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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
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
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
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
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
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,
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
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
for each 
. On the other hand, by Theorem 2.2, there exists a strictly increasing continuous convex function 
with 
such that
for all 
, 
, 
with 
and nonexpansive mapping 
of 
into 
. Thus we get
for every 
, which implies 
as 
. From condition (I), we get 
, that is,
By (3.1) and (3.5), we have
Next we show that (ii) implies (i). Let 
be a sequence in 
such that
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
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
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
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:
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
for every 
, where 
. Thus we obtain 
. Let 
. Similarly, we have
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
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
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
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
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
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
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
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
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
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
for every 
from basic properties of 
and 
. Therefore, for each theorem we have
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
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
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
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