Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 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 Zhao
1, 2
and Songnian He
1, 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,
Received 23 June 2010; Accepted 19 October 2010
Academic Editor: W. A. Kirk
Copyright q 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 asymptotically κ-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 that H is a real Hilbert space with inner product
·, · and norm ·.and → denote weak and strong convergence, respectively. ω
for all y ∈ C. P
C
is called the metric projection of H onto C. P
C
is a nonexpansive mapping of
H onto C and satisfies
x − y, P
C
x − P
C
y
≥
P
C
x − P
C
y
2
, ∀x, y ∈ H.
1.2
2 Fixed Point Theory and Applications
Let T : C → C be a mapping. In this paper, we denote the fixed point set of T by FT.
Recall that T is said to be uniformly L-Lipschitzian if there exists a constant L>0, such that
} in 1, ∞ with
lim
n →∞
k
n
1, such that
T
n
x − T
n
y
≤ k
n
x − y
, ∀x, y ∈ C, ∀n ≥ 1. 1.5
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk 1
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
n
x − T
n
y
−
x − y
,
1.7
then τ
n
→ 0asn →∞. It follows that 1.6 is reduced to
T
n
x − T
n
y
≤
I − Tx − I − Ty
2
, ∀x, y ∈ C.
1.9
Fixed Point Theory and Applications 3
T is said to be an asymptotically κ-strict pseudocontraction with sequence {γ
n
} if there exist
a constant κ ∈ 0, 1 and a sequence {γ
n
}⊂0, ∞ with γ
n
→ 0asn →∞, such that
T
n
x − T
n
y
2
≤
1 γ
n
Recently, Sahu et al. 7 introduced a class of new mappings: asymptotically κ-
strict pseudocontractive mappings in the intermediate sense. Recall that T 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
→ 0asn →∞,
such that
lim sup
n →∞
sup
x,y∈C
T
n
x − T
n
y
2
−
1 γ
n
0, sup
x,y∈C
T
n
x − T
n
y
2
−
1 γ
n
x − y
2
− κ
I − T
1 γ
n
x − y
2
κ
I − T
n
x − I −T
n
y
2
c
n
, ∀x, y ∈ C.
1.13
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; see 7 for more
details.
In this paper, we consider the problem of convergence of Ishikawa iterative processes
for the class of asymptotically κ-strict pseudocontractive mappings in the intermediate sense.
γ
n
< ∞,thenlim
n →∞
δ
n
exists.
4 Fixed Point Theory and Applications
Lemma 1.2 see 10. Let {x
n
} be a bounded sequence in a reflexive Banach space X.Ifω
w
x
n
{x},thenx
n
x.
Lemma 1.3 see 11. Let C be a nonempty closed convex subset of a real Hilbert space H.Given
x ∈ H and z ∈ C,thenz P
C
x if and only if x − z, y − z≤0, for all y ∈ C.
Lemma 1.4 see 11. For a real Hilbert space H, the following identities hold:
i x −y
2
x
2
−y
2
− 2x − y, y, for all x, y ∈ H,
z − y
2
, ∀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
asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γ
n
}.Then
T
n
x − T
n
y
≤
1
1 − κ
κ
x − y
n
}.Letn ∈ N.Ifγ
n
< 1,
then
T
n
x − T
n
y
≤
1
1 − κ
κ
√
2 − κ
x − y
√
c
1 − κ
γ
n
x − y
2
1 − κ
c
n
≤
1
1 − κ
κ
x − y
x − y
√
c
n
2
1
1 − κ
κ
√
2 − κ
x − y
√
c
n
.
1.18
x∈ C and lim sup
m →∞
lim sup
n →∞
x
n
− T
m
x
n
0,thenI − Tx 0.
Lemma 1.9 see 7. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C
a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense. Then FT
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
a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense
with sequence {γ
n
} such that FT
/
∅.Let{x
n
}
∞
n1
be a sequence in C generated by the following
Ishikawa iterative process:
x
1
x
n
, ∀n ≥ 1,
2.1
where {α
n
} and {β
n
} are sequences in 0, 1. Assume that the following restrictions are satisfied:
i
∞
n1
α
n
c
n
< ∞ and
∞
n1
1 γ
n
2
− 1 < ∞,
ii 0 <a≤ α
n
≤ β
n
β
n
T
n
x
n
− p1 − β
n
x
n
− p
2
β
n
T
n
x
n
− p
2
1 γ
n
x
n
− p
2
κ
x
n
− T
n
x
n
2
c
n
1 − β
n
n
− p
2
− β
n
1 − β
n
− κ
x
n
− T
n
x
n
2
β
n
c
n
.
2.2
6 Fixed Point Theory and Applications
Without loss of generality, we may assume that γ
n
n
x
n
− T
n
x
n
2
,
2.3
it follows from Lemma 1.6 that
y
n
− T
n
y
n
2
β
n
T
n
2
1 − β
n
x
n
− T
n
y
n
2
− β
n
1 − β
n
x
n
− T
2
1 − β
n
x
n
− T
n
y
n
2
− β
n
1 − β
n
x
n
− T
n
x
n
1 − β
n
x
n
− T
n
y
n
2
− β
n
1 − β
n
x
n
− T
n
x
n
y
n
− T
n
y
n
2
c
n
≤
1 γ
n
2
x
n
− p
2
− β
n
1 γ
n
2
x
n
− T
n
x
n
2
2κβ
n
c
n
1 − κ
2
κ
1 − β
n
x
n
− T
x
n
− p
2
− β
n
⎡
⎣
1 γ
n
1 − β
n
− κ
− 2κβ
2
n
κ
√
2 − κ
1 − κ
2
κ
2
c
n
M
1
,
2.5
Fixed Point Theory and Applications 7
where M
1
sup
n≥1
{β
n
1 γ
n
2κβ
n
/1 − κ
2
1}. It follows from 2.5 and α
n
≤ β
n
that
x
n1
2
1 − α
n
x
n
− p
2
− α
n
1 − α
n
T
n
y
n
− x
n
− κ
− 2κβ
2
n
κ
√
2 − κ
1 − κ
2
κ
1 − β
n
⎤
⎦
×
x
n
− T
n
x
n
2
α
x
n
− p
2
− α
n
1 − α
n
T
n
y
n
− x
n
2
≤
1 γ
n
2
2
− κβ
n
⎤
⎦
×
x
n
− T
n
x
n
2
− α
n
1 − α
n
− κ
1 − β
n
x
n
n
⎡
⎣
1 γ
n
1 − β
n
− κγ
n
− 2κβ
2
n
κ
√
2 − κ
1 − κ
2
− κβ
n
⎤
⎦
×
x
n
n
κ
√
2 − κ
1 − κ
2
− κβ
n
≥ 1 − β
n
− κγ
n
− 2β
2
n
κ
√
2 − κ
1 − κ
2
− κβ
n
≥ 1 − 2β
n
− κγ
n
√
2 − κ
1 − κ
2
⎞
⎠
> 0, ∀n ≥ n
0
.
2.7
8 Fixed Point Theory and Applications
By 2.6, we have
x
n1
− p
2
≤
1 γ
n
2
x
κ
√
2 − κ
1 − κ
2
⎞
⎠
x
n
− T
n
x
n
2
≤
1 γ
n
2
x
n
− p
0.
2.10
Note that
x
n1
− x
n
α
n
T
n
y
n
− x
n
≤ α
n
T
n
y
n
− T
√
c
n
α
n
T
n
x
n
− x
n
α
n
β
n
1 − κ
κ
√
2 − κ
x
n1
− x
n
0.
2.12
Since T is uniformly continuous, we obtain from 2.10, 2.12 and Lemma 1.7 that
lim
n →∞
x
n
− Tx
n
0.
2.13
By the boundedness of {x
n
}, there exist a subsequence {x
n
k
} of {x
n
} such that x
n
k
x.
Observe that T is uniformly continuous and x
n
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
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 to x by Lemma 1.2.
Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C
a uniformly continuous asymptotically κ-strict pseudocontractive mapping with sequence {γ
n
} such
n
T
n
y
n
1 − α
n
x
n
, ∀n ≥ 1,
2.15
where {α
n
} and {β
n
} are sequences in 0, 1. Assume that the following restrictions are satisfied:
i
∞
n1
1 γ
n
2
− 1 < ∞,
ii 0 <a≤ α
n
n
} and {β
n
} are sequences in 0, 1.Let
{x
n
}
∞
n1
be a sequence in C generated by the modified Ishikawa iterative process:
x
1
∈ C,
y
n
β
n
T
n
x
n
1 − β
n
x
n
,
z
− z
2
θ
n
− ρ
n
x
n
− T
n
x
n
2
,
Q
n
{
z ∈ C :
x
n
− z, x
1
− x
n
, M
1
sup
n≥1
{β
n
1 γ
n
2κβ
n
/1 − κ
2
1}, Δ
n
sup{x
n
− z
2
: z ∈ FT} < ∞ and ρ
n
α
n
β
n
1−2β
n
−κγ
n
−2β
√
2 − κ
2
.Then
the sequence {x
n
} given by 2.16 converges strongly to an element of FT.
Proof. We break the proof into six steps.
Step 1 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. Note that the defining inequality in C
n
is equivalent to the
inequality
2
x
n
− z
n
,z
≤
∩Q
n
is closed and convex for each
n ≥ 1.
Step 2 FT ⊂ C
n
∩ Q
n
for each n ≥ 1.Letp ∈ FT. Following 2.6, 2.7 and the algorithm
2.16, we have
z
n
− p
2
≤
1 γ
n
2
x
n
− p
⎦
×
x
n
− T
n
x
n
2
α
n
c
n
M
1
≤
1 γ
n
2
x
n
− p
n
2
α
n
c
n
M
1
x
n
− p
2
− ρ
n
x
n
− T
n
x
n
2
α
− ρ
n
x
n
− T
n
x
n
2
θ
n
,
2.18
where θ
n
α
n
c
n
M
1
2γ
n
γ
2
n
Δ
n
2
n
κ
√
2 − κ/1 − κ
2
for
each n ≥ 1. Hence p ∈ C
n
for each n ≥ 1.
Next, we show that FT ⊂ Q
n
for each n ≥ 1. We prove this by induction. For n 1,
we have FT ⊂ C Q
1
. Assume that FT ⊂ Q
n
for some n>1. Since x
n1
is the projection
of x
1
onto C
n
∩ Q
n
, we have
x
n1
. By the principle of mathematical induction,
we get FT ⊂ Q
n
and hence FT ⊂ C
n
∩ Q
n
for all n ≥ 1. This means that the iteration
algorithm 2.16 is well defined.
Step 3 lim
n →∞
x
n
−x
1
exists and {x
n
} is bounded.Inviewof2.16,weseethatx
n
P
Q
n
x
1
and x
n1
P
C
n
∩Q
Q
n
x
1
, we have
x
1
− x
n
≤
x
1
− p
, ∀p ∈ F
T
. 2.22
This shows that the sequence {x
n
−x
1
} is bounded. Therefore, the limit of {x
n
n
≤ 0. 2.23
Using Lemma 1.4,weobtain
x
n1
− x
n
2
x
n1
− x
1
−
x
n
− x
1
2
1
2
−
x
n
− x
1
2
.
2.24
Hence, we obtain that x
n1
− x
n
→ 0asn →∞.
Step 5 x
n
− Tx
n
→ 0asn →∞.Inviewofx
n1
∈ C
n
, we have
z
n
n
− x
n1
2
z
n
− x
n
x
n
− x
n1
2
z
n
− x
n
2
x
n
− x
α
2
n
T
n
y
n
− x
n
2
2
α
n
T
n
y
n
− x
n
,x
n
− x
n1
1 − κ
2
≥
1
2
⎛
⎝
1 − 2b − 2b
2
κ
√
2 − κ
1 − κ
2
⎞
⎠
> 0, ∀n ≥ n
0
.
2.28
For any n ≥ n
0
, it follows from the definition of ρ
n
and 2.27 that
a
2
n
y
n
− x
n
·
x
n
− x
n1
.
2.29
Noting that θ
n
→ 0asn →∞and Step 4,weobtainthat
lim
n →∞
x
n
− T
n
x
n
0.
n
i
x∈ C. Observe that T is uniformly
continuous and x
n
− Tx
n
→0asn →∞, for any m ∈ N we have x
n
− T
m
x
n
→0as
n →∞.FromLemma 1.8,weseethatx ∈ ω
w
{x
n
} ⊂ FT.
Since x
n1
P
C
n
∩Q
n
x
1
,weobtainthat
− P
FT
x
1
≤
x
1
− x
≤ lim inf
n →∞
x
1
− x
n
i
≤ lim sup
n →∞
x
1
− x
n
i
, 2.33
lim
n →∞
x
1
− x
n
i
x
1
− P
FT
x
1
.
2.34
Hence x P
FT
x
1
by the uniqueness of the nearest point projection of x
1
as n →∞. This completes
the proof.
Acknowledgments
This research is supported by Fundamental Research Funds for the Central Universities
ZXH2009D021 and supported by the Science Research Foundation Program in Civil
Aviation University of China no. 09CAUC-S05 as well.
References
1 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”
Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
2 R. E. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive
mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no.
2, pp. 169–179, 1993.
3 W. A. Kirk, “Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive
type,” Israel Journal of Mathematics, vol. 17, pp. 339–346, 1974.
4 L. Qihou, “Convergence theorems of the sequence of iterates for asymptotically demicontractive and
hemicontractive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 26, no. 11, pp.
1835–1842, 1996.
5 Y. X. Tian, S S. Chang, J. Huang, X. Wang, and J. K. Kim, “Implicit iteration process for common fixed
points of strictly asymptotically pseudocontractive mappings in Banach spaces,” Fixed Point Theory
and Applications, vol. 2008, Article ID 324575, 12 pages, 2008.
6 T H. Kim and H K. Xu, “Convergence of the modified Mann’s iteration method for asymptotically
strict pseudo-contractions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 9, pp. 2828–
2836, 2008.
7 D. R. Sahu, H K. Xu, and J C. Yao, “Asymptotically strict pseudocontractive mappings in the
intermediate sense,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 10, pp. 3502–3511,
2009.
8 M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of
asymptotically nonexpansive mappings,” Mathematical and Computer Modelling, vol. 32, no. 10, pp.
1181–1191, 2000.
9 K K. Tan and H K. Xu, “The nonlinear ergodic theorem for asymptotically nonexpansive mappings
Copyright: Tài liệu đại học ©