 obtained some weak and strong convergence theorems for λ-strict pseudocontractions in
Hilbert spaces which extend the corresponding results in2
Recently, Hu and Wang 4 obtained that the Mann iterative sequence converges weakly to a fixed point ofλ-strict pseudocontractions with respect to p in p-uniformly convex
Banach spaces
In this paper, we first introduce the modified Mann iterative sequence Let C be a nonempty closed convex subset of E, and let f : C → C be a k-contraction For any x1 ∈ C, the
sequence {x n } is defined by
x n1 α n x n 1 − α n T n
β n fx n 1− β n
x n
Trang 3where T n x : 1 − μ n x μ n Tx, for all x ∈ C, {α n }, {β n }, and {μ n }in 0, 1 The iterative sequence
1.7 is a natural generalization of the Mann iterative sequences 1.6 If we take β n ≡ 0, for alln ≥ 1, in 1.7, then 1.7 is reduced to the Mann iteration
The purpose in this paper is to show strong convergence theorems of the modified Mann iteration scheme for λ-strict pseudocontractions with respect to p in p-uniformly
convex Banach spaces with uniformly Gateaux differentiable norm by using viscosity approximation methods Our theorems improve and extend the comparable results in the following four aspects: 1 in contrast to weak convergence results in 2 4, strong convergence theorems of the modified Mann iterative sequence are obtained inp-uniformly
convex Banach spaces;2 in contrast to the results in 7,8, these results with respect to nonexpansive mappings are extended to λ-strict pseudocontractions with respect to p; 3
the restrictions∞
n1 |α n1 − α n | < ∞ and∞
n1 |β n1 − β n | < ∞ in 8, Theorem 3.1 are removed;
4 our results partially answer the open question
2 Preliminaries
The modulus of convexity ofE is the function δ E: 0, 2 → 0, 1 defined by
δ E inf
1−
x y2 : x 1, y 1, x − y ≥ , 0 ≤ ≤ 2. 2.1
E is uniformly convex if and only if, for all 0 < ≤ 2 such that δ E > 0 E is said to be
p-uniformly convex if there exists a constant a > 0 such that δ E ≥ a p Hilbert spaces,L por
l p spaces 1 < p < ∞ and Sobolev spaces W m p 1 < p < ∞ are p-uniformly convex Hilbert
spaces are 2-uniformly convex, while
L p , l p , W m p are
⎧
⎨
⎩
2-uniformly convex if 1< p ≤ 2,
A Banach spaceE is said to have Gateaux differentiable norm if the limit
lim
t → 0
x ty − x
exists for eachx, y ∈ U, where U {x ∈ E : x 1} The norm of E is a uniformly Gateaux differentiable if for each y ∈ U, the limit is attained uniformly for x ∈ U It is well known that
ifE is a uniformly Gateaux differentiable norm, then the duality mapping J is single valued
and norm-to-weak∗uniformly continuous on each bounded subset ofE.
Lemma 2.1 see 4 Let E be a real p-uniformly convex Banach space, and let C be a nonempty
closed convex subset of E Let T : C → C be a λ-strict pseudocontraction with respect to p, and let
{ξ n } be a real sequence in 0, 1 If T n: C → C is defined by T n x : 1 − ξ n x ξ n Tx, for all x ∈ C, then for all x, y ∈ C, the inequality holds
T n x − T n yp≤x − y p−w p ξ n c p − ξ n k I − Tx − I − Ty p , 2.4
Trang 4where c p is a constant in [ 9 , Theorem 1] In addition, if 0 ≤ λ < min{1, 2 −p−2 c p }, ξ 1−λ·2 p−2 /c p , and ξ n ∈ 0, ξ, then T n x − T n y ≤ x − y, for all x, y ∈ C.
Lemma 2.2 see 10 Let {x n } and {y n } be bounded sequences in a Banach space E such that
where {α n } is a sequence in 0, 1 such that 0 < lim inf n → ∞ α n≤ lim supn → ∞ α n < 1 Assuming
lim sup
n → ∞ y n1 − y n − x n1 − x n≤ 0, 2.6
then lim n → ∞ x n − y n 0.
Lemma 2.3 Let E be a real Banach space Then, for all x, y ∈ E and jxy ∈ Jxy, the following
inequality holds:
x y2
Lemma 2.4 see 11 Let {a n } be a sequence of nonnegative real number such that
a n1 ≤ 1 − δ n a n δ n η n , ∀n ≥ 0, 2.8
where {δ n } is a sequence in 0, 1 and {η n } is a sequence in R satisfying the following conditions:
i ∞
n1 δ n ∞; ii lim sup n → ∞ η n ≤ 0 or∞
n1 δ n |η n | < ∞ Then, lim n → ∞ a n 0.
3 Main Results
Theorem 3.1 Let E be a real p-uniformly convex Banach space with a uniformly Gateaux
differentiable norm, and let C be a nonempty closed convex subset of E which has the fixed point property for nonexpansive mappings Let T : C → C be a λ-strict pseudocontraction with respect to
p, λ ∈ 0, min{1, 2 −p−2 c p } and FixT / ∅ Let f : C → C be a k-contraction with k ∈ 0, 1.
Assume that real sequences {α n }, {β n }, and {ξ n } in 0, 1 satisfy the following conditions:
i 0 < lim inf n → ∞ α n≤ lim supn → ∞ α n < 1,
ii limn → ∞ β n 0 and∞
n1 β n ∞,
iii 0 < inf n ξ n ≤ ξ and lim n → ∞ |ξ n1 − ξ n | 0, where ξ 1 − λ · 2 p−2 /c p
For any x1∈ C, the sequence {x n } is generated by
x n1 α n x n 1 − α n T n
β n fx n 1− β nx n, n ≥ 1, 3.1
where T n x : 1 − ξ n x ξ n Tx, for all x ∈ C Then, the sequence {x n } converges strongly to a fixed
point of T.
Trang 5Proof Equation3.1 can be expressed as follows:
where
y n β n fx n 1− β nx n , ∀n ≥ 1. 3.3 Takingp ∈ FixT, we obtain fromLemma 2.1
x n1 − p ≤ α n x n − p 1 − α nTn y n − p
≤ α n x n − p 1 − α nβ n fx n − p 1 − β n x n − p
≤ α n x n − p 1 − α nβ n k x n − p β n fp − p 1 − β n x n − p
1− 1 − α n β n 1 − kx n − p 1 − α n β n 1 − k 1
1− k fp − p
≤ max x
1− p, 1
1− k fp − p
3.4
Therefore, the sequence{x n } is bounded, and so are the sequences {fx n }, {T n y n }, and {y n} SinceT n y n 1 − ξ n y n ξ n Ty nand the conditioniii, we know that {Ty n} is bounded We estimate from3.3 that
y n1 − y n ≤ β n1 fx n1 − fx n 1 − βn1
x n1 − x n
β n1 − β n fx n − x n
≤1− β n1 1 − kx n1 − x n β n1 − β n fx n − x n . 3.5
SinceT n: 1 − ξn I ξ n T, where I is the identity mapping, we have
T n1 y n1 − T n y n ≤ 1−ξ n1 y n1 ξ n1 Ty n1 −1−ξ n1 y n −ξ n1 Ty n
|ξ n1 −ξ n|yn −Ty n
≤y n1 − y n |ξ n1 − ξ n|yn − Ty n . 3.6
limn → ∞ β n 0 and limn → ∞ |ξ n1 − ξ n| 0 imply from 3.5 and 3.6 that
lim sup
n → ∞ T n1 y n1 − T n y n − x n1 − x n ≤ 0. 3.7
Hence, byLemma 2.2, we obtain
lim
n → ∞ T n y n − x n 0. 3.8
Trang 6From3.3, we get
lim
n → ∞ y n − x n limn → ∞ β n fx n − x n 0, 3.9
and so it follows from3.8 and 3.9 that limn → ∞ y n −T n y n 0 Since y n −T n y n ξ n y n −Ty n and infn ξ n > 0, we have
lim
n → ∞ y n − Ty n lim
n → ∞
y n − T n y n
For anyδ ∈ 0, ξ, defining T δ: 1 − δI δT, we have
lim
n → ∞ y n − T δ y n limn → ∞ δ y n − Ty n 0. 3.11
Since T δ is a nonexpansive mapping, we have from 12, Theorem 4.1 that the net {x t} generated byx t tfx t 1 − tT δ x tconverges strongly toq ∈ FixT δ FixT, as t → 0.
Clearly,
x t − y n 1 − tT δ x t − y n tfx t − y n. 3.12
In view ofLemma 2.3, we find
x t − y n2≤ 1 − t2T δ x t − y n2 2tfx t − y n , Jx t − y n
≤1− 2t t2x
t − y n T δ y n − y n2 2tfx t − x t , Jx t − y n
2tx t − y n2,
3.13
and hence
fx t − x t , Jy n − x t≤ t
2x t − y n2
1 t2y n − T δ y n
2t
2x t − y n y n − T δ y n .
3.14
Since the sequences{y n }, {x t }, and {T δ y n} are bounded and limn → ∞ y n − T δ y n /2t 0, we
obtain
lim sup
n → ∞
fx t − x t , Jy n − x t≤ t
whereM sup n≥1,t∈0,1 {x t − y n2} We also know that
fq− q, Jy n − qfx t − x t , Jy n − x tfq− fx t x t − q, Jy n − x t
fq− q, Jy n − q− Jy n − x t
Trang 7From the facts thatx t → q ∈ FixT, as t → 0, {y n } is bounded, and the duality mapping J is
norm-to-weak∗uniformly continuous on bounded subset ofE, it follows that
fq− q, Jy n − q− Jy n − x t−→ 0, as t −→ 0,
fq− fx t x t − q, Jy n − x t
Combining3.15, 3.16, and the two results mentioned above, we get
lim sup
n → ∞
From3.9 and the fact that the duality mapping J is norm-to-weak∗uniformly continuous
on bounded subset ofE, it follows that
lim
n → ∞ fx n − fq, Jy n − q− Jx n − q 0. 3.19
Writing
x n1 − q α n
x n − q 1 − α n T n
and fromLemma 2.3, we find
x n1 − q2≤ α n x n − q2
1 − α nβn
fx n − q1− β nx n − q2
≤ α n x n − q2
1 − α n1− β n2x n − q2
21 − α n β nfx n − q, Jy n − q
≤ α n x n − q2
1 − α n1− β n2x n − q2
21 − α n β n k x n − q2
21 − α n β nfq− q, Jy n − q
21 − α n β n
fx n − fq, Jy n − q− Jx n − q
≤1− 21 − α n 1 − kβ n x n − q2 21 − α n β n
×β n x n −qfx n −fq, Jy n −q−Jx n −qfq−q,Jy n −q
1 − 1 − kδ nxn − q2 δ n η n ,
3.21 where
δ n 21 − α n β n ,
η n β n x n − q fx n − fq, Jy n − q− Jx n − q fq − q,Jy n − q. 3.22
Trang 8From 3.18, 3.19, and the conditions i, ii, it follows that ∞n1 δ n ∞ and lim supn → ∞ η n ≤ 0 Consequently, applying Lemma 2.4 to 3.21, we conclude that limn → ∞ x n − q 0.
Corollary 3.2 Let E, C, T, {α n }, {β n }, and {ξ n } be as in Theorem 3.1 For any u, x1 ∈ C, the
sequence {x n } is generated by
x n1 α n x n 1 − α n T nβ n u 1− β nx n, n ≥ 1, 3.23
where T n x : 1 − ξ n x ξ n Tx, for all x ∈ C Then the sequence {x n } converges strongly to a fixed
point of T.
Remark 3.3. Theorem 3.1andCorollary 3.2improve and extend the corresponding results in
2 4,7,8 essentially since the following facts hold
1 Theorem 3.1 and Corollary 3.2 give strong convergence results in p-uniformly
convex Banach spaces for the modification of Mann iteration scheme in contrast
to the weak convergence result in2, Theorem 3.1, 3, Theorem 3.1 and Corollary
3.3, and 4, Theorems 3.2 and 3.3.
2 In contrast to the results in 7, Theorem 2, and 8, Theorem 3.1, these results with
respect to nonexpansive mappings are extended to λ-strict pseudocontraction in p-uniformly convex Banach spaces.
3 In contrast to the results in 8, Theorem 3.1, the restrictions∞
n1 |α n1 − α n | < ∞
and∞
n1 |β n1 − β n | < ∞ are removed.
Acknowledgments
The authors would like to thank the referees for the helpful suggestions Liang-Gen Hu was supported partly by Ningbo Natural Science Foundation 2010A610100, the NNSFC
60872095, the K C Wong Magna Fund of Ningbo University and the Scientific Research Fund of Zhejiang Provincial Education Department Y200906210 Wei-Wei Lin was supported partly by the Fundamental Research Funds for the Central Universities, SCUT20092M0103 Jin-Ping Wang were supported partly by the NNSFC60872095 and Ningbo Natural Science Foundation2008A610018
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