On The Stability Of The Vector Variational Inequality And Its Dual Problem

In this thesis, we study three problems: the stability of the solution set map of the dual weak vector variational inequality problem, the semicontinuity for a parametric generalized vector quasivariational inequality and the Painleve-Kuratowski convergences of the sequence of solution sets for the perturbed set-valued weak vector variational inequality problem. The detailed contents are listed below:Firstly, we study the dual problem of the weak vector variational inequality problem in Banach spaces. We obtain the upper semicontinuity and closedness of the solution set map. By virtue of the nonlinear scalarization function, we introduce the gap function for the dual problem. Furthermore, we prove the lower semicontinuity of the solution set map by the gap function and a constraint qualification. We give some examples to explain the necessity of our research on duality.Secondly, in locally convex Hausdorff topological vector spaces, we study the parametric generalized vector quasivariational inequality, which is the generalization of the parametric set-valued weak vector variational inequalities and the perturbed weak vector variational inequalitiy problems. We study the Hausdorff upper semicontinuity of the solution set map. Then, by introducing a parametric gap function and a constraint qualification, we obtain the Hausdorff lower semicontinuity of the solution set map. Since the constraint set varying with the decision variable, we notice that the condition of the theorem on the H-lower semicontinuity of the solution set map is not the sufficient condition for the H-continuuity, which is different from the Hausdorff continuity of the parametric set-valued weak vector variational inequality problem. We also give some examples to explain this case.Thirdly, we discuss the perturbed set-valued weak vector variational inequality with the sequence of mappings satisfying Painleve-Kuratowski converging. Since the solutions is a sequence of sets, we study the Painleve-Kuratowski convergence of the sequence of solution sets. We obtain the closedness and the Painleve-Kuratowski upper convergence of the sequence of solution sets. Then, we construct a sequence of functions as the gap functions for the perturbed set-valued weak vector variational inequality problem. By virtue of the lower semicontinuity of the gap function and constraint qualification, we establish the Painleve-Kuratowski lower convergence of the sequence of solution sets. We also give some examples to illustrate our results.