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# Some Characterizations for a Family of Nonexpansive Mappings and Convergence of a Generated Sequence to Their Common Fixed Point

## 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

(1.1)

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

(1.2)

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

(2.1)

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

(2.2)

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

(2.3)

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,

(2.4)

for . In this setting, we may show that is a single-valued one-to-one mapping onto . For more details, see [1416].

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.

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 [2325].

(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

(3.1)

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

(3.2)

for each . On the other hand, by Theorem 2.2, there exists a strictly increasing continuous convex function with such that

(3.3)

for all , , with and nonexpansive mapping of into . Thus we get

(3.4)

for every , which implies as . From condition (I), we get , that is,

(3.5)

By (3.1) and (3.5), we have

(3.6)

Next we show that (ii) implies (i). Let be a sequence in such that

(3.7)

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

(3.8)

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

(3.9)

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

(4.1)

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:

(4.2)

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

(4.3)

for every , where . Thus we obtain . Let . Similarly, we have

(4.4)

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

(4.5)

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

(4.6)

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

(4.7)

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

(5.1)

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

(5.2)

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

(5.3)

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

(5.4)

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

(5.5)

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

(5.6)

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

(6.1)

for every from basic properties of and . Therefore, for each theorem we have

(6.2)

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

(6.3)

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

(6.4)

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

(6.5)

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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Correspondence to Yasunori Kimura.

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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