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Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces
Fixed Point Theory and Applications volume 2014, Article number: 171 (2014)
Abstract
The purpose of this paper is to introduce and study a general split variational inclusion problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split variational inclusion problem. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces.
1 Introduction
Let C and Q be nonempty closed convex subsets of real Hilbert spaces and , respectively. The split feasibility problem is formulated as
where is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been found that the can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [3–5]. The in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10]. For comprehensive literature, bibliography and a survey on SFP, we refer to [11].
Assuming that the is consistent, it is not hard to see that solves if and only if it solves the fixed point equation
where and are the metric projection from onto C and from onto Q, respectively, is a positive constant, and is the adjoint of A.
A popular algorithm to be used to solves the (1.1) is due to Byrne’s CQ-algorithm [2]:
where with λ being the spectral radius of the operator .
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 H is the proximal point algorithm first introduced by Martinet [12] and generated by Rockafellar [13]. 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 . Rockafellar [13] proved that 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 sub-differential ∂f of a proper convex and lower semicontinuous function , then (1.2) is reduced to
In this case, converges weakly to a minimizer of f. Later, many researchers have studied the convergence problems of the proximal point algorithm in Hilbert spaces (see [14–21] and the references therein).
Motivated by the works in [14–17] and related literature, the purpose of this paper is to introduce and consider the following general split variational inclusion problem.
Let and be two real Hilbert spaces, and , be two families of set-valued maximal monotone mappings, be a linear and bounded operator, and be the adjoint of A. The so-called general split variational inclusion problem is
The following examples are special cases of (GSVIP) (1.4).
Classical split variational inclusion problem. Let and be set-valued maximal monotone mappings. The so-called classical split variational inclusion problem (CSVIP) is
which was introduced by Moudafi [17]. It is obvious that problem (1.5) is a special case of (GSVIP) (1.4). In [17], Moudafi proved that the iteration process
converges weakly to a solution of problem (1.5), where λ and γ are given positive numbers.
Split optimization problem. Let , be two proper convex and lower semicontinuous functions. The so-called split optimization problem (SOP) is
Denote by and , then B and K both are maximal monotone mappings, and problem (1.6) is equivalent to the following classical split variational inclusion problem, i.e.:
Split feasibility problem. As in (1.1), let C and Q be two nonempty closed convex subsets of real Hilbert spaces and , respectively and A be the same as above. The split feasibility problem is
It is well known that this kind of problems was first introduced by Censor and Elfving [1] for modeling inverse problems arising from phase retrievals and in medical image reconstruction [2]. Also it can be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning.
Let () be the indicator function of C (Q), i.e.,
Then and both are proper convex and lower semicontinuous functions, and its subdifferentials and are maximal monotone operators. Consequently problem (1.8) is equivalent to the following ‘split optimization problem’ and ‘Moudafi’s classical split variational inclusion problem’, i.e.,
For solving (GSVIP) (1.4), in our paper we propose the following iterative algorithms:
where is a contraction mapping with a contractive constant , , and are sequence in satisfying some conditions. Under suitable conditions, some strong convergence theorems for the sequence proposed by (1.11) to a solution for (GSVIP) (1.4) in Hilbert spaces are proved. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces. Our results extend and improve the related results of Censor and Elfving [1], Byrne [2], Censor et al. [3–5], Rockafellar [13], Moudafi [14, 17], Eslamian and Latif [15], Eslamian [21], and Chuang [22].
2 Preliminaries
Throughout the paper, we denote by H a real Hilbert space, C be a nonempty closed and convex subset of H. denote by the set of fixed points of a mapping T. Let be a sequence in H and . Strong convergence of to x is denoted by , and weak convergence of to x is denoted by . For every point , there exists a unique nearest point in C, denoted by . This point satisfies.
The operator is called the metric projection. The metric projection is characterized by the fact that and
Recall that a mapping is said to be nonexpansive, if for every . T is said to be quasi-nonexpansive, 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, if
Lemma 2.1 (demi-closed principle)
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 .
Lemma 2.2 [23]
Let H be a (real) Hilbert space. Then for all ,
Lemma 2.3 [24]
Let H be a Hilbert space and let be a sequence in H. Then, for any given sequence with and for any positive integers i, j with ,
Lemma 2.4 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 real numbers with . If
then .
Lemma 2.5 [25]
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 2.6 [22]
Let H be a real Hilbert space, be a set-valued maximal monotone mapping, , and let be the resolvent mapping of B.
-
(i)
For each , is a single-valued and firmly nonexpansive mapping;
-
(ii)
and ;
-
(iii)
is a firmly nonexpansive mapping for each ;
-
(iv)
suppose that , then for each , each and each
-
(v)
suppose that . Then for each and each , and each .
Lemma 2.7 Let , be two real Hilbert spaces, be a linear bounded operator and be the adjoint of A. Let be a set-valued maximal monotone mapping, , and let be the resolvent mapping of B, then
-
(i)
;
-
(ii)
;
-
(iii)
if , then is a nonexpansive mapping.
Proof By Lemma 2.6(iii), the mapping is firmly nonexpansive, hence the conclusions (i) and (ii) are obvious.
Now we prove the conclusion (iii).
In fact, for any , it follows from the conclusions (i) and (ii) that
This completes the proof of Lemma 2.7. □
3 Main results
The following lemma will be used in proving our main results.
Lemma 3.1 Let and be two real Hilbert spaces, be a linear and bounded operator, and be the adjoint of A. Let , and , , be two families of set-valued maximal monotone mappings, and let and . If (the solution set of (GSVIP) (1.4)), then is a solution of (GSVIP) (1.4) if and only if for each , for each and for each
Proof Indeed, if is a solution of (GSVIP) (1.4), then for each , and ,
This implies that .
Conversely, if solves (3.1), by Lemma 2.6(v), we have
Hence we have
On the other hand, by Lemma 2.6(v) again
Adding up (3.2) and (3.3), we have
Simplifying it, we have
By the assumption that . Taking , hence for each and . In (3.4), taking and , then we have
This implies that , and so for each . Hence from (3.1), , i.e., . Hence is a solution of (GSVIP)(1.4).
This completes the proof of Lemma 3.1. □
We are now in a position to prove the following main result.
Theorem 3.2 Let , , A, , , , Ω be the same as in Lemma 3.1. Let be a contractive mapping with contractive constant . Let , , be the sequences in with , for each . Let be a sequence in , and be a sequence in . Let be the sequence defined by (1.11). If and the following conditions are satisfied:
-
(i)
, and ;
-
(ii)
for each ;
-
(iii)
,
then where , where is the metric projection from onto Ω.
Proof (I) First we prove that is bounded.
In fact, letting , by Lemma 3.1, for each ,
Hence it follows from Lemma 2.7(iii) that for each and each we have
By induction, we can prove that
This implies that is bounded, so is .
(II) Now we prove that for each
Indeed, it follows from Lemma 2.3 that for any positive
Simplifying it, (3.6) is proved.
By the assumption that , and it is easy to prove that Ω is closed and convex. This implies that is well defined. Again since is a contraction mapping with contractive constant , there exists a unique such that . Since , it solves (GSVIP) (1.4). By Lemma 3.1,
-
(III)
Now we prove that .
In order to prove that (as ), we consider two cases.
Case 1. Assume that is a monotone sequence. In other words, for large enough, is either nondecreasing or non-increasing. Since is bounded, is convergence. Again since , and is bounded, from (3.6) we get
By condition (ii), we obtain
Now we prove that
To show this inequality, we choose a subsequence of such that , for each , and
It follows from (3.8) that
For each , is a nonexpansive mapping. Thus from Lemma 2.1, . By Lemma 3.1 , i.e., w is a solution of (GSVIP) (1.4). Consequently we have
-
(IV)
Finally, we prove that .
In fact, from Lemma 2.2 we have
Simplifying it, we have
where , , and . It is easy to see that , , and . Hence by Lemma 2.4, the sequence converges strongly to .
Case 2. Assume that is not a monotone sequence. Then, by Lemma 2.3, we can define a sequence of positive integers: , (where large enough) by
Clearly is a nondecreasing sequence such that as , and for all
Therefore is a nondecreasing sequence. According to Case (1), and . Hence we have
This implies that and is a solution of (GSVIP) (1.4).
This completes the proof of Theorem 3.2. □
In Theorem 3.2, if and , for each , where and are two set-valued maximal monotone mappings, then from Theorem 3.2 we have the following.
Theorem 3.3 Let , , A, , B, K, Ω, f be the same as in Theorem 3.2. Let , , be the sequence in with for each . Let be any given positive number, and be a sequence in . Let be the sequence defined by
If and the following conditions are satisfied:
-
(i)
, and ;
-
(ii)
;
-
(iii)
,
then where .
4 Applications
In this section we shall utilize the results presented in Theorem 3.2 and Theorem 3.3 to study some problems.
4.1 Application to split optimization problem
Let and be two real Hilbert spaces. Let and be two proper, convex and lower semicontinuous functions, and be a linear and bounded operators. The so-called split optimization problem (SOP) is
Denote by and . It is know that (resp. ) is a maximal monotone mapping, so we can define the resolvent and , where . Since and is a minimum of h on and g on , respectively, for any given , we have
This implies that the (SOP) (4.1) is equivalent to the split variational inclusion problem (SVIP) (4.2). From Theorem 3.3 we have the following.
Theorem 4.1 Let , , A, B, K, h, g be the same as above. Let f, , , be the same as in Theorem 3.3. Let be any given positive number, and be a sequence in . Let be a sequence generated by
If , the solution set of the split optimization problem (4.1), and the following conditions are satisfied:
-
(i)
, and ;
-
(ii)
;
-
(iii)
,
then where .
Proof Since , , and , we have
This implies that
Similarly, from (4.3), we have
From (4.3)-(4.5), we have
Therefore (4.3) can be rewritten as
The conclusion of Theorem 4.1 can be obtained from Theorem 3.3 immediately. □
4.2 Application to split feasibility problem
Let and be two nonempty closed convex subsets and be a bounded linear operator. Now we consider the following split feasibility problem, i.e.: to find
Let and be the indicator functions of C and Q defined by (1.9). Let be the normal cone at defined by
Since and both are proper convex and lower semicontinuous functions on and , respectively, and the subdifferential of (resp. of ) is a maximal monotone operator, we can define the resolvents of and of by
where . By definition, we know that
Hence, for each , we have
This implies that . Similarly . Taking and in (4.1), then the (SFP) (4.8) is equivalent to the following split optimization problem:
Hence, the following result can be obtained from Theorem 4.1 immediately.
Theorem 4.2 Let , , A, , , be the same as above. Let f, , , be the same as in Theorem 4.1. Let be a sequence in . Let be the sequence defined by
If the solution set of the split optimization problem (4.4) , and the following conditions are satisfied:
-
(i)
, and ;
-
(ii)
;
-
(iii)
,
then where .
Remark 4.3 Theorem 4.2 extends and improves the main results in Censor and Elfving [1] and Byrne [2].
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The authors would like to express their thanks to the referees and the editors for their kind and helpful comments and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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Chang, Ss., Wang, L. Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces. Fixed Point Theory Appl 2014, 171 (2014). https://doi.org/10.1186/1687-1812-2014-171
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DOI: https://doi.org/10.1186/1687-1812-2014-171