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.