Solution sensitivity of generalized nonlinear parametric -proximal operator system of equations in Hilbert spaces
© Kim et al.; licensee Springer. 2014
Received: 25 March 2014
Accepted: 18 August 2014
Published: 24 September 2014
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.
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.
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 1.3 ()
holds for all .
On the other hand, Lan  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.  and Verma  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.  and Jeong  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].
In the sequel, let Λ be a nonempty open subset of a real Hilbert space ℋ in which the parameter λ take values.
- (i)m-relaxed monotone in the first argument if there exists a positive constant m such that
- (ii)s-cocoercive in the first argument if there exists a constant such that
- (iii)γ-relaxed cocoercive with respect to A in the first argument if there exists a positive constant γ such that
- (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.
In a similar way, we can define Lipschitz continuity of the operator in the second and third argument.
In a similar way, we can define -Lipschitz continuity of the operator in the second argument.
Lemma 2.1 ()
where and are fixed point sets of and , respectively.
M is m-relaxed η-monotone,
for every .
Proposition 2.1 ()
Let be a r-strongly η-monotone operator, be an -monotone operator. Then the operator is single-valued.
Proposition 2.2 ()
where is a constant.
In connection with the -proximal operator equations system (1.1), we consider the following generalized parametric -monotone variational inclusion system:
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.
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.
where and are the corresponding resolvent operator in first argument of an -monotone operator , -monotone operator , respectively, is an -strongly monotone operator for and .
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).
with and , i.e. with is a solution of problem (1.1).
i.e., with is a solution of problem (2.1). □
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).
the solution set of problem (1.1) is nonempty;
is a closed subset in .
for all .
For any , since , , , , , E, F, , are continuous, we have . Now, for each fixed , we prove that is a multi-valued contractive operator.
for all , i.e., is a multi-valued contractive operator, which is uniform with respect to . By a fixed point theorem of Nadler , 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 .
Hence, we have and . Therefore, is a closed subset of . □
for any , is --Lipschitz continuous (or continuous);
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 .
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].
In this section, we give an application.
Lemma 4.1 ()
Let be a proper convex lower semi-continuous function. Then is nonexpansive for any constant .
- (1)is the unique solution of the following nonlinear problem:(4.1)
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.
then it is easy to see that is a Banach space (see ). 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. □
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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