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Existence of Fixed Points of Firmly Nonexpansive-Like Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 512751 (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.
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
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
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
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
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
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
for all 
If 
is a Hilbert space, then 
and hence 
is of type (P) if and only if 
is 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]).
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
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
for all 
, 
and 
Let 
be a sequence defined by
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
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
This implies that
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
for all 
.
Proof.
By Theorem 2.3, there exists 
such that 
This implies that 
for all 
Fix 
Then we have
Since 
is of type (P), we also know that
Using (3.2) and (3.3), we obtain
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
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
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
for all 
and hence 
The part 
is a direct consequence of (1).
We finally show 
By 
we have
On the other hand, if 
then, by the assumption that 
is of type (P), we have
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
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
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
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
for all 
By Theorem 2.2, we have 
and 
Thus it follows from (3.13) that
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
for all 
Since 
and 
for all 
we have
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
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
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
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
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
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
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
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