- Research Article
- Open Access

# Existence of Fixed Points of Firmly Nonexpansive-Like Mappings in Banach Spaces

- Koji Aoyama
^{1}and - Fumiaki Kohsaka
^{2}Email author

**2010**:512751

https://doi.org/10.1155/2010/512751

© K. Aoyama and F. Kohsaka. 2010

**Received:**27 November 2009**Accepted:**12 February 2010**Published:**23 February 2010

## Abstract

The aim of this paper is to obtain some existence theorems related to a hybrid projection method and a hybrid shrinking projection method for firmly nonexpansive-like mappings (mappings of type (P)) in a Banach space. The class of mappings of type (P) contains the classes of resolvents of maximal monotone operators in Banach spaces and firmly nonexpansive mappings in Hilbert spaces.

## Keywords

- Hilbert Space
- Banach Space
- Equilibrium Problem
- Nonexpansive Mapping
- Existence Theorem

## 1. Introduction

*firmly nonexpansive mapping*in the Hilbert space setting. In fact, if is a Hilbert space and : is a maximal monotone operator, then the resolvent of is a single-valued firmly nonexpansive mapping of onto the domain of that is, : is onto and

for all Further, the set of fixed points of coincides with that of solutions to (1.1); see, for example, [5].

In 2000, Solodov and Svaiter [9] proved the following strong convergence theorem for maximal monotone operators in Hilbert spaces.

Theorem 1.1 (see [9]).

for all where is a sequence of positive real numbers such that and denotes the metric projection of onto for all . Then converges strongly to

This method is sometimes called a hybrid projection method; see also Bauschke and Combettes [10] on more general results for a class of nonlinear operators including that of resolvents of maximal monotone operators in Hilbert spaces. Ohsawa and Takahashi [11] obtained a generalization of Theorem 1.1 for maximal monotone operators in Banach spaces.

Many authors have investigated several types of hybrid projection methods since then; see, for example, [12–30] and references therein. In particular, Kamimura and Takahashi [17] obtained another generalization of Theorem 1.1 for maximal monotone operators in Banach spaces. Bauschke and Combettes [16] and Otero and Svaiter [25] also obtained generalizations of Theorem 1.1 with Bregman functions in Banach spaces. Matsushita and Takahashi [20] obtained a generalization of Ohsawa and Takahashi's theorem [11] and some existence theorems for their iterative method.

Recently, Aoyama et al. [31] discussed some properties of *mappings of type*
*and*
in Banach spaces. These are all generalizations of firmly nonexpansive mappings in Hilbert spaces. It is known that the classes of mappings of type (P), (Q), and (R) correspond to three types of resolvents of monotone operators in Banach spaces, respectively, [31, 32].

The aim of this paper is to investigate a hybrid projection method and a hybrid shrinking projection method introduced in [30] for a single mapping of type (P) in a Banach space; see (2.2) for the definition of mappings of type (P). Using the techniques in [12, 20, 21] we show that the sequences generated by these methods are well defined without assuming the existence of fixed points. We also show that the boundedness of the generated sequences is equivalent to the existence of fixed points of mappings of type (P).

## 2. Preliminaries

for all
. The space
is said to be *smooth* if
exists for all
, where
denotes the unit sphere of
. The space
is also said to be *strictly convex* if
whenever
and
. It is also said to be *uniformly convex* if for all
there exists
such that
and
imply
The space
is said to have the *Kadec-Klee* property if
whenever
is a sequence of
such that
and
. We know the following (see, e.g., [4, 33, 34]).

(i) is smooth if and only if is single-valued. In this case, is demicontinuous, that is, norm-to-wea continuous.

(ii)If is smooth, strictly convex, and reflexive, then is single-valued, one-to-one, and onto.

(iii)If is uniformly convex, then is a strictly convex and reflexive Banach space which has the Kadec-Klee property.

Let
be a strictly convex and reflexive Banach space,
a nonempty closed convex subset of
and
Then there exists a unique
such that
The mapping
defined by
for all
is called the *metric projection* of
onto
We know that
if and only if
and
for all

Let
be a nonempty subset of a Banach space
and
a mapping. Then the set of fixed points of
is denoted by
A point
is said to be an *asymptotic fixed point* of
[35] if there exists a sequence
of
such that
and
The set of asymptotic fixed points of
is denoted by
The mapping
is said to be *nonexpansive* if
for all
. The *identity mapping* on
is denoted by

*of type (P)*if

*firmly nonexpansive*, that is,

for all . We know that if is a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space then the metric projection of onto is of type (P) and

Let
be a smooth, strictly convex, and reflexive Banach space and
a mapping. The graph of
, the domain of
, and the range of
are defined by
and
respectively. The mapping
is said to be *monotone* if
for all
and
. It is known that if
is monotone, then the resolvent
of
is a single-valued mapping of
onto
and of type (P), and moreover,
see [31]. A monotone operator
is said to be *maximal monotone* if
whenever
is a monotone operator and
. By Rockafellar's result [6], if
is maximal monotone, then the resolvent
of
is a mapping of
onto
; see [4, 36] for more details.

We know the following.

Lemma 2.1 (see [14]).

Let be a smooth Banach space, a nonempty subset of and a mapping of type (P). Then the following hold.

(1)If is closed and convex, then so is

(2)

Theorem 2.2 (see [31]).

Let be a smooth, strictly convex, and reflexive Banach space, a nonempty subset of and a mapping of type (P). Then the following hold.

(1)If is a sequence of such that then and

(2)If has the Kadec-Klee property, then is norm-to-norm continuous.

(3) If is uniformly convex, then is uniformly norm-to-norm continuous on each nonempty bounded subset of

Theorem 2.3 (see [31]).

Let be a smooth, strictly convex, and reflexive Banach space, a nonempty bounded closed convex subset of and a mapping of type (P). Then is nonempty. Furthermore, if is a self-mapping, then is nonempty.

Lemma 2.4 (see [14]).

Let be a smooth and uniformly convex Banach space, and sequences of nonempty closed convex subsets of , and a sequence of such that and for all Then the following hold.

(1)If is bounded, then

(2)If for all and is nonempty, then is bounded.

Theorem 2.5 (see [14]).

for all . Then converges strongly to

Let be a reflexive Banach space and a sequence of nonempty closed convex subsets of Then subsets and of are defined as follows.

(i) if there exists a sequence of such that for all and

(ii) if there exists a subsequence of and a sequence of such that for all and

The sequence
is said to be *Mosco convergent* to a subset
of
if
holds. We represent this by
We know that if
is a sequence of nonempty closed convex subsets of
such that
for all
and
is nonempty, then
We also know the following theorem.

Theorem 2.6 (see [37]).

Let be a strictly convex and reflexive Banach space and a sequence of nonempty closed convex subsets of such that exists and nonempty. Then converges weakly to for all Furthermore, if has the Kadec-Klee property, then converges strongly to for all .

Kimura et al. [18] obtained the following strong convergence theorem by using Theorem 2.6; see also Kimura and Takahashi [19] for related results which were obtained by using Mosco convergence.

Theorem 2.7 (see [18]).

for all Then the following hold.

(1) for all and is well defined.

(2)If has the Kadec-Klee property, , and satisfies the condition that whenever is a sequence of such that and , then converges strongly to .

Using Theorems 2.2 and 2.7, we obtain the following strong convergence theorem for mappings of type (P).

Corollary 2.8.

for all Then converges strongly to

Proof.

Hence satisfies (2.5) with given by for all .

We next show that satisfies the assumption in of Theorem 2.7. Let be a sequence of such that and Since is demicontinuous by of Theorem 2.2, . Hence we have . Therefore, Theorem 2.7 implies the conclusion.

## 3. Existence Theorems

Using the techniques in [12, 20, 21], we show the following two lemmas.

Lemma 3.1.

for all .

Proof.

Since by putting we obtain the desired result.

Lemma 3.2.

for all Then the following hold.

(1) and is nonempty for all

(2) is well defined.

(3) .

Proof.

for all and hence The part is a direct consequence of (1).

for all Thus Therefore we obtain the desired result.

Similarly, we can also show the following lemma.

Lemma 3.3.

for all Then the following hold.

(1) is nonempty for all

(2) is well defined.

(3)

Proof.

for all and hence . Part is a direct consequence of Part follows from the assumption that is of type (P).

Using Lemmas 2.4, 3.2, and Theorem 2.5, we can prove the following existence theorem.

Theorem 3.4.

Let be a smooth and uniformly convex Banach space, a nonempty closed convex subset of and a mapping of type (P). Let and be defined by (3.5). Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) is bounded.

(4) converges strongly.

In this case, converges strongly to

Proof.

By of Lemma 3.2, we know that implies We first show that implies Suppose that is nonempty and let and for all It is clear that and for all . By of Lemma 3.2 and assumption, the equality holds and this set is nonempty. Thus, of Lemma 2.4 implies that is bounded.

This gives us that By of Lemma 2.1, we get

It follows from Theorem 2.5 that implies that converges strongly to . Thus implies It is obvious that implies This completes the proof.

Using Lemmas 2.4, 3.3, and Corollary 2.8, we can also show the following existence theorem. We employ the methods, based on Mosco convergence, which were developed by Kimura et al. [18] and Kimura and Takahashi [19].

Theorem 3.5.

Let be a smooth, strictly convex, and reflexive Banach space which has the Kadec-Klee property, a nonempty closed convex subset of , and a mapping of type (P). Let and be defined by (3.10). Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) converges strongly.

In this case, converges strongly to Moreover, if is uniformly convex, then these conditions are also equivalent to the following.

(4) is bounded.

Proof.

This gives us that and hence is nonempty.

Using Corollary 2.8, we know that implies that converges strongly to Hence implies

We next show that implies Suppose that converges strongly to Let Then we have for all Since is closed and we have This gives us that and hence is nonempty.

This gives us that By of Lemma 2.1, we get Thus is nonempty. It is obvious that implies This completes the proof.

## 4. Deduced Results

In this section, we obtain some corollaries of Theorems 3.4 and 3.5. We first deduce the following corollary from Theorem 3.4.

Corollary 4.1.

for all Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) is bounded.

(4) converges strongly.

In this case, converges strongly to

Proof.

By assumption, we know that : is a mapping of type (P) and Therefore, Theorem 3.4 implies the conclusion.

We can similarly deduce the following corollary from Theorem 3.5; see Kimura and Takahashi [19] for related results.

Corollary 4.2.

for all . Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) converges strongly.

In this case, converges strongly to Moreover, if is uniformly convex, then these conditions are also equivalent to the following.

(4) is bounded.

As direct consequences of Theorems 3.4 and 3.5, we also obtain the following corollaries.

Corollary 4.3.

for all Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) is bounded.

(4) converges strongly.

In this case, converges strongly to .

Corollary 4.4 (see [12]).

for all Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) is bounded.

(4) converges strongly.

In this case, converges strongly to

Using Corollary 4.3, we next show the following result; see also [21, 24].

Corollary 4.5.

for all Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) is bounded.

(4) converges strongly.

In this case, converges strongly to .

Proof.

for all This implies that for all Thus Corollary 4.3 implies the conclusion.

Using Corollary 4.4, we can similarly show the following result.

Corollary 4.6.

for all . Then is well defined and the following are equivalent.

(1) is nonempty.

(2) is nonempty.

(3) is bounded.

(4) converges strongly.

In this case, converges strongly to

## Declarations

### Acknowledgments

The authors would like to express their sincere appreciation to an anonymous referee for valuable comments on the original version of the manuscript. This work is dedicated to Professor Wataru Takahashi on the occasion of his 65th birthday.

## Authors’ Affiliations

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