Levitin-Polyak Well-posedness Of Generalized Variational-hemivariational Inequalities

Variational inequality is an important nonlinear problem.As an important generalization of variational inequality,hemivariational inequality is a class of inequality problems with nonconvex and nonsmooth functions.Hemivariational inequality has been widely studied and applied to many mechanical and engineering problems,such as nonlinear elastic friction contact problems and nonconvex semipermeable problems etc.In the present thesis,we study the Levitin-Polyak well-posedness of a generalized variational-hemivariational inequality.The objectives of this thesis are threefold: First of all,we study metric characterizations for the Levitin-Polyak well-posedness of the generalized variational-hemivariational inequality;Then,we give some conditions under which the generalized variational-hemivariational inequality is Levitin-Polyak well-posed;Last,we establish the relationships among the Levitin-Polyak well-posedness of the generalized variational-hemivariational inequality,the corresponding inclusion problem and the corresponding constrained minimization problem.The details of the study are as follows:First,we introduce some kinds of concepts of Levitin-Polyak well-posedness for the generalized variational-hemivariational inequality,and give some metric characterizations for the strong Levitin-Polyak well-posedness of the generalized variational-hemivariational inequality by defining an approximate solution set.Second,based on some weaker conditions,we establish the equivalence between the Levitin-Polyak well-posedness of the generalized variational-hemivariational inequality and its unique solvability,and further give some sufficient conditions under which the generalized variational-hemivariational inequality is strongly Levitin-Polyak well-posed in the generalized sense.Finally,we study the equivalence of the Levitin-Polyak well-posedness between the generalized variationalhemivariational inequality and its corresponding inclusion problem.Meanwhile,by defining a gap function,we establish the links of Levitin-Polyak well-posedness between the generalized variational-hemivariational inequality and its corresponding constrained minimization problem constructed by the gap function.