- Research Article
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Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 579725 (2010)
Abstract
For a countable family of strictly pseudo-contractions, a strong convergence of viscosity iteration is shown in order to find a common fixed point of
in either a p-uniformly convex Banach space which admits a weakly continuous duality mapping or a p-uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper we apply our results to the problem of finding a zero of accretive operators. The main result extends various results existing in the current literature.
1. Introduction
Let be a real Banach space and
a nonempty closed convex subset of
. A mapping
is called
-contraction if there exists a constant
such that
for all
,
. We use
to denote the collection of all contractions on
. That is,
. A mapping
is said to be
-strictly pseudo-contractive mapping (see, e.g., [1]) if there exists a constant
, such that

for all ,
. Note that the class of
-strict pseudo-contractions strictly includes the class of nonexpansive mappings which are mapping
on
such that
, for all
,
. That is,
is nonexpansive if and only if
is a 0-strict pseudo-contraction. A mapping
is said to be
-strictly pseudo-contractive mapping with respect to
if, for all
,
, there exists a constant
such that

A countable family of mapping is called a family of uniformly
-strict pseudo-contractions with respect to
, if there exists a constant
such that

We denote by the set of fixed points of
, that is,
.
In order to find a fixed point of nonexpansive mapping , Halpern [2] was the first to introduce the following iteration scheme which was referred to as Halpern iteration in a Hilbert space:
,
,
,

He pointed out that the control conditions (C1) and (C2)
are necessary for the convergence of the iteration scheme (1.4) to a fixed point of
. Furthermore, the modified version of Halpern iteration was investigated widely by many mathematicians. Recently, for the sequence of nonexpansive mappings
with some special conditions, Aoyama et al. [3] introduced a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings
satisfying some conditions. Let
and

for all where
is a nonempty closed convex subset of a uniformly convex Banach space
whose norm is uniformly Gâteaux differentiable, and
is a sequence in
. They proved that
defined by (1.5) converges strongly to a common fixed point of
Very recently, Song and Zheng [4] also studied the strong convergence theorem of Halpern iteration (1.5) for a countable family of nonexpansive mappings
satisfying some conditions in either a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm or a reflexive Banach space
with a weakly continuous duality mapping. Other investigations of approximating common fixed points for a countable family of nonexpansive mappings can be found in [3, 5–10] and many results not cited here.
On the other hand, in the last twenty years or so, there are many papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudo-contractive mappings by using the Mann and Ishikawa iteration process. Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and a more general class of mappings (see, e.g., [1, 11–13] and the references therein).
In 2007, Marino and Xu [12] proved that the Mann iterative sequence converges weakly to a fixed point of -strict pseudo-contractions in Hilbert spaces, which extend Reich's theorem [14, Theorem
] from nonexpansive mappings to
-strict pseudo-contractions in Hilbert spaces.
Recently, Zhou [13] obtained some weak and strong convergence theorems for -strict pseudo-contractions in Hilbert spaces by using Mann iteration and modified Ishikawa iteration which extend Marino and Xu's convergence theorems [12].
More recently, Hu and Wang [11] obtained that the Mann iterative sequence converges weakly to a fixed point of -strict pseudo-contractions with respect to
in
-uniformly convex Banach spaces. To be more precise, they obtained the following theorem.
Theorem HW
Let be a real
-uniformly convex Banach space which satisfies one of the following:
(i) has a Fréchet differentiable norm;
(ii) satisfies Opial's property.
Let a nonempty closed convex subset of
. Let
be a
-strict pseudo-contractions with respect to
,
and
. Assume that a real sequence
in
satisfy the following conditions:

Then Mann iterative sequence defined by

converges weakly to a fixed point of .
Very recently, Hu [15] obtained strong convergence theorems on a mixed iteration scheme by the viscosity approximation methods for -strict pseudo-contractions in
-uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. To be more precise, Hu [15] obtained the following theorem.
Theorem H.
Let be a real
-uniformly convex Banach space with uniformly Gâteaux differentiable norm, and
a nonempty closed convex subset of
which has the fixed point property for nonexpansive mappings. Let
be a
-strict pseudo-contractions with respect to
,
and
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
,
(ii) and
,
(iii), where
.
Let be the sequence generated by the following:

Then the sequence converges strongly to a fixed point of
.
In this paper, motivated by Hu and Wang [11], Hu [15], Aoyama et al. [3] and Song and Zheng [4], we introduce a viscosity iterative approximation method for finding a common fixed point of a countable family of strictly pseudo-contractions which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in either -uniformly convex Banach space which admits a weakly continuous duality mapping or
-uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The results presented in this paper improve and extend the corresponding results announced by Hu and Wang [11], Hu [15], Aoyama et al. [3] Song and Zheng [4], and many others.
2. Preliminaries
Throughout this paper, let be a real Banach space and
its dual space. We write
(resp.,
) to indicate that the sequence
weakly (resp., weak*) converges to
; as usual
will symbolize strong convergence. Let
denote the unit sphere of a Banach space
. A Banach space
is said to have
(i)a Gâteaux differentiable norm (we also say that is smooth), if the limit

exists for each ,
,
(ii)a uniformly Gâteaux differentiable norm, if for each in
, the limit (2.1) is uniformly attained for
,
(iii)a Fréchet differentiable norm, if for each , the limit (2.1) is attained uniformly for
,
(iv)a uniformly Fréchet differentiable norm (we also say that is uniformly smooth), if the limit (2.1) is attained uniformly for
.
The modulus of convexity of is the function
defined by

is uniformly convex if and only if, for all
such that
.
is said to be
-uniformly convex, if there exists a constant
such that
.
The following facts are well known which can be found in [16, 17]:
(i)the normalized duality mapping in a Banach space
with a uniformly Gâteaux differentiable norm is single-valued and strong-weak* uniformly continuous on any bounded subset of
;
(ii)each uniformly convex Banach space is reflexive and strictly convex and has fixed point property for nonexpansive self-mappings;
(iii)every uniformly smooth Banach space is a reflexive Banach space with a uniformly Gâteaux differentiable norm and has fixed point property for nonexpansive self-mappings.
Now we collect some useful lemmas for proving the convergence result of this paper.
Lemma 2.1 (see [11]).
Let be a real
-uniformly convex Banach space and
a nonempty closed convex subset of
. let
be a
-strict pseudo-contraction with respect to
, and
a real sequence in
. If
is defined by
,
, then for all
,
, the inequality holds

where is a constant in [18, Theorem
]. In addition, if
,
, and
, then
, for all
,
.
Let be a nonempty closed convex subset of a Banach space
which has uniformly Gâteaux differentiable norm,
a nonexpansive mapping with
and
a
-contraction. Assume that every nonempty closed convex bounded subset of
has the fixed points property for nonexpansive mappings. Then there exists a continuous path:
,
satisfying
, which converges to a fixed point of
as
.
Lemma 2.3 (see [21]).
Let and
be bounded sequences in Banach space
such that

where is a sequence in
such that
. Assume

Then .
Definition 2.4 (see [3]).
Let be a family of mappings from a subset
of a Banach space
into
with
. We say that
satisfies the AKTT-condition if for each bounded subset
of
,

Remark 2.5.
The example of the sequence of mappings satisfying AKTT-condition is supported by Lemma 4.1.
Lemma 2.6 (see [3, Lemma ]).
Suppose that satisfies AKTT-condition. Then, for each
,
converses strongly to a point in
. Moreover, let the mapping
be defined by

Then for each bounded subset of
,
Lemma 2.7 (see [22]).
Assume that is a sequence of nonnegative real numbers such that

where is a sequence in
and
is a sequence such that
(a)
(b) or
Then
By a gauge function we mean a continuous strictly increasing function
such that
and
as
. Let
be the dual space of
. The duality mapping
associated to a gauge function
is defined by

In particular, the duality mapping with the gauge function , denoted by
, is referred to as the normalized duality mapping. Clearly, there holds the relation
for all
(see [23]). Browder [23] initiated the study of certain classes of nonlinear operators by means of the duality mapping
. Following Browder [23], we say that a Banach space
has a
if there exists a gauge
for which the duality mapping
is single-valued and continuous from the weak topology to the weak* topology, that is, for any
with
, the sequence
converges weakly* to
. It is known that
has a weakly continuous duality mapping with a gauge function
for all
. Set

then

where denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function
at
is the set
.
The following lemma is an immediate consequence of the subdifferential inequality. The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [24].
Lemma 2.8 (see [24]).
Assume that a Banach space has a weakly continuous duality mapping
with gauge
.
(i)For all ,
, the following inequality holds:

In particular, in a smooth Banach space , for all
,
,

(ii)Assume that a sequence in
converges weakly to a point
. Then the following identity holds:

3. Main Results
For a nonexpansive mapping,
and
,
defines a contraction mapping. Thus, by the Banach contraction mapping principle, there exists a unique fixed point
satisfying

For simplicity we will write for
provided no confusion occurs. Next, we will prove the following lemma.
Lemma 3.1.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
. Let
be a nonempty closed convex subset of
,
a nonexpansive mapping with
and
. Then the net
defined by (3.1) converges strongly as
to a fixed point
of
which solves the variational inequality:

Proof.
We first show that the uniqueness of a solution of the variational inequality (3.2). Suppose both and
are solutions to (3.2), then

Adding (3.3), we obtain

Noticing that for any ,
,

From (3.4), we conclude that . This implies that
and the uniqueness is proved. Below we use
to denote the unique solution of (3.2). Next, we will prove that
is bounded. Take a
; then we have

It follows that

Hence is bounded, so are
and
. The definition of
implies that

If follows from reflexivity of and the boundedness of sequence
that there exists
which is a subsequence of
converging weakly to
as
. Since
is weakly sequentially continuous, we have by Lemma 2.8 that

Let

It follows that

Since

we obtain

On the other hand, however,

It follows from (3.13) and (3.14) that

This implies that . Next we show that
as
. In fact, since
and
is a gauge function, then for
,
and

Following Lemma 2.8, we have

This implies that

Now observing that implies
, we conclude from the last inequality that

Hence as
. Next we prove that
solves the variational inequality (3.2). For any
, we observe that

Since

we can derive that

Thus

Noticing that

Now replacing in (3.23) with
and letting
, we have

So, is a solution of the variational inequality (3.2), and hence
by the uniqueness. In a summary, we have shown that each cluster point of
(at
) equals
. Therefore,
as
. This completes the proof.
Theorem 3.2.
Let be a real
-uniformly convex Banach space with a weakly continuous duality mapping
, and
a nonempty closed convex subset of
. Let
be a family of uniformly
-strict pseudo-contractions with respect to
,
and
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii), where
.
Let be the sequence generated by the following:

Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
and suppose that
. Then the sequence
converges strongly to
which solves the variational inequality:

Proof.
Rewrite the iterative sequence (3.26) as follows:

where ,
and
,
is the identity mapping. By Lemma 2.1,
is nonexpansive such that
for all
. Taking any
, from (3.28), it implies that

Therefore, the sequence is bounded, and so are the sequences
,
. Since
and
, we know that
is bounded. We note that for any bounded subset
of
,

From and
satisfing AKTT-condition, we obtain that

that is, the sequence satisfies AKTT-condition. Applying Lemma 2.6, we can take the mapping
defined by

Moreover, we have is nonexpansive and

It is easy to see that . Hence
The iterative sequence (3.28) can be expressed as follows:

where

We estimate from (3.35)

Hence

Since , and
, we have from (3.37) that

Hence, by Lemma 2.3, we obtain

From (3.35), we get

and so it follows from (3.39) and (3.40) that

It follows from Lemma 2.6 and (3.41), we have

Since is a nonexpansive mapping, we have from Lemma 3.1 that the net
generated by

converges strongly to , as
. Next, we prove that

Let be a subsequence of
such that

If follows from reflexivity of and the boundedness of sequence
that there exists
which is a subsequence of
converging weakly to
as
. Since
is weakly continuous, we have by Lemma 2.8 that

Let

It follows that

From (3.42), we obtain

On the other hand, however,

It follows from (3.49) and (3.50) that

This implies that , that is,
. Since the duality map
is single-valued and weakly continuous, we get that

as required. Finally, we show that as
.

It follows that from condition (i) and (3.44) that

Apply Lemma 2.7 to (3.53) to conclude as
; that is,
as
. This completes the proof.
If is a family of nonexpansive mappings, then we obtain the following results.
Corollary 3.3.
Let be a real
-uniformly convex Banach space with a weakly continuous duality mapping
, and
a nonempty closed convex subset of
. Let
be a family of nonexpansive mappings such that
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii).
Let be the sequence generated by the following:

Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
and suppose that
. Then the sequence
converges strongly
which solves the variational inequality:

Corollary 3.4.
Let be a real
-uniformly convex Banach space with a weakly continuous duality mapping
, and
a nonempty closed convex subset of
. Let
be a
-strict pseudo-contraction with respect to
,
and
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii), where
.
Let be the sequence generated by the following

Then the sequence converges strongly to
which solves the following variational inequality:

Theorem 3.5.
Let be a real
-uniformly convex Banach space with uniformly Gâteaux differentiable norm, and
a nonempty closed convex subset of
which has the fixed point property for nonexpansive mappings. Let
be a family of uniformly
-strict pseudo-contractions with respect to
,
and
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii), where
.
Let be the sequence generated by the following:

Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
and suppose that
. Then the sequence
converges strongly to a common fixed point
of
.
Proof.
It follows from the same argumentation as Theorem 3.2 that is bounded and
, where
is a nonexpansive mapping defined by (3.32). From Lemma 2.2 that the net
generated by
converges strongly to
, as
. Obviously,

In view of Lemma 2.8, we calculate

and therefore

Since ,
and
are bounded and
, we obtain

where . We also know that

From the fact that , as
,
is bounded and the duality mapping
is norm-to-weak
uniformly continuous on bounded subset of
, it follows that as
,

Combining (3.63), (3.64) and two results mentioned above, we get

From (3.28) and Lemma 2.8, we get

Hence applying in Lemma 2.7 to (3.67), we conclude that .
Corollary 3.6.
Let be a real
-uniformly convex Banach space with uniformly Gâteaux differentiable norm, and
a nonempty closed convex subset of
which has the fixed point property for nonexpansive mappings. Let
be a family of nonexpansive mappings such that
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii)
Let be the sequence generated by the following:

Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
and suppose that
. Then the sequence
converges strongly to a common fixed point
of
.
Corollary 3.7.
Let be a real
-uniformly convex Banach space with uniformly Gâteaux differentiable norm, and
a nonempty closed convex subset of
which has the fixed point property for nonexpansive mappings. Let
be a
-strict pseudo-contractions with respect to
,
and
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii), where
.
Let be the sequence generated by the following:

Then the sequence converges strongly to a common fixed point
of
.
4. Some Applications for Accretive Operators
We consider the problem of finding a zero of an accretive operator. An operator is said to be accretive if for each
and
, there exists
such that
An accretive operator
is said to satisfy the range condition if
for all
where
is the domain of
is the identity mapping on
,
is the range of
and
is the closure of
. If
is an accretive operator which satisfies the range condition, then we can define, for each
a mapping
by
which is called the resolvent of
. We know that
is nonexpansive and
for all
We also know the following [25]: For each
,
and
, it holds that

By the proof of Theorem in [3], we have the following lemma.
Lemma 4.1.
Let be a Banach space and
a nonempty closed convex subset of
Let
be an accretive operator such that
and
Suppose that
is a sequence of
such that
and
Then
(i)The sequence satisfies the AKTT-condition.
(ii) for all
and
where
as
By Corollary 3.3, we obtain the following result.
Theorem 4.2.
Let be a real
-uniformly convex Banach space with a weakly continuous duality mapping
, and
a nonempty closed convex subset of
. Let
is an
-accretive operator in
such that
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii);
(iv) is a sequence of
such that
and
Let be the sequence generated by the following:

Then the sequence converges strongly
which solves the following variational inequality:

By Corollary 3.6, we obtain the following result.
Theorem 4.3.
Let be a real
-uniformly convex Banach space with uniformly Gâteaux differentiable norm, and
a nonempty closed convex subset of
. Let
is an
-accretive operator in
such that
. Let
be a
-contraction with
. Assume that real sequences
,
and
in
satisfy the following conditions:
(i) for all
;
(ii) and
;
(iii);
(iv) is a sequence of
such that
and
Let be the sequence generated by the following:

Then the sequence converges strongly
in
.
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Acknowledgments
The first author is supported by the Thailand Research Fund under Grant TRG5280011 and the second author is supported by grant from the program of Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand.
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Wangkeeree, R., Kamraksa, U. Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces. Fixed Point Theory Appl 2010, 579725 (2010). https://doi.org/10.1155/2010/579725
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DOI: https://doi.org/10.1155/2010/579725
Keywords
- Banach Space
- Variational Inequality
- Nonexpansive Mapping
- Common Fixed Point
- Nonempty Closed Convex Subset