- Research Article
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Some Characterizations for a Family of Nonexpansive Mappings and Convergence of a Generated Sequence to Their Common Fixed Point
Fixed Point Theory and Applications volume 2010, Article number: 732872 (2009)
Abstract
Motivated by the method of Xu (2006) and Matsushita and Takahashi (2008), we characterize the set of all common fixed points of a family of nonexpansive mappings by the notion of Mosco convergence and prove strong convergence theorems for nonexpansive mappings and semigroups in a uniformly convex Banach space.
1. Introduction
Let be a nonempty bounded closed convex subset of a Banach space and
a nonexpansive mapping; that is,
satisfies
for any
, and consider approximating a fixed point of
. This problem has been investigated by many researchers and various types of strong convergent algorithm have been established. For implicit algorithms, see Browder [1], Reich [2], Takahashi and Ueda [3], and others. For explicit iterative schemes, see Halpern [4], Wittmann [5], Shioji and Takahashi [6], and others. Nakajo and Takahashi [7] introduced a hybrid type iterative scheme by using the metric projection, and recently Takahashi et al. [8] established a modified type of this projection method, also known as the shrinking projection method.
Let us focus on the following methods generating an approximating sequence to a fixed point of a nonexpansive mapping.Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space
and let
be a nonexpansive mapping of
into itself. Xu [9] considered a sequence
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ1_HTML.gif)
for each , where
is the closure of the convex hull of
,
is the generalized projection onto
, and
is a sequence in
with
as
. Then, he proved that
converges strongly to
. Matsushita and Takahashi [10] considered a sequence
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ2_HTML.gif)
for each , where
is the metric projection onto
and
is a sequence in
with
as
. They proved that
converges strongly to
.
In this paper, motivated by these results, we characterize the set of all common fixed points of a family of nonexpansive mappings by the notion of Mosco convergence and prove strong convergence theorems for nonexpansive mappings and semigroups in a uniformly convex Banach space.
2. Preliminaries
Throughout this paper, we denote by a real Banach space with norm
. We write
to indicate that a sequence
converges weakly to
. Similarly,
will symbolize strong convergence. Let
be the family of all strictly increasing continuous convex functions
satisfying that
. We have the following theorem [11, Theorem
] for a uniformly convex Banach space.
Theorem 2.1 (Xu [11]).
is a uniformly convex Banach space if and only if, for every bounded subset
of
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ3_HTML.gif)
for all and
.
Bruck [12] proved the following result for nonexpansive mappings.
Theorem 2.2 (Bruck [12]).
Let be a bounded closed convex subset of a uniformly convex Banach space
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ4_HTML.gif)
for all ,
,
with
and nonexpansive mapping
of
into
.
Let be a sequence of nonempty closed convex subsets of a reflexive Banach space
. We denote the set of all strong limit points of
by
, that is,
if and only if there exists
such that
converges strongly to
and that
for all
. Similarly the set of all weak subsequential limit points by
;
if and only if there exist a subsequence
of
and a sequence
such that
converges weakly to
and that
for all
. If
satisfies that
, then we say that
converges to
in the sense of Mosco and we write
. By definition, it always holds that
. Therefore, to prove
, it suffices to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ5_HTML.gif)
One of the simplest examples of Mosco convergence is a decreasing sequence with respect to inclusion. The Mosco limit of such a sequence is
. For more details, see [13].
Suppose that is smooth, strictly convex, and reflexive. The normalized duality mapping of
is denoted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ6_HTML.gif)
for . In this setting, we may show that
is a single-valued one-to-one mapping onto
. For more details, see [14–16].
Let be a nonempty closed convex subset of a strictly convex and reflexive Banach space
. Then, for an arbitrarily fixed
, a function
has a unique minimizer
. Using such a point, we define the metric projection
by
for every
. The metric projection has the following important property:
if and only if
and
for all
.
In the same manner, we define the generalized projection [17] for a nonempty closed convex subset
of a strictly convex, smooth, and reflexive Banach space
as follows. For a fixed
, a function
has a unique minimizer and we define
by this point. We know that the following characterization holds for the generalized projection [17, 18]:
if and only if
and
for all
.
Tsukada [19] proved the following theorem for a sequence of metric projections in a Banach space.
Theorem 2.3 (Tsukada [19]).
Let be a reflexive and strictly convex Banach space and let
be a sequence of nonempty closed convex subsets of
. If
exists and nonempty, then, for each
,
converges weakly to
, where
is the metric projection onto a nonempty closed convex subset
of
. Moreover, if
has the Kadec-Klee property, the convergence is in the strong topology.
On the other hand, Ibaraki et al. [20] proved the following theorem for a sequence of generalized projections in a Banach space.
Theorem 2.4 (Ibaraki et al. [20]).
Let be a strictly convex, smooth, and reflexive Banach space and let
be a sequence of nonempty closed convex subsets of
. If
exists and nonempty, then, for each
,
converges weakly to
, where
is the generalized projection onto a nonempty closed convex subset
of
. Moreover, if
has the Kadec-Klee property, the convergence is in the strong topology.
Kimura [21] obtained the further generalization of this theorem by using the Bregman projection; see also [22].
Theorem 2.5 (Kimura [21]).
Let be a nonempty closed convex subset of a reflexive Banach space
and let
be a Bregman function on
; that is, (i)
is lower semicontinuous and strictly convex; (ii)
is contained by the interior of the domain of
; (iii)
is Gâteaux differentiable on
; (iv) the subsets
and
of
are both bounded for all
and
, where
for all
and
. Let
be a sequence of nonempty closed convex subsets of
such that
exists and is nonempty. Suppose that
is sequentially consistent; that is, for any bounded sequence
of
and
of the domain of
,
implies
. Then, the sequence
of Bregman projections converges strongly to
for all
.
We note that the generalized duality mapping coincides with
if the function
is defined by
for
. In this case, the Bregman projection
with respect to
becomes the generalized projection
for any nonempty closed convex subset
of
.
3. Main Results
Let be a nonempty closed convex subset of
and let
be a sequence of mappings of
into itself such that
. We consider the following conditions.
(I)For every bounded sequence in
,
implies
, where
is the set of all weak cluster points of
; see [23–25].
(II)for every sequence in
and
,
and
imply
.
We know that condition (I) implies condition (II). Then, we have the following results.
Theorem 3.1.
Let be a nonempty bounded closed convex subset of a uniformly convex Banach space
and let
be a family of nonexpansive mappings of
into itself with
. Let
for each
, where
. Then, the following are equivalent:
(i) satisfies condition (I);
(ii)for every with
as
,
.
Proof.
First, let us prove that (i) implies (ii). Let with
as
. It is obvious that
and
is closed and convex for all
. Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ7_HTML.gif)
Let . Then, there exists a sequence
such that
for all
and
as
. Let
be a sequence in
such that
for every
and that
for all
. Fix
. From the definition of
, there exist
,
, and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ8_HTML.gif)
for each . On the other hand, by Theorem 2.2, there exists a strictly increasing continuous convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ9_HTML.gif)
for all ,
,
with
and nonexpansive mapping
of
into
. Thus we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ10_HTML.gif)
for every , which implies
as
. From condition (I), we get
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ11_HTML.gif)
By (3.1) and (3.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ12_HTML.gif)
Next we show that (ii) implies (i). Let be a sequence in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ13_HTML.gif)
and define by
for each
. Suppose that a subsequence
of
converges weakly to
. Then since
for all
and
, we have
; that is, condition (I) holds.
For a sequence of mappings satisfying condition (II), we have the following characterization.
Theorem 3.2.
Let be a nonempty bounded closed convex subset of a uniformly convex Banach space
and let
be a family of nonexpansive mappings of
into itself with
. Let
and
for each
, where
. Then, the following are equivalent:
(i) satisfies condition (II);
(ii)for every with
as
,
.
Proof.
Let us show that (i) implies (ii). Let with
as
. We have
for all
. Thus we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ14_HTML.gif)
Let . We have
for all
. As in the proof of Theorem 3.1, we get
. By condition (II), we obtain
, which implies
. Hence we have
.
Suppose that condition (ii) holds. Let be a sequence in
and
such that
and that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ15_HTML.gif)
for each , we have
. Letting
for each
, we have
for every
and
as
, which implies
. Hence (i) holds, which is the desired result.
Remark 3.3.
In Theorem 3.2, it is obvious by definition that is a decreasing sequence with respect to inclusion. Therefore, conditions
and
are also equivalent to
for every with
as
,
,
where is the Painlevé-Kuratowski limit of
; see, for example, [13] for more details.
In the next section, we will see various types of sequences of nonexpansive mappings which satisfy conditions (I) and (II).
4. The Sequences of Mappings Satisfying Conditions (I) and (II)
First let us show some known results which play important roles for our results.
Theorem 4.1 (Browder [1]).
Let be a nonempty closed convex subset of a uniformly convex Banach space
and
a nonexpansive mapping on
with
. If
converges weakly to
and
converges strongly to
, then
is a fixed point of
.
Theorem 4.2 (Bruck [26]).
Let be a nonempty closed convex subset of a strictly convex Banach space
and
a nonexpansive mapping for each
. Let
be a sequence of positive real numbers such that
. If
is nonempty, then the mapping
is well defined and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ16_HTML.gif)
Theorems 4.3, 4.5(i), 4.6–4.9 show the examples of a family of nonexpansive mappings satisfying condition (I). Theorems 4.5(ii), 4.11, and 4.12 show those satisfying condition (II).
Theorem 4.3.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be a nonexpansive mapping of
into itself with
. Let
for all
. Then,
is a family of nonexpansive mappings of
into itself with
and satisfies condition (I).
Proof.
This is a direct consequence of Theorem 4.1.
Remark 4.4.
In the previous theorem, if is bounded, then
is guaranteed to be nonempty by Kirk's fixed point theorem [27].
Let be a Banach space and
a set-valued operator on
.
is called an accretive operator if
for every
and
with
and
.
Let be an accretive operator and
. We know that the operator
has a single-valued inverse, where
is the identity operator on
. We call
the resolvent of
and denote it by
. We also know that
is a nonexpansive mapping with
for any
, where
. For more details, see, for example, [15].
We have the following result for the resolvents of an accretive operator by [25]; see also [15, Theorem ], and [16, Theorem
] .
Theorem 4.5.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be an accretive operator with
and
. Let
for every
, where
for all
. Then,
is a family of nonexpansive mappings of
into itself with
and the following hold:
(i)if , then
satisfies condition (I),
(ii)if there exists a subsequence of
such that
, then
satisfies condition (II).
Proof.
It is obvious that is a nonexpansive mapping of
into itself and
for all
.
For (i), suppose and let
be a bounded sequence in
such that
. By [25, Lemma
], we have
. Using Theorem 4.1 we obtain
.
Let us show (ii). Let be a subsequence of
with
and let
be a sequence in
and
such that
and
. As in the proof of (i), we get
and
.
Let be a nonempty closed convex subset of
. Let
be a family of mappings of
into itself and let
be a sequence of real numbers such that
for every
with
. Takahashi [16, 28] introduced a mapping
of
into itself for each
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ17_HTML.gif)
Such a mapping is called the W-mapping generated by
and
. We have the following result for the W-mapping by [29, 30]; see also [25, Lemma
].
Theorem 4.6.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be a family of nonexpansive mappings of
into itself with
. Let
be a sequence of real numbers such that
for every
with
and let
be the W-mapping generated by
and
. Let
for every
. Then,
is a family of nonexpansive mappings of
into itself with
and satisfies condition (I).
Proof.
It is obvious that is a family of nonexpansive mappings of
into itself. By [29, Lemma
],
for all
, which implies
. Let
be a bounded sequence in
such that
. We have
. Let
. From Theorem 2.1, for a bounded subset
of
containing
and
, there exists
, where
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ18_HTML.gif)
for every , where
. Thus we obtain
. Let
. Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ19_HTML.gif)
As in the proof of [30, Theorem ], we get
for each
. Using Theorem 4.1 we obtain
.
We have the following result for a convex combination of nonexpansive mappings which Aoyama et al. [31] proposed.
Theorem 4.7.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be a family of nonexpansive mappings of
into itself such that
. Let
be a family of nonnegative numbers with indices
with
such that
(i) for every
,
(ii) for each
,
and let for all
, where
for some
with
. Then,
is a family of nonexpansive mappings of
into itself with
and satisfies condition (I).
Proof.
It is obvious that is a family of nonexpansive mappings of
into itself. By Theorem 4.2, we have
and thus
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ20_HTML.gif)
Let be a bounded sequence in
such that
. Let
,
, and
for
. By Theorem 2.1, for a bounded subset
of
containing
and
, there exists
with
which satisfies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ21_HTML.gif)
for , where
. Since
for all
and
, we get
and hence
for each
. Therefore, using Theorem 4.1 we obtain
.
Let be a nonempty closed convex subset of a Banach space
and let
be a semigroup. A family
is said to be a nonexpansive semigroup on
if
(i)for each ,
is a nonexpansive mapping of
into itself;
(ii) for every
.
We denote by the set of all common fixed points of
, that is,
. We have the following result for nonexpansive semigroups by [25, Lemma
]; see also [32, 33].
Theorem 4.8.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be a semigroup. Let
be a nonexpansive semigroup on
such that
and let
be a subspace of
such that
contains constants,
is
-invariant (i.e.,
) for each
, and the function
belongs to
for every
and
. Let
be a sequence of means on
such that
as
for all
and let
for each
. Then,
is a family of nonexpansive mappings of
into itself with
and satisfies condition (I).
Proof.
It is obvious that is a family of nonexpansive mappings of
into itself. By [25, Lemma
], we have
. Let
be a bounded sequence in
such that
. Then we get
for every
. Using Theorem 4.1 we have
.
Let be a nonempty closed convex subset of a Banach space
. A family
of mappings of
into itself is called a one-parameter nonexpansive semigroup on
if it satisfies the following conditions:
(i) for all
;
(ii) for every
;
(iii) for each
and
;
(iv)for all ,
is continuous.
We have the following result for one-parameter nonexpansive semigroups by [25, Lemma ].
Theorem 4.9.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be a one-parameter nonexpansive semigroup on
with
. Let
satisfy
and let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ22_HTML.gif)
for all and
. Then,
is a family of nonexpansive mappings of
into itself satisfying that
and condition (I).
Remark 4.10.
If is bounded, then
is guaranteed to be nonempty; see [34].
Proof.
It is obvious that is a family of nonexpansive mappings of
into itself. By [25, Lemma
], we have
. Let
be a bounded sequence in
such that
. We get
for every
. Hence, using Theorem 4.1 we have
.
Motivated by the idea of [23, page 256], we have the following result for nonexpansive mappings.
Theorem 4.11.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be a countable index set. Let
be an index mapping such that, for all
, there exist infinitely many
satisfying
. Let
be a family of nonexpansive mappings of
into itself satisfying
and let
for all
. Then,
is a family of nonexpansive mappings of
into itself with
and satisfies condition (II).
Proof.
It is obvious that . Let
be a sequence in
and
such that
and
. Fix
. There exists a subsequence
of
such that
for all
. Thus we have
. Therefore, using Theorem 4.1
for every
and hence we get
.
From Theorem 4.11, we have the following result for one-parameter nonexpansive semigroups.
Theorem 4.12.
Let be a nonempty closed convex subset of a uniformly convex Banach space
and let
be a one-parameter nonexpansive semigroup on
such that
. Let
for every
with
and
as
and
for all
, where
is an an index mapping satisfing, for all
, there exist infinitely many
such that
. Then,
is a family of nonexpansive mappings of
into itself with
and satisfies condition (II).
Remark 4.13.
If is bounded, it is guaranteed that
. See [34].
Proof.
We have by [35, Lemma
]; see also [36]. By Theorem 4.11, we obtain the desired result.
5. Strong Convergence Theorems
Throughout this section, we assume that is a nonempty bounded closed convex subset of a uniformly convex Banach space
and
is a family of nonexpansive mappings of
into itself with
. Then, we know that
is closed and convex.
We get the following results for the metric projection by using Theorems 2.3, 3.1, and 3.2.
Theorem 5.1.
Let and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ23_HTML.gif)
for each , where
such that
as
, and
is the metric projection onto
. If
satisfies condition (I), then
converges strongly to
.
Theorem 5.2.
Let and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ24_HTML.gif)
for each , where
such that
as
. If
satisfies condition (II), then
converges strongly to
.
On the other hand, we have the following results for the Bregman projection by using Theorems 2.5, 3.1, and 3.2.
Theorem 5.3.
Let and let
be a Bregman function on
and let
be sequentially consistent. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ25_HTML.gif)
for each , where
such that
as
and
is the Bregman projection onto
. If
satisfies condition (I), then
converges strongly to
.
Theorem 5.4.
Let , let
be a Bregman function on
, and let
be sequentially consistent. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ26_HTML.gif)
for each , where
such that
as
. If
satisfies condition (II), then
converges strongly to
.
In a similar fashion, we have the following results for the generalized projection by using Theorems 2.4, 3.1, and 3.2.
Theorem 5.5.
Suppose that is smooth. Let
and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ27_HTML.gif)
for each , where
such that
as
and
is the generalized projection onto
. If
satisfies condition (I), then
converges strongly to
.
Theorem 5.6.
Suppose that is smooth. Let
and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ28_HTML.gif)
for each , where
with
as
. If
satisfies condition (II), then
converges strongly to
.
Combining these theorems with the results shown in the previous section, we can obtain various types of convergence theorems for families of nonexpansive mappings.
6. Generalization of Xu's and Matsushita-Takahashi's Theorems
At the end of this paper, we remark the relationship between these results and the convergence theorems by Xu [9] and Matsushita and Takahashi [10] mentioned in the introduction.
Let us suppose the all assumptions in their results, respectively. Let be a countable family of nonexpansive mappings of
into itself such that
and suppose that it satisfies condition (I). Let us define
for
. Then, by definition, we have that
for every
. On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ29_HTML.gif)
for every from basic properties of
and
. Therefore, for each theorem we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ30_HTML.gif)
for every by using mathematical induction. Since
is bounded, a sequence
converges to
for any
in
whenever
converges to
. Thus, using Theorem 3.1 we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ31_HTML.gif)
and therefore . Consequently, by using Theorems 2.3 and 2.4, we obtain the following results generalizing the theorems of Xu, and Matsushita and Takahashi, respectively.
Theorem 6.1.
Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space
and
a sequence of nonexpansive mappings of
into itself such that
and suppose that it satisfies condition (I). Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ32_HTML.gif)
for each , where
is a sequence in
with
as
. Then,
converges strongly to
.
Theorem 6.2.
Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space
and
a sequence of nonexpansive mappings of
into itself such that
and suppose that it satisfies condition (I). Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F732872/MediaObjects/13663_2009_Article_1334_Equ33_HTML.gif)
for each , where
is a sequence in
with
as
. Then,
converges strongly to
.
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Acknowledgment
The first author is supported by Grant-in-Aid for Scientific Research no. 19740065 from Japan Society for the Promotion of Science. This work is Dedicated to Professor Wataru Takahashi on his retirement.
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Kimura, Y., Nakajo, K. Some Characterizations for a Family of Nonexpansive Mappings and Convergence of a Generated Sequence to Their Common Fixed Point. Fixed Point Theory Appl 2010, 732872 (2009). https://doi.org/10.1155/2010/732872
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DOI: https://doi.org/10.1155/2010/732872