Open Access

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

Fixed Point Theory and Applications20092010:732872

DOI: 10.1155/2010/732872

Received: 7 October 2009

Accepted: 19 October 2009

Published: 27 October 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
(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.

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 [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 .

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology
(2)
Faculty of Engineering, Tamagawa University

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© Y. Kimura and K. Nakajo. 2010

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