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Mathematical programming with multiple sets split monotone variational inclusion constraints
Fixed Point Theory and Applications volume 2014, Article number: 20 (2014)
Abstract
In this paper, we first study a hierarchical problem of Baillon’s type, and we study a strong convergence theorem of this problem. For the special case of this convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon’s type. Our result of the ergodic theorem of Baillon’s type improves and generalizes many existence theorems of this type of problem. Two numerical examples are given to demonstrate our results.
As applications of our convergence theorem of the hierarchical problem, we study the unique solution for the following problems: mathematical programming with multiply sets split variational inclusion and fixed point set constraints; mathematical programming with multiple sets split variational inequalities and fixed point set constraints; the variational inequality problem with a system of mixed type equilibria and fixed point set constraints; the variational inequality problem with multiple sets split system of mixed type equilibria and fixed point set constraints; mathematical programming with a system of mixed type equilibria and fixed point set constraints. We give iteration processes for these types of problems and establish the strong convergence for the unique solution of these problems. For our special case, our results can be reduced to the following problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibria problem; the minimum norm solution of the system of mixed type equilibria problem. Our results will have many applications in diverse fields of science.
1 Introduction
Let be nonempty closed convex subsets of a Hilbert space . The well-known convex feasiblity problem (CFP) is to find such that
The split feasibility problem (SFP) is to find a point
where C is a nonempty closed convex subset of a Hilbert space , Q is a nonempty closed convex subset of a Hilbert space , and is an operation. The split feasibility problem (SFP) in the finite dimensional Hilbert spaces was first introduced by Censor et al. [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. Since then, the convex feasibility problem and 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 example, one can refer to [1–5] and related literature.
Let be nonempty closed convex subsets of a Hilbert space , let be nonempty closed convex sets and let be linear operator operators. The well-known multiple sets split feasibility problem studied by Censor et al. [6]. Xu [7] and Lopez et al. [8] (MSSFP) is to find such that
In 2011, Moudafi [9] introduced and studied the following split monotone variational inclusion (SMVI):
and
where and are real Hilbert spaces, is a bounded linear operator, and are given operators, and are given multivalued mappings.
Moudafi [9] proved a weakly convergence theorem for the solution of the split monotone variational inclusion (SMVI) with an iteration process.
In 2011, Maruyama et al. [10] proved the following ergodic theorem of Baillon’s type [11].
Theorem 1.1 [10]
Let C be a nonempty closed convex subset of a real Hilbert space H, be a 2-generalized hybrid mapping with and let be the metric projection of onto . Then, for any ,
converges weakly to an element p of , where .
In this paper, we first study a hierarchical problem of Baillon’s type, and we study a strong convergence theorem of this problem. For the special case of this convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon’s type. Our result of the ergodic theorem of Baillon’s type improves and generalizes many existence theorems of this type of problem. Two numerical examples are given to demonstrate our results.
As applications of our convergence theorem of the hierarchical problem, we study the unique solution for the following problems: mathematical programming with multiply sets split variational inclusion and a fixed point set constraints; mathematical programming with multiple sets split variational inequalities and fixed point set constraints; the variational inequality problem with a system of mixed type equilibria and fixed point set constraints; the variational inequality problem with multiple sets split system of mixed type equilibria and a fixed point set constraints; mathematical programming with system of mixed type equilibrium and a fixed point set constraints. We give iteration processes for these types of problems and establish the strong convergence for the unique solution of these problems. For the special case of our results, our results can be reduced to the following problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibrium problem; the minimum norm solution of the system of the mixed type equilibria problem. Our results will have many applications in diverse fields of science.
2 Preliminaries
Throughout this paper, let ℕ be the set of positive integers and let ℝ be the set of real numbers, be a (real) Hilbert space with inner product and norm , respectively, and C be a nonempty closed convex subset of . We denote the strongly convergence and the weak convergence of to by and , respectively.
Let be a mapping, and let denote the set of fixed points of T. A mapping is called
-
(i)
a 2-generalized hybrid mapping [10] if there exist such that
for all .
We know that the class of 2-generalized hybrid mapping contains the classes of nonexpansive mappings, nonspreading mappings, and a -generalized hybrid [12] in a Hilbert space. We give an example for a 2-generalized hybrid mapping.
Example 2.1 [13]
Let be defined as
Then T is a 2-generalized hybrid mapping and .
Proof Let , .
Case 1: If , , then , and . We know that
and
Therefore,
Case 2: If , , then , . We know that
Case 3: If , then , , , . We know that
and
Therefore,
By the above case, we know that T is a 2-generalized hybrid. □
A mapping is called
-
(i)
strongly monotone if there exists such that for all ;
-
(ii)
α-inverse-strongly monotone if for all and .
We also know that if V is a α-inverse-strongly monotone mapping and , then is nonexpansive.
Let be a multivalued mapping. The effective domain of G is denoted by , that is, .
Then is called
-
(i)
a monotone operator on if for all , , and ;
-
(ii)
a maximal monotone operator on if G is a monotone operator on and its graph is not properly contained in the graph of any other monotone operator on .
Lemma 2.1 [14]
Let C be a nonempty closed convex subset of a real Hilbert space . Let T be a nonexpansive mapping of C into itself, and let be a sequence in C. If and , then .
In 2012, Hojo et al. [15] also gave an example for a 2-generalized hybrid mapping which is not a generalized hybrid mapping with . We shall prove that this example for a 2-generalized hybrid mapping does not satisfy the demiclosed property as in Lemma 2.1.
Example 2.2 Let and be defined as
Hojo et al. [15] showed that T is a 2-generalized hybrid mapping, but T is not a generalized hybrid mapping. Note that T does not have the demiclosed property. Indeed, there exists a sequence such that and , but w in .
Proof Let , for all , then and . We also have , but . □
Lemma 2.2 [16]
Let be a -strongly monotone and L-Lipschitzian continuous operator with and . Let , and such that . Then is a -strongly monotone and L-Lipschitzian continuous mapping. Furthermore, there is a unique fixed point in C satisfying . This point is also a unique solution of the hierarchical variational inequality , for all .
Let C be a nonempty subset of a real Hilbert space . Then is a firmly nonexpansive mapping if for every , that is, for every .
Lemma 2.3 [17]
Let be a maximal monotone mapping on . Let be the resolvent of defined by for each . Then the following holds:
for all and . In particular,
for all and .
A mapping is said to be averaged if , where and is nonexpansive. In this case, we also say that T is α-averaged. A firmly nonexpansive mapping is -averaged.
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a mapping. Then the following are satisfied:
-
(i)
T is nonexpansive if and only if the complement is -ism.
-
(ii)
If S is υ-ism, then for , γS is -ism.
-
(iii)
S is averaged if and only if the complement is υ-ism for some .
-
(iv)
If S and T are both averaged, then the product (composite) ST is averaged.
-
(v)
If the mappings are averaged and have a common fixed point, then .
Lemma 2.5 [19]
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 the following properties are satisfied by all (sufficiently large) numbers :
In fact, .
Lemma 2.6 [20]
Let be a sequence of nonnegative real numbers, a sequence of real numbers in with , a sequence of nonnegative real numbers with , a sequence of real numbers with . Suppose that for each . Then .
3 Convergence theorems of hierarchical problems
Let be a real Hilbert space and let C be a nonempty closed convex subset of . For each , and , let be a -inverse-strongly monotone mapping of C into . For each , let be a maximal monotone mapping on such that the domain of is included in C and define the set as . Let and for each , and . Let be a sequence. Let V be a -strongly monotone and L-Lipschitzian continuous operator with and . Throughout this paper, we use these notations and assumptions unless specified otherwise.
The following strong convergence theorem for hierarchical problems is one of our main results in this paper.
Theorem 3.1 Let be a 2-generalized hybrid mapping with . Take as follows:
Let be defined by
for each , , , , and . Assume that:
-
(i)
;
-
(ii)
, and ;
-
(iii)
, and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the hierarchical variational inequality:
Proof Take any and let be fixed. Then we have and . Let . For each , by the same argument as the proof of Theorem 3.1 [16], we have
and
By equations (3) and (4), we have
Since T is a 2-generalized hybrid mapping with , we know that T is a quasi-nonexpansive, and
By the same argument as in the proof of Theorem 3.1 [16], we find that the sequence is bounded. Furthermore, , , , and are bounded. We also have
and
We will divide the proof into two cases.
Case 1: there exists a natural number N such that for each . Therefore, exists. Hence, it follows from equation (7), (i), and (ii) that
By equations (6), (8), (i), and (ii), we have
We also have
By equation (10), (iv), and (ii) we have
By the same argument as in the proof of Theorem 3.1 [16], we have
and
Since is a nonempty closed convex subset of , by Lemma 2.2, we can take such that
This point is also a unique solution of the hierarchical variational inequality:
We shall show that
Without loss of generality, there exists a subsequence of such that
for some and
By equations (12) and (13), we have
and . On the other hand, since , there exists a subsequence of such that converges to a number . By equation (14) and Lemma 2.3, we have
By equation (18), , and Lemma 2.1, .
Since , there exists a subsequence of such that converges to a number . By the same argument as for equation (18), we have
By equation (13) and , we have .
By equation (19), and Lemma 2.1, we have .
Since T is a 2-generalized hybrid mapping, there exist such that
for all . Replacing x by in equation (20), we have, for all and ,
This implies that
Summing up these inequalities (21) with respect to to and dividing by n, we have
Replacing n by and let . Then from equation (11), (16), and (22), we have , and
Taking in the above inequality, we have
This implies that . Hence, . Therefore, we have from equations (15) and (17)
By the same argument as the proof of Theorem 3.1 [16], we have
By equations (23), (24), assumptions, and Lemma 2.6, we know that , where
Case 2: Suppose that there exists of such that for all . By Lemma 2.5, there exists a nondecreasing sequence in ℕ such that and
Hence, it follows from equations (7) and (25) that
for each . Hence, it follows from equation (26), (i), and (ii) that
We show that
Without loss of generality, there exists a subsequence of such that for some and
By a similar argument as in the proof of Case 1, we have . Therefore, we have from equations (28) and (15)
Following a similar argument as in the proof of Case 1, we have
From , we have
Since , we have
By equations (29), (32), and the assumptions, we know that
By (6), (27), and the assumptions, we know that
Thus, we have
Thus, the proof is completed. □
Remark 3.1
-
(i)
The assumptions, method, conclusion, and applications of Theorem 3.1 are different from Theorem 3.1 in [21] and [22]. In Theorem 3.1, Lemma 2.5 is used to prove the result, but in [21] and [22] we did not use this lemma.
-
(ii)
The assumptions, method, and conclusion of Theorem 3.1 are different from Theorem 3.1 [16]. In Theorem 3.1 [16], T is a quasi-nonexpansive with the demiclosed property, but in Theorem 3.1, T is a 2-generalized hybrid mapping, and by Example 2.2, we know that T does not satisfy the demiclosed property. Therefore Theorem 3.1 [16] cannot apply for a 2-generalized hybrid mapping.
Example 3.1 Let T be the same as Example 2.1. Let , , , , , , , . Then V, , , , satisfy all conditions of Theorem 3.1 and , and if we let , we see the following numerical results and graph (see Figure 1) demonstrating Theorem 3.1: Besides, we know the following.
If , then ; if , then ; if , then ; if , then .
For , let , , and for all in Theorem 3.1. Furthermore, put , and for all ; we obtain the following theorem which generalizes Theorem 4.1 in [10].
Theorem 3.2 Let be a 2-generalized hybrid mapping such that . Let , and be defined by
for each , , and . Assume that and , and . Then , where .
Example 3.2 Let T be the same as Example 2.1. Let , , . Then the following numerical results and graph (see Figure 2) demonstrate Theorem 3.2: Besides, we know the following.
If , then ; if , then ; if , then ; if , then .
4 Mathematical programming with multiple sets split feasibility constraints
Let be a Hilbert space, let f be a proper lower semicontinuous convex function of into . The subdifferential ∂f of f is defined as follows:
for all . From Rockafellar [23], we know that ∂f is a maximal monotone operator. Let C be a nonempty closed convex subset of a real Hilbert space , and be the indicator function of C, i.e.
Furthermore, 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. Thus, we can define the resolvent of for , i.e.
for all . Since
for all , we have
The equilibrium problem is to find such that
The solutions set of the equilibrium problem (EP) is denoted by . For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:
(A1) for each ;
(A2) g is monotone, i.e., for any ;
(A3) for each , ;
(A4) for each , the scalar function is convex and lower semicontinuous.
Let be a bifunction which satisfies conditions (A1)-(A4). Let and . Then there exists such that
Furthermore, if
then we have:
-
(i)
is single-valued;
-
(ii)
is a firmly nonexpansive mapping;
-
(iii)
is a closed convex subset of C;
-
(iv)
.
We call such the resolvent of g for . Throughout these section, we use these notations and assumptions unless specified otherwise.
Takahashi et al. [26] gave the following lemma.
Lemma 4.2 [26]
Let be a bifunction satisfying the conditions (A1)-(A4). Define as follows:
Then, and is a maximal monotone operator with the domain of . Furthermore, for any and , the resolvent of g coincides with the resolvent of , i.e., .
Let C, Q, and be nonempty closed convex subsets of real Hilbert spaces , , and , respectively, let be a maximal monotone mapping on such that the domains of is included in C for each . Let and for each and , let be a -inverse-strongly monotone mapping of C into , let be a -inverse-strongly monotone mapping of C into , let B be a ν-inverse-strongly monotone mapping of Q into and let be a -inverse-strongly monotone mapping of into . Let G be a maximal monotone mappings on such that the domain of G is included in Q and let be maximal a monotone mappings on such that the domain of is included in . Let and for each and . Let be bounded linear operators, and the adjoints of and A respectively, a bounded linear operator, and the adjoint of . Let be the spectral radius of the operator for , respectively, and R the spectral radius of the operator . Let I, , be the identity mappings of , , , respectively. We use these notations throughout this section unless specified otherwise.
In order to study the convergence theorems for the solutions set of the multiple sets split monotone variational inclusion problem, we study the following essential problem (SFP-1):
Theorem 4.1 Given any we have the following.
-
(i)
If is a solution of (SFP-1), then , where , and .
-
(ii)
Suppose that , , . Then is a mapping, , and are averaged for some . Suppose further that solution set of (SFP-1) is nonempty and . Then is a solution of (SFP-1).
Proof (i) Suppose that is a solution of (SFP-1). Then , . It is easy to see that .
(ii) Since the solutions set of (SFP-1) is nonempty, there exists such that . Since B is a ν-inverse-strongly monotone mapping of Q into , it follows from Lemma 2.4(iii) and (iv) that
By Lemma 2.4(iii), for some , we know that
In Theorem 3.1 [9], Moudafi showed that
By Lemma 2.4(iii) and , we know that
for some . Since
This implies
We know that , with , which combined with the fact that yields
Since is a nonexpansive mapping and we have equation (41), we have .
This implies that
This shows that is a solution of (SFP-1). □
In the following theorem, we consider the multiple set split monotonic variational inclusion problem (MSSMVIP-1):
That is,
Let be the solutions set of (MSSMVIP-1).
Theorem 4.2 Let be a 2-generalized hybrid mapping. Suppose that is the solutions set of (MSSMVIP-1) with . Take as follows:
Let be defined by
where , , , , and . We have
-
(i)
;
-
(ii)
, and ;
-
(iii)
, , and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the hierarchical variational inequality:
Proof Let , , and for all in Theorem 3.1. It follow from Theorem 4.1(ii) that is for some and each . Then algorithm (3.1) in Theorem 3.1 follows immediately from algorithm (4.2) in Theorem 4.2.
Since is nonempty, there exists such that
This implies that
That is,
Hence,
It follows from Theorem 3.1 that , where
This point is also a unique solution of the hierarchical variational inequality:
If
By equations (44), (45), and (46), we know that
By , equation (48), and Lemma 2.4(v), we have
It follows from Theorem 4.1(ii) that w is a solution of (MSSMVIP-1). Therefore, and
Conversely, if , by equations (43), (44), (45), and (46), we know that and
Therefore, and the proof is completed. □
Remark 4.1 Moudafi [9] studied a weak convergence theorem for the split monotone variational inclusion problem, while Theorem 4.2 is a strong convergence theorem for the multiply sets split monotone variational inclusion problem.
By Theorem 4.2, we study the mathematical programming problem with (MSSMVIP-1) and fixed point set constraints.
Theorem 4.3 In Theorem 4.2, let be a convex Gâteaux differential function with Gâteaux derivative V, and the assumption (iv) is replaced by . Then , where . This point is also a unique solution of the mathematical programming problem with (MSSMVIP-1) and fixed point constraints: .
Proof Let in Theorem 4.2, by Theorem 4.2, we see that
Since is a convex Gâteaux differential function with Gâteaux dirivitive V, we obtain
for all . By equations (50) and (51), it is easy to see that for all . □
If we put in Theorem 4.3, then , and we have the following minimum norm of common solutions for (MSSMVIP-1) and .
Theorem 4.4 In Theorem 4.3, let the iteration process be replaced by
Then , where . This point is also a unique minimum solution of : .
The multiple sets split variational inequality problem (MSSMVIP-2) is defined as follows:
and
That is,
and
By Theorem 4.2, we can study a variational inequality problem with the split variational inequality (MSSMVIP-2) and fixed point set constraints.
Theorem 4.5 In Theorem 4.2, let , be replaced by , , respectively. Suppose that the set of solutions for (MSSMVIP-2) is and . Then , where . This point is also a unique solution of the hierarchical variational inequality:
Proof Let and in Theorem 4.2, then, by equation (34), we have , . Since , there exists such that we can find
and
This implies that
Therefore, . It follows from Theorem 4.2 that , where . This point is also a unique solution of the hierarchical variational inequality:
By equations (57), (58), and (59), , where . This point is also a unique solution of the hierarchical variational inequality:
□
Remark 4.2 Censor et al. [27] studied a weak convergence theorem for the split variational inequalities problem with the additional assumption, while Theorem 4.5 studies a strong convergence theorem for multiply sets split variational inequalities problem without this additional assumption.
By Theorem 4.5, we study a mathematical programming problem with (MSSMVIP-2) and fixed point set constraints.
Theorem 4.6 In Theorem 4.5, let be a convex Gâteaux differential function with Gâteaux derivative V, and the assumption (iv) is replaced by . Then , where . This point is also a unique solution of the following mathematical programming problem with (MSSMVIP-2) and fixed point constraints:
Proof By Theorem 4.5 and following the same argument as in the proof of Theorem 4.3, we see that the proof is complete. □
In the following theorem, we consider the following split monotonic variational inclusion problem (MSSMVIP-3):
That is,
Let be the solutions set of (MSSMVIP-3).
Theorem 4.7 Let be a 2-generalized hybrid mapping. Suppose that is the solutions set of (MSSMVIP-3) with . Take as follows:
Let be defined by
where , , , and .
-
(i)
;
-
(ii)
, and ;
-
(iii)
, and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the hierarchical variational inequality:
Proof Let , and for all in Theorem 3.1. It follows from Theorem 4.1(ii) that is for some . Then algorithm (3.1) in Theorem 3.1 follows immediately from algorithm (4.7) in Theorem 4.7.
Since is nonempty, there exists such that
This implies that
That is,
Hence,
It follows from Theorem 3.1 that , where
This point is also a unique solution of the hierarchical variational inequality:
If
That is,
By , equation (65) and Lemma 2.4(v), we have
By , equation (66), and Theorem 4.1(ii), we see that w is a solution of (MSSMVIP-3). Therefore, and . Conversely, if , by equations (60), (61), (62), and (63), we know that and
Therefore, and the proof is completed. □
Remark 4.3 Theorem 4.7 also improve Theorem 3.1 [9].
For each , let be a bifunction satisfying the conditions (A1)-(A4). The system of mixed type equilibria problem (MSSMVIP-4) is defined as follows.
for all .
By Theorem 3.1 and Lemma 4.2, we study a variational inequality problem with (MSSMVIP-4) and fixed point set constraints.
Theorem 4.8 Let be a 2-generalized hybrid mapping. For each , let be a bifunction satisfying the conditions (A1)-(A4), and , , defined as Lemma 4.2. Suppose that is the solutions set of (MSSMVIP-4) with . Let be defined by
where , , and . Assume that:
-
(i)
;
-
(ii)
, and ;
-
(iii)
, ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the hierarchical variational inequality:
Proof For each , let be as in Lemma 4.2. By Lemma 4.2, we see that is a maximal monotone operator with the domain of . Furthermore, for any and , the resolvent of coincides with the resolvent of , i.e.,
For , let in Theorem 3.1. By equation (67), we have
Then algorithm (3.1) in Theorem 3.1 follows immediately from algorithm (4.8) in Theorem 4.8.
By equation (68), we have and .
It follows from Theorem 3.1 that , where
This point is also a unique solution of the hierarchical variational inequality:
Here . That is, . That is, . That is,
and
for all . Therefore, and the proof is complete. □
By Theorem 4.2, we study a mathematical programming problem with (MSSMVIP-4) and fixed point set constraints.
Theorem 4.9 In Theorem 4.8, let be a convex Gâteaux differential function with Gâteaux derivative V, and let the assumption (iv) be replaced by . Then , where . This point is also a unique solution of the mathematical programming problem with (MSSMVIP-2) constraints:
For each , let and , be bifunctions satisfying conditions (A1)-(A4). The multiple sets split system of mixed type equilibrium problems (MSSMVIP-5) is defined as follows.
for all , .
By Theorem 4.2, we can study a variational inequality problem with (MSSMVIP-5) and fixed point set constraints.
Theorem 4.10 For each , let and , be bifunctions satisfying the conditions (A1)-(A4). Let be a 2-generalized hybrid mapping, and let , , , be defined as in Lemma 4.2. Suppose that is the solutions set of (MSSMVIP-4) with . Let be defined by
where , . Then , where . This point is also a unique solution of the hierarchical variational inequality:
Proof For , let , , and in Theorem 4.2. By Theorem 4.2 and following the same argument as in the proof of Theorem 4.8, we prove Theorem 4.10. □
For each , let and be bifunctions satisfying conditions (A1)-(A4). The split mixed type equilibrium problem (MSSMVIP-6) is defined as follows.
for all , .
Applying Theorem 4.7 and following a similar argument as in Theorem 4.10, we can study a variational inequality problem with (MSSMVIP-6) and with fixed point set constraints.
Theorem 4.11 For each , let and be bifunctions satisfying conditions (A1)-(A4). Let be a 2-generalized hybrid mapping, and , , , defined as in Lemma 4.2. Suppose that is the solutions set of (MSSMVIP-6) with . Let be defined by
where , , , and . Assume further that
-
(i)
;
-
(ii)
, and ;
-
(iii)
, and ;
-
(iv)
for some .
Then , where . This point is also a unique solution of the hierarchical variational inequality:
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The authors declare no competing interests, except Prof. Z-TY was supported by Nan Kai University of Technology, and Prof. C-SC was supported by the National Science Council of Republic of China while he work on the publish.
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Z-TY carry out the project, draft and revise the manuscript. L-JL design this research project, coordination, revise the paper. C-SC coordinate and the project and revise the paper, and give the numerical results. All authors read and approved the final manuscript.
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Yu, ZT., Lin, LJ. & Chuang, CS. Mathematical programming with multiple sets split monotone variational inclusion constraints. Fixed Point Theory Appl 2014, 20 (2014). https://doi.org/10.1186/1687-1812-2014-20
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DOI: https://doi.org/10.1186/1687-1812-2014-20