On the lower semicontinuity of the solution map- ping for parametric vector mixed quasivariational inequality problem of the Minty type

In this paper, we study a class of parametric vector mixed quasivariational inequality problem of the Minty type (in short, (MQVIP)). Afterward, we establish some sufficient conditions for the stability properties such as the inner-openness, lower semicontinuity and Hausdorff lower semicontinuity of the solution mapping for this problem. The results presented in this paper is new and wide to the corresponding results in the literature

On the upper semicontinuity of the solution mapping for parametric vector mixed quasivariational inequality

In this paper, we first study a class of parametric generalized vector mixed quasivariational inequality problem of the Minty type in locally convex Hausdorff topological vector spaces, this problem contains many problems as special cases, such as optimization problems, traffic network problems, Nash equilibrium problems, fixed point problems, variational inequality problems and complementarity problems, economic equibrium problems. Then, we establishe the conditions sufficient for stability properties such as: the upper semicontinuity, closedness, outer-continuity, outer-openness of the solution mapping for parametric generalized vector mixed quasivariational inequality problem of the Minty type. The results of the upper semi-continuity and the closeness of the solution mapping for parametric generalized vector mixed quasivariational inequality problem of the Minty type are improve and extend some of the results given by Lalitha and Bhatia. An example is given to demonstrate our results.The results of the outer continuity and the outer-openness of the solution mapping for the parametric generalized vector mixed quasivariational inequality problem of the Minty type are new. We also give some examples to show the relationship between upper semi-continuity, closedness outer continuity and outer-openness.

A class of parabolic evolutionary quasivariational inequalities in contact mechanics

In this paper, we obtain an existence and uniqueness of the solution for a class of parabolic evolutionary quasivariational inequalities in contact mechanics under some mild conditions. We also study an error estimate for the parabolic evolutionary quasivariational inequality by employing the forward Euler difference scheme and the element-free Galerkin spatial approximation.

Differential quasivariational inequalities in contact mechanics

We consider a new class of differential quasivariational inequalities, i.e. a nonlinear system that couples a differential equation with a time-dependent quasivariational inequality, both defined on abstract Banach spaces. We state and prove a general fixed principle that provides the existence and the uniqueness of the solution of the system. Then we consider a relevant particular setting for which our abstract result holds. We proceed with two examples that arise in Contact Mechanics. For each example, we describe the physical setting, the mathematical model and the assumption on the data. Then we state the variational formulation of each model, which is in the form of a differential quasivariational inequality. Finally, we apply our abstract results to provide the unique weak solvability of the corresponding contact problems.

A Stackelberg quasi-equilibrium problem via quasi-variational inequalities

In this paper, we consider a class of Stackelberg quasi-equilibrium problem with two players in finite dimensional spaces. Existence and location of the Stackelberg quasi-equilibrium is discussed by employing the quasivariational inequality techniques and the fixed point arguments. The results presented in this paper generalize some corresponding ones due to Nagy [Nagy, S., Stackelberg equilibia via variational inequalities and projections, J. Global Optim., 57 (2013), 821–828].

The GAP function of a parametric mixed strong vector quasivariational inequality problem

The parametric mixed strong vector quasivariational inequality problem contains many problems such as, variational inequality problems, fixed point problems, coincidence point problems, complementary problems etc. There are many authors who have been studied the gap functions for vector variational inequality problem. This problem plays an important role in many fields of applied mathematics, especially theory of optimization. In this paper, we study a parametric gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem (in short, (SQVIP)) in Hausdorff topological vector spaces. (SQVIP) Find x ̅ ∈ K(x ̅ ,γ) and z ̅ ∈ T(x ̅ ,γ) such that < z ̅ , y-x ̅ >+ f(y, x ̅ ,γ) ∈ Rn+ ∀ y ∈ K(x ̅ ,γ), where we denote the nonnegative of Rn by Rn+= {t=(t1 ,t2,…,tn )T ∈ Rn |ti >0, i = 1,2, ...,n}. Moreover, we also discuss the lower semicontinuity, upper semicontinuity and the continuity for the parametric gap function for this problem. To the best of our knowledge, until now there have not been any paper devoted to the lower semicontinuity, continuity of the gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem in Hausdorff topological vector spaces. Hence the results presented in this paper (Theorem 1.3 and Theorem 1.4) are new and different in comparison with some main results in the literature.