Abstract
In this paper, we first study the set of common solutions for two variational inclusion problems in a real Hilbert space and establish a strong convergence theorem of this problem. As applications, we study unique minimum norm solutions of the following problems: multiple sets split feasibility problems, system of convex constrained linear inverse problems, convex constrained linear inverse problems, split feasibility problems, convex feasibility problems. We establish iteration processes of these problems and show the strong convergence theorems of these iteration processes.
MSC:47J20, 47J25, 47H05, 47H09.
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1 Introduction
Let be nonempty closed convex subsets of a real Hilbert space ℋ. The well-known convex feasibility problem (CFP) is to find such that
The convex feasibility problem has received a lot of attention due to its diverse applications in mathematics, approximation theory, communications, geophysics, control theory, biomedical engineering. One can refer to [1, 2].
If and are closed vector spaces of a real Hilbert space ℋ, in 1933, von Neumann showed that any sequence is generated from the method of alternating projections: , , , , … . Then converges strongly to some . If and are nonempty closed convex subsets of ℋ, Bregman [3] showed that the sequence generated from the method of alternating projection converges weakly to a point in . Hundal [4] showed that the strong convergence fails if and are nonempty closed convex subsets of ℋ. Recently, Boikanyo et al. [5] proposed the following process:
and
where are given arbitrarily, and are two set-valued maximal monotone operators with , and , , , , , are sequences. Boikanyo et al. [5] proved that the sequence converges strongly to a point under suitable conditions.
The split feasibility problem (SFP) is to find a point
where C, Q are nonempty closed convex subsets of real Hilbert spaces , , respectively. is a bounded linear operator. The split feasibility problem (SFP) in finite dimensional real Hilbert spaces was first introduced by Censor and Elfving [6] for modeling inverse problems which arise from medical image reconstruction. Since then, the split feasibility problem (SFP) 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, 2, 6–17] and related literature. A special case of problem (SFP) is the convexly constrained linear inverse problem in a finite dimensional real Hilbert space [18]:
where C is a nonempty closed convex subset of a real Hilbert space and b is a given element of a real Hilbert space , which has extensively been investigated by using the Landweber iterative method [19]:
Let be nonempty closed convex subsets of , let be nonempty closed convex subsets of , and let be bounded linear operators. The well-known multiple sets split feasibility problem (MSSFP) is to find such that
The multiple sets split feasibility problem (MSSFP) contains convex feasibility problem (CFP) and split feasibility problem (SFP) as special cases [6, 10, 13]. Indeed, Censor et al. [11] first studied this type of problem. Xu [16] and Lopez et al. [13] also studied this type of problem. In 2011, Boikanyo and Moroşanu [20] gave the following algorithm:
where u is given in ℋ, and , are two set-valued maximal monotone mappings on ℋ, , , , , and are sequences in . Boikanyo and Moroşanu [20] proved that in (1.3) converges strongly to some under suitable conditions.
Motivated by the above works, we consider the following algorithm:
where , are two set-valued maximal monotone mappings on a real Hilbert space , are two mappings, , , , , , and are sequences in . We show that the sequence generated by (1.4) converges strongly to some under suitable conditions.
Algorithm (1.4) contains as special cases the inexact proximal point algorithm (e.g., [21, 22]) and the generalized contraction proximal point algorithm (e.g., [23]). Our conclusion extends and unifies Boikanyo and Moroşanu’s result in [20], Wang and Cui’s result in [24] becomes a special case. For details, one can see Section 3.
In this paper, we first study the set of common solutions for two variational inclusion problems in a Hilbert space and establish a strong convergence theorem of this problem. As applications, we study unique minimum norm solutions of the following problems: multiple sets split feasibility problems, system of convex constrained linear inverse problems, convex constrained linear inverse problems, split feasibility problems, convex feasibility problems. We establish iteration processes of these problems and show strong convergence theorems of these iteration processes.
2 Preliminaries
Throughout this paper, let ℋ, , , denote real Hilbert spaces with the inner product and the norm ; let ℕ be the set of all natural numbers and be the set of all positive real numbers. A set-valued mapping A with domain on ℋ is called monotone if for any , and for all . A monotone operator A is called maximal monotone if its graph is not properly contained in the graph of any other monotone mapping. The set of all zero points of A is denoted by , i.e., . In what follows, we denote the strong convergence and the weak convergence of to by and , respectively. In order to facilitate our discussion, in the next section, we recall some facts. The following equality is easy to check:
for each and with . Besides, we also have
for each . Let C be a nonempty closed convex subset of ℋ, and a mapping . We denote the set of all fixed points of T by . A mapping is said to be nonexpansive if for every . A mapping is said to be quasi-nonexpansive if and for all and . A mapping is said to be firmly nonexpansive if
for every . Besides, it is easy to see that is a closed convex subset of C if is a quasi-nonexpansive mapping. A mapping is said to be α-inverse-strongly monotone (α-ism) if
for all and .
The following lemmas are needed in this paper.
Lemma 2.1 [25]
Let be a bounded linear operator, and let be the adjoint of A. Suppose that C is a nonempty closed convex subset of , and that is a firmly nonexpansive mapping. Then is -ism, that is,
for all .
Lemma 2.2 Let C be a nonempty closed convex subset of ℋ, and let be a firmly nonexpansive mapping. Suppose that is nonempty. Then for each and each .
Let C be a nonempty closed convex subset of 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 [26]
Let C be a nonempty closed convex subset of ℋ, and let be the metric projection from ℋ onto C. Then for each and each .
For a set-valued maximal monotone operator G on ℋ and , we may define an operator with which is called the resolvent mapping of G for r.
Let be a set-valued maximal monotone mapping. Then we have that
-
(i)
for each , is a single-valued and firmly nonexpansive mapping.
-
(ii)
and .
-
(iii)
for each and all with .
Lemma 2.5 [26]
Let G be a set-valued maximal monotone operator on ℋ. For , we define the resolvent . Then the following holds:
for all and . In fact,
for all and .
Lemma 2.6 [28]
Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of such that for all . Consider the sequence defined by = for some sufficiently large number . Then is a nondecreasing sequence with as , and for all ,
In fact, .
Lemma 2.7 [5]
Let and be two sequences in . Let be a sequence of nonnegative real numbers. Let and be two sequences of real numbers. Let be a sequence of nonnegative real numbers with
for each . Assume that:
Then .
A mapping is said to be averaged if , where is a nonexpansive mapping and .
Lemma 2.8 [29]
Let C be a nonempty closed convex subset of ℋ and be a mapping. Then the following hold:
-
(i)
T is a nonexpansive mapping if and only if is -inverse-strongly monotone (-ism).
-
(ii)
If S is ν-ism, then γS is -ism.
-
(iii)
S is averaged if and only if is ν-ism for some .
Indeed, S is α-averaged if and only if is -ism for .
-
(iv)
If S and T are averaged, then the composition ST is also averaged.
-
(v)
If the mappings are averaged and have a common fixed point, then for each .
Lemma 2.9 [30]
Let T be a nonexpansive self-mapping on a nonempty closed convex subset C of ℋ, and let be a sequence in C. If and , then .
Let C be a nonempty closed convex subset of ℋ. The indicator function defined by
is a proper lower semicontinuous convex function and its subdifferential defined by
is a maximal monotone operator [31]. Furthermore, we also define the normal cone of C at u as follows:
We can define the resolvent of for , i.e.,
for all . Since
for all , we have that
3 Main results
Let C, Q and be nonempty closed convex subsets of , and , respectively. For each and , let be a -inverse-strongly monotone mapping of C into , and let be a set-valued maximal monotone mapping on such that the domain of is included in C for each . Let be a firmly nonexpansive mapping of into and be a firmly nonexpansive mapping of into . Note and for each and . Let be a bounded linear operator, be a bounded linear operator, and be the adjoint of for . Throughout this paper, we use these notations unless specified otherwise.
Theorem 3.1 Suppose that is nonempty, and , , , , , are sequences in such that , , , and for each . For an arbitrarily fixed , define a sequence by
Then provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
and for each and for some ;
-
(iv)
, .
Proof Given any fixed . Then and . Since is -ism and , it follows from Lemma 2.8 that is averaged. Hence is nonexpansive. By Lemma 2.8, and are averaged. Hence and are nonexpansive. By condition (iv), we may assume that there exist positive real numbers c and h such that and for each . For each , we have that
and
By the mathematical induction method, we know that , and are bounded sequences. By Lemma 2.4, we have that
for each . Hence,
where
Similarly, we have
Consequently, it follows from (3.2) and (3.3) that
For each , set . Then and (3.4) become
Case 1: is eventually decreasing, i.e., there exists a natural number N such that for each . So, is convergent and exists. For all , , and (3.5), we have that
Noting via condition (i) and the fact that is bounded that
we conclude from (3.6) that
Therefore,
Since is a bounded sequence in ℋ, there is a subsequence of such that and
On the other hand, , there exists a subsequence of such that converges to a number . By Lemma 2.5, we have
By (3.8) and Lemma 2.9, . Since , there exists a subsequence of such that converges to a number . We have that
and
Since is bounded, we conclude from (3.7), (3.9), (3.10), and conditions (i), (ii) that
By Lemma 2.5, we have
Since , we know that . By (3.12) and Lemma 2.9, we know that . So, . This shows that . It follows from and Lemma 2.3 that
By (3.11) and (3.13),
Applying Lemma 2.7 to inequality (3.5) with and , we obtain from (3.13) and (3.14) and conditions (i), (ii) that . That is, . And then it follows from (3.11) that . Thus, .
Case 2: Suppose that is not an eventually decreasing sequence. Let be a subsequence of such that for all , also consider the sequence of integers , defined by , for some ( is a sufficiently large number). Then is a nondecreasing sequence as , and for all , one has that
That is, . For such , it follows from (3.15) that
and
From (3.15) and (3.5), we obtain
Just as the argument of Case 1, we have
By (3.15) and (3.18), we have
for all . This implies that
for all , where . Furthermore, we have
Hence, it follows from (3.19) and (3.20) that
By (3.15) and (3.21), we know that . Then, just as the argument in the proof of Case 1, we obtain . Therefore, the proof is completed. □
Remark 3.1
-
(i)
If we put in Theorem 3.1, then Theorem 3.1 is reduced to Theorem 7 in [20].
-
(ii)
Boikanyo et al. [20] showed that strong convergence theorem of a proximal point algorithm with error can be obtained from strong convergence of a proximal point algorithm without errors. Therefore, in Theorem 3.1, we study strong convergence of variational inclusion problems without error.
As a simple consequence of Theorem 3.1, we have the following theorem.
Theorem 3.2 Suppose that is nonempty, and , , are sequences in such that , for each . For an arbitrary fixed , define a sequence by
Then if the following conditions are satisfied:
-
(i)
, ;
-
(ii)
for each and for some ;
-
(iii)
.
Proof Set , , , and are sequences in , and . Define a sequence by
and
Then
and
Since
it is easy to see that
Then Theorem 3.2 follows from Theorem 3.1. □
Remark 3.2 Following the same argument as in Remark 12 in [20], we see that Theorems 3.1 and 3.2 contain [[23], Theorem 3.3], [[24], Theorem 1], [[32], Theorems 1-4] and many other recent results as special cases.
4 Applications
Now, we recall the following multiple sets split feasibility problem (MSSFP-A1):
Theorem 4.1 [33]
Given any , and are sequences in .
-
(i)
If is a solution of (MSSFP-A1), then for each .
-
(ii)
Suppose that with , for each and the solution set of (MSSFP-A1) is nonempty. Then is a solution of (MSSFP-A1).
In order to study the convergence theorems for the solution set of multiple split feasibility problem (MSSFP-A1), we need the following problems and the following essential tool which is a special case of Theorem 3.2 in [33]:
Lemma 4.1 Given any .
-
(i)
If is a solution of (SFP-1), then for each .
-
(ii)
Suppose that with for each , and the solution set of (SFP-1) is nonempty. Then and are averaged and is a solution of (SFP-1).
Proof (i) Suppose that is a solution of (SFP-1). Then , for each . It is easy to see that
(ii) Since the solution set of (SFP-1) is nonempty, there exists such that , . Then . If we put and , we get that the solution set of (MSSFP-A1) is nonempty. By Lemma 2.1 we have that
By (4.1), , and Lemma 2.8(ii), (iii), we know that
On the other hand, for each , is a firmly nonexpansive mappings, it is easy to see that
Hence, by (4.2), (4.3) and Lemma 2.8(iv) and (v), we see that
Since
so
Then Lemma 4.1 follows from Theorem 4.1 by taking , and . □
Remark 4.1 From the following result, we know that Lemma 4.1 is more useful than Theorem 4.1.
Theorem 4.2 Suppose that the solution set of (MSSFP-A1) is nonempty and , , , , , are sequences in such that , , , and for each . For an arbitrarily fixed , a sequence is defined by
Then is a unique solution of the following optimization problem:
provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
, for each , and for some ;
-
(iv)
, .
Proof Since is firmly nonexpansive, it follows from Lemma 4.1 that is -ism for each . For each , put in Lemma 4.1. Then algorithm in Theorem 3.1 follows immediately from algorithm in Theorem 4.2. Since is nonempty, by Lemma 4.1, we have that
for each . This implies that
for each . Hence,
By Theorem 3.1, , where . That is,
and
for all . Since
we know that
That is,
and
By Lemma 4.1, we get that . Similarly, if , then . Therefore . This shows that is a unique solution of the optimization problem
Therefore, the proof is completed. □
In the following theorem, we study the following multiple sets split feasibility problem (MSSMVIP-A2):
Let denote the solution set of (MSSMVIP-A2). The following theorem is a special case of Theorem 4.3. Hence, it is also a special case of Theorem 4.2.
Theorem 4.3 Suppose that is nonempty, and that , , , , , are sequences in with , , , and for each . For an arbitrary fixed , a sequence is defined by
Then is a unique solution of the following optimization problem:
provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
, for each and for some ;
-
(iv)
, .
Proof Put , , and . Then , are two set-valued maximum monotone mappings, and are firmly nonexpansive mappings. Since and , we have , , and . Then Theorem 4.3 follows from Theorem 4.2. □
In the following theorem, we study the following split feasibility problem (MSSMVIP-A3):
Let denote the solution set of problem (MSSMVIP-A3). The following is also a special case of Theorem 4.3.
Theorem 4.4 Suppose that is a nonempty closed convex subset of , is nonempty, and , , , , , are sequences in with , , , and for each . For an arbitrary fixed , a sequence is defined by
Then is a unique solution of the following optimization problem:
provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
, for each and for some ;
-
(iv)
, .
Proof Put and = in Theorem 4.3. Then Theorem 4.4 follows from Theorem 4.3. □
In the following theorem, we study the following convex feasibility problem (MSSMVIP-A4):
Let denote the solution set of (MSSMVIP-A4). The following is a special case of Theorem 4.3.
Theorem 4.5 Suppose that Q and are nonempty closed convex subsets of , is nonempty, and , , , , , are sequences in with , , , and for each . For an arbitrary fixed , a sequence is defined by
Then is a unique solution of the following optimization problem:
provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
, for each and for some ;
-
(iv)
, .
Proof Put and = = in Theorem 4.3. Then Theorem 4.5 follows from Theorem 4.3. □
In the following theorem, we study the following convex feasibility problem (MSSMVIP-A5):
Let denote the solution set of (MSSMVIP-A5).
The following existent theorem of a convex feasibility problem follows immediately from Theorem 4.5.
Theorem 4.6 Suppose that Q and are nonempty closed convex subsets of , is nonempty, and , , , , , are sequences in with , , , and for each . For an arbitrary fixed . Define a sequence by
Then is a unique solution of the following optimization problem:
provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
, for each and for some ;
-
(iv)
, .
Proof Put , then . Then Theorem 4.6 follows from Theorem 4.5. □
In the following theorem, we study the following system of convexly constrained linear inverse problem (SCCLIP):
Let denote the solution set of (SCCLIP).
Theorem 4.7 Suppose that is nonempty, and , . Let , , , , , and be sequences in with , , , and for each . For an arbitrary fixed , a sequence is defined by
Then is a unique solution of the following optimization problem:
provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
, for each and for some ;
-
(iv)
, .
Proof Put and . Then Theorem 4.7 follows from Theorem 4.2. □
In the following theorem, we study the following convexly constrained linear inverse problem (CCLIP):
Let denote the solution set of (CCLIP).
Theorem 4.8 Suppose that is a nonempty closed convex subset of . is nonempty, , and , , , , , are sequences in with , , , and for each . For an arbitrary fixed , a sequence is defined by
Then is a unique solution of the following optimization problem:
provided the following conditions are satisfied:
-
(i)
;
-
(ii)
either or ;
-
(iii)
, for each , and for some ;
-
(iv)
, .
Remark 4.2 The iteration in Theorem 4.8 is different from the Landweber iterative method [19]:
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Acknowledgements
Prof. C-S Chuang was supported by the National Science Council of Republic of China while he work on the publish, and Y-D Chen was supported by Southern Taiwan University of Science and Technology.
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L-JL designed and coordinated this research project and revised the paper. Y-DC carried out the project, drafted and revised the manuscript. C-SC coordinated the project and revised the paper.
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Lin, LJ., Chen, YD. & Chuang, CS. Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems. Fixed Point Theory Appl 2013, 333 (2013). https://doi.org/10.1186/1687-1812-2013-333
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DOI: https://doi.org/10.1186/1687-1812-2013-333