#### Abstract

We establish strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space.

#### 1. Introduction

Let be a real Banach space with and let be a nonempty closed convex subset of . A mapping of into itself is called *nonexpansive* if for all . We use to denote the set of fixed points of ; that is, . A mapping of into itself is called *quasinonexpansive* if is nonempty and for all and . For two mappings and of into itself, Das and Debata [1] considered the following iteration scheme: and
where and are sequences in . In this case of , such an iteration process was considered by Ishikawa [2]; see also Mann [3]. Das and Debata [1] proved the strong convergence of the iterates defined by (1.1) in the case when is strictly convex and , are quasinonexpansive mappings. Fixed point iteration processes for nonexpansive mappings in a Hilbert space and a Banach space including Das and Debata's iteration and Ishikawa's iteration have been studied by many researchers to approximating a common fixed point of two mappings; see, for instance, Takahashi and Tamura [4].

Let be a maximal monotone operator from to , where is the dual space of . It is well known that many problems in nonlinear analysis and optimization can be formulated as follows. Find a point satisfying We denote by the set of all points such that Such a problem contains numerous problems in economics, optimization, and physics. A well-known method to solve this problem is called the proximal point algorithm: and where and are the resovents of . Many researchers have studied this algorithm in a Hilbert space; see, for instance, [5–8] and in a Banach space; see, for instance, [9–11].

Next, we recall that for all and , we denote the value of at by . Then, the normalized duality mapping on is defined by We know that if is smooth, then the duality mapping is single valued. Next, we assume that is a smooth Banach space and define the function by

A point is said to be an *asymptotic* fixed point of [12] if contains a sequence which converges weakly to and . We denote the set of all asymptotic fixed points of by . A mapping is said to be *relatively nonexpansive* [13–15] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [13–15].

In 2004, Matsushita and Takahashi [15] proposed the following modification of Mann's iteration for a relatively nonexpansive mapping by using the hybrid method in a Banach space. Four years later, Qin and Su [16] have adapted Matsushita and Takahashi's idea [15] to modify Halpern's iteration and Ishikawa's iteration for a relatively nonexpansive mapping in a Banach space. In particular, in a Hilbert space Mann's iteration, Halpern's iteration, and Ishikawa's iteration were considered by many researchers.

Very recently, Inoue et al. [17] proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method.

Theorem 1.1 (Inoue et al. [17]). *Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a maximal monotone operator satisfying and let for all . Let be a relatively nonexpansive mapping such that . Let be a sequence generated by and
**
for all , where is the duality mapping on , , and for some . If , then converges strongly to , where is the generalized projection of onto .*

The purpose of this paper is to employ the idea of Inoue et al. [17] and Das and Debata [1] to introduce a new hybrid method for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings. We prove a strong convergence theorem of the new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space.

#### 2. Preliminaries

Throughout this paper, all linear spaces are real. Let and be the sets of all positive integers and real numbers, respectively. Let be a Banach space and let be the dual space of . For a sequence of and a point , the *weak* convergence of to and the *strong* convergence of to are denoted by and , respectively.

Let be the unit sphere centered at the origin of . Then the space is said to be *smooth* if the limit
exists for all . It is also said to be *uniformly smooth* if the limit exists uniformly in . A Banach space is said to be *strictly convex* if whenever and . It is said to be *uniformly convex* if for each , there exists such that whenever and . We know the following [18]:

A Banach space is said to have the *Kadec-Klee* property if for a sequence of satisfying that and , . It is known that if is uniformly convex, then has the Kadec-Klee property; see [18, 19] for more details. Let be a smooth, strictly convex, and reflexive Banach space and let be a closed convex subset of . Throughout this paper, define the function by
Observe that, in a Hilbert space , (2.2) reduces to , for all . It is obvious from the definition of the function that, for all ,

Following Alber [20], the generalized projection from onto is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem Existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping . In a Hilbert space, is the metric projection of onto . We need the following lemmas for the proof of our main results.

Lemma 2.1 (Kamimura and Takahashi [6]). *Let be a uniformly convex and smooth Banach space and let and be two sequences in such that either or is bounded. If , then .*

Lemma 2.2 (Matsushita and Takahashi [15]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space and let be a relatively nonexpansive mapping from into itself. Then is closed and convex.*

Lemma 2.3 (Alber [20] and Kamimura and Takahashi [6]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space, and let . Then, if and only if for all .*

Lemma 2.4 (Alber [20] and Kamimura and Takahashi [6]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space. Then
*

Let be a smooth, strictly convex, and reflexive Banach space, and let be a set-valued mapping from to with graph , domain , and range . We denote a set-valued operator from to by . is said to be *monotone* if , for all A monotone operator is said to be *maximal monotone* if its graph is not properly contained in the graph of any other monotone operator. We know that if is a maximal monotone operator, then is closed and convex. The following theorem is well known.

Lemma 2.5 (Rockafellar [21]). *Let be a smooth, strictly convex, and reflexive Banach space and let be a monotone operator. Then is maximal if and only if for all .*

Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying Then we can define the resolvent of by We know that consists of one point. For , the Yosida approximation is defined by for all .

Lemma 2.6 (Kohsaka and Takahashi [22]). *Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying
**
Let and let and be the resolvent and the Yosida approximation of , respectively. Then, the following hold:*(i)*, for all , ;*(ii)*, for all ;*(iii)*.*

Lemma 2.7 (Zălinescu [23] and Xu [24]). *Let be a uniformly convex Banach space and let . Then there exists a strictly increasing, continuous, and convex function such that and
**
for all and , where .*

#### 3. Main Results

In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using the hybrid method.

Theorem 3.1. *Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a maximal monotone operator satisfying and let for all . Let and be relatively nonexpansive mappings from into itself such that . Let be a sequence generated by and
**
for all , where is the duality mapping on , and for some . If and , then converges strongly to , where is the generalized projection of onto .*

*Proof. *We first show that and are closed and convex for each . From the definitions of and , it is obvious that is closed and is closed and convex for each . Next, we prove that is convex. Since is equivalent to
which is affine in , and hence is convex. So, is a closed and convex subset of for all . Next, we show that for all . Indeed, let and for all . Since are relatively nonexpansive mappings, we have
It follows that
So, for all , which implies that . Next, we show that for all . We prove by induction. For , we have . Assume that . Since is the projection of onto , by Lemma 2.3 we have
As by the induction assumptions, we have
This together with definition of implies that and hence for all . So, we have that for all . This implies that is well defined. From definition of that and , we have
Therefore, is nondecreasing. It follows from Lemma 2.4 and that
for all . Therefore, is bounded. Moreover, by definition of , we know that is bounded. So, we have and are bounded. So, the limit of exists. From and Lemma 2.4, we have
for all . This implies that . From , we have
Therefore, we have .

Since and is uniformly convex and smooth, we have from Lemma 2.1 that
So, we have Since is uniformly norm-to-norm continuous on bounded sets, we have
On the other hand, we have
This follows that
From (3.12) and , we obtain that .

Since is uniformly norm-to-norm continuous on bounded sets, we have
From
we have
Since and are bounded, we also obtain that and are bounded. So, there exists such that . Therefore Lemma 2.7 is applicable and we observe that
where is a continuous, strictly increasing, and convex function with . That is

Let be any subsequence of . Since is bounded, there exists a subsequence of such that
where . By (2) and (3), we have
Since and hence , it follows that
We also have from (3.3) that
and hence
Since , it follows from (3.19) that . By properties of the function , we have . Since is also uniformly norm-to-norm continuous on bounded sets, we obtain and then
So, we have . Since
it follows that , and hence
From (3.3), we have
Using and Lemma 2.6, we have
It follows that
Since , we have that . So, we have Since is uniformly convex and smooth, we have from Lemma 2.1 that
Since
from (3.17), (3.25), (3.27), and (3.31), we obtain that
Since is bounded, there exists a subsequence of such that . From and , we have and . Since and are relatively nonexpansive, we have that . Next, we show . Since is uniformly norm-to-norm continuous on bounded sets, from (3.31) we have
From , we have
Therefore, we have
For , from the monotonicity of , we have for all . Replacing by and letting , we get . From the maximallity of , we have , that is, .

Finally, we show that . Let . From and , we obtain that
Since the norm is weakly lower semicontinuous, we have
From the definition of , we obtain . This implies that
Therefore we have
Since has the Kadec-Klee property, we obtain that . Since is an arbitrary weakly convergent subsequence of , we can conclude that converges strongly to . This completes the proof.

As direct consequences of Theorem 3.1, we can obtain the following corollaries.

Corollary 3.2. *Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a maximal monotone operator satisfying and let for all . Let be a relatively nonexpansive mapping from into itself such that . Let be a sequence generated by and
**
for all , where is the duality mapping on , , and for some . If and , then converges strongly to , where is the generalized projection of onto .*

*Proof. *Putting in Theorem 3.1, we obtain Corollary 3.2.

Corollary 3.3. *Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a maximal monotone operator satisfying and let for all . Let be a relatively nonexpansive mapping such that . Let be a sequence generated by and
**
for all , where is the duality mapping on , , and for some . If , then converges strongly to , where is the generalized projection of onto .*

*Proof. *Putting and in Theorem 3.1, we obtain Corollary 3.3.

Let be a Banach space and let be a proper lower semicontinuous convex function. Define the subdifferential of as follows: for each . Then, we know that is a maximal monotone operator; see [18] for more details.

Corollary 3.4. *Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let and be relatively nonexpansive mappings from into itself such that . Let be a sequence generated by and
**
for all , where is the duality mapping on and . If and , then converges strongly to , where is the generalized projection of onto .*

*Proof. *Set in Theorem 3.1, where is the indicator function; that is,
Then, we have that is a maximal monotone operator and for . In fact, for any and , we have from Lemma 2.3 that
So, from Theorem 3.1, we obtain Corollary 3.4.

#### 4. Applications

In this section, we discuss the problem of strong convergence concerning a maximal monotone operator and two nonexpansive mappings in a Hilbert space. Using Theorem 3.1, we obtain the following results.

Theorem 4.1. *Let be a nonempty closed convex subset of a Hilbert space . Let be a monotone operator satisfying and let for all . Let and be nonexpansive mappings from into itself such that . Let be a sequence generated by and
**
for all , where and for some . If and , then converges strongly to , where is the metric projection of onto .*

*Proof. *We know that every nonexpansive mapping with a fixed point is a relatively nonexpansive one. We also know that for all . Using Theorem 3.1, we are easily able to obtain the desired conclusion by putting . This completes the proof.

The following corollary follows from Theorem 4.1.

Corollary 4.2. *Let be a nonempty closed convex subset of a Hilbert space . Let be a monotone operator satisfying and let for all . Let be a nonexpansive mapping from into itself such that . Let be a sequence generated by and
**
for all , where and for some . If and , then converges strongly to , where is the metric projection of onto .*

*Proof. *Putting in Theorem 4.1, we obtain Corollary 4.2.

Corollary 4.3. *Let be a nonempty closed convex subset of a Hilbert space . Let be a maximal monotone operator satisfying and let for all . Let be a nonexpansive mapping from into itself such that . Let be a sequence generated by and
**
for all , where and for some . If then converges strongly to , where is the metric projection of onto .*

*Proof. *Putting and in Theorem 4.1, we obtain Corollary 4.3.

Corollary 4.4. *Let be a nonempty closed convex subset of a Hilbert space . Let and be nonexpansive mappings from into itself such that . Let be a sequence generated by and
**
for all , where . If and , then converges strongly to , where is the metric projection of onto .*

*Proof. *Set in Theorem 4.1, where is the indicator function; that is,
Then, we have that is a maximal monotone operator and for . In fact, for any and , we have that
So, from Theorem 4.1, we obtain Corollary 4.4.

#### Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve this manuscript and the Thailand Research Fund (RGJ Project) and Commission on Higher Education for their financial support during the preparation of this paper. The first author was supported by the Royal Golden Jubilee Grant PHD/0018/2550 and the Graduate School, Chiang Mai University, Thailand. The third author is supported by Grant-in-Aid for Scientific Research no. 19540167 from Japan Society for the Promotion of Science.