- Research Article
- Open access
- Published:
New Iterative Approximation Methods for a Countable Family of Nonexpansive Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2011, Article number: 671754 (2011)
Abstract
We introduce new general iterative approximation methods for finding a common fixed point of a countable family of nonexpansive mappings. Strong convergence theorems are established in the framework of reflexive Banach spaces which admit a weakly continuous duality mapping. Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator. The results presented in this paper mainly improve on the corresponding results reported by many others.
1. Introduction
In recent years, the existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [1–4] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [5, 6]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [2, 7]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [7, 8]).
Let be a normed linear space. Recall that a mapping
is called nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ1_HTML.gif)
We use to denote the set of fixed points of
, that is,
. A self mapping
is a contraction on
if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ2_HTML.gif)
One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [9–11]. More precisely, take and define a contraction
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ3_HTML.gif)
where is a fixed point. Banach's contraction mapping principle guarantees that
has a unique fixed point
in
. It is unclear, in general, what is the behavior of
as
, even if
has a fixed point. However, in the case of
having a fixed point, Browder [9] proved that if
is a Hilbert space, then
converges strongly to a fixed point of
. Reich [10] extended Browder's result to the setting of Banach spaces and proved that if
is a uniformly smooth Banach space, then
converges strongly to a fixed point of
and the limit defines the (unique) sunny nonexpansive retraction from
onto
. Xu [11] proved Reich's results hold in reflexive Banach spaces which have a weakly continuous duality mapping.
The iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [12–14] and the references therein. Let be a real Hilbert space, whose inner product and norm are denoted by
and
, respectively. Let
be a strongly positive bounded linear operator on
; that is, there is a constant
with property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ4_HTML.gif)
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ5_HTML.gif)
where is a given point in
. In 2003, Xu [13] proved that the sequence
defined by the iterative method below, with the initial guess
chosen arbitrarily
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ6_HTML.gif)
converges strongly to the unique solution of the minimization problem (1.5) provided the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [15] introduced the following iterative process for nonexpansive mappings (see [16] for further developments in both Hilbert and Banach spaces). Let
be a contraction on
. Starting with an arbitrary initial
, define a sequence
recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ7_HTML.gif)
where is a sequence in
. It is proved [15, 16] that under certain appropriate conditions imposed on
, the sequence
generated by (1.7) strongly converges to the unique solution
in
of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ8_HTML.gif)
Recently, Marino and Xu [17] mixed the iterative method (1.6) and the viscosity approximation method (1.7) and considered the following general iterative method:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ9_HTML.gif)
where is a strongly positive bounded linear operator on
. They proved that if the sequence
of parameters satisfies the following conditions:
(C1),
(C2),
(C3),
then the sequence generated by (1.9) converges strongly to the unique solution
in
of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ10_HTML.gif)
which is the optimality condition for the minimization problem: , where
is a potential function for
for
).
On the other hand, in order to find a fixed point of nonexpansive mapping , Halpern [18] was the first who introduced the following iteration scheme which was referred to as Halpern iteration in a Hilbert space:
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ11_HTML.gif)
He pointed out that the control conditions (C1) and (C2)
are necessary for the convergence of the iteration scheme (1.11) 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. [1] studied the strong convergence of the following modified version of Halpern iteration for
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ12_HTML.gif)
where is a nonempty closed convex subset of a uniformly convex Banach space
whose norm is uniformly Gáteaux differentiable,
is a sequence in
satisfying (C1)
, (C2)
, and either (C3)
or (C3')
for all
and
. Very recently, Song and Zheng [19] also introduced the conception of the condition
on a countable family of nonexpansive mappings and proved strong convergence theorems of the modified Halpern iteration (1.12) and the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ13_HTML.gif)
in a reflexive Banach space with a weakly continuous duality mapping and in a reflexive strictly convex Banach space with a uniformly Gáteaux differentiable norm.
Other investigations of approximating common fixed points for a countable family of nonexpansive mappings can be found in [1, 20–24] and many results not cited here.
In a Banach space having a weakly continuous duality mapping
with a gauge function
, an operator
is said to be strongly positive [25] if there exists a constant
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ15_HTML.gif)
where is the identity mapping. If
is a real Hilbert space, then the inequality (1.14) reduces to (1.4).
In this paper, motivated by Aoyama et al. [1], Song and Zheng [19], and Marino and Xu [17], we will combine the iterative method (1.12) with the viscosity approximation method (1.9) and consider the following three new general iterative methods in a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ17_HTML.gif)
where is strongly positive defined by (1.15),
is a countable family of nonexpansive mappings, and
is an
-contraction. We will prove in Section 3 that if the sequence
of parameters satisfies the appropriate conditions, then the sequences
, and
converge strongly to the unique solution
of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ18_HTML.gif)
Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator.
2. Preliminaries
Throughout this paper, let be a real Banach space, and
be its dual space. We write
(resp.,
) to indicate that the sequence
weakly (resp., weak*) converges to
; as usual
will symbolize strong convergence. Let
. A Banach space
is said to uniformly convex if, for any
, there exists
such that, for any
,
implies
. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [26]). A Banach space
is said to be smooth if the limit
exists for all
. It is also said to be uniformly smooth if the limit is attained uniformly for
.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ19_HTML.gif)
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 [27]). Browder [27] initiated the study of certain classes of nonlinear operators by means of the duality mapping
. Following Browder [27], we say that a Banach space
has a  weakly  continuous  duality  mapping  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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ20_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ21_HTML.gif)
where denotes the subdifferential in the sense of convex analysis.
Now, we collect some useful lemmas for proving the convergence result of this paper.
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 [28].
Lemma 2.1 (see [28]).
Assume that a Banach space has a weakly continuous duality mapping
with gauge
.
(i)For all , the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ22_HTML.gif)
In particular, for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ23_HTML.gif)
(ii)Assume that a sequence in
converges weakly to a point
,
then the following identity holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ24_HTML.gif)
Lemma 2.2 (see [1, Lemma  2.3]).
Let be a sequence of nonnegative real numbers such that satisfying the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ25_HTML.gif)
where satisfying the restrictions
(i);    (ii)
;    (iii)
.
Then, .
Definition 2.3 (see [1]).
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ26_HTML.gif)
Remark 2.4.
The example of the sequence of mappings satisfying AKTT-condition is supported by Lemma 4.6.
Lemma 2.5 (see [1, Lemma  3.2]).
Suppose that satisfies AKTT-condition, then, for each
,
converses strongly to a point in
. Moreover, let the mapping
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ27_HTML.gif)
Then, for each bounded subset of
,
.
The next valuable lemma was proved by Wangkeeree et al. [25]. Here, we present the proof for the sake of completeness.
Lemma 2.6.
Assume that a Banach space has a weakly continuous duality mapping
with gauge
. Let
be a strongly positive bounded linear operator on
with coefficient
and
, then
.
Proof.
From (1.15), we obtain that . Now, for any
with
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ28_HTML.gif)
That is, is positive. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ29_HTML.gif)
Let be a Banach space which admits a weakly continuous duality
with gauge
such that
is invariant on
that is,
. Let
be a nonexpansive mapping,
,
an
-contraction, and
a strongly positive bounded linear operator with coefficient
and
. Define the mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ30_HTML.gif)
Then, is a contraction mapping. Indeed, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ31_HTML.gif)
Thus, by Banach contraction mapping principle, there exists a unique fixed point in
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ32_HTML.gif)
Remark 2.7.
We note that space has a weakly continuous duality mapping with a gauge function
for all
. This shows that
is invariant on
.
Lemma 2.8 (see [25, Lemma  3.3]).
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
such that
is invariant on
. Let
be a nonexpansive mapping with
,
an
-contraction, and
a strongly positive bounded linear operator with coefficient
and
. Then, the net
defined by (2.14) converges strongly as
to a fixed point
of
which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ33_HTML.gif)
3. Main Results
We now state and prove the main theorems of this section.
Theorem 3.1.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
such that
is invariant on
. Let
be a countable family of nonexpansive mappings satisfying
. Let
be an
-contraction and
a strongly positive bounded linear operator with coefficient
and
. Let the sequence
be generated by (1.16), where
is a sequence in
satisfying the following conditions:
(C1),
(C2),
(C3).
Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
, and suppose that
. Then,
converges strongly to
which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ34_HTML.gif)
Proof.
Applying Lemma 2.8, there exists a point which solves the variational inequality (3.1). Next, we observe that
is bounded. Indeed, pick any
to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ35_HTML.gif)
It follows from induction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ36_HTML.gif)
Thus, is bounded, and hence so are
and
. Now, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ37_HTML.gif)
We observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ38_HTML.gif)
for all , where
is a constant satisfying
. Putting
. From AKTT-condition and (C3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ39_HTML.gif)
Therefore, it follows from Lemma 2.2 that . Since
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ40_HTML.gif)
Using Lemma 2.5, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ41_HTML.gif)
Next, we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ42_HTML.gif)
Let be a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ43_HTML.gif)
If follows from reflexivity of and the boundedness of a sequence
that there exists
which is a subsequence of
converging weakly to
as
. Since
is weakly continuous, we have by Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ44_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ45_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ46_HTML.gif)
Then, from , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ47_HTML.gif)
On the other hand, however,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ48_HTML.gif)
It follows from (3.14) and (3.15) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ49_HTML.gif)
Therefore, , and hence
. Since the duality map
is single valued and weakly continuous, we obtain, by (3.1), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ50_HTML.gif)
Next, we show that as
. In fact, since
, and
is a gauge function, then for
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ51_HTML.gif)
Finally, we show that as
. Following Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ52_HTML.gif)
It then follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ53_HTML.gif)
where . Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ54_HTML.gif)
It follows that from condition (C1), and (3.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ55_HTML.gif)
Applying Lemma 2.2 to (3.20), we conclude that as
; that is,
as
. This completes the proof.
Setting , where
is the identity mapping and
for all
in Theorem 3.1, we have the following result.
Corollary 3.2.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
. Suppose that
is a countable family of nonexpansive mappings satisfying
. Assume that
is defined by, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ56_HTML.gif)
where is a sequence in
satisfying the following conditions:
(C1),
(C2),
(C3).
Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
, and suppose that
, then
converges strongly to
of
which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ57_HTML.gif)
Applying Theorem 3.1, we can obtain the following two strong convergence theorems for the iterative sequences and
defined by (1.17).
Theorem 3.3.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
such that
is invariant on
. Let
be a countable family of nonexpansive mappings satisfying
. Let
be an
-contraction and
a strongly positive bounded linear operator with coefficient
and
. Let the sequence
be generated by (1.17), where
is a sequence in
satisfying the following conditions:
(C1),
(C2),
(C3).
Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
, and suppose that
, then
converges strongly to
which solves the variational inequality (3.1).
Proof.
Let be the sequence given by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ58_HTML.gif)
Form Theorem 3.1, . We claim that
. Applying Lemma 2.6, we estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ59_HTML.gif)
It follows from ,
, and Lemma 2.2 that
. Consequently,
as required.
Theorem 3.4.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
such that
is invariant on
. Let
be a countable family of nonexpansive mappings satisfying
. Let
be an
-contraction and
a strongly positive bounded linear operator with coefficient
and
. Let the sequence
be generated by (1.17), where
is sequence in
satisfying the following conditions:
(C1),
(C2),
(C3).
Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
, and suppose that
, then
converges strongly to
which solves the variational inequality (3.1).
Proof.
Let the sequences and
be given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ60_HTML.gif)
Taking , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ61_HTML.gif)
It follows from induction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ62_HTML.gif)
Thus, both and
are bounded. We observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ63_HTML.gif)
Thus, Theorem 3.1 implies that converges strongly to some point
. In this case, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ64_HTML.gif)
Hence, the sequence converges strongly to
. This competes the proof.
Setting , where
is the identity mapping and
for all
in Theorem 3.4, we have the following result.
Corollary 3.5.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
. Suppose that
is a countable family of nonexpansive mappings satisfying
. Assume that
is defined by for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ65_HTML.gif)
where is a sequence in
satisfying the following conditions:
(C1),
(C2),
(C3).
Suppose that satisfies the AKTT-condition. Let
be a mapping of
into itself defined by
for all
, and suppose that
, then
converges strongly to
of
which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ66_HTML.gif)
4. Applications
4.1.
-Mappings
Let be infinite mappings of
into itself, and let
be a nonnegative real sequence with
,
. For any
, define a mapping
of
into itself as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ67_HTML.gif)
Nonexpansivity of each ensures the nonexpansivity of
. The mapping
is called a
-mapping generated by
and
.
Throughout this section, we will assume that . Concerning
defined by (4.1), we have the following useful lemmas.
Lemma 4.1 (see [4]).
Let be a nonempty closed convex subset of a a strictly convex, reflexive Banach space
,
a family of infinitely nonexpansive mapping with
, and
a real sequence such that
,
, then:
(1) is nonexpansive and
for each
;
(2)for each and for each positive integer
, the limit
exists;
(3)the mapping define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ68_HTML.gif)
is a nonexpansive mapping satisfying , and it is called the
-mapping generated by
and
.
From Remark 3.1 of Peng and Yao [29], we obtain the following lemma.
Lemma 4.2.
Let be a strictly convex, reflexive Banach space,
a family of infinitely nonexpansive mappings with
, and
a real sequence such that
,
. Then sequence
satisfies the
-condition.
Applying Lemma 4.2 and Theorem 3.1, we obtain the following result.
Theorem 4.3.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
such that
is invariant on
. Let
be a countable family of nonexpansive mappings with
and
an
-contraction and
a strongly positive bounded linear operator with coefficient
and
. Let the sequence
be generated by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ69_HTML.gif)
where is defined by (4.1) and
is a sequence in
satisfying the conditions (C1), (C2), and (C3). Then
converges strongly to
in
.
Applying Lemma 4.2 and Theorem 3.3, we obtain the following result.
Theorem 4.4.
Let ,
,
,
,
, and
be as in Theorem 4.3. Let the sequence
be generated by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ70_HTML.gif)
then converges strongly to
in
.
Applying Lemma 4.2 and Theorem 3.4, we obtain the following result.
Theorem 4.5.
Let ,
,
,
,
, and
be as in Theorem 4.3. Let the sequence
be generated by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ71_HTML.gif)
then converges strongly to
in
.
4.2. 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 [30]: for each
and
, it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ72_HTML.gif)
From the Resolvent identity, we have the following lemma.
Lemma 4.6.
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 AKTT-condition,
(ii) for all
and
, where
as
.
Proof.
By the proof of Theorem 4.3 in [1] and applying Lemma 4.6 and Theorem 3.1, we obtain the following result.
Theorem 4.7.
Let be a reflexive Banach space which admits a weakly continuous duality mapping
with gauge
such that
is invariant on
. Let
be an accretive operator such that
. Assume that
is a nonempty closed convex subset of
such that
and
is an
-contraction. Let
be a strongly positive bounded linear operator with coefficient
and
. Suppose that
is a sequence of
such that
. Let the sequence
be generated by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ73_HTML.gif)
where is a sequence in
satisfying the following conditions (C1), (C2), and (C3), then
converges strongly to
in
.
Applying Lemma 4.6 and Theorem 3.3, we obtain the following result.
Theorem 4.8.
Let ,
,
,
,
,
, and
be as in Theorem 4.7. Let
be generated by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ74_HTML.gif)
then converges strongly to
in
.
Applying Lemma 4.6 and Theorem 3.4, we obtain the following result.
Theorem 4.9.
Let ,
,
,
,
,
, and
be as in Theorem 4.7. Let
be generated by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ75_HTML.gif)
Then converges strongly to
in
.
4.3. The Equilibrium Problems
Let be a real Hilbert space, and let  
be a bifunction of
, where
is the set of real numbers. The equilibrium problem for
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ76_HTML.gif)
The set of solutions of (4.10) is denoted by . Given a mapping
, let
for all
. Then,
if and only if
for all
, that is,
is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (4.10). Some methods have been proposed to solve the equilibrium problem; see, for instance, Blum and Oettli [31] and Combettes and Hirstoaga [32]. For the purpose of solving the equilibrium problem for a bifunction
, let us assume that
satisfies the following conditions:
(A1) for all
,
(A2) is monotone, that is,
for all
,
(A3) for each ,
(A4) for each is convex and lower semicontinuous.
The following lemmas were also given in [31, 32], respectively.
Lemma 4.10 (see [31, Corollary  1]).
Let be a nonempty closed convex subset of
, and let
be a bifunction of
satisfying
. Let
and
, then there exists
such that
.
Lemma 4.11 (see [32, Lemma  2.12]).
Assume that satisfies
. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ77_HTML.gif)
then, the following hold:
(1) is single valued,
(2) is firmly nonexpansive, that is, for any
,
(3),
(4) is closed and convex.
Theorem 4.12.
Let be a real Hilbert space. Let
be a bifunction from
satisfying (A1)–(A4) and
. Let
be an
-contraction,
a strongly positive bounded linear operator with coefficient
and
. Let the sequences
be generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ78_HTML.gif)
for all , where
is a sequence in
and
satisfying the following conditions:
(C1),
(C2),
(C3),
(C4) and
.
then and
converge strongly to
.
Proof.
Following the proof technique of Theorem 3.1, we only need, show that , for all
. From (4.12), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ79_HTML.gif)
On the other hand, from the definition of we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ80_HTML.gif)
Putting and
in (4.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ81_HTML.gif)
So, from (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ82_HTML.gif)
and hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ83_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ84_HTML.gif)
and hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ85_HTML.gif)
where is a constant satisfying
. Substituting (4.19) in (4.13) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ86_HTML.gif)
for some with
(the definition
). By the assumptions on
and
and using Lemma 2.2, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ87_HTML.gif)
From the definition of and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ88_HTML.gif)
Combining (4.21) and (4.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ89_HTML.gif)
From the definition of , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ90_HTML.gif)
Putting in (4.14) and
in (4.24), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ91_HTML.gif)
So, from (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ92_HTML.gif)
and hence, for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ93_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ94_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ95_HTML.gif)
then for each , we have from (4.23)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ96_HTML.gif)
This completes the proof.
Applying Theorem 4.12, we can obtain the following result.
Corollary 4.13.
Let be a real Hilbert space. Let
be a bifunction from
satisfying (A1)–(A4) and
. Let
be an
-contraction,
a strongly positive bounded linear operator with coefficient
and
. Let the sequences
be generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ97_HTML.gif)
for all , where
is a sequence in
and
satisfying the following conditions:
(C1),
(C2),
(C3),
(C4) and
,
then and
converge strongly to
.
Proof.
We observe that for all
. Then we rewrite the iterative sequence (4.31) by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ98_HTML.gif)
Let be the sequence given by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ99_HTML.gif)
Form Theorem 4.12, in
. We claim that
. Applying Lemma 2.6, we estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F671754/MediaObjects/13663_2010_Article_1420_Equ100_HTML.gif)
It follows from ,
, and Lemma 2.2 that
as
. Consequently,
as required.
References
Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(8):2350–2360. 10.1016/j.na.2006.08.032
Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996,202(1):150–159. 10.1006/jmaa.1996.0308
Shang M, Su Y, Qin X: Strong convergence theorems for a finite family of nonexpansive mappings. Fixed Point Theory and Applications 2007, 2007:-9.
Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001,5(2):387–404.
Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Review 1996,38(3):367–426. 10.1137/S0036144593251710
Combettes PL: Foundations of set theoretic estimation. Proceedings of the IEEE 1993,81(2):182–208.
Youla DC: Mathematical theory of image restoration by the method of convex projections. In Image Recovery: Theory and Applications. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:29–77.
Iusem AN, De Pierro AR: On the convergence of Han's method for convex programming with quadratic objective. Mathematical Programming. Series B 1991,52(2):265–284.
Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. Journal of Mathematical Analysis and Applications 1980,75(1):287–292. 10.1016/0022-247X(80)90323-6
Xu H-K: Strong convergence of an iterative method for nonexpansive and accretive operators. Journal of Mathematical Analysis and Applications 2006,314(2):631–643. 10.1016/j.jmaa.2005.04.082
Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numerical Functional Analysis and Optimization 1998,19(1–2):33–56.
Xu H-K: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589
Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society. Second Series 2002,66(1):240–256. 10.1112/S0024610702003332
Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028
Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Song Y, Zheng Y: Strong convergence of iteration algorithms for a countable family of nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):3072–3082. 10.1016/j.na.2009.01.219
Jung JS: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 2005,302(2):509–520. 10.1016/j.jmaa.2004.08.022
O'Hara JG, Pillay P, Xu H-K: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2003,54(8):1417–1426. 10.1016/S0362-546X(03)00193-7
O'Hara JG, Pillay P, Xu H-K: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2006,64(9):2022–2042. 10.1016/j.na.2005.07.036
Wangkeeree R: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-17.
Wangkeeree R, Kamraksa U: A general iterative method for solving the variational inequality problem and fixed point problem of an infinite family of nonexpansive mappings in Hilbert spaces. Fixed Point Theory and Applications 2009, 2009:-23.
Wangkeeree R, Petrot N, Wangkeeree R: The general iterative methods for nonexpansive mappings in Banach spaces. Journal of Global Optimization. In press
Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
Browder FE: Convergence theorems for sequences of nonlinear operators in Banach spaces. Mathematische Zeitschrift 1967, 100: 201–225. 10.1007/BF01109805
Lim T-C, Xu H-K: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 1994,22(11):1345–1355. 10.1016/0362-546X(94)90116-3
Peng J-W, Yao J-C: A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):6001–6010. 10.1016/j.na.2009.05.028
Eshita K, Takahashi W: Approximating zero points of accretive operators in general Banach spaces. JP Journal of Fixed Point Theory and Applications 2007,2(2):105–116.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.
Acknowledgments
The authors would like to thank the Centre of Excellence in Mathematics, Thailand for financial support. Finally, They would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Nammanee, K., Wangkeeree, R. New Iterative Approximation Methods for a Countable Family of Nonexpansive Mappings in Banach Spaces. Fixed Point Theory Appl 2011, 671754 (2011). https://doi.org/10.1155/2011/671754
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/671754