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Solution sensitivity of generalized nonlinear parametric -proximal operator system of equations in Hilbert spaces
Journal of Inequalities and Applications volume 2014, Article number: 362 (2014)
Abstract
By using the new parametric resolvent operator technique associated with -monotone operators, the purpose of this paper is to analyze and establish an existence theorem for a new class of generalized nonlinear parametric -proximal operator system of equations with non-monotone multi-valued operators in Hilbert spaces. The results presented in this paper generalize the sensitivity analysis results of recent work on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions, and nonlinear mixed quasi-variational inclusion systems in Hilbert spaces.
MSC:49J40, 47H05, 90C33.
1 Introduction
Recently, since the study of the sensitivity (analysis) of solutions for variational inclusion (operator equation) problems involving strongly monotone and relaxed cocoercive mappings under suitable second order and regularity assumptions is an increasing interest, there are many motivated researchers basing their work on the generalized resolvent operator (equation) techniques, which is used to develop powerful and efficient numerical techniques for solving (mixed) variational inequalities, related optimization, control theory, operations research, transportation network modeling, and mathematical programming problems. It is well known that the project technique and the resolvent operator technique can be used to establish an equivalence between (mixed) variational inequalities, variational inclusions, and resolvent equations. See, for example, [1–35] and the references therein.
In this paper, we consider the following system of -proximal operator equations: For each fixed , find such that and
where Ω and Λ are two nonempty open subsets of real Hilbert spaces and , in which the parameter ω and λ take values, respectively, is a set-valued operator, , , , , , and are nonlinear single-valued operators, , , and are any nonlinear operators such that for all , is an -monotone operator with and for all , is an -monotone operator with , respectively, , I is the identity operator, , , , and for all , , and .
For appropriate and suitable choices of S, E, F, M, N, f, g, , , and for , one sees that problem (1.1) is a generalized version of some problems, which includes a number (systems) of (parametric) quasi-variational inclusions, (parametric) generalized quasi-variational inclusions, (parametric) quasi-variational inequalities, (parametric) implicit quasi-variational inequalities studied by many authors as special cases; see, [1, 2, 5, 6, 8, 10–13, 15–21, 25, 28, 32–35] and the references therein.
Example 1.1 If is a single-valued operator, then for each fixed , problem (1.1) reduces to the following problem of finding such that:
Example 1.2 If , , , , , and , then problem (1.2) reduces to finding such that
Problem (1.3) is equivalent to the following nonlinear equation:
which can be rewritten as the following generalized strongly monotone mixed quasi-variational inclusion:
and studied by Verma [34, 35] when M is A-monotone and -monotone with respect to first variable.
Example 1.3 ([36])
Let ℋ be a real Hilbert space and be an operator on ℋ such that M is monotone and . Then based on the Yosida approximation , for each given , there exists exactly one continuous function such that the following first-order evolution equation:
where the derivative exists in the sense of weak convergence, that is,
holds for all .
On the other hand, Lan [27] introduced a new concept of -monotone operators, which generalizes the -monotonicity and A-monotonicity in Hilbert spaces and other existing monotone operators as special cases, and studied some properties of -monotone operators and applied resolvent operators associated with -monotone operators to approximate the solutions of a new class of nonlinear -monotone operator inclusion problems with relaxed cocoercive operators in Hilbert spaces. Lan et al. [29] and Verma [34] introduced and studied a new class of parametric generalized relaxed cocoercive implicit quasi-variational inclusions with A-monotone operators, respectively. By using the parametric implicit resolvent operator technique for A-monotone, we analyzed solution sensitivity for this kind of generalized relaxed cocoercive inclusions in Hilbert spaces. In [31, 35], based on the -resolvent operator technique, Verma and Lan introduced and investigated a sensitivity analysis for a class of generalized strongly monotone variational inclusions in Hilbert spaces, respectively. Furthermore, using the concept and technique of resolvent operators, Agarwal et al. [2] and Jeong [19] introduced and studied a new system of parametric generalized nonlinear mixed quasi-variational inclusions in a Hilbert space and in () spaces, respectively.
In this paper, we shall generalize the resolvent equations by introducing -proximal operator equations in Hilbert spaces and establish a relationship between a class of parametric -monotone variational inclusion systems and a class of generalized nonlinear parametric -proximal operator system of equations. Further, we study sensitivity analysis of the solution set for the system (1.1) of -proximal operator equations with non-monotone set-valued operators in Hilbert spaces.
Our results improve and generalize the results on the sensitivity analysis for generalized nonlinear mixed quasi-variational inclusions [2, 9, 22, 29, 33–35] and others. For more details, we recommend [4, 7, 10, 13, 14, 16, 17, 23, 24, 26, 32].
2 Preliminaries
In the sequel, let Λ be a nonempty open subset of a real Hilbert space ℋ in which the parameter λ take values.
Definition 2.1 An operator is said to be
-
(i)
m-relaxed monotone in the first argument if there exists a positive constant m such that
for all ;
-
(ii)
s-cocoercive in the first argument if there exists a constant such that
for all ;
-
(iii)
γ-relaxed cocoercive with respect to A in the first argument if there exists a positive constant γ such that
for all ;
-
(iv)
-relaxed cocoercive with respect to A in the first argument if there exist positive constants ϵ and α such that
for all .
In a similar way, we can define (relaxed) cocoercivity of the operator in the second argument.
Definition 2.2 An operator is said to be μ-Lipschitz continuous in the first argument if there exists a constant such that
In a similar way, we can define Lipschitz continuity of the operator in the second and third argument.
Definition 2.3 Let be a multi-valued operator. Then F is said to be τ--Lipschitz continuous in the first argument if there exists a constant such that
where is the Hausdorff metric, i.e.,
In a similar way, we can define -Lipschitz continuity of the operator in the second argument.
Lemma 2.1 ([37])
Let be a complete metric space and be two set-valued contractive operators with same contractive constant , i.e.,
Then
where and are fixed point sets of and , respectively.
Definition 2.4 Let , be two single-valued operators. Then a multi-valued operator is called -monotone (so-called -monotonicity [27, 35], -maximal relaxed monotonicity [3]) if
-
(i)
M is m-relaxed η-monotone,
-
(ii)
for every .
Remark 2.1 For appropriate and suitable choices of , and ℋ, it is easy to see that Definition 2.4 includes a number of definitions of monotone operators and monotone mappings (see [14, 27, 29, 30]).
Proposition 2.1 ([27])
Let be a r-strongly η-monotone operator, be an -monotone operator. Then the operator is single-valued.
Definition 2.5 Let be a strictly η-monotone operator and be an -monotone operator. The resolvent operator is defined by
Proposition 2.2 ([27])
Let ℋ be a q-uniformly smooth Banach space and be τ-Lipschitz continuous, be a r-strongly η-monotone operator and be an -monotone operator. Then the resolvent operator is -Lipschitz continuous, i.e.,
where is a constant.
In connection with the -proximal operator equations system (1.1), we consider the following generalized parametric -monotone variational inclusion system:
For each fixed , find such that and
Remark 2.2 For appropriate and suitable choices of E, F, M, N, S, , , and for , it is easy to see that problem (2.1) includes a number (systems) of (parametric) quasi-variational inclusions, (parametric) generalized quasi-variational inclusions, (parametric) quasi-variational inequalities, (parametric) implicit quasi-variational inequalities studied by many authors as special cases; see, for example, [1–35] and the references therein.
Now, for each fixed , the solution set of problem (2.1) is denoted by
In this paper, our aim is to study the behavior of the solution set and the conditions on these operators S, E, F, M, N, , , , under which the function is continuous or Lipschitz continuous with respect to the parameter .
3 Sensitivity analysis results
In the sequel, we first transfer problem (2.1) into a problem of finding the parametric fixed point of the associated -resolvent operator.
Lemma 3.1 For each fixed , an element is a solution of problem (2.1) if and only if there are and such that
where and are the corresponding resolvent operator in first argument of an -monotone operator , -monotone operator , respectively, is an -strongly monotone operator for and .
Proof For each fixed , by the definition of the resolvent operators of and of , respectively, we know that there exist , , and such that (3.1) holds if and only if
i.e.,
It follows from the definition of that is a solution of problem (2.1) if and only if there exist , and such that equation (3.1) holds. □
Now, we show that problem (1.1) is equivalent to problem (2.1).
Lemma 3.2 The problem (1.1) has a solution with if and only if problem (2.1) has a solution with , where
and
Proof Let with be a solution of problem (2.1). Then, by Lemma 3.1, it is a solution of the following system of equations:
By using the fact , and (3.1), we have
and
which imply that
with and , i.e. with is a solution of problem (1.1).
Conversely, letting with is a solution of problem (1.1), then
It follows from (3.2) and (3.3) that
which imply that
and so
i.e., with is a solution of problem (2.1). □
Alternative proof Let
Then, by (3.2), we know
and
Since and , we have
the required problem (1.1). □
We now invoke Lemmas 3.1 and 3.2 to suggest the following sensitivity analysis results for the system of -proximal operator equations (1.1).
Theorem 3.1 Let be -strongly monotone and -Lipschitz continuous for all , be κ--Lipschitz continuous in the first variable, be -monotone with constant in the first variable, and be -monotone with constant in the first variable. Let be -Lipschitz continuous, be -Lipschitz continuous, be -relaxed cocoercive with respect to and -Lipschitz continuous in the first variable, be -relaxed cocoercive with respect to and -Lipschitz continuous in the second variable, and let E be -Lipschitz continuous in the second variable, and F be -Lipschitz continuous in the first variable. If
with for and there exist constants , such that
then, for each , the following results hold:
-
(1)
the solution set of problem (1.1) is nonempty;
-
(2)
is a closed subset in .
Proof In the sequel, from (3.1), we first define operators and as follows:
for all .
Now, define a norm on by
It is easy to see that is a Banach space (see [14]). By (3.7), for any given and , define an operator by
For any , since , , , , , E, F, , are continuous, we have . Now, for each fixed , we prove that is a multi-valued contractive operator.
In fact, for any and , there exists such that
Note that , it follows from Nadler’s result [38] that there exists such that
Setting
then we have . It follows from (3.4) and Proposition 2.2 that
By the assumptions of E, , we have
and
Combining (3.9)-(3.11), we have
where
Similarly, by the assumptions of S, , F, and (3.8), we obtain
where
It follows from (3.12) and (3.13) that
where
It follows from condition (3.6) that . Hence, from (3.14), we get
Since is arbitrary, we obtain
By the same argument, we can prove
It follows from the definition of the Hausdorff metric on that
for all , i.e., is a multi-valued contractive operator, which is uniform with respect to . By a fixed point theorem of Nadler [38], for each , has a fixed point , i.e., . By the definition of G, we know that there exists such that (3.1) holds. Thus, it follows from Lemma 3.1 that with is a solution of problem (2.1). Hence, it follows from Lemma 3.2 that with is a solution of problem (1.1). Therefore, for all .
Next, we prove the conclusion (2). For each , let and , , , as . Then we know that there exists and
and
By the proof of conclusion (1), we have
It follows that
Hence, we have and . Therefore, is a closed subset of . □
Theorem 3.2 Under the hypotheses of Theorem 3.1, further assume that
-
(i)
for any , is --Lipschitz continuous (or continuous);
-
(ii)
for any , , , , and both are Lipschitz continuous (or continuous) with Lipschitz constants , , , and , respectively.
Then the solution set of problem (1.1) is Lipschitz continuous (or continuous) from to .
Proof From the hypotheses and Theorem 3.1, for any , we know that and are nonempty closed subsets of . By the proof of Theorem 3.1, and are both multi-valued contractive operators with the same contraction constant and have fixed points and , respectively. It follows from Lemmas 2.1 and 3.2 that
Setting , there exists such that
Since , it follows from Nadler’s result [38] that there exists such that
Let
Then we have . It follows from the assumptions on , E, , and S that
where and are the constants of (3.12) and
Similarly, by the assumptions on g, , F, and S,
where and are the constants of (3.13) and
It follows from (3.17), (3.18) and (3.1) that
where σ is the constant of (3.13), which implies that
where
Hence, from (3.19), we obtain
By using a similar argument as above, we get
It follows that
for all . Thus, (3.15) implies
This proves that is Lipschitz continuous in . If each operator under conditions (i) and (ii) is assumed to be continuous in , then by a similar argument as above, we can show that is continuous in . □
Remark 3.1 In Theorems 3.1 and 3.2, if E, F are strongly monotone in the first and second variable, i.e., when () in Theorems 3.1 and 3.2, respectively, then we can obtain the corresponding results. Our results improve and generalize the well-known results in [2, 29, 33–35].
4 Application
In this section, we give an application.
Lemma 4.1 ([39])
Let be a proper convex lower semi-continuous function. Then is nonexpansive for any constant .
Theorem 4.1 Let be a real Hilbert space and be a proper convex lower semi-continuous function for . Suppose that is -strongly monotone and -Lipschitz continuous in the first variable, and is -Lipschitz continuous in the second variable, is -strongly monotone and -Lipschitz continuous in the second variable, and is -Lipschitz continuous in the first variable. If there exist positive constants ρ and ϱ such that
then, for each :
-
(1)
is the unique solution of the following nonlinear problem:
(4.1) -
(2)
Moreover, the solution of problem (4.1) is continuous (or Lipschitz continuous) from to , if in addition, for any , , , , and both are Lipschitz continuous (or continuous) with Lipschitz constants , , , and , respectively.
Proof Letting
for all and defining on by
then it is easy to see that is a Banach space (see [14]). Further, one can show that is a contractive operator and the rest of proof can be carried out by Theorems 3.1 and 3.2, and so it is omitted. □
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Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the republic of Korea (2013R1A1A2054617), and partially supported by the Open Foundation of Artificial Intelligence Key Laboratory of Sichuan Province (2012RYY04). The authors are grateful to the referee for the helpful suggests and comments.
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The main idea of this paper was proposed by JKK. JKK and HYL prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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Kim, J.K., Lan, Hy. & Cho, Y.J. Solution sensitivity of generalized nonlinear parametric -proximal operator system of equations in Hilbert spaces. J Inequal Appl 2014, 362 (2014). https://doi.org/10.1186/1029-242X-2014-362
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DOI: https://doi.org/10.1186/1029-242X-2014-362