SENSITIVITY ANALYSIS FOR A PARAMETRIC GENERALIZED MULTI-VALUED IMPLICIT QUASI-VARIATIONAL-LIKE INCLUSION

In this paper, using proximal-point mapping technique and the property of a fixed-point set of multivalued contractive mapping, we study the behavior and sensitivity analysis of a solution set for a parametric generalized multi-valued implicit quasi-variational-like inclusion in real Hilbert space. Further, under some suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameter. Our results can be viewed as a refinement and improvement of some known results in the literature.


Introduction
Variational inequality theory has become very effective and powerful tool for studying a wide range of problems arising in mechanics, optimization, operation research, equilibrium problems and boundary value problems etc.Variational inequalities have been generalized and extended in different directions using novel and innovative techniques.A useful and important generalization of variational inequality is called the variational inclusion.Hassouni and Moudafi [9], Agarwal et al. [2,3], Ding [5,6], Ding and Luo [7], Fang and Huang [8], Huang [10] and Noor [17,18] have used the resolvent operator technique to obtain some important extensions and generalizations in existence results for the various classes of variational inequalities (inclusions).
In recent years, much attention has been given to develop general techniques for the sensitivity analysis of solution set of the various classes of variational inequalities (inclusions).From the mathematical and engineering point of view, sensitivity properties of various classes of variational inequalities can provide new insight concerning the problem being studied and stimulate ideas for solving problems.The sensitivity analysis of solution set for variational inequalities have been studied extensively by many authors using quite different techniques.By using the projection technique, Dafermos [4], Ding and Luo [7], Mukherjee and Verma [15] and Yen [21] studied the sensitivity analysis of solution set for some classes of variational inequalities with single-valued mappings.By using the implicit function approach that makes use of so-called normal mappings, Robinson [20] studied the sensitivity analysis of solutions for variational inequalities in finite-dimensional spaces.By using the projection and resolvent operator techniques, Adly [1], Agarwal et al. [2,3], Ding [5,6], Lim [13], Liu et al. [14], Noor [17,18], Peng and Long [19] and Zeng et al. [22] studied the behavior and sensitivity analysis of solution set for the various classes of parametric generalized variational inclusions involving single and multi-valued mappings.
The technique based on proximal-point mapping is a generalization of projection technique and has been widely used to study the existence of solutions and to develop iterative schemes for the various classes of variational (-like) inclusions.Recently Fang and Huang [8], Huang [10], Kazmi and Alvi [11] and Kazmi and Khan [12] has introduced the notion of η-proximal point mapping, P-proximal point mapping and P-η-proximal point mapping and used these to study the behavior and sensitivity analysis of solution set for some classes of parametric generalized variational (-like) inclusions involving single and multi-valued mappings.
Motivated by recent work going in this direction, we define strongly P-η-proximal mapping for strongly maximal P-η-monotone mapping and discuss some of its properties.Further, we consider a parametric generalized multi-valued implicit quasi-variational-like inclusion problem (in short, PGMIQVLIP) in real Hilbert space.Further, using proximal-point mapping technique and the property of a fixed-point set of multi-valued contractive mapping, we study the behavior and sensitivity analysis of a solution set for the PGMIQVLIP.Furthermore, under some suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameter.The results presented in this paper generalize and improve the results given by many authors, see for example [5,6,8,[10][11][12][13][14]18,19,22].

Preliminaries
Let H be a real Hilbert space equipped with inner product •, • and norm • ; 2 H is the power First, we review and define the following known concepts: Definition 2.1 [11].Let η : H × H → H be a mapping.Then a mapping P : H → H is said to be: (ii) strictly η-monotone if P(x) − P(y), η(x, y) > 0, ∀ x, y ∈ H, and equality holds if and only if x = y; (iii) δ -strongly η-monotone if there exists a constant δ > 0 such that [11,12].A mapping η : H × H → H is said to be τ-Lipschitz continuous if there exists a constant τ > 0 such that η(x, y) ≤ τ x − y , ∀ x, y ∈ H. Definition 2.3 [11,12].Let η : H × H → H be a single-valued mapping.Then a multi-valued mapping M : H → 2 H is said to be: and equality holds if and only if x = y; (iii) γ-strongly η-monotone if there exists a constant γ > 0 such that (iv) maximal η-monotone if M is η-monotone and (I + ρM)(H) = H for any ρ > 0, where I stands for identity mapping.
The following theorems give some important properties of γ-strongly maximal P-η-monotone mappings.
By Theorem 2.1, we define strongly P-η-proximal mapping for a γ-strongly maximal η-monotone mapping M as follows: where ρ > 0 is a constant, η : H × H → H is a mapping and P : H → H is a strictly η-monotone mapping.
Lemma 2.2 [13].Let (X, d) be a complete metric space and let T 1 , T 2 : X → C(X) be θ -Hcontraction mappings, then where F(T 1 ) and F(T 2 ) are the sets of fixed points of T 1 and T 2 , respectively.

Formulation of problem
Let Λ and Ω be open subsets of a real Hilbert space H such that (Λ, d 1 ) and (Ω, d 2 ) are metric spaces, in which the parameters λ and µ takes values, respectively.

Sensitivity analysis of the solution set S(λ , µ)
First, we define the following concepts.
Definition 4.1 [11,12].A mapping g : H × Λ → H is said to be: (i) (L g , l g )-mixed Lipschitz continuous if there exist constants L g , l g > 0 such that (ii) s-strongly monotone if there exists a constant s > 0 such that Definition 4.2 [11,12].A multi-valued mapping A : Definition 4.3 [11,12].Let A, B,C : H × Ω → C(H) be multi-valued mappings.A single-valued mapping N : H × H × H × Ω → H is said to be: (i) α-strongly mixed monotone with respect to A, B and C if there exists a constant α > 0 such that Now, we transfer the PGMIQVLIP (3.1) into a parametric fixed-point problem.

Now, for any
set of H; CB(H) is the family of all nonempty closed and bounded subsets of H; C(H) is the family of all nonempty compact subsets of H; H (•, •) is the Hausdorff metric on C(H) defined by H (A, B) = max sup x∈A inf y∈B d(x, y), sup y∈B inf x∈A d(x, y) , A, B ∈ C(H).

Theorem 4 . 1 .
Let the multi-valued mappings A, B,C : H × Ω → C(H) and F : H × Λ → C(H) be H -Lipschitz continuous in the first arguments with constant L A , L B , L C and L F , respectively.Let the mappings η : H × H → H be τ-Lipschitz continuous and P : H → H be δ -strongly ηmonotone.Let the mappings g, m : H × Λ → H be such that (g − m) is s-strongly monotone and L (g−m) -Lipschitz continuous in the first argument and P • (g − m) be r-strongly monotone and L P•(g−m) -Lipschitz continuous in the first argument.Let the mapping N : H × H × H × Ω → H be α-strongly mixed monotone with respect to A, B and C and (L (N,1) , L (N,2) , L (N,3) , l N )-mixed