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.
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1. Introduction
Many problems in optimization, such as convex minimization problems, variational inequality problems, minimax problems, and equilibrium problems, can be formulated as the problem of solving the inclusion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ1_HTML.gif)
for a maximal monotone operator defined in a Banach space
see, for example, [1–5] for convex minimization problems, [3, 5, 6] for variational inequality problems, [3, 5, 7] for minimax problems, and [8] for equilibrium problems. It is also known that the problem can be regarded as a fixed point problem for a 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ2_HTML.gif)
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]).
Let be a Hilbert space,
a maximal monotone operator such that
is nonempty, and
the resolvent of
defined by
for all
Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ3_HTML.gif)
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 typeand
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
Throughout the present paper, every linear space is real. We denote the set of positive integers by Let
be a Banach space with norm
Then the dual space of
is denoted by
The norm of
is also denoted by
For
and
we denote
by
For a sequence
of
and
, strong convergence of
and weak convergence of
to
are denoted by
and
respectively. The normalized duality mapping
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ4_HTML.gif)
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
Let be a smooth Banach space,
a nonempty subset of
, and
a mapping. Following [31], we say that
is of type (P) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ5_HTML.gif)
for all If
is a Hilbert space, then
and hence
is of type (P) if and only if
is firmly nonexpansive, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ6_HTML.gif)
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]).
Let be a smooth and uniformly convex Banach space,
a nonempty closed convex subset of
and
a sequence of mappings of type (P) of
into itself such that
is nonempty. Suppose that each weak subsequential limit of
belongs to
whenever
is a bounded sequence of
such that
and
Let
be a sequence defined by
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ7_HTML.gif)
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]).
Let be a smooth, strictly convex, and reflexive Banach space,
a nonempty closed convex subset of
and
a sequence of mappings of
into itself such that
is nonempty. Suppose that there exists a sequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ8_HTML.gif)
for all ,
and
Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ9_HTML.gif)
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.
Let be a smooth, strictly convex, and reflexive Banach space which has the Kadec-Klee property,
a nonempty closed convex subset of
and
a mappings of type (P) such that
is nonempty. Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ10_HTML.gif)
for all Then
converges strongly to
Proof.
We first show that defined by
for all
satisfies (2.5). Let
and
be given. Since
is of type (P), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ11_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ12_HTML.gif)
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.
Let be a smooth, strictly convex, and reflexive Banach space,
a nonempty subset of
,
a mapping of type (P), and
a nonempty bounded closed convex subset of
such that
. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ13_HTML.gif)
for all .
Proof.
By Theorem 2.3, there exists such that
This implies that
for all
Fix
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ14_HTML.gif)
Since is of type (P), we also know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ15_HTML.gif)
Using (3.2) and (3.3), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ16_HTML.gif)
Since by putting
we obtain the desired result.
Lemma 3.2.
Let be a smooth, strictly convex, and reflexive Banach space,
a nonempty closed convex subset of
and
a mapping of type (P). Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ17_HTML.gif)
for all Then the following hold.
(1) and
is nonempty for all
(2) is well defined.
(3).
Proof.
We first show by induction on
It is obvious that
and
Thus
Fix
and suppose that
for all
Then
are defined. Note that
by the definitions of
and
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ18_HTML.gif)
We next show that is nonempty. Let
be a positive real number such that
for all
and put
. It is clear that
for all
By Lemma 3.1, we have
such that
for all
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ19_HTML.gif)
for all and hence
The part
is a direct consequence of (1).
We finally show By
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ20_HTML.gif)
On the other hand, if then, by the assumption that
is of type (P), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ21_HTML.gif)
for all Thus
Therefore we obtain the desired result.
Similarly, we can also show the following lemma.
Lemma 3.3.
Let be a smooth, strictly convex, and reflexive Banach space,
a nonempty closed convex subset of
and
a mapping of type (P). Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ22_HTML.gif)
for all Then the following hold.
(1) is nonempty for all
(2) is well defined.
(3)
Proof.
We first show It is obvious that
and hence
is nonempty. Fix
and suppose that
is nonempty for all
Then
are defined. We next show that
is nonempty. Let
be a positive real number such that
for all
and put
. It is clear that
for all
. By Lemma 3.1, we have
such that
for all
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ23_HTML.gif)
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.
We next show that implies
Suppose that
is bounded. Then
of Lemma 2.4 implies that
Since
is reflexive and
is weakly closed, there exists a subsequence
of
such that
By
and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ24_HTML.gif)
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.
By of Lemma 3.3, we know that
implies
We first show that
implies
Suppose that
is nonempty. By this assumption and
for all
, we know that
By Theorem 2.6,
converges strongly to
This implies that
converges strongly to
Put
Since
for all
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ25_HTML.gif)
for all By Theorem 2.2, we have
and
Thus it follows from (3.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ26_HTML.gif)
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.
We next show that implies
Suppose that
is uniformly convex and
is bounded and let
and
for all
Then it is clear that
and
By
of Lemma 2.4, we know that
Since
is reflexive and
is weakly closed, there exists a subsequence
of
such that
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ27_HTML.gif)
for all Since
and
for all
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ28_HTML.gif)
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.
Let be a smooth and uniformly convex Banach space and
a monotone operator such that there exists a nonempty closed convex subset
of
satisfying
Let
be the mapping defined by
for all
and
a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ29_HTML.gif)
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.
Let be a smooth, strictly convex, and reflexive Banach space which has the Kadec-Klee property and
a monotone operator such that there exists a nonempty closed convex subset
of
satisfying
. Let
be the mapping defined by
for all
and
a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ30_HTML.gif)
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.
Let be a Hilbert space,
a nonempty closed convex subset of
, and
a firmly nonexpansive mapping. Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ31_HTML.gif)
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]).
Let be a Hilbert space,
a nonempty closed convex subset of
and
a firmly nonexpansive mapping. Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ32_HTML.gif)
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.
Let be a Hilbert space,
a nonempty closed convex subset of
, and
a nonexpansive mapping. Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ33_HTML.gif)
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.
Let be a mapping defined by
Then
:
is a firmly nonexpansive mapping and
. We also know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ34_HTML.gif)
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.
Let be a Hilbert space,
a nonempty closed convex subset of
and
a nonexpansive mapping. Let
be a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F512751/MediaObjects/13663_2009_Article_1296_Equ35_HTML.gif)
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
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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.
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Aoyama, K., Kohsaka, F. Existence of Fixed Points of Firmly Nonexpansive-Like Mappings in Banach Spaces. Fixed Point Theory Appl 2010, 512751 (2010). https://doi.org/10.1155/2010/512751
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DOI: https://doi.org/10.1155/2010/512751