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Strong convergence theorems for the split variational inclusion problem in Hilbert spaces
Fixed Point Theory and Applications volume 2013, Article number: 350 (2013)
Abstract
In this paper, we first consider a split variational inclusion problem and give several strong convergence theorems in Hilbert spaces, like the Halpern-Mann type iteration method and the regularized iteration method. As applications, we consider the algorithms for a split feasibility problem and a split optimization problem and give strong convergence theorems for these problems in Hilbert spaces. Our results for the split feasibility problem improve the related results in the literature.
MSC:47H10, 49J40, 54H25.
1 Introduction
In 1994, the split feasibility problem in finite dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from medical image reconstruction. Since then, the split feasibility problem has received much attention due to its applications in signal processing, image reconstruction, approximation theory, control theory, biomedical engineering, communications, and geophysics. For examples, one can refer to [1–5] and related literature.
We know that the split feasibility problem can be formulated as the following problem:
where C and Q are nonempty closed convex subsets of Hilbert spaces and , respectively, and is an operator. It is worth noting that a special case of problem (SFP) is the convexly constrained linear inverse problem in the finite dimensional Hilbert space [6]:
Originally, problem (SFP) was considered in Euclidean spaces. (Note that if and are two Euclidean spaces, then A is a matrix.) In 1994, problem (SFP) in finite dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from medical image reconstruction. Since then, many researchers have studied (SFP) in finite dimensional or infinite dimensional Hilbert spaces. For example, one can see [2, 7–16] and related literature.
In 2002, Byrne [2] first introduced the following recursive procedure:
where the stepsize is chosen in the interval , and and are the metric projections onto and , respectively. This algorithm is called CQ algorithm. Note that A may be not invertible. In 2010, Wang and Xu [11] modified Byrne’s CQ algorithm and gave a weak convergence theorem in infinite dimensional Hilbert spaces.
In 2004, motivated by the works on CQ algorithm (1.1), Yang [14] considered (SFP) under the following conditions:
where and are convex and lower semicontinuous functions. In fact, Yang [14] studied the following problem, and we call this problem the relaxed split feasibility problem:
In 2010, Xu [13] modified and extended Yang’s algorithm and gave a weak convergence theorem in infinite dimensional Hilbert spaces.
On the other hand, let H be a real Hilbert space, and B be a set-valued mapping with domain . Recall that B is called monotone if for any and ; B 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 B. 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 [17] and generated by Rockafellar [18]. This is an iterative procedure, which generates by and
where , B is a maximal monotone mapping in a real Hilbert space, and is the resolvent mapping of B defined by for each . In 1976, Rockafellar [18] proved the following in the Hilbert space setting: If the solution set is nonempty and , then the sequence in (1.2) converges weakly to an element of . In particular, if B is the subdifferential ∂f of a proper lower semicontinuous and convex function , then (1.2) is reduced to
In this case, converges weakly to a minimizer of f. Later, many researchers have studied the convergence theorems of the proximal point algorithm in Hilbert spaces. For examples, one can refer to [19–24] and references therein.
Let and be two real Hilbert spaces, and be two set-valued maximal monotone mappings, be a linear and bounded operator, and be the adjoint of A. In this paper, motivated by the works in [13, 14] and related literature, we consider the following split variational inclusion problem:
Clearly, we know that the following split variational inclusion problem (SFVIP) is a generalization of variational inclusion problem. Further, we observed that problem (SFVIP) was introduced by Moudafi [25], and Moudafi [25] gave a weak convergence theorem for problem (SFVIP). The following is an iteration process given by Moudafi [25]:
It is worth noting that λ and γ are fixed numbers. Hence, it is important to establish generalized iteration processes and the related strong convergence theorems for problem (SFVIP).
Besides, we know that the following problems are special cases of problem (SFVIP).
(SFOP) Find such that and , where and are two proper, lower semicontinuous, and convex functions.
(SFP) Find such that and , where C and Q are two nonempty closed convex subsets of real Hilbert spaces and , respectively.
In this paper, we first consider a split variational inclusion problem and give several strong convergence theorems in Hilbert spaces, like the Halpern-Mann type iteration method, the regularized iteration method. As applications, we consider algorithms for a split feasibility problem and a split optimization problem and give strong convergence theorems for these problems in Hilbert spaces. Our results for the split feasibility problem improve the related results in the literature.
2 Preliminaries
Throughout this paper, let ℕ be the set of positive integers and let ℝ be the set of real numbers. Let H be a (real) Hilbert space with the inner product and the norm , respectively. We denote the strong convergence and the weak convergence of to by and , respectively. From [26], for each and , we have
Hence, we also have
for all . Furthermore, we know that
for each and with [27].
Lemma 2.1 [28]
Let H be a (real) Hilbert space, and let . Then .
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a mapping. Let . Then T is said to be a nonexpansive mapping if for every . T is said to be a quasi-nonexpansive mapping if and for every and . It is easy to see that is a closed convex subset of C if T is a quasi-nonexpansive mapping. Besides, T is said to be a firmly nonexpansive mapping if for every , that is, for every .
Lemma 2.2 [29]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive mapping, and let be a sequence in C. If and , then .
Let C be a nonempty closed convex subset of a real Hilbert space H. Then, for each , there is a unique element such that . Here, we set and is said to be the metric projection from H onto C.
Lemma 2.3 [30]
Let C be a nonempty closed convex subset of a Hilbert space H. Let be the metric projection from H onto C. 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 [31].
Lemma 2.4 Let H be a real Hilbert space. Let be a set-valued maximal monotone mapping, , and let be a resolvent mapping of B.
-
(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 and each , and each .
Lemma 2.5 Let and be two real Hilbert spaces, be a linear operator, and be the adjoint of A, and let be fixed. Let be a set-valued maximal monotone mapping, and let be a resolvent mapping of B. Let be defined by for each . Then
-
(i)
for all ;
-
(ii)
for all .
Proof (i) By Lemma 2.4,
for all . (ii) Further, we have
for all . Therefore, the proof is completed. □
Lemma 2.6 Let and be two real Hilbert spaces, be a linear operator, and be the adjoint of A, and 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.
Proof For all , we have
Hence, it follows from (2.1) and Lemma 2.4 that
for all . Therefore, the proof is completed. □
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 got many generalized results. For examples, one can see [32, 33]. In this paper, we also use this result to get our strong convergence theorems, and our results for the split feasibility problem improve the results in the literature.
Lemma 2.7 [34]
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 2.8 [35]
Let be a sequence of nonnegative real numbers, be a sequence of real numbers in with , be a sequence of nonnegative real numbers with , be a sequence of real numbers with . Suppose that for each . Then .
3 Halpern-Mann type algorithm with perturbations
In this section, we first give the following result.
Lemma 3.1 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let , and be two set-valued maximal monotone mappings, and let and . Given any .
-
(i)
If is a solution of (SFVIP), then .
-
(ii)
Suppose that and the solution set of (SFVIP) is nonempty. Then is a solution of (SFVIP).
Proof (i) Suppose that is a solution of (SFVIP). Then and . By Lemma 2.4, it is easy to see that
(ii) Suppose that is a solution of (SFVIP) and . By Lemma 2.4,
That is,
By (3.1) and is the adjoint of A,
On the other hand, by Lemma 2.4 again,
By (3.2) and (3.3),
for each and each . That is,
for each and each . Since is a solution of (SFVIP), and . So, it follows from (3.5) that . So, . Further,
Then . Therefore, is a solution of (SFVIP). □
Theorem 3.1 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two set-valued maximal monotone mappings. Let , , , and be sequences of real numbers in with and for each . Let be a sequence in . Let be a bounded sequence in H. Let be fixed. Let . Let Ω be the solution set of (SFVIP) and suppose that . Let be defined by
for each . Assume that:
-
(i)
; ; ;
-
(ii)
, , .
Then , where .
Proof Let , where is the metric projection from onto Ω. Then, for each , it follows from Lemma 2.6 that
This implies that is a bounded sequence. Besides, by Lemmas 2.4 and 2.6, we have
Hence, it follows from Lemma 2.1 that
where , , . Further, by (3.6) and (3.7), we have
Since , we may assume that for each . Next, we consider two cases.
Case 1: There exists a natural number N such that for each . So, exists. Hence, it follows from (3.8) and (i) that
Clearly, . Since , we have
By (3.9) and Lemma 2.4,
Similarly, we know that
Further, there exists a subsequence of such that for some and
Clearly, . By (3.10), Lemmas 2.2 and 2.4, we know that . Besides, it follows from Lemma 2.4 that
By (3.9) and (3.13),
By (3.11) and (3.14),
By (3.15) and Lemma 2.4,
Then it follows from (3.16) and Lemma 2.2 that . So, z is a solution of (SFVIP). By (3.12) and Lemma 2.3,
By assumptions, (3.8), (3.17), and Lemma 2.8, we know that .
Case 2: Suppose that there exists of such that for all . By Lemma 2.7, there exists a nondecreasing sequence in ℕ such that ,
for all . By (3.8) and (3.18), we have
Following a similar argument as the proof of Case 1, we have
and
By (3.19),
By assumption, (3.22), and (3.23),
Besides, we have
By assumptions, (3.20), and (3.25),
By (3.24) and (3.26),
By (3.18) and (3.27),
Therefore, the proof is completed. □
In Theorem 3.1, if we set and for each , then we get the following result.
Corollary 3.1 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two set-valued maximal monotone mappings. Let , , and be sequences of real numbers in with and for each . Let be a sequence in . Let be fixed. Let . Let Ω be the solution set of (SFVIP) and suppose that . Let be defined by
for each . Assume that , , , , and . Then , where .
Further, we can get the following result by Corollary 3.1 and Lemma 2.8. In fact, Corollary 3.1 and Theorem 3.2 are equivalent.
Theorem 3.2 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two set-valued maximal monotone mappings. Let , , and be sequences of real numbers in with and for each . Let be a sequence in . Let be a bounded sequence in H. Let be fixed. Let . Let Ω be the solution set of (SFVIP) and suppose that . Let be a bounded sequence. Let be defined by
for each . Assume that , , , , , and . Then , where .
Proof Let be defined by
By Corollary 3.1, , where . Besides, we know that
By (3.28) and Lemma 2.8, . So, , where . Therefore, the proof is completed. □
4 Regularized method for (SFVIP)
Lemma 4.1 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two set-valued maximal monotone mappings. Let , , and . Then
for all .
Proof For each , it follows from Lemma 2.4 and Lemma 2.5 that
If , then . This implies that the conclusion of Lemma 4.1 holds. □
Theorem 4.1 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two set-valued maximal monotone mappings. Let be a sequence in , , and . Let Ω be the solution set of (SFVIP) and suppose that . Let be defined by
for each . Assume that:
Then , where , i.e., is the minimal norm solution of (SFVIP).
Proof Let . Take any and let w be fixed. Then we know that
for each . Then is a bounded sequence. Further, we have
for each . By (4.1) and Lemma 2.6,
for each . By (4.1)-(4.2), Lemma 2.4, we know that
for each . Next, we know that
for each , and
for each . Further, we have
for each . Hence,
for each . This implies that
for each .
Case 1: There exists a natural number N such that for each . So, exists.
Hence, it follows from exists and (4.2) that
Clearly,
By assumption, (4.8), and (4.9),
Without loss of generality, we may assume that for each . By (4.10) and Lemma 2.4,
By assumption, (4.5), and (4.10),
By assumption, exists, is a bounded sequence, and (4.3), we know that
Clearly,
for each . By assumption, (4.10), and (4.14),
By (4.15),
By (4.4), (4.13), (4.15), and (4.16), we know that
By (4.12) and (4.17),
Since is a bounded sequence, there exists a subsequence of such that for some and
Then . By (4.11), (4.18), Lemma 2.2, and Lemma 2.4, we know that and . That is, . By Lemma 2.3,
By (4.7), (4.19), and Lemma 2.8, we know that , where .
Case 2: Suppose that there exists of such that for all . By Lemma 2.7, there exists a nondecreasing sequence in ℕ such that ,
for each . By (4.2), we have
for each . By (4.20) and (4.21),
for each . Then following the same argument as the above, we know that
By (4.3),
for each . This implies that
Following the same argument as the above, we know that
and
By (4.7) and (4.20),
for each . This implies that
for each . By (4.29) and (4.30),
By (4.28) and (4.31),
By (4.20) and (4.32),
Therefore, the proof is completed. □
5 Applications: (SFOP) and (SFP)
We get the following results by Theorems 3.1 and 3.2, respectively.
Theorem 5.1 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two proper lower semicontinuous and convex functions. Let , , , and be sequences of real numbers in with and for each . Let be a sequence in . Let be a bounded sequence in H. Let be fixed. Let . Let Ω be the solution set of (SFOP) and suppose that . Let be defined by
Assume that:
-
(i)
; ; ;
-
(ii)
; ; .
Then , where .
Theorem 5.2 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two proper lower semicontinuous and convex functions. Let , , and be sequences of real numbers in with and for each . Let be a sequence in . Let be a bounded sequence in H. Let be fixed. Let . Let Ω be the solution set of (SFOP) and suppose that . Let be defined by
Assume that , , , , , . Then , where .
By Theorem 4.1, we get the following result.
Theorem 5.3 Let and be two real Hilbert spaces, be a linear and bounded operator, and let denote the adjoint of A. Let and be two proper lower semicontinuous and convex functions. Let be a sequence in , , and . Let Ω be the solution set of (SFOP) and suppose that . Let be defined by
Assume that , , , and . Then , where , i.e., is the minimal norm solution of (SFOP).
Let H be a Hilbert space and let g be a proper lower semicontinuous convex function of H into . Then the subdifferential ∂g of g is defined as follows:
for all . Let C be a nonempty closed convex subset of a real Hilbert space H, and be the indicator function of C, i.e.,
Further, we also define the normal cone of C at u as follows:
Then is a proper lower semicontinuous convex function on H, and the subdifferential of is a maximal monotone operator. So, we can define the resolvent of for , i.e.,
for all . By definitions, we know that
for all . Hence, for each , we have that
Hence, we have the following result by Theorem 3.2.
Theorem 5.4 Let C and Q be two nonempty closed convex subsets of and , respectively. Let be a linear and bounded operator, and let denote the adjoint of A. Let , , and be sequences of real numbers in with and for each . Let be a sequence in . Let be a bounded sequence in H. Let be fixed. Let . Let Ω be the solution set of (SFP) and suppose that . Let be defined by
for each . Assume that , , , and . Then , where .
By Theorem 4.1, we get the following result.
Theorem 5.5 Let C and Q be two nonempty closed convex subsets of and , respectively. Let be a linear and bounded operator, and let denote the adjoint of A. Let be a sequence in , , and . Let Ω be the solution set of (SFP) and suppose that . Let be defined by
for each . Assume that , , , and . Then , where , i.e., is the minimal norm solution of (SFP).
Remark 5.1 Theorem 5.5 improves some conditions of [[13], Theorem 5.5].
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The author was supported by the National Science Council of Republic of China. Also, the author is grateful to an anonymous referee for his fruitful comments.
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Chuang, CS. Strong convergence theorems for the split variational inclusion problem in Hilbert spaces. Fixed Point Theory Appl 2013, 350 (2013). https://doi.org/10.1186/1687-1812-2013-350
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DOI: https://doi.org/10.1186/1687-1812-2013-350