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New general systems of set-valued variational inclusions involving relative -maximal monotone operators in Hilbert spaces
Journal of Inequalities and Applications volume 2014, Article number: 407 (2014)
Abstract
The purpose of this paper is to introduce and study a class of new general systems of set-valued variational inclusions involving relative -maximal monotone operators in Hilbert spaces. By using the generalized resolvent operator technique associated with relative -maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms. The results presented in this paper improve and extend some known results in the literature.
1 Introduction
Recently, some systems of variational inequalities, variational inclusions, complementarity problems, and equilibrium problems have been studied by many authors because of their close relations to some problems arising in economics, mechanics, engineering science and other pure and applied sciences. Among these methods, the resolvent operator technique is very important. Huang and Fang [1] introduced a system of order complementarity problems and established some existence results for the system using fixed point theory. Verma [2] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of the systems of variational inequalities. Cho et al. [3] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. Further, the authors proved some existence and uniqueness theorems of solutions for the systems, and also constructed some iterative algorithms for approximating the solution of the systems of nonlinear variational inequalities, respectively.
Moreover, Fang et al. [4], Yan et al. [5], Fang and Huang [6] introduced and studied some new systems of variational inclusions involving H-monotone operators and -monotone operators in Hilbert space, respectively. Using the corresponding resolvent operator technique associated with H-monotone operators, -monotone operators, the authors proved the existence of solutions for the variational inclusion systems and constructed some algorithms for approximating the solutions of the systems and discussed convergence of the iteration sequences generated by the algorithms, respectively. Very recently, Lan et al. [7] introduced and studied a new system of nonlinear A-monotone multivalued variational inclusions in Hilbert spaces. By using the concept and properties of A-monotone operators, and the resolvent operator technique associated with A-monotone operators due to Verma [8], the authors constructed a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions with A-monotone operators in Hilbert spaces and proved the existence of solutions for the nonlinear multivalued variational inclusion systems and the convergence of iterative sequences generated by the algorithm. For some related work, see, for example, [1–32] and the references therein.
On the other hand, Cao [33] introduced and studied a new system of generalized quasi-variational-like-inclusions applying the η-proximal mapping technique. Further, Agarwal and Verma [34] introduced and studied relative -maximal monotone operators and discussed the approximation solvability of a new system of nonlinear (set-valued) variational inclusions involving -maximal relaxed monotone and relative -maximal monotone operators in Hilbert spaces based on a generalized hybrid iterative algorithm and the general -resolvent operator method.
Inspired and motivated by the above works, the purpose of this paper is to consider the following new general system of set-valued variational inclusions involving relative -maximal monotone operators in Hilbert spaces: Find and for any such that
where m is a given positive integer, , , and are single-valued operators, is a set-valued operator and is relative -maximal monotone.
We note that for appropriate and suitable choices of positive integer m, the operators , , , , , , and for , one can know that the problem (1.1) includes a number of known general problems of variational character, including variational inequality (system) problems, variational inclusion (system) problems as special cases. For more details, see [1–31, 35] and the following examples.
Example 1.1 For , if is single-valued operator, the problem (1.1) reduces to finding , such that
Example 1.2 For , if and , an identity operator, and , where is proper and lower semi-continuous -subdifferentiable functional and denotes -subdifferential operator, then the problem (1.1) reduces to finding and for such that
The problem (1.3) is called a set-valued nonlinear generalized quasi-variational-like-inclusion system, which was considered and studied by Cao [33].
Example 1.3 When and for , then the problem (1.1) is equivalent to the following nonlinear set-valued variational inclusion system problem: Find and , such that
which was studied by Agarwal and Verma [34].
Example 1.4 If and , where is proper, convex, and lower semi-continuous functional and denotes the subdifferential operator of for all , , then the problem (1.4) reduces to the following system of set-valued mixed variational inequalities: Find , and such that
If , then the problem (1.5) reduces to finding such that
which is called the system of nonlinear variational inequalities considered by Cho et al. [3]. Some specializations of the problem (1.6) are dealt by Kim and Kim [35].
Example 1.5 If and , then the problem (1.1) reduces to the problem of finding such that
which was introduced and studied by Fang et al. [4].
Moreover, by using the generalized resolvent operator technique associated with relative -maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove convergence of the sequences generated by the algorithms.
2 Preliminaries
Throughout, let H and () be real Hilbert spaces and endowed with the norm and inner product . Let and denote the family of all the nonempty subsets of H and the family of all closed subsets of H, respectively.
Definition 2.1 Let be a single-valued operator. Then the map T is said to be
-
(i)
r-strongly monotone, if there exists a constant such that
-
(ii)
β-Lipschitz continuous, if there exists a constant such that
Definition 2.2 Let and be single-valued operators, be set-valued operator. Then
-
(i)
η is said to be t-strongly monotone, if there exists a constant such that
-
(ii)
η is said to be τ-Lipschitz continuous, if there exists a constant such that
-
(iii)
A is said to be η-monotone, if
-
(iv)
A is said to be strictly η-monotone, if A is η-monotone and
-
(v)
A is said to be -strongly monotone, if there exists a constant such that
-
(vi)
M is said to be η-monotone with respect to A (or relative -monotone) if
-
(vii)
M is said to be relative -maximal monotone, if M is η-monotone with respect to A (or relative -monotone) and , where is an arbitrary constant.
Definition 2.3 For , let be a Hilbert space, be single-valued operator, be set-valued operator. Then nonlinear operator is said to be
-
(i)
-relaxed cocoercive with respect to (or relative -relaxed cocoercive) in the j th argument, if there exist constants such that for all , and for any , ,
-
(ii)
-Lipschitz continuous in the j th argument, if there exists constant such that for all ,
Remark 2.1
-
(i)
When and , then (i) and (ii) of Definition 2.3 reduce to corresponding concept of the relative relaxed cocoerciveness and Lipschitz continuity, respectively.
-
(ii)
If is single-valued operator for , then is -relaxed cocoercive with respect to in the j th argument reduce to -relaxed cocoercive with respect to in the j th argument, that is, if there exist constants such that for all ,
Lemma 2.1 ([34])
Let be a single-valued mapping, be a strictly η-monotone mapping and be a relative -maximal monotone mapping. Then the mapping is single-valued, where is arbitrary constant.
Definition 2.4 Let be a single-valued mapping, be a strictly η-monotone mapping and be a relative -maximal monotone mapping. Then generalized resolvent operator is defined by
where is a constant.
Lemma 2.2 ([34])
Let be a t-strongly monotone and τ-Lipschitz continuous mapping, be an r-strongly monotone mapping, and be a relative -maximal monotone mapping. Then generalized resolvent operator is -Lipschitz continuous, that is,
Definition 2.5 A set-valued operator is said to be D-γ-Lipschitz continuous, if there exists a constant such that
where is called the Hausdorff pseudo-metric defined as follows:
Furthermore, the Hausdorff pseudo-metric D reduces to the Hausdorff metric when is restricted to closed bounded subsets of the family .
Lemma 2.3 Let be a constant. Then function for is nonnegative and strictly decrease and . Further, if , then .
Lemma 2.4 ([36])
Let and be two nonnegative real sequences satisfying
with and . Then .
3 Iterative algorithm and convergence analysis
In this section, we construct a class of new iterative algorithms for finding approximate solutions of the problems (1.1) and (1.2), respectively. Then the convergence criterion for the algorithms is also discussed.
Lemma 3.1 Let and for , then (denoted by ) is a solution of the problem (1.1) if and only if satisfy
where and is a constant for .
Proof It follows from the definition of generalized resolvent operator that the proof can be obtained directly, and so it is omitted. □
Algorithm 3.1
Step 1. Setting and choose for .
Step 2. Let
for all and , where is a constant.
Step 3. By the results of Nadler [37], we can choose such that
where is the Hausdorff pseudo-metric on and .
Step 4. If and for satisfy (3.2) to sufficient accuracy, stop. Otherwise, set and return to Step 2.
Remark 3.1 If reduces to , where is proper and lower semi-continuous -subdifferentiable functional, for and , then Algorithm 3.1 reduces to Algorithm (I) of Cao [33].
When and is single-valued operator for , then Algorithm 3.1 reduces to the following algorithm for the problem (1.2).
Algorithm 3.2 For any given , we compute as follows:
for and , where is error to take into account a possible inexact computation of the resolvent operator point satisfying conditions .
Remark 3.2
-
(i)
Let , , for , then Algorithm 3.1 reduces to Algorithm 4.3 of Agarwal and Verma [34].
-
(ii)
If for appropriate and suitable choices of positive integer m and mappings , , , , M, , and for , one can know that Algorithms 3.1-3.2 are extending a number of known algorithms.
In the sequel, we provide main result concerning the problem (1.1) with respect to Algorithm 3.1.
Theorem 3.1 For , let be -Lipschitz continuous and -strongly monotone operator, be -Lipschitz continuous and -strongly monotone operator, be -Lipschitz continuous and -strongly monotone operator and be relative -maximal monotone. Suppose that is --Lipschitz continuous, is -relaxed cocoercive with respect to in the ith argument and -Lipschitz continuous in the jth for . If there exists constant for such that
for all , then the problem (1.1) admits a solution , i.e. , where and for . Moreover, iterative sequences and generated by Algorithm 3.1 strongly converge to and for , respectively.
Proof For , applying Algorithm 3.1 and Lemma 2.2, we have
By -Lipschitz continuity and -strongly monotonicity of , we get
Since is -Lipschitz continuous, is -relaxed cocoercive with respect to in the i th argument and is -Lipschitz continuous in the j th argument, then we have
By --Lipschitz continuity of the and (3.3), we get
and
Combining (3.8) and (3.10), we have
It follows from (3.6)-(3.9), and (3.11), that
which implies that
where
and
By condition (3.5), we know that sequence is monotone decreasing and as . Thus,
Since for , we get , by Lemma 2.3, we have . From (3.12), it follows that is a Cauchy sequence and there exists such that as for .
Next, we show that as for .
It follows from (3.9) and (3.10) that are also Cauchy sequences. Hence, there exists such that as for . Furthermore,
Since is closed for , we have for . Using continuity, and for satisfy (3.1) and so in light of Lemma 3.1, is a solution to the problem (1.1). This completes the proof. □
Remark 3.3 If the generalized resolvent operator reduces to , where is proper and lower semi-continuous -subdifferentiable functional, for , and -relaxed cocoerciveness with respect to in the i th argument of reduces to --strongly monotonicity (right now, , ), then Theorem 3.1 reduces to Theorem 3.1 of Cao [33].
Theorem 3.2 Assume that , , , are the same as in the Theorem 3.1 for . Suppose that is -Lipschitz continuous, is -relaxed cocoercive with respect to in the ith argument and -Lipschitz continuous in the jth for . If there exists constant for such that
for , then the problem (1.2) has a unique solution . Moreover, the iterative sequences generated by Algorithm 3.2 strongly converge to for .
Proof Define the norm on product space by
It is easy to see that is a Banach space. Set
Let be defined by
For any , it follows from Lemma 2.2 that
By -Lipschitz continuity and -strongly monotonicity of , we get
Since is -Lipschitz continuous, is -relaxed cocoercive with respect to in the i th argument and is -Lipschitz continuous in the j th argument and is -Lipschitz continuous, then we have
and
From (3.13)-(3.16), we have
where . It follows from assumption (3.5) that . This shows that is a contractive operator, and so there exists a unique such that . Thus, is the unique solution of the problem (1.2).
Now we prove that as for . In fact, it follows from (3.4) and Lemma 2.2 that
Following very similar arguments from (3.14)-(3.16), we have
which implies that
where , . The condition of Algorithm 3.2 yields . Now Lemma 2.4 implies that , and so as for . This completes the proof. □
Remark 3.4 If , (right now, for ), then Theorem 3.1 reduces to Theorem 4.5 based on Algorithm 4.3 of Agarwal and Verma [34]. Our presented results improve and extend some known results in the literature.
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Acknowledgements
This work was supported by the Cultivation Project of Sichuan University of Science and Engineering (2011PY01) and the Open Research Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2013WZJ01).
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TX carried out the proof of the corollaries and gave some examples to show the main results. HL conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Xiong, Tj., Lan, Hy. New general systems of set-valued variational inclusions involving relative -maximal monotone operators in Hilbert spaces. J Inequal Appl 2014, 407 (2014). https://doi.org/10.1186/1029-242X-2014-407
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DOI: https://doi.org/10.1186/1029-242X-2014-407