Volume 2010, Article ID 186874, 14 pagesdoi:10.1155/2010/186874 Research Article Convergence Theorems on Asymptotically Pseudocontractive Mappings in the Intermediate Sense 1 Department
Trang 1Volume 2010, Article ID 186874, 14 pages
doi:10.1155/2010/186874
Research Article
Convergence Theorems on
Asymptotically Pseudocontractive Mappings in
the Intermediate Sense
1 Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2 Department of Mathematics, Gyeongsang National University, Jinju 660-701, South Korea
3 Department of Mathematics Education, Kyungnam University, Masan 631-701, South Korea
Correspondence should be addressed to Jong Kyu Kim,jongkyuk@kyungnam.ac.kr
Received 15 October 2009; Revised 8 January 2010; Accepted 23 February 2010
Academic Editor: Tomonari Suzuki
Copyrightq 2010 Xiaolong Qin 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
A new nonlinear mapping is introduced The convergence of Ishikawa iterative processes for the class of asymptotically pseudocontractive mappings in the intermediate sense is studied Weak convergence theorems are established A strong convergence theorem is also established without any compact assumption by considering the so-called hybrid projection methods
1 Introduction and Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space, whose inner product
and norm are denoted by ·, · and · The symbols → and are denoted by strong convergence and weak convergence, respectively ω w x n {x : ∃x n i x} denotes the weak
w-limit set of {x n } Let C be a nonempty closed and convex subset of H and T : C → C
a mapping In this paper, we denote the fixed point set of T by FT.
Recall that T is said to be nonexpansive if
Tx − Ty ≤ x − y, ∀x,y ∈ C. 1.1
T is said to be asymptotically nonexpansive if there exists a sequence {k n } ⊂ 1, ∞ with k n → 1
as n → ∞ such that
T n x − T n y ≤ k nx − y, ∀n ≥ 1, ∀x, y ∈ C. 1.2
Trang 2The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk
1 as a generalization of the class of nonexpansive mappings They proved that if C is a
nonempty closed convex and bounded subset of a real uniformly convex Banach space and
T is an asymptotically nonexpansive mapping on C, then T has a fixed point.
T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and
the following inequality holds:
lim sup
x,y ∈C
T n x − T n y − x − y ≤ 0. 1.3
Observe that if we define
τ n max
0, sup
x,y ∈C
T n x − T n y − x − y, 1.4
then τ n → 0 as n → ∞ It follows that 1.3 is reduced to
T n x − T n y ≤ x − y τ n , ∀n ≥ 1, ∀x, y ∈ C. 1.5
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 close convex subset of a uniformly convex Banach space E and T is asymptotically nonexpansive in the intermediate sense, then T 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 that T is said to be strictly pseudocontractive if there exists a constant k ∈ 0, 1
such that
Tx − Ty ≤ x − y2 kI − Tx − I − Ty2
, ∀x, y ∈ C. 1.6
The class of strict pseudocontractions was introduced by Browder and Petryshyn 4
in a real Hilbert space Marino and Xu 5 proved that the fixed point set of strict pseudocontractions is closed convex, and they also obtained a weak convergence theorem for strictly pseudocontractive mappings by Mann iterative process; see5 for more details
Recall that T is said to be a asymptotically strict pseudocontraction if there exist a constant
k ∈ 0, 1 and a sequence {k n } ⊂ 1, ∞ with k n → 1 as n → ∞ such that
T n x − T n y2≤ k nx − y2 kI − T n x − I − T n y2
, ∀x, y ∈ C. 1.7
The class of asymptotically strict pseudocontractions was introduced by Qihou6 in 1996
see also 7 Kim and Xu 8 proved that the fixed point set of asymptotically strict pseudocontractions is closed convex They also obtained that the class of asymptotically strict pseudocontractions is demiclosed at the origin; see8,9 for more details
Trang 3Recently, Sahu et al.10 introduced a class of new mappings: asymptotically strict
pseudocontractive mappings in the intermediate sense Recall that T is said to be an
asymptotically strict pseudocontraction in the intermediate sense if
lim sup
x,y ∈C
T n
x − T n y2− k nx − y2− kI − T n x − I − T n y2
≤ 0, 1.8 where k ∈ 0, 1 and {k n } ⊂ 1, ∞ such that k n → 1 as n → ∞ Put
ξ n max
0, sup
T n
x − T n y2− k nx − y2− kI − T n x − I − T n y2
. 1.9
It follows that ξ n → 0 as n → ∞ Then, 1.8 is reduced to the following:
T n x − T n y2≤ k nx − y2 kI − T n x − I − T n y2 ξ n , ∀x, y ∈ C. 1.10
They obtained a weak convergence theorem of modified Mann iterative processes for the class
of mappings Moreover, a strong convergence theorem was also established in a real Hilbert space by considering the so-called hybrid projection methods; see10 for more details
Recall that T is said to be asymptotically pseudocontractive if there exists a sequence k n⊂
1, ∞ with k n → 1 as n → ∞ such that
T n x − T n y, x − y ≤ k nx − y2
, ∀x, y ∈ C. 1.11
The class of asymptotically pseudocontractive mapping was introduced by Schu 11 see also 12 In 13, Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see 13 for more details In 1991, Schu 11 established the following classical results
Theorem JS Let H be a Hilbert space: ∅ / A ⊂ H closed bounded and covnex; L > 0; T : A → A
completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence
{k n } ⊂ 1, ∞; q n 2k n − 1 for all n ≥ 1; ∞n1q n − 1 < ∞; {α n }, {β n } are sequences in 0, 1;
≤ α n ≤ β n ≤ b for all n ≥ 1, some > 0 and some b ∈ 0, L−2√1 L2− 1; x1 ∈ A; for all n ≥ 1,
define
z n β n T n x n1− β n
x n ,
x n1 α n T n z n 1 − α n y n , ∀n ≥ 1, 1.12
then {x n } converges strongly to some fixed point of T.
Recently, Zhou 14 showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point Moreover, the fixed point set is closed and convex
In this paper, we introduce and consider the following mapping
Trang 4Definition 1.1 A mapping T : C → C is said to be a asymptotically pseudocontractive mapping
in the intermediate sense if
lim sup
≤ 0, 1.13
where{k n } is a sequence in 1, ∞ such that k n → 1 as n → ∞ Put
ν n max
0, sup
. 1.14
It follows that ν n → 0 as n → ∞ Then, 1.13 is reduced to the following:
T n x − T n y, x − y ≤ k nx − y2 ν n , ∀n ≥ 1, x, y ∈ C. 1.15
In real Hilbert spaces, we see that1.15 is equivalent to
T n x − T n y2≤ 2k n− 1x − y2I − T n x − I − T n y2 2ν n , ∀n ≥ 1, x, y ∈ C.
1.16
We remark that if ν n 0 for each n ≥ 1, then the class of asymptotically pseudocontractive
mappings in the intermediate sense is reduced to the class of asymptotically pseudocontrac-tive mappings
In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of mappings which are asymptotically pseudocontractive in the intermediate sense
In order to prove our main results, we also need the following lemmas
Lemma 1.2 see 15 Let {r n }, {s n }, and {t n } be three nonnegative sequences satisfying the
following condition:
r n1≤ 1 s n r n t n , ∀n ≥ n0, 1.17
where n0is some nonnegative integer If ∞
n1s n < ∞ and ∞
n1t n < ∞, then lim n→ ∞r n exists.
Lemma 1.3 In a real Hilbert space, the following inequality holds:
ax 1 − ay2
ax2 1 − ay2
− a1 − ax − y2
, ∀a ∈ 0, 1, x, y ∈ C 1.18 From now on, we always use M to denotes diam C2
Lemma 1.4 Let C be a nonempty close convex subset of a real Hilbert space H and T : C → C a
uniformly L-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense with sequences {k n } and {ν n } as defined in 1.15 Then FT is a closed convex subset of C.
Trang 5Proof To show that F T is convex, let f1 ∈ FT and f2∈ FT Put f tf1 1 − tf2, where
t ∈ 0, 1 Next, we show that f Tf Choose α ∈ 0, 1/1L and define y α,n 1−αfαT n f
for each n ≥ 1 From the assumption that T is uniformly L-Lipschitz, we see that
f − y α,n ,
f − T n f
−y α,n − T n y α,n
≤ 1 Lf − y α,n2
. 1.19
For any g ∈ FT, it follows that
f − T n f2 f − T n f, f − T n f
1
α f − y α,n , f − T n f
1
α f − y α,n ,
f − T n f
−y α,n − T n y α,n
1
α f − y α,n , y α,n − T n y α,n
1
α f − y α,n ,
f − T n f
−y α,n − T n y α,n
1
α f − g, y α,n − T n y α,n
1
α g − y α,n , y α,n − g 1
α g − y α,n , g − T n y α,n
≤ α1 Lf − T n f2 1
α f − g, y α,n − T n y α,n
k n− 1g − y α,n2 ν n
1.20 This implies that
α 1 − α1 Lf − T n f2≤ f − g, y α,n − T n y α,n k n − 1M ν n , ∀g ∈ FT. 1.21
Letting g f1and g f2in1.21, respectively, we see that
α 1 − α1 Lf − T n f2≤ f − f1, y α,n − T n y α,n
k n − 1M ν n ,
α 1 − α1 Lf − T n f2≤ f − f2, y α,n − T n y α,n
k n − 1M ν n
1.22
It follows that
α 1 − α1 Lf − T n f2
≤ k n − 1M ν n 1.23
Letting n → ∞ in 1.23, we obtain that T n f → f Since T is uniformly L-Lipschitz, we see that f Tf This completes the proof of the convexity of FT From the continuity of T, we can also obtain the closedness of FT The proof is completed.
Lemma 1.5 Let C be a nonempty close convex subset of a real Hilbert space H and T : C → C a
uniformly L-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that F T is nonempty Then I − T is demiclosed at zero.
Trang 6Proof Let {x n } be a sequence in C such that x n x and x n − Tx n → 0 as n → ∞ Next, we show that x ∈ C and x Tx Since C is closed and convex, we see that x ∈ C It is sufficient
to show that x Tx Choose α ∈ 0, 1/1 L and define y α,m 1 − αx αT m x for arbitrary
but fixed m ≥ 1 From the assumption that T is uniformly L-Lipschitz, we see that
x n − T m x n ≤ x n − Tx n Tx
n − T2x n · · · T m−1x n − T m x n
≤ 1 m − 1Lx n − Tx n .
1.24
It follows from the assumption that
lim
n→ ∞x n − T m x n 0. 1.25
Note that
x − y α,m , y α,m − T m y α,m x − x n , y α,m − T m y α,m x n − y α,m , y α,m − T m y α,m
x − x n , y α,m − T m y α,m x n − y α,m , T m x n − T m y α,m
− x n − y α,m , x n − y α,m x n − y α,m , x n − T m x n
≤ x − x n , y α,m − T m y α,m k mx n − y α,m2 ν m
−x n − y α,m2x n − y α,m x n − T m x n
≤ x − x n , y α,m − T m y α,m k m − 1M ν m
x n − y α,m x n − T m x n .
1.26
Since x n x and1.25, we arrive at
x − y α,m , y α,m − T m y α,m ≤ k m − 1M ν m 1.27
On the other hand, we have
x − y α,m , x − T m x −y α,m − T m y α,m
≤ 1 Lx − y α,m2
1 Lα2x − T m x2 1.28
Note that
x − T m x2 x − T m x, x − T m x 1
α x − y α,m , x − T m x
1
α x − y α,m , x − T m x −y α,m − T m y α,m
1
α x−, y α,m , y α,m − T m y α,m .
1.29
Trang 7Substituting1.27 and 1.28 into 1.29, we arrive at
x − T m x2 ≤ 1 Lαx − T m x2k m − 1M ν m
α . 1.30 This implies that
α 1 − 1 Lαx − T m x2≤ k m − 1M ν m , ∀m ≥ 1. 1.31
Letting m → ∞ in 1.31, we see that T m x → x Since T is uniformly L-Lipschitz, we can obtain that x Tx This completes the proof.
2 Main Results
Theorem 2.1 Let C be a nonempty closed convex bounded subset of a real Hilbert space H and T :
C → C a uniformly L-Lipschitz and asymptotically pseudocontractive mapping in the intermediate
sense with sequences {k n } ⊂ 1, ∞ and {ν n } ⊂ 0, ∞ defined as in 1.15 Assume that FT is
nonempty Let {x n } be a sequence generated in the following manner:
x1∈ 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,
∗
where {α n } and {β n } are sequences in 0, 1 Assume that the following restrictions are satisfied:
a ∞n1ν n < ∞, ∞n1q2
n − 1 < ∞, where q n 2k n − 1 for each n ≥ 1;
b a ≤ α n ≤ β n ≤ b for some a > 0 and some b ∈ 0, L−2√1 L2− 1,
then the sequence {x n } generated by ∗ converges weakly to fixed point of T.
Proof Fix x∗∈ FT From 1.16 andLemma 1.3, we see that
y n − x∗2β n T n x n − x∗ 1 − β n x n − x∗2
β n T n x n − x∗21− β n
x n − x∗2− β n
1− β n
T n x n − x n2
≤ β n
q n x n − x∗2 x n − T n x n 2ν n
1− β n
x n − x∗2
− β n
1− β n
T n x n − x n2
≤ q n x n − x∗2 β2
2.1
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 nT n x n − T n y n21− β nx n − T n y n2− β n
1− β n
T n x n − x n2
≤ β3
n L2x n − T n x n21− β nx n − T n y n2− β n
1− β n
T n x n − x n2.
2.2
Trang 8From2.1 and 2.2, we arrive at
T n y n − x∗2≤ q ny n − x∗2y n − T n y n2 2ν n
≤ q2
1− q n β n − β2
T n x n − x n2
2q n 1ν n1− β nx n − T n y n2
.
2.3
It follows that
x n1− x∗2 α n T n y n − x∗ 1 − α n x n − x∗2
α nT n y n − x∗2 1 − α n x n − x∗2− α n 1 − α nT n y n − x n2
≤ α n q2n x n − x∗2− α n β n
1− q n β n − β2
T n x n − x n2 2q n 1ν n
α n
1− β nx n − T n y n2
1 − α n x n − x∗2− α n 1 − α nT n y n − x n2
≤ q2
n x n − x∗2− α n β n
1− q n β n − β2
T n x n − x n2 2q n 1ν n
2.4 From conditionb, we see that there exists n0such that
1− q n β n − β2
n L2− β n≥ 1− 2b − L2b2
Note that
x n1− x∗2≤ 1q2n− 1 x n − x∗2 2q n 1ν n , ∀n ≥ n0.
2.6
In view ofLemma 1.2, we see that limn→ ∞x n − x∗ exists For any n ≥ n0, we see that
a2
1− 2b − L2b2
2 T n x n − x n2 ≤q2n− 1x n − x∗2 x n − x∗2− x n1− x∗2 2q n 1ν n ,
2.7 from which it follows that
lim
n→ ∞T n x n − x n 0. 2.8
Trang 9Note that
x n1− x n ≤ α nT n y n − x n ≤ α nT n y n − T n x n T n x n − x n
≤ α n
Ly n − x n T n x n − x n≤ α n
1 β n L
T n x n − x n . 2.9
Thanks to2.8, we obtain that
lim
n→ ∞x n1− x n 0. 2.10
Note that
x n − Tx n ≤ x n − x n1 x
n1− T n1x n1 T n1x n1− T n1x n T n1x n − Tx n
≤ 1 Lx n − x n1 x
n1− T n1x n1 LT n x n − x n .
2.11 From2.8 and 2.10, we obtain that
lim
n→ ∞Tx n − x n 0. 2.12
Since{x n } is bounded, we see that there exists a subsequence {x n i } ⊂ {x n } such that x n i x.
FromLemma 1.5, we see that x ∈ FT.
Next we prove that{x n } converges weakly to x Suppose the contrary Then we see
that there exists some subsequence{x n j } ⊂ {x n } such that {x n j } converges weakly to x ∈ C
and x / x FromLemma 1.5, we can also prove thatx ∈ FT Put d lim n→ ∞x n − x Since
H satisfies Opial property, we see that
d lim inf
n i→ ∞x n i − x < lim inf
n i→ ∞ x n i − x
lim inf
x n j − x < lim inf
x n j − x
lim inf
n i→ ∞x n i − x d.
2.13
This derives a contradiction It follows that x x This completes the proof.
Next, we modify Ishikawa iterative processes to obtain a strong convergence theorem without any compact assumption
Trang 10Theorem 2.2 Let C be a nonempty closed convex bounded subset of a real Hilbert space H, P C
the metric projection from H onto C, and T : C → C a uniformly L-Lipschitz and asymptotically
pseudocontractive mapping in the intermediate sense with sequences {k n } ⊂ 1, ∞ and {ν n } ⊂ 0, ∞
as defined in1.15 Let q n 2k n − 1 for each n ≥ 1 Assume that FT is nonempty Let {α n } and {β n } be sequences in 0, 1 Let {x n } be a sequence generated in the following manner:
x1∈ C, chosen arbitrarily,
z n1− β n
x n β n T n x n ,
y n 1 − α n x n α n T n z n ,
C nu ∈ C :y n − u2≤ x n − u2 α n θ n α n β n
q n β n β2
Q n {u ∈ C : x1− x n , x n − u ≥ 0},
x n1 P C n ∩Q n x1,
∗∗
where θ n q n 1 β n q n − 1 − 1M 2q n 1ν n for each n ≥ 1 Assume that the control
sequences {α n } and {β n } are chosen such that a ≤ α n ≤ β n ≤ b for some a > 0 and some b ∈
0, L−2√1 L2− 1 Then the sequence {x n } generated in ∗∗ converges strongly to a fixed point
of T.
Proof The proof is divided into seven steps.
Step 1 Show that C n ∩ Q n is closed and convex for each n ≥ 1.
It is obvious that Q n is closed and convex and C n is closed for each n≥ 1 We, therefore,
only need to prove that C n is convex for each n≥ 1 Note that
C nu ∈ C :y n − u2
≤ x n − u2 α n θ n α n β n
q n β n β2
2.14
is equivalent to
Cnu ∈ C : 2 x n −y n , u
≤x n2−y n2α n θ n α n β n
q n β n β2
.
2.15
It is easy to see that Cn is convex for each n ≥ 1 Hence, we obtain that C n ∩ Q nis closed and
convex for each n ≥ 1 This completesStep 1
... and asymptotically< /i>pseudocontractive mapping in the intermediate sense with sequences {k n } ⊂ 1, ∞ and {ν n } ⊂ 0, ∞
as defined in< /i>1.15... contradiction It follows that x x This completes the proof.
Next, we modify Ishikawa iterative processes to obtain a strong convergence theorem without any compact assumption... class="page_container" data-page="10">
Theorem 2.2 Let C be a nonempty closed convex bounded subset of a real Hilbert space H, P C
the metric projection from H onto