Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 783502, 12 pages
doi:10.1155/2011/783502
Research Article
General Viscosity Approximation Methods for
Common Fixed Points of Nonexpansive Semigroups
in Hilbert Spaces
Xue-song Li,
1
Nan-jing Huang,
1
and Jong Kyu Kim
2
1
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2
Department of Mathematics, Kyungnam University, Masan, Kyungnam 631-701, Republic of Korea
Correspondence should be addressed to Jong Kyu Kim, [email protected]
Received 12 November 2010; Accepted 17 December 2010
Academic Editor: Jen Chih Yao
Copyright q 2011 Xue-song Li 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.
This paper is devoted to the strong convergence of two kinds of general viscosity iteration
processes for approximating common fixed points of a nonexpansive semigroup in Hilbert spaces.
The results presented in this paper improve and generalize some corresponding results in X. Li et
al., 2009, S. Li et al., 2009, and Marino and Xu, 2006.
1. Introduction
Let H be a real Hilbert space and A be a linear bounded operator on H. Throughout this
paper, we always assume that A is strongly positive; that is, there exists a constant
Tx − Ty
≤ L
x − y
, ∀x, y ∈ H; 1.3
iii pseudocontractive if
Tx − Ty,x − y
≤
x − y
2
, ∀x, y ∈ H;
1.4
iv φ-strongly pseudocontractive if there exists a strictly increasing function φ :
0, ∞ → 0, ∞ with φ00suchthat
Tx − Ty,x − y
≤
s
: s ≥ 0
}
1.6
of self-mappings on H such that
1 T0I,whereI is the identity mapping on H;
2 Ts tx T
sTtx for all x ∈ H and s, t ≥ 0;
3 Ts is nonexpansive for each s ≥ 0;
4 for each x ∈ H, the mapping T·x from R
into H is continuous.
We denote by FΓ the common fixed points set of nonexpansive semigroup Γ,thatis,
F
Γ
s≥0
F
T
s
{
x ∈ H : T
, 1.8
Fixed Point Theory and Applications 3
where t ∈ 0, 1 such that t<A
−1
and 0 <γ<γ/α.MarinoandXu6 proved that the
sequence {x
t
} generated by 1.8 converges strongly as t → 0 to the unique solution of the
variational inequality
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F, 1.9
which is the optimality condition for the minimization problem
min
x∈F
1
2
Ax, x
h
x
γf
x
n
, ∀n ≥ 1,
1.11
where {α
n
}⊂0, 1 and {t
n
} are two sequences satisfying certain conditions. S. Li et al. 5
claimed that the sequence {x
n
} generated by 1.11 converges strongly as t
n
→∞to x
∗
∈
FΓ which solves the following variational inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F
γf
x
n
, ∀n ≥ 1, 1.13
where α
n
∈ 0, 1, γ>0, Tt
n
∈ Γ and f is a φ-strongly pseudocontractive mapping
pseudocontractive mapping, resp.. The results presented in this paper improve and
generalize some corresponding results in 4–6.
4 Fixed Point Theory and Applications
2. Preliminaries
A mapping T with domain DT and range RT in H is said to be demiclosed at a point
p ∈ H if {x
n
} is a sequence in DT which converges weakly to x ∈ DT and {Tx
n
} converges
strongly to p,thenTx p.
For the sake of convenience, we restate the following lemmas that will be used.
Lemma 2.1 see 6. Let A be a strongly positive and linear bounded operator on a real Hilbert space
H with coefficient
γ>0 and 0 <ρ≤A
−1
.ThenI − ρA≤1 − ργ.
Lemma 2.2 see 16. Let E be a Banach space and T : E → E be a φ-strongly pseudocontractive
and continuous mapping. Then T has a unique fixed point in E.
t
0
T
s
xds
0. 2.1
3. Main Results
We first discuss the convergence of general implicit viscosity iteration process 1.11
constructed from a nonexpansive semigroup Γ : {Ts : s ≥ 0}.
Theorem 3.1. Let Γ : {Ts : s ≥ 0} be a nonexpansive semigroup on H and f : H → H be an
L
f
-Lipschitzian φ-strongly pseudocontractive mapping with lim
t → ∞
φt∞.LetA be a strongly
positive and linear bounded operator on H with coefficient
γ. Then for any 0 <γ≤ γ, the sequence
{x
n
} generated by 1.11 is well defined. Suppose that
lim
t →∞
. 3.2
Proof. Since lim
n →∞
α
n
0, we may assume without loss of generality that α
n
< A
−1
,for
any n ≥ 1. Let us define a mapping T
n
: H → H provided by
T
n
x : α
n
γf
x
I − α
n
A
1
t
n
t
n
0
T
s
x − T
s
y
ds, x − y
α
n
γ
f
x
− f
y
,x− y
x − y
≤
1 − α
n
γ − γ
x − y
2
− α
n
γφ
x − y
is φ-strongly pseudocontractive and strongly continuous. It follows from
Lemma 2.2 that T
n
has a unique fixed point say x
n
∈ H,thatis,{x
n
} generated by 1.11 is
well defined.
Taking p ∈ FΓ,wehave
x
n
− p
2
α
n
γf
x
n
− Ap, x
n
− p
γf
x
n
− γf
p
,x
n
− p
α
n
γf
p
− Ap, x
n
− p
I − α
n
A
n
− p
x
n
− p
α
n
γf
p
− Ap
x
n
− p
. 3.6
This implies that x
n
− p≤φ
−1
γfp − Ap/γ and {x
n
} is bounded.
We denote z
n
1/t
n
t
n
0
Tsx
n
ds and have z
n
− p≤x
n
− p,foranyp ∈ FΓ.
Since {x
n
} and {z
n
x − p
≤ φ
−1
γf
p
− Ap
γ
. 3.8
Since t
n
→∞, C is a nonempty bounded closed convex subset and Ts-invariant i.e., TsC
is a subset of C, it follows from Lemma 2.4 that
lim
n →∞
z
n
− T
n
− T
s
z
n
T
s
z
n
− T
s
x
n
≤ 2
x
n
− z
n
} is bounded, there exists a subsequence {x
n
k
}⊂{x
n
} which converges weakly to
some x
∗
.ItisknownfromLemma 2.3 that I − Ts is demiclosed at zero for each s ≥ 0, where
I is the identity mapping on H.Thus,x
∗
∈ FΓ follows readily.
In addition, by 1.11 and Lemma 2.1, we observe
x
n
− x
∗
2
α
n
γf
x
n
− Ax
∗
n
− x
∗
≤ α
n
γf
x
n
− γf
x
∗
,x
n
− x
∗
α
n
γf
x
∗
∗
2
− α
n
γφ
x
n
−x
∗
x
n
−x
∗
α
n
γf
x
∗
− Ax
∗
,x
n
− x
. 3.13
This means that {x
n
k
} converges strongly to x
∗
. If there exists another subsequence {x
n
j
}⊂
{x
n
} which converges weakly to y
∗
,thenfrom3.11 and 3.13 we know that {x
n
j
} converges
strongly to y
∗
∈ FΓ.Foranyp ∈ FΓ, it follows from 1.11 that
Az
n
− γf
x
n
T
s
x
n
− p
ds, x
n
− p
−
x
n
− p
2
≤ 0.
3.14
The convergence of sequences {x
n
k
} and {x
n
j
Fixed Point Theory and Applications 7
Thus,
γ
x
∗
− y
∗
2
≤
A
x
∗
− y
∗
,x
∗
− y
∗
≤ γ
f
∗
x
∗
− y
∗
.
3.16
This implies that x
∗
y
∗
. Therefore, {x
n
} converges strongly to x
∗
∈ FΓ.From3.14 and
the deduction above, we know that x
∗
is also the unique solution to VI 3.2. This completes
the proof.
Theorem 3.2. Let Γ : {Ts : s ≥ 0} be a nonexpansive semigroup on H and f : H → H be an
L
f
− p
≤
1
γ − γ
γf
p
− Ap
for a fixed p ∈ F
Γ
,
lim
n →∞
x
n
− T
s
x
x
∗
− Ax
∗
,x
n
− x
∗
.
3.19
This means that {x
n
k
} converges strongly to x
∗
.Therestoftheproofisalmostthesameas
Theorem 3.1. This completes the proof.
Remark 3.3. 1 Theorems 3.1 and 3.2 improve and generalize Theorem 3.1 of 5 from
contractive mapping to φ-strongly pseudocontractive mapping and pseudocontractive
mapping, respectively. 2 Theorems 3.1 and 3.2 also improve and generalize Theorem 3.2 of
6 from nonexpansive mapping to nonexpansive semigroup, and from contractive mapping
to φ-strongly pseudocontractive mapping and pseudocontractive mapping, respectively.
A strong mean convergence theorem for nonexpansive mappings was first established
by Baillon 17, and later generalized to that for nonlinear semigroup see, e.g., 8.Itisclear
8 Fixed Point Theory and Applications
that Theorems 3.1 and 3.2 are valid for nonexpansive mappings. Thus, we have the following
mean ergodic assertions of general viscosity iteration process for nonexpansive mappings in
α
n
γf
x
n
, ∀n ≥ 0.
3.20
Moreover, the sequence {x
n
} generated by 3.20 converges strongly as n →∞toacommonfixed
point x
∗
∈ F that is the unique solution in F to variational inequality (VI):
γf
x
∗
− Ax
∗
,x
∗
− p
≥ 0, ∀ p ∈ F. 3.21
Corollary 3.5. Let H, f, A be as in Theorem 3 .2, T : H → H be a nonexpansive mapping such that
thefixedpointssetF of T is nonempty. Let {α
s → 0
sup
x∈K
T
s
x − x
0,
3.22
lim
n →∞
t
n
lim
n →∞
α
n
t
n
0.
3.23
Then the sequence {x
n
} converges strongly as n →∞toacommonfixedpointx
∗
∈ FΓ that is the
unique solution in FΓ to VI 3.2.
x, ∀n ≥ 1 .
3.24
Fixed Point Theory and Applications 9
By Lemma 2.2,weknow
T
f
n
x − T
f
n
y, x − y
I − α
n
A
T
t
n
x − T
t
n
2
α
n
γ
x − y
2
− φ
x − y
x − y
≤
x − y
} generated by 1.13 is
well defined.
Taking p ∈ FΓ,wenote
x
n
− p
2
α
n
γf
x
n
− Ap, x
n
− p
I − α
n
A
α
n
γf
p
− Ap, x
n
− p
I − α
n
A
x
n
− p
2
≤
1 − α
n
n
γf
p
− Ap
x
n
− p
,
3.26
and so x
n
− p≤φ
−1
γfp − Ap/γ,thesequence{x
n
} is bounded. It follows from the
Lipschitzian conditions of Γ and f that {ATt
n
x
n
n
−→ 0. 3.27
For any given t>0,
x
n
− T
t
x
n
t/t
n
−1
k0
T
k 1
t
n
x
n
≤
t
t
n
x
n
− T
t
n
x
n
T
t −
− γf
x
n
max
{
T
s
x
n
− x
n
:0≤ s ≤ t
n
}
,
3.28
where t/t
n
is the integral part of t/t
n
. Since lim
n →∞
α
∗
.ByLemma 2.3, we know that x
∗
∈ FΓ.
10 Fixed Point Theory and Applications
In addition, by 1.13 and Lemma 2.1, we observe
x
n
− x
∗
2
α
n
γf
x
n
− Ax
∗
,x
n
− x
∗
∗
,x
n
− x
∗
α
n
γf
x
∗
− Ax
∗
,x
n
− x
∗
I − α
n
A
x
n
∗
α
n
γf
x
∗
− Ax
∗
,x
n
− x
∗
,
3.30
which implies that
γφ
x
n
− x
∗
x
n
− x
∗
n
− p
1
α
n
T
t
n
x
n
− x
n
,x
n
− p
1
α
n
T
t
x, ∀x ∈ H. 3.33
Then it is easy to check that Γ : {Tt : t ≥ 0} is a nonexpansive semigroup satisfying 3.22
and FΓ is a singleton {θ},whereθ is the zero point in H.
Combining the proofs of Theorems 3.2 and 3.6, we can easily conclude the following
result.
Theorem 3.8. Let f : H → H be an L
f
-Lipschitzian pseudocontractive mapping and Γ : {Tt :
t ≥ 0} be a nonexpansive semigroup on H such that 3.22 holds. Let A be a strongly positive and
linear bounded operator with coefficient
γ. Then for any 0 <γ<γ , the sequence {x
n
} generated by
1.13 is well defined. Suppose that
lim
n →∞
t
n
lim
n →∞
α
n
t
n
0.
3.34
Fixed Point Theory and Applications 11
Then the sequence {x
n
} converges strongly as n →∞toacommonfixedpointx
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