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

(1.1)

for a maximal monotone operator defined in a Banach space see, for example, [15] 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

(1.2)

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

(1.3)

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, [1230] 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

(2.1)

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

(2.2)

for all If is a Hilbert space, then and hence is of type (P) if and only if is firmly nonexpansive, that is,

(2.3)

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

(2.4)

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

(2.5)

for all , and Let be a sequence defined by

(2.6)

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

(2.7)

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

(2.8)

This implies that

(2.9)

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

(3.1)

for all .

Proof.

By Theorem 2.3, there exists such that This implies that for all Fix Then we have

(3.2)

Since is of type (P), we also know that

(3.3)

Using (3.2) and (3.3), we obtain

(3.4)

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

(3.5)

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

(3.6)

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

(3.7)

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

We finally show By we have

(3.8)

On the other hand, if then, by the assumption that is of type (P), we have

(3.9)

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

(3.10)

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

(3.11)

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

(3.12)

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

(3.13)

for all By Theorem 2.2, we have and Thus it follows from (3.13) that

(3.14)

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

(3.15)

for all Since and for all we have

(3.16)

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

(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.

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

(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.

Let be a Hilbert space, a nonempty closed convex subset of , and a firmly nonexpansive mapping. Let be a sequence defined by

(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]).

Let be a Hilbert space, a nonempty closed convex subset of and a firmly nonexpansive mapping. Let be a sequence defined by

(4.4)

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

(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.

Let be a mapping defined by Then : is a firmly nonexpansive mapping and . We also know that

(4.6)

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

(4.7)

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