General Split Variational Inclusion Problem in Hilbert Spaces

Abstract

We consider a general split variational inclusion problem (GSFVIP) and propose an algorithm for finding the solutions of GSFVIP in Hilbert space. We establish the strong convergence of the proposed algorithm to a solution of GSFVIP. Our results extend and improve the related results in the literature.

1. Introduction

Let be a real Hilbert space, and let be a set-valued mapping with domain . Recall that is called monotone if for any and ; is maximal monotone if its graph is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find such that . Here, is called a zero point of . A well-known method for approximating a zero point of a maximal monotone mapping defined in a real Hilbert space is the proximal point algorithm first introduced by Martinet [1] and generated by Rockafellar [2]. This is an iterative procedure, which generates by and where , is a maximal monotone mapping in a real Hilbert space, and is the resolvent mapping of defined by for each . In 1976, Rockafellar [2] proved the following in the Hilbert space setting. If the solution set is nonempty and , then the sequence in (1) converges weakly to an element of . Later, many researchers have studied the convergence theorems of the proximal point algorithm in Hilbert spaces. For example, one can refer to [3–8] and references therein.

Let and be two real Hilbert spaces, and two set-valued maximal monotone mappings, a linear and bounded operator, and the adjoint of . Chuang [9] considers the following split variational inclusion problem:

which was introduced by Moudafi [10]. In this paper, motivated by the works in Chuang [9] and related literature, we consider the following general split variational inclusion problem.

Definition 1. Let and be two real Hilbert spaces, and two families of set-valued maximal monotone mappings, a linear and bounded operator, and the adjoint of . The general split variational inclusion problem (GSFVIP) is formulated as the following problem:

In this paper, we propose an algorithm for finding the solutions of GSFVIP in a Hilbert space and prove that the sequence generated by the proposed method converges strongly to a solution of GSFVIP. Our results extend and improve the related results in the literature.

2. Preliminaries

Throughout this paper, let be the set of positive integers. Let be a real Hilbert space with the inner product and the norm , respectively. We also use “” to stand for strong convergence and “” to stand for weak convergence.

Lemma 2 (see [11]). Let be a real Hilbert space, and let , . Then

Lemma 3 (see [12]). Let be a Hilbert space, and let be a sequence in . Then, for any given sequence with and for any positive integer with ,

Let be a nonempty closed convex subset of a real Hilbert space , and let be a mapping. Then is said to be a nonexpansive mapping if for every . It is easy to see that is a closed convex subset of if is a nonexpansive mapping. Besides, is said to be a firmly nonexpansive mapping if for every , .

Lemma 4 (see [13]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping, and let be a sequence in . If and , then .

Let be a nonempty closed convex subset of . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . It is known that is a nonexpansive mapping of onto .

Lemma 5 (see [14]). Let be a nonempty closed convex subset of a Hilbert space . Let be the metric projection from onto . Then, for each and , we know that if and only if for all .

The following result is an important tool in this paper. For similar results, one can see [15].

Lemma 6. Let be a real Hilbert space. Let be a set-valued maximal monotone mapping, , and let be a resolvent mapping of .(i)For each , is a single-valued and firmly nonexpansive mapping.(ii) and .(iii) for all and for all .(iv) is a firmly nonexpansive mapping for each .(v)Suppose that . Then for each , each , and each .(vi)Suppose that . Then for each , each , and each .

Lemma 7 (see [9]). Let and be two real Hilbert spaces, let be a linear operator, let be the adjoint of , let be fixed, and let . Let be a set-valued maximal monotone mapping and let be a resolvent mapping of . Then for all , . Furthermore, is a nonexpansive mapping.

The following is a very important result for various strong convergence theorems. Recently, many researchers have studied Halpern’s type strong convergence theorems by using the following lemma and get many generalized results. For example, one can see [9, 16, 17]. In this paper, we also use this result to get our strong convergence theorems.

Lemma 8 (see [18]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that , , and are satisfied by all (sufficiently large) numbers . In fact, .

Lemma 9 (see [19]). Let be a sequence of nonnegative real numbers, a sequence of real numbers in with , a sequence of nonnegative real numbers with , and a sequence of real numbers with . Suppose that for each . Then .

3. Main Results

In this section, we first give the following result.

Lemma 10. Let and be two real Hilbert spaces, let be a linear and bounded operator, and let denote the adjoint of . Let and be two families of set-valued maximal monotone mappings, and let and for all . Given any , we have the following.(i)If is a solution of GSFVIP, then , for all .(ii)Suppose that , for all , and that the solution set of GSFVIP is nonempty; then is a solution of GSFVIP.

Proof. (i) Suppose that is a solution of GSFVIP. Then and . By Lemma 6(ii), it is easy to see that
(ii) Suppose that is a solution of GSFVIP and , for all . By Lemma 6(vi), That is, By (10) and the fact that is the adjoint of , On the other hand, by Lemma 6(vi) again, By (11) and (12), for each and each , for all . That is, for each and each , for all . Since is a solution of GSFVIP, and . So, it follows from (14) that , for all . Then, , for all . Therefore . Further, Then , for all . So, . Therefore, is a solution of GSFVIP.

Theorem 11. Let and be two real Hilbert spaces, let be a linear and bounded operator, and let denote the adjoint of . Let and be two families of set-valued maximal monotone mappings. Let be sequences of real numbers in with . Let be a sequence in and for each . Let be the solution set of GSFVIP and suppose that . Let be a self -contraction mapping of , . Let be defined by
If the sequences , , , , and satisfy the following conditions: (i), ,(ii)for each , , , , and ,then the sequence converges strongly to .

Proof. First, we show that is bounded. In fact, let ; it follows from Lemmas 6(i) and 7 that , for all , are nonexpansive, and by Lemma 10 we have which implies that is bounded, and we also obtain that is bounded.
Next, we show that there exists a unique such that .
Since, for all , , we may assume that for each . Since, for all , is bounded, there exists a converge subsequence. Without loss of generality, we can assume that for each .
It follows from Lemma 10 that solves the GSFVIP if and only if solves the fixed point equation that is, the solution sets of fixed point equation (18) and GSFVIP are the same. By Lemmas 6(i) and 7, the operators , for all , are nonexpansive. Since the fixed point set of nonexpansive operators is closed and convex, the projection onto the solution set is well defined whenever . We observe that is a contraction of into itself. Indeed, since is nonexpansive and is a self -contraction mapping , Hence, there exists a unique element such that .
In order to prove that as , we consider two possible cases.

Case 1. There exists a natural number such that for each . Since is bounded, we have is convergent.
Next, we show that, for each , By Lemmas 6(i) and 7, for every and , we have By using Lemma 3 and (21), for every and , we have Hence, for each , we have
Since and is bounded, from (23) we get that By assuming that, for all , , it follows from (24) that Further, for all , we have Clearly, for all , . Since, for all , , it follows from (25) that and it follows from Lemma 6(iii) that Besides, by Lemma 6(i) and (28), for all , we have By (27) and (30), for all , we obtain It follows from Lemma 6(iii) that By Lemma 6(i) and (29), for all , we have By (32) and (33), for all , we obtain
Now, we show that To show this inequality, we choose a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . Notice that, for each , is nonexpansive. Thus, from Lemma 4 and (34), we have . Therefore, it follows from Lemma 5 that
Finally, we show that . Applying Lemma 2, we have that This implies that where , , and . It is easy to see that , , and . Hence, by Lemma 9, the sequence converges strongly to .

Case 2. Assume that is not a monotone sequence. Then, we can define an integer sequence for all (for some large enough) by Clearly, is a nondecreasing sequence such that as for all :
Following a similar argument as the proof of Case 1, we have And by similar argument, we have where , , and . Hence, by Lemma 9, we obtain and . Now, from Lemma 8, we have Therefore, the sequence converges strongly to . This completes the proof.

Corollary 12. Let and be two real Hilbert spaces, let be a linear and bounded operator, and let denote the adjoint of . Let and be two families of set-valued maximal monotone mappings for all . Let be sequences of real numbers in with . Let be a sequence in and for each . Let be the solution set of GSFVIP and suppose that . Let be defined by
If the sequences , , , and satisfy the following conditions: (i), ,(ii)for each , , , , and ,then the sequence converges strongly to a point which is the minimum norm solution of GSFVIP.

Proof. Let in Theorem 11. We have proved that the sequence converges strongly to . Then, for all , we have Since , then , for all . Hence, we obtain that is, . Thus, is the minimum norm solution of GSFVIP. This completes the proof.

In Theorem 11, if we set , then we get the following result.

Corollary 13. Let and be two real Hilbert spaces, let be a linear and bounded operator, and let denote the adjoint of . Let and be two families of set-valued maximal monotone mappings for all . Let be sequences of real numbers in with . Let be a sequence in and for each . Let be fixed. Let be the solution set of GSFVIP and suppose that . Let be defined by
If the sequences , , , , and satisfy the following conditions: (i), ,(ii)for each , , , , and ,then the sequence converges strongly to .

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgment

This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department of China (no. 08ZA008).