Volume 2010, Article ID 281070, 13 pagesdoi:10.1155/2010/281070 Research Article Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermed
Trang 1Volume 2010, Article ID 281070, 13 pages
doi:10.1155/2010/281070
Research Article
Weak and Strong Convergence Theorems for
Asymptotically Strict Pseudocontractive Mappings
in the Intermediate Sense
Jing Zhao1, 2 and Songnian He1, 2
1 College of Science, Civil Aviation University of China, Tianjin 300300, China
2 Tianjin Key Laboratory For Advanced Signal Processing, Civil Aviation University of China,
Tianjin 300300, China
Correspondence should be addressed to Jing Zhao,zhaojing200103@163.com
Received 23 June 2010; Accepted 19 October 2010
Academic Editor: W A Kirk
Copyrightq 2010 J Zhao and S He 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 study the convergence of Ishikawa iteration process for the class of asymptoticallyκ-strict
pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian Weak convergence theorem is established We also obtain a strong convergence theorem by using hybrid projection for this iteration process Our results improve and extend the corresponding results announced by many others
1 Introduction and Preliminaries
Throughout this paper, we always assume thatH is a real Hilbert space with inner product
·, · and norm · and → denote weak and strong convergence, respectively ω w x n denotes the weakω-limit set of {x n }, that is, ω w x n {x ∈ H : ∃x n j x} Let C be a
nonempty closed convex subset ofH It is well known that for every point x ∈ H, there exists
a unique nearest point inC, denoted by P C x, such that
x − P C x ≤ x − y, 1.1
for ally ∈ C P Cis called the metric projection ofH onto C P Cis a nonexpansive mapping of
H onto C and satisfies
Trang 2
LetT : C → C be a mapping In this paper, we denote the fixed point set of T by FT.
Recall thatT is said to be uniformly L-Lipschitzian if there exists a constant L > 0, such that
T is said to be nonexpansive if
Tx − Ty ≤ x − y, ∀x,y ∈ C. 1.4
limn → ∞ k n 1, such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk1
as a generalization of the class of nonexpansive mappings.T is said to be asymptotically
nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
lim sup
n → ∞ sup
x,y∈C T n x − T n y − x − y ≤ 0. 1.6
Observe that if we define
τ n max
x,y∈C T n x − T n y − x − y, 1.7
thenτ n → 0 as n → ∞ It follows that 1.6 is reduced to
The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al.2 It is known 3 that if C is a nonempty closed convex bounded
subset of a uniformly convex Banach spaceE and T is asymptotically nonexpansive in the
intermediate sense, thenT has a fixed point It is worth mentioning that the class of mappings
which are asymptotically nonexpansive in the intermediate sense contains properly the class
of asymptotically nonexpansive mappings
Recall thatT is said to be a κ-strict pseudocontraction if there exists a constant κ ∈
0, 1, such that
Tx − Ty2≤x − y2 κI − Tx − I − Ty2, ∀x, y ∈ C. 1.9
Trang 3T is said to be an asymptotically κ-strict pseudocontraction with sequence {γ n} if there exist
a constantκ ∈ 0, 1 and a sequence {γ n } ⊂ 0, ∞ with γ n → 0 as n → ∞, such that
1.10
The class of asymptoticallyκ-strict pseudocontractions was introduced by Qihou 4 in 1996
see also 5 Kim and Xu 6 studied weak and strong convergence theorems for this class
of mappings It is important to note that every asymptotically κ-strict pseudocontractive
mapping with sequence {γ n } is a uniformly L-Lipschitzian mapping with L sup{κ
1 1 − κγ n /1 κ : n ∈ N}.
Recently, Sahu et al 7 introduced a class of new mappings: asymptotically
κ-strict pseudocontractive mappings in the intermediate sense Recall thatT is said to be an
asymptotically κ-strict pseudocontraction in the intermediate sense with sequence {γ n} if there exist a constant κ ∈ 0, 1 and a sequence {γ n } ⊂ 0, ∞ with γ n → 0 as n → ∞,
such that
lim sup
n → ∞ sup
x,y∈C T n x − T n y2−1 γ n x − y2− κI − T n x − I − T n y2
≤ 0. 1.11
Throughout this paper, we assume that
c n max
x,y∈C T n x − T n y2−1 γ n x − y2− κI − T n x − I − T n y2
.
1.12
It follows thatc n → 0 as n → ∞ and 1.11 is reduced to the relation
They obtained a weak convergence theorem of modified Mann iterative processes for the class
of mappings which is not necessarily Lipschitzian Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection methods; see7 for more details
In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of asymptoticallyκ-strict pseudocontractive mappings in the intermediate sense.
In order to prove our main results, we also need the following lemmas
Lemma 1.1 see 8, 9 Let {δ n }, {β n }, and {γ n } be three sequences of nonnegative numbers
satisfying the recursive inequality
δ n1 ≤ β n δ n γ n , ∀n ≥ 1. 1.14
n1 β n − 1 < ∞ and ∞
Trang 4Lemma 1.2 see 10 Let {x n } be a bounded sequence in a reflexive Banach space X If ω w x n
{x}, then x n x.
Lemma 1.3 see 11 Let C be a nonempty closed convex subset of a real Hilbert space H Given
Lemma 1.4 see 11 For a real Hilbert space H, the following identities hold:
i x − y2 x2− y2− 2x − y, y, for all x, y ∈ H,
ii tx 1 − ty2 tx21−ty2−t1−tx − y2, for all t ∈ 0, 1, for all x, y ∈ H;
iii (Opial condition) If {x n } is a sequence in H weakly convergent to z, then
lim sup
n → ∞ x n − y2 lim sup
n → ∞ x n − z2z − y2, ∀y ∈ H. 1.15
Lemma 1.5 see 7 Let C be a nonempty subset of a Hilbert space H and T : C → C an
1 − κc n
,
∀x, y ∈ C, ∀n ∈ N.
1.16
Lemma 1.6 Let C be a nonempty subset of a Hilbert space H and T : C → C an asymptotically
then
1− κ κ
√
2− κ x − y √c n
1− κ κ x − y 1 1 − κγ n x − y2 1 − κc n
≤ 1
1− κ κ x − y 2 − κx − y2 c n
≤ 1
1− κ
κ x − y 2− κx − y √c n
2
1
1− κ κ
√
2− κ x − y √c n
.
1.18
Lemma 1.7 see 7 Let C be a nonempty subset of a Hilbert space H and T : C → C a uniformly
{γ n } Let {x n } be a sequence in C such that x n − x n1 → 0 and x n − T n x n → 0 as n → ∞,
then x n − Tx n → 0 as n → ∞.
Trang 5Lemma 1.8 see 7, Proposition 3.1 Let C be a nonempty closed convex subset of a Hilbert
Lemma 1.9 see 7 Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C
is closed and convex.
2 Main Results
Theorem 2.1 Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C
Ishikawa iterative process:
y n β n T n x n1− β n
x n ,
x n1 α n T n y n 1 − α n x n , ∀n ≥ 1,
2.1
i ∞
n1 1 γ n2− 1 < ∞,
ii 0 < a ≤ α n ≤ β n ≤ b for some a > 0 and b ∈ 0, −1 − κ2
1 − κ4 2κ √2− κ21 − κ2/2κ √2− κ2.
y n − p2β n T n x n − p 1 − β n x n − p2
β n T n x n − p21− β n x n − p2− β n
1− β n
x n − T n x n2
≤ β n 1 γ n x n − p2 κx n − T n x n2 c n
1− β n x n − p2− β n1− β nx n − T n x n2
≤1 γ n x n − p2− β n
1− β n − κx n − T n x n2 β n c n
2.2
Trang 6Without loss of generality, we may assume thatγ n < 1 for all n ∈ N Since
x n − y n2x n − β n T n x n − 1 − β n x n2 β2
n x n − T n x n2, 2.3
it follows fromLemma 1.6that
y n − T n y n2β n T n x n − T n y n 1 − β n x n − T n y n2
β n T n x n − T n y n21− β n x n − T n y n2− β n
1− β n
x n − T n x n2
≤ β n
1 − κ2 κ √2− κ x
n − y n √c n
2
1− β n x n − T n y n2− β n
1− β n
x n − T n x n2
≤ 2β3
n
1− κ
2
x n − T n x n2 2β n c n
1 − κ2
1− β n x n − T n y n2− β n1− β nx n − T n x n2.
2.4
By2.2 and 2.4, we obtain that
T n y n − p2
≤1 γ n y n − p2 κy n − T n y n2 c n
≤1 γ n2x n − p2− β n
1 γ n
1− β n − κx n − T n x n2
β n1 γ nc n 2κβ3
n
1− κ
2
x n − T n x n2 2κβ n c n
1 − κ2
κ1− β n x n − T n y n2− κβ n1− β nx n − T n x n2 c n
1 γ n2x n − p2− β n
⎡
⎣1 γ n1− β n − κ− 2κβ2
n
1− κ
2
κ1− β n
⎤
⎦
× x n − T n x n2 κ1− β n x n − T n y n2 c n M1,
2.5
Trang 7whereM1 supn≥1 {β n 1 γ n 2κβ n /1 − κ2 1} It follows from 2.5 and α n ≤ β nthat
x n1 − p2
α n T n y n − p 1 − α n x n − p2
α n T n y n − p2
1 − α nxn − p2− α n 1 − α nTn y n − x n2
≤ α n
1 γ n2x n − p2− α n β n
⎡
⎣1 γ n
1− β n − κ− 2κβ2
n
1− κ
2
κ1− β n⎤⎦
× x n − T n x n2 α n κ1− β n x n − T n y n2
α n c n M1 1 − α nxn − p2− α n 1 − α nTn y n − x n2
≤1 γ n2x n − p2− α n β n
⎡
⎣1 γ n
1− β n− κγ n − 2κβ2
n
1− κ
2
− κβ n
⎤
⎦
× x n − T n x n2− α n
1− α n − κ1− β n x n − T n y n2 α n c n M1
≤1 γ n2x n − p2− α n β n
⎡
⎣1 γ n
1− β n
− κγ n − 2κβ2
n
1− κ
2
− κβ n
⎤
⎦
× x n − T n x n2 α n c n M1.
2.6 From the conditionii and γ n → 0, we see that there exists n0such that
1 γ n
1− β n
− κγ n − 2κβ2
n
1− κ
2
− κβ n
≥ 1 − β n − κγ n − 2β2
n
1− κ
2
− κβ n
≥ 1 − 2β n − κγ n − 2β2
n
1− κ
2
≥ 1 − 2b − 2b2
1− κ
2
− κγ n
≥ 1 2
⎛
⎝1 − 2b − 2b2
1− κ
2⎞
⎠ > 0, ∀n ≥ n0.
2.7
Trang 8By2.6, we have
In view ofLemma 1.1and the conditioni, we obtain that limn → ∞ x n − p exists For any
a2
2
⎛
⎝1 − 2b − 2b2
1− κ
2⎞
⎠x n − T n x n2
≤1 γ n2x n − p2−x n1 − p2 α n c n M1,
2.9
which implies that
lim
Note that
x n1 − x n α n T n y n − x n
≤ α n T n y n − T n x n α n T n x n − x n
≤ α n
1− κ κ
√
2− κ x
n − y n √c n
α n T n x n − x n
α n β n
1− κ κ
√
2− κ x n − T n x n α n √c n
1− κ α n T n x n − x n .
2.11
From2.10, we have
lim
SinceT is uniformly continuous, we obtain from 2.10, 2.12 andLemma 1.7that
lim
By the boundedness of {x n }, there exist a subsequence {x n k } of {x n } such that x n k x.
Observe thatT is uniformly continuous and x n − Tx n → 0 as n → ∞, for any m ∈ N we
havex n − T m x n → 0 as n → ∞ FromLemma 1.8, we see thatx ∈ FT.
To complete the proof, it suffices to show that ωw {x n} consists of exactly one point, namely,x Suppose there exists another subsequence {x n } of {x n } such that {x n} converges
Trang 9weakly to somez ∈ C and z / x As in the case of x, we can also see that z ∈ FT It follows
that limn → ∞ x n − x and lim n → ∞ x n − z exist Since H satisfies the Opial condition, we have
lim
n → ∞ x n − x lim
k → ∞ x n k − x < lim
k → ∞ x n k − z lim n → ∞ x n − z,
lim
n → ∞ x n − z lim
j → ∞
x n j − z < lim
j → ∞
x n j − x lim
n → ∞ x n − x, 2.14
which is a contradiction We see x z and hence ω w {x n } is a singleton Thus, {x n} converges weakly tox byLemma 1.2
Corollary 2.2 Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C
y n β n T n x n1− β n
x n ,
x n1 α n T n y n 1 − α n x n , ∀n ≥ 1,
2.15
i ∞n1 1 γ n2− 1 < ∞,
ii 0 < a ≤ α n ≤ β n ≤ b for some a > 0 and b ∈ 0, −1 − κ2
1 − κ4 2κ √2− κ21 − κ2/2κ √2− κ2.
Next, we modify Ishikawa iterative process to get a strong convergence theorem
Theorem 2.3 Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C
y n β n T n x n1− β nx n ,
z n α n T n y n 1 − α n x n ,
,
Q n {z ∈ C : x n − z, x1− x n ≥ 0},
x n1 P C n ∩Q n x1,
2.16
nΔn , M1 supn≥1 {β n 1 γ n 2κβ n /1 − κ2 1}, Δ n sup{xn − z2:z ∈ FT} < ∞ and ρ n α n β n 1−2β n −κγ n −2β2
n κ√2− κ/1−κ2 for each
Trang 10n ≥ 1 Assume that the control sequences {α n } and {β n } are chosen such that 0 < a ≤ α n ≤ β n ≤ b for
Proof We break the proof into six steps.
andC nis closed for eachn ≥ 1 Note that the defining inequality in C n is equivalent to the inequality
it is easy to see thatC nis convex for eachn ≥ 1 Hence, C n ∩ Q nis closed and convex for each
n ≥ 1.
2.16, we have
z n − p2 ≤1 γ n2x n − p2
− α n β n
⎡
⎣1 γ n1− β n− κγ n − 2κβ2
n
1− κ
2
− κβ n
⎤
⎦
× x n − T n x n2 α n c n M1
≤1 γ n2x n − p2− α n β n
⎡
⎣1 − 2β n − 2β2
n
1− κ
2
− κγ n
⎤
⎦
× x n − T n x n2 α n c n M1
x n − p2− ρ n x n − T n x n2 α n c n M1 2γ n γ2
n
x
n − p2
≤x n − p2− ρ n x n − T n x n2 θ n ,
2.18
where θ n α n c n M1 2γ n γ2
nΔn, M1 supn≥1 {β n 1 γ n 2κβ n /1 − κ2 1}, Δn sup{xn − z2 :z ∈ FT} < ∞ and ρ n α n β n 1 − 2β n − κγ n − 2β2
n κ √2− κ/1 − κ2 for
Next, we show thatFT ⊂ Q nfor eachn ≥ 1 We prove this by induction For n 1,
we haveFT ⊂ C Q1 Assume thatFT ⊂ Q nfor somen > 1 Since x n1is the projection
ofx1ontoC n ∩ Q n, we have
x n1 − z, x1− x n1 ≥ 0, ∀z ∈ C n ∩ Q n 2.19
Trang 11By the induction consumption, we know thatFT ⊂ C n ∩ Q n In particular, for anyp ∈ FT
we have
x n1 − p, x1− x n1
This implies thatp ∈ Q n1 That is,FT ⊂ Q n1 By the principle of mathematical induction,
we get FT ⊂ Q n and henceFT ⊂ C n ∩ Q n for alln ≥ 1 This means that the iteration
algorithm2.16 is well defined
andx n1 P C n ∩Q n x1 ∈ Q n It follows that
x n − x1 ≤ x n1 − x1 2.21
for eachn ≥ 1 We, therefore, obtain that the sequence {x n − x1} is nondecreasing Noticing
x1− x n ≤x1− p, ∀p ∈ FT. 2.22
This shows that the sequence{x n − x1} is bounded Therefore, the limit of {x n − x1} exists and{x n} is bounded
x n1 − x n , x1− x n ≤ 0. 2.23 UsingLemma 1.4, we obtain
x n1 − x n2 x n1 − x1 − x n − x12
x n1 − x12− x n − x12− 2x n1 − x n , x n − x1
≤ x n1 − x12− x n − x12.
2.24
Hence, we obtain thatx n1 − x n → 0 as n → ∞.
z n − x n12 ≤ x n − x n12 θ n − ρ n x n − T n x n2. 2.25
On the other hand, we see that
z n − x n12 z n − x n x n − x n12
z n − x n2 x n − x n12 2z n − x n , x n − x n1 . 2.26
Trang 12Combing2.25 and 2.26 and noting z n α n T n y n 1 − α n x n, we obtain that
α2
n T n y n − x n2 2α nT n y n − x n, x n − x n1≤ θ n − ρ n x n − T n x n2. 2.27
From the assumption and2.7, we see that there exists n0∈ N such that
1− 2β n − κγ n − 2β2
n
1− κ
2
≥ 1 2
⎛
⎝1 − 2b − 2b2
1− κ
2⎞
⎠ > 0, ∀n ≥ n0.
2.28
For anyn ≥ n0, it follows from the definition ofρ nand2.27 that
a2
2
⎛
⎝1 − 2b − 2b2
1− κ
2⎞
⎠x n − T n x n2≤ θ n 2α n T n y n − x n · x n − x n1 .
2.29 Noting thatθ n → 0 as n → ∞ andStep 4, we obtain that
lim
It follows fromStep 4,2.30 andLemma 1.7thatx n − Tx n → 0 as n → ∞.
bounded, we get that ω w {x n } is nonempty First, we show that ω w {x n} is a singleton Assume that{x n i } is subsequence of {x n } such that x n i x ∈ C Observe that T is uniformly
continuous andx n − Tx n → 0 as n → ∞, for any m ∈ N we have x n − T m x n → 0 as
Sincex n1 P C n ∩Q n x1, we obtain that
x1− x n1 ≤x1− P FT x1, 2.31
for eachn ≥ 1 Observe that x1− x n i x1− x as n → ∞ By the weak lower semicontinuity
of norm, we have
x1− P FT x1 ≤ x1− x ≤ lim inf
n → ∞ x1− x n i ≤ lim sup
n → ∞ x1− x n i ≤x1− P FT x1.
2.32
... Trang 8By2.6, we have
In view ofLemma 1. 1and the conditioni, we obtain that limn... n} converges
Trang 9weakly to somez ∈ C and z / x As in the case of x, we can also see that... 2.26
Trang 12Combing2.25 and 2.26 and noting z n α n