On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings

Abstract

We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

1. Introduction

Let be a subset of a Banach space . A mapping is nonexpansive if for all In this paper, the fixed-point set of the mapping is denoted by , that is, In 1973, Bruck [1] proved that for a given countable family of nonexpansive mappings in a strictly convex Banach space there exists a single nonexpansive mapping whose fixed-point set is identical to the set of common fixed points of the family. More precisely, the following is obtained.

Theorem 1.1.

Let be a closed convex subset of a strictly convex Banach space and let be a sequence of nonexpansive mappings such that Suppose that is a sequence in such that and is defined by

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Then is nonexpansive and

Recall that is strictly convex if whenever and are norm-one elements in satisfying it follows that It is worth mentioning that Bruck's result above remains true for the class of quasi-nonexpansive mappings, that is, the set of common fixed points of a countable family of quasi-nonexpansive mappings is identical to the fixed-point set of a single quasi-nonexpansive mapping. A mapping : is quasi-nonexpansive if and for all and

In 2004, Matsushita and Takahashi [2–4] introduced the so-called relatively nonexpansive mappings in Banach spaces. This class of mappings includes the resolvent of a maximal monotone operator and Alber's generalized projection. For more examples, we refer to [2–6]. Recently, Kohsaka and Takahashi [7] proved an analogue version of Bruck's result for a family of relatively nonexpansive mappings and they asked the following question.

Question 1.

For a given countable family of relatively nonexpansive mappings, is there a single strongly relatively nonexpansive mapping such that its fixed-point set is identical to the set of common fixed points of the family?

A positive answer to this question is given in [7] for a finite family of mappings. The purpose of this paper is to give the answer of Kohsaka and Takahashi's question in positive. We also introduce a concept of strongly generalized nonexpansive mappings and present the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. Finally, inspired by [8], we prove the duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings.

2. Preliminaries

We collect together some definitions and preliminaries which are needed in this paper. The strong and weak convergences of a sequence in a Banach space to an element are denoted by and , respectively. A Banach space is uniformly convex if whenever and are sequences in satisfying and it follows that It is known that if is uniformly convex, then it is reflexive and strictly convex. We say that is uniformly smooth if the dual space of is uniformly convex. A Banach space is smooth if the limit exists for all norm-one elements and in . It is not hard to show that if is reflexive, then is smooth if and only if is strictly convex. The value of at is denoted by The duality mapping is defined by

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for all The following facts are known (e.g., see [9, 10]).

(a)If is smooth, then is single valued.

(b)If is strictly convex, then is one-to-one, that is, implies that .

(c)If is reflexive, then is onto.

(d) If is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

For a smooth Banach space , Alber [5] considered the functional defined by

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Using this functional, Matsushita and Takahashi [2–4] studied and investigated the following mappings in Banach spaces. Suppose that is a subset of a smooth Banach space . A mapping is relatively nonexpansive if the following properties are satisfied.

(R1)  

(R2)  for all and

(R3)   is demiclosed at zero, that is; whenever a sequence in converges weakly to and converges strongly to it follows that

In a Hilbert space , the duality mapping is an identity mapping and for all . Hence, if is relatively nonexpansive, then it is quasi-nonexpansive and is demiclosed at zero.

Recently, Kohsaka and Takahashi [7] proved an analogue version of Bruck's result for a family of relatively nonexpansive mappings. More precisely, they obtained the following.

Theorem 2.1 (see [7, Theorem  3.4]).

Let be a closed convex subset of a uniformly convex and uniformly smooth Banach space and let be a finite family of relatively nonexpansive mappings such that . Suppose that and are finite sequences such that and is defined by

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Then is strongly relatively nonexpansive and

Recall that a relatively nonexpansive mapping [6] is strongly relatively nonexpansive if whenever is a bounded sequence in such that for some it follows that

To obtain the result for a countable family of relatively nonexpansive mappings, the same authors proved the following result.

Theorem 2.2 (see [7, Theorem  3.3]).

Let and be as in Theorem 2.1 and let be a sequence of relatively nonexpansive mappings such that . Suppose that and are sequences such that and is defined by

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Then is relatively nonexpansive and .

Remark 2.3.

They also asked the question of whether the mapping in Theorem 2.2 is strongly relatively nonexpansive (see [7, Problem  3.5]).

The following lemmas are needed in proving the result.

Lemma 2.4 (see [11, Theorem  2]).

Let be a uniformly convex Banach space and let Then there exists a strictly increasing, continuous and convex function such that and

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for all and

Lemma 2.5.

Let be a uniformly convex Banach space and let . Then there exists a strictly increasing, continuous and convex function such that and

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for all with and .

Proof.

We note that both series and converge. For let be a function satisfying the properties of Lemma 2.4. Using the convexity of , we have

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This completes the proof.

Lemma 2.6 (see [12, Lemma  2.10]).

Let be a strictly convex Banach space and let with If is a sequence in such that both series and converge, and

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then is a constant sequence.

Lemma 2.7 (see [13, Proposition  2]).

Let be a smooth and uniformly convex Banach space. Suppose that either or is a bounded sequence in and Then

3. Relatively Nonexpansive Mappings and Quasi-Nonexpansive Mappings

We first start with some observation which is a tool for proving Theorem 3.2.

Theorem 3.1.

Let be a closed convex subset of a uniformly convex and uniformly smooth Banach space and let be a sequence of mappings such that and

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Suppose that is a sequence in such that and is defined by

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Let be a bounded sequence in Then the following are equivalent.

(a)

(b) for each .

In particular,

Proof.

For fixed and we have

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In particular, for all and Hence, for each the series converges (absolutely). This implies that the mapping is well defined.

Let be a bounded sequence in Suppose that

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By the boundedness of we put

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Then for all We now consider the following estimates for each such that and for any :

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where is the function given in Lemma 2.5 associated with the uniform convexity of and the number . Notice that . Consequently, for ,

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This implies that

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We next prove that

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Let be given. We choose an integer such that Since as for all , we now choose an integer such that

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for all and Then, if

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This implies that (3.9) holds. In particular, since is uniformly norm-to-norm continuous on each bounded set, we can conclude from (3.8) that

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and from (3.9) that

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This together with (3.4) gives

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Assertion follows immediately from (3.12) and (3.14).

Conversely, we assume that for each Since is uniformly norm-to-norm continuous on each bounded set,

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We show that

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Let Then there exist positive integers such that and

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for all and If , then

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By the uniform norm-to-norm continuity of on each bounded set, we can conclude assertion from (3.16). This completes the proof.

Theorem 3.2.

Let be a closed convex subset of a uniformly convex and uniformly smooth Banach space and let be a countable family of relatively nonexpansive mappings such that Suppose that is a sequence in such that and is defined by

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Then is relatively nonexpansive and

Proof.

To show that is relatively nonexpansive, we prove only that is demiclosed at zero. Suppose that is a sequence in such that and . From Theorem 3.1, we have for each . Since each is demiclosed at zero, . Consequently, as desired.

We now give an answer of Kohsaka and Takahashi's question in positive.

Theorem 3.3.

The mapping in Theorem 2.2 is strongly relatively nonexpansive.

Proof.

The mapping can be rewritten as

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where is the identity mapping, for all , and It follows from Theorem 3.2 that is relatively nonexpansive, where Consequently, by Theorem 2.1 with the mapping

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is strongly relatively nonexpansive.

Using the same idea as in Theorem 3.1, we also have the following result whose proof is left to the reader to verify.

Theorem 3.4.

Let be a closed convex subset of a uniformly convex Banach space and let be a sequence of quasi-nonexpansive mappings such that . Suppose that is a sequence in such that and is defined by

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Then is demi-closed at zero if and only if each mapping is demi-closed at zero.

4. Ibaraki-Takahashi's Generalized Nonexpansive Mappings

Let be a subset of a smooth Banach space . In 2007, Ibaraki and Takahashi [14] introduced the following mapping. A mapping is generalized nonexpansive if the following properties are satisfied:

(G1)

(G2) for all and .

A mapping satisfies property (G3) if whenever is a sequence in such that and it follows that Here denotes the convergence in the dual space.

The generalized resolvent of the maximal monotone operator where is a smooth and uniformly convex Banach space, and the sunny generalized nonexpansive retraction from a strictly convex, smooth, and reflexive Banach space onto its closed subset are examples of generalized nonexpansive mappings satisfying property (G3) (see [15]). The relation between two classes of relatively nonexpansive mappings and of generalized nonexpansive mappings is recently obtained in [8].

The property of the mapping and the demiclosedness of are related as shown in the following remark.

Remark 4.1.

Let be a subset of a smooth Banach space and Then the following assertions hold true.

If is uniformly smooth, the duality mapping is weakly sequentially continuous, and satisfies property then is demiclosed at zero.

If is uniformly convex, is weakly sequentially continuous, and is demi-closed at zero, then satisfies property

Theorem 4.2.

Let be a closed convex subset of a smooth Banach space and let be a sequence of generalized nonexpansive mappings such that Suppose that is a sequence in such that and is defined by

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Then the mapping is well defined and the following assertions hold true.

If is strictly convex, then and is generalized nonexpansive.

If is uniformly convex and is a bounded sequence in then the following statements are equivalent:

(a)

(b) for each

The mapping is demi-closed at zero if and only if each mapping is demi-closed at zero.

Suppose that is uniformly convex and uniformly smooth. Then the mapping satisfies property if and only if each mapping satisfies property

Proof.

Using some basic properties of the functional , we have for all , Since the sequence is bounded for each and, hence the series converges (absolutely). This implies that is well defined. For fixed and we have the following expressions:

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(i)The inclusion is obvious. To see the reverse inclusion, let By the convexity of and the expressions of (4.2), we have

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It follows from Lemma 2.6 that is a constant sequence, and hence for all This implies that , that is, . Now Again, using the convexity of we can show that satisfies property and hence it is generalized nonexpansive, as desired.

(ii)Since the proof of this assertion is very similar to that of Theorem 3.1, it is omitted.

(iii) and (iv) follow directly from .

Remark 4.3.

Theorem 4.2 generalizes [16, Theorem  3.3] from a finite family to a countable one.

Following Reich [6], we introduced the following concept. A generalized nonexpansive mapping is strongly generalized nonexpansive if whenever is a bounded sequence in such that for some it follows that .

Lemma 4.4.

Let be a closed convex subset of a strictly convex and smooth Banach space . Suppose that is a generalized nonexpansive mapping and a strongly generalized nonexpansive mapping, respectively, and suppose that For let the mapping be defined by

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Then . If, in addition, is uniformly convex, then is strongly generalized nonexpansive.

Proof.

The first assertion follows from Theorem 4.2 We now assume that is uniformly convex. Suppose that is a bounded sequence in such that for some It is clear that the sequences and are both bounded. By the uniform convexity of , we have

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where is a function given by Lemma 2.4. Since and are generalized nonexpansive,

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Consequently, and hence . This implies that . Since is strongly generalized nonexpansive, It follows from Lemma 2.7 that and hence This implies that and is strongly generalized nonexpansive, as desired.

The following is an analogue version of Kohsaka and Takahashi's question for a countable family of generalized nonexpansive mappings.

Theorem 4.5.

Let be a closed convex subset of a smooth and uniformly convex Banach space and let be a countable family of generalized nonexpansive mappings such that Then there exists a strongly generalized nonexpansive mapping such that

Proof.

Suppose that is a sequence in such that and We define by

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Notice that is generalized nonexpansive and by Theorem 4.2 Moreover, by Lemma 4.4 and the fact that the identity is strongly generalized nonexpansive, the conclusion is satisfied by the mapping

5. Duality between Strongly Relatively Nonexpansive Mappings and Strongly Generalized Nonexpansive Mappings

Let be a subset of a smooth, strictly convex and reflexive Banach space and let be a mapping. We can define the duality of by (see [8])

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We now consider a functional from into still denoted by , by

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where is the duality mapping from onto It is clear that Then, whenever are elements in and are elements in satisfying and it follows that

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Remark 5.1.

The following assertions hold (see [8]).

If and then In particular, Moreover, if is a sequence in and then

(i) if and only if

(ii) if and only if

If and then

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The following duality theorem is proved in [8].

Theorem 5.2.

Let be a subset of a smooth, strictly convex and reflexive Banach space and let be a mapping. Suppose that is the duality of . Then the following assertions hold true.

(1)If is relatively nonexpansive, then is generalized nonexpansive with property

(2) If is generalized nonexpansive with property then is relatively nonexpansive.

We now prove the duality theorem for strongly relatively nonexpansive mappings and strongly generalized nonexpansive mappings.

Theorem 5.3.

Let be a subset of a smooth, strictly convex and reflexive Banach space and let : be a mapping. Suppose that : is the duality of . Then the following assertions hold true.

(1)If is strongly relatively nonexpansive, then is strongly generalized nonexpansive with property

(2)If is strongly generalized nonexpansive with property then is strongly relatively nonexpansive.

Proof.

We prove only and leave for the reader to verify. Suppose that is a bounded sequence in such that for some We assume that is a sequence in such that and is a point in such that Clearly, is bounded. Moreover, by Remark 5.1, we have and Consequently, It follows from the strongly relative nonexpansiveness that This completes the proof.