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Parallel algorithms for variational inclusions and fixed points with applications
Fixed Point Theory and Applications volume 2014, Article number: 174 (2014)
Abstract
In this paper, we introduce two parallel algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive mapping in Hilbert spaces and prove some strong convergence theorems of the proposed algorithms. As special cases, we can approach the minimum-norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive mapping without using the metric projection. Further, we give some applications of our main results.
MSC:49J40, 47J20, 47H09, 65J15.
1 Introduction
Let H be a real Hilbert space. Let be a single-valued nonlinear mapping and be a set-valued mapping.
Now, we are concerned with the following variational inclusion:
Find a zero of the sum of two monotone operators A and B such that
where 0 is the zero vector in H.
The set of solutions of the problem (1.1) is denoted by . If , then the problem (1.1) becomes the generalized equation introduced by Robinson [1]. If , then the problem (1.1) becomes the inclusion problem introduced by Rockafellar [2]. It is well known that the problem (1.1) is among the most interesting and intensively studied classes of mathematical problems and has wide applications in the fields of optimization and control, economics and transportation equilibrium, engineering science, and many others. For the past years, many existence results and iterative algorithms for various variational inequality and variational inclusion problems have been extended and generalized in various directions using novel and innovative techniques. A useful and important generalization is called the general variational inclusion involving the sum of two nonlinear operators. Moudafi and Noor [3] studied the sensitivity analysis of variational inclusions by using the technique of resolvent equations. Recently much attention has been given to developing iterative algorithms for solving the variational inclusions. Dong et al. [4] analyzed the solution’s sensitivity for variational inequalities and variational inclusions by using a resolvent operator technique. By using the concept and technique of resolvent operators, Agarwal et al. [5] and Jeong [6] introduced and studied a new system of parametric generalized nonlinear mixed quasi-variational inclusions in a Hilbert space. Lan [7] introduced and studied a stable iteration procedure for a class of generalized mixed quasi-variational inclusion systems in Hilbert spaces. Recently, Zhang et al. [8] introduced a new iterative scheme for finding a common element of the set of solutions to the problem (1.1) and the set of fixed points of nonexpansive mappings in Hilbert spaces. Peng et al. [9] introduced another iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping. For some related work, see [9–23] and the references therein.
Recently, Takahashi et al. [24] introduced the following iterative algorithm for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive mapping:
for all . Under some assumptions, they proved that the sequence converges strongly to a point of .
Remark 1.1 We note that the algorithm (1.2) cannot be used to find the minimum-norm element due to the facts that and S is a self-mapping of C. However, there exist a large number of problems for which one needs to find the minimum-norm solution (see, for example, [25–29]). A useful path to circumvent this problem is to use projection. Bauschke and Browein [30] and Censor and Zenios [31] provide reviews of the field. The main difficulty is in the computation. Hence it is an interesting problem to find the minimum-norm element without using the projection.
Motivated and inspired by the works in this field, we first suggest the following two algorithms without using projection:
for all and
for all . Notice that these two algorithms are indeed well defined (see the next section). We show that the suggested algorithms converge strongly to a point which solves the following variational inequality:
for all .
As special cases, we can approach the minimum-norm element in without using the metric projection and give some applications.
2 Preliminaries
Let H be a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H.
-
(1)
A mapping is said to be nonexpansive if
for all . We denote by the set of fixed points of S.
-
(2)
A mapping is said to be α-inverse strongly monotone if there exists such that
for all .
It is well known that, if A is α-inverse strongly monotone, then for all .
Let B be a mapping from H into . The effective domain of B is denoted by , that is, .
-
(3)
A multi-valued mapping B is said to be a monotone operator on H if
for all , , and .
-
(4)
A monotone operator B on H is said to be maximal if its graph is not strictly contained in the graph of any other monotone operator on H.
Let B be a maximal monotone operator on H and . For a maximal monotone operator B on H and , we may define a single-valued operator , which is called the resolvent of B for λ. It is well known that the resolvent is firmly nonexpansive, i.e.,
for all and for all . The following resolvent identity is well known: for any and , the following identity holds:
for all .
We use the notation that stands for the weak convergence of to x and stands for the strong convergence of to x, respectively.
We need the following lemmas for the next section.
Lemma 2.1 ([32])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an α-inverse strongly monotone mapping and be a constant. Then we have
for all . In particular, if , then is nonexpansive.
Lemma 2.2 ([33])
Let C be a closed convex subset of a Hilbert space H. Let be a nonexpansive mapping. Then is a closed convex subset of C and the mapping is demiclosed at 0, i.e. whenever is such that and , then .
Lemma 2.3 ([1])
Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that the mapping is monotone and weakly continuous along segments, that is, weakly as . Then the variational inequality
for all is equivalent to the dual variational inequality
for all .
Lemma 2.4 ([34])
Let , be bounded sequences in a Banach space X and be a sequence in with
Suppose that for all and
Then .
Lemma 2.5 ([35])
Assume that is a sequence of nonnegative real numbers such that
for all , where is a sequence in and is a sequence such that
-
(a)
;
-
(b)
or .
Then .
3 Main results
In this section, we prove our main results.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping from C into H. Let be a ρ-contraction and γ be a constant such that . Let B be a maximal monotone operator on H such that the domain of B is included in C. Let be the resolvent of B for any and S be a nonexpansive mapping from C into itself such that . Let λ and κ be two constants satisfying , where and . For any , let be a net generated by
Then the net converges strongly as to a point , which solves the following variational inequality:
for all .
Proof First, we show that the net is well defined. For any , we define a mapping . Note that , S, and (see Lemma 2.1) are nonexpansive. For any , we have
which implies the mapping W is a contraction on C. We use to denote the unique fixed point of W in C. Therefore, is well defined. Set and for all . Taking , it is obvious that for all and so
for all . From (3.1), it follows that
Hence we get . Since is nonexpansive, we have
Thus it follows that
Therefore, is bounded. We deduce immediately that , , , , and are also bounded. By using the convexity of and the α-inverse strong monotonicity of A, from (3.2), we derive
and so
By the assumption, we have for all and so we obtain
Next, we show . By using the firm nonexpansivity of , we have
Thus it follows that
By the nonexpansivity of , we have
and thus
Hence it follows that
Since , we deduce , which implies that
From (3.2), we have
Note that . Then we obtain
Thus it follows that
where M is some constant such that
Next, we show that is relatively norm-compact as . Assume that is such that as . Put . From (3.6), we have
Since is bounded, without loss of generality, we may assume that . Hence because of . From (3.5), we have
We can use Lemma 2.2 to (3.8) to deduce . Further, we show that is also in . Let . Note that for all . Then we have
Since B is monotone, we have, for all ,
Thus it follows that
Since , , and , it follows that . We also observe that and . Then, from (3.9), we can derive , that is, . Since B is maximal monotone, we have . This shows that . Hence we have . Therefore, substituting for z in (3.7), we get
Consequently, the weak convergence of to actually implies that . This proved the relative norm-compactness of the net as .
Now, we return to (3.7) and, taking the limit as , we have
for all . In particular, solves the following variational inequality:
for all or the equivalent dual variational inequality (see Lemma 2.3):
for all . Hence . Clearly, this is sufficient to conclude that the entire net converges to . This completes the proof. □
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping from C into H. Let be a ρ-contraction and γ be a constant such that . Let B be a maximal monotone operator on H such that the domain of B is included in C. Let be the resolvent of B for any and S be a nonexpansive mapping from C into itself such that . For any , let be a sequence generated by
for all , where , and satisfy the following conditions:
-
(a)
, and ;
-
(b)
, where and .
Then the sequence converges strongly to a point , which solves the following variational inequality:
for all .
Proof Set , for all . Pick up . It is obvious that
for all . Since , S, and are nonexpansive for all and , we have
Hence we have
By induction, we have
Therefore, is bounded. Since A is α-inverse strongly monotone, it is -Lipschitz continuous. We deduce immediately that , , , , and are also bounded. By using the convexity of and the α-inverse strong monotonicity of A, it follows from (3.11) that
By the condition (c), we get for all . Then, from (3.11) and (3.12), we obtain
From (3.10), it follows that
Next, we estimate . In fact, we have
and
By the resolvent identity (2.1), we have
Thus it follows that
and so
By the assumptions, we know that and . Then, from Lemma 2.5, we get
Thus, from (3.13) and (3.14), it follows that
and so
Since , , and , we have
Next, we show . By using the firm nonexpansivity of , we have
From the condition (c) and the α-inverse strongly monotonicity of A, we know that is nonexpansive. Hence it follows that
and thus
that is,
This together with (3.14) implies that
and hence
Since , , and (by (3.16)), we deduce
This implies that
Combining (3.10), (3.15), and (3.17), we get
Put , where is the net defined by (3.1).
Finally, we show that . Taking in (3.16), we get . First, we prove . We take a subsequence of such that
There exists a subsequence of which converges weakly to a point . Hence also converges weakly to w because of . By the demi-closedness principle of the nonexpansive mapping (see Lemma 2.2) and (3.18), we deduce . Furthermore, by a similar argument to that of Theorem 3.1, we can show that w is also in . Hence we have . This implies that
Note that . Then we have
for all . Therefore, it follows that
From (3.10), we have
It is clear that and
Therefore, we can apply Lemma 2.5 to conclude that . This completes the proof. □
Remark 3.3 One quite often seeks a particular solution of a given nonlinear problem, in particular, the minimum-norm element. For instance, given a closed convex subset C of a Hilbert space and a bounded linear operator , where is another Hilbert space. The C-constrained pseudoinverse of W, , is then defined as the minimum-norm solution of the constrained minimization problem
which is equivalent to the fixed point problem
where is the adjoint of W and is a constant, and is such that . From Theorems 3.1 and 3.2, we get the following corollaries which can find the minimum-norm element in ; that is, find such that
Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping from C into H. Let B be a maximal monotone operator on H such that the domain of B is included in C. Let be the resolvent of B for any and S be a nonexpansive mapping from C into itself such that . Let λ and κ be two constants satisfying , where and . For any , let be a net generated by
Then the net converges strongly as to a point which is the minimum-norm element in .
Corollary 3.5 Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping from C into H and let B be a maximal monotone operator on H such that the domain of B is included in C. Let be the resolvent of B for any such that . Let λ be a constant satisfying , where . For any , let be a net generated by
Then the net converges strongly as to a point , which is the minimum-norm element in .
Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping from C into H. Let B be a maximal monotone operator on H such that the domain of B is included in C. Let be the resolvent of B for any and let S be a nonexpansive mapping from C into itself such that . For any , let be a sequence generated by
for all , where , , and satisfy the following conditions:
-
(a)
, , and ;
-
(b)
, where and .
Then the sequence converges strongly to a point , which is the minimum-norm element in .
Corollary 3.7 Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping from C into H and let B be a maximal monotone operator on H such that the domain of B is included in C. Let be the resolvent of B for any such that . For any , let be a sequence generated by
for all , where , , and satisfy the following conditions:
-
(a)
, , and ;
-
(b)
, where and .
Then the sequence converges strongly to a point , which is the minimum-norm element in .
Remark 3.8 The present paper provides some methods which do not use projection for finding the minimum-norm solution problem.
4 Applications
Next, we consider the problem for finding the minimum-norm solution of a mathematical model related to equilibrium problems. Let C be a nonempty closed convex subset of a Hilbert space and be a bifunction satisfying the following conditions:
(E1) for all ;
(E2) G is monotone, i.e., for all ;
(E3) for all , ;
(E4) for all , is convex and lower semicontinuous.
Then the mathematical model related to the equilibrium problem (with respect to C) is as follows:
Find such that
for all . The set of such solutions is denoted by .
The following lemma appears implicitly in Blum and Oettli [36].
Lemma 4.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let G be a bifunction from into R satisfying the conditions (E1)-(E4). Then, for any and , there exists such that
for all .
The following lemma was given in Combettes and Hirstoaga [37].
Lemma 4.2 Assume that G is a bifunction from into R satisfying the conditions (E1)-(E4). For any and , define a mapping as follows:
for all . Then the following hold:
-
(a)
is single-valued;
-
(b)
is a firmly nonexpansive mapping, i.e., for all ,
-
(c)
;
-
(d)
is closed and convex.
We call such a the resolvent of G for any . Using Lemmas 4.1 and 4.2, we have the following lemma (see [38] for a more general result).
Lemma 4.3 Let C be a nonempty closed convex subset of a Hilbert space H. Let G be a bifunction from into R satisfying the conditions (E1)-(E4). Let be a multi-valued mapping from H into itself defined by
Then and is a maximal monotone operator with . Further, for any and , the resolvent of G coincides with the resolvent of , i.e.,
Form Lemma 4.3 and Theorems 3.1 and 3.2, we have the following results.
Theorem 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from into R satisfying the conditions (E1)-(E4) and be the resolvent of G for any . Let S be a nonexpansive mapping from C into itself such that . For any , let be a net generated by
Then the net converges strongly as to a point , which is the minimum-norm element in .
Proof From Lemma 4.3, we know is maximal monotone. Thus, in Theorem 3.1, we can set . At the same time, in Theorem 3.1, we can choose and , and (3.1) reduces to
Consequently, from Theorem 3.1, we get the desired result. This completes the proof. □
Corollary 4.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from into R satisfying the conditions (E1)-(E4) and be the resolvent of G for any . Suppose that . For any , let be a net generated by
Then the net converges strongly as to a point , which is the minimum-norm element in .
Theorem 4.6 Let C be a nonempty closed and convex subset of a real Hilbert space H. Let G be a bifunction from into R satisfying the conditions (E1)-(E4) and be the resolvent of G for any . Let S be a nonexpansive mapping from C into itself such that . For any , let be a sequence generated by
for all , where , , and satisfy the conditions:
-
(a)
, , and ;
-
(b)
, where and .
Then the sequence converges strongly to a point , which is the minimum-norm element in .
Proof From Lemma 4.3, we know is maximal monotone. Thus, in Theorem 3.2, we can set . At the same time, in Theorem 3.2, we can choose and , and (3.10) reduces to
for all . Consequently, from Theorem 3.2, we get the desired result. This completes the proof. □
Corollary 4.7 Let C be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from into R satisfying the conditions (E1)-(E4) and be the resolvent of G for any . Suppose . For any , let be a sequence generated by
for all , where , , and satisfy the following conditions:
-
(a)
, , and ;
-
(b)
, where and .
Then the sequence converges strongly to a point , which is the minimum-norm element in .
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant No. (31-130-35-HiCi) and the fifth author was supported in part by NNSF of China (61362033) and NGY2012097.
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Abdou, A.A., Alamri, B.A., Cho, Y.J. et al. Parallel algorithms for variational inclusions and fixed points with applications. Fixed Point Theory Appl 2014, 174 (2014). https://doi.org/10.1186/1687-1812-2014-174
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DOI: https://doi.org/10.1186/1687-1812-2014-174