Tikhonov Regularization For A Class Of Hemivariational Inequality

In many complicated physical processes and engineering applications,mathematical models of the problems are formulated as two kinds of inequalities in addition to the more commonly seen equations.One is the variational inequality,and the other is the hemivariational inequality.As an important generalization of variational inequalities,hemivariational inequalities are characterized by the fact that the involved functions may be nonconvex and nonsmooth.The theory of hemivariational inequalities is of great importance in solving practical problems involving nonconvex and nonsmooth energy functions,and is an effective mathematical tool in the study of various complex phenomena arising in physics and engineering sciences.Therefore,in recent years,many scholars at home and abroad have devoted themselves to the study of hemivariational inequalities and have obtained many important academic achievements.This thesis considers a class of generalized hemivariational inequality problem involving a set-valued mapping and a nonlinear perturbation.The aim of this thesis consists of two folds: One is to study the solvability and the properties of the solution set for the generalized hemivariational inequality by introducing some new coercivity conditions and hemivariational inequality property? and the other is to study the convergence of the regularized problem for the generalized hemivariational inequality problem by using the Tikhonov regularization.In the thesis,Tikhonov regularization method and Ky Fan’s lemma are used to carry out the study of the generalized hemivariational inequality problem.The main research work is as follows.Firstly,in this thesis,we obtain the solvability of the generalized hemivariational inequality problem by using a mild coercivity condition and hemivariational inequality property,based on which the boundedness,weak compactness and compactness of the solution set of the generalized hemivariational inequality problem are proved under different coercivity conditions.Then,by means of the normalized duality mapping,this thesis establishes the regularized problem for the generalized hemivariational inequality problem by using the Tikhonov regularization,obtains its solvability under some mild coercivity conditions and then constructs an approximating sequence for the solution of the generalized hemivariational inequality problem.Finally,by proving the boundedness of the approximating sequence,we obtain in this thesis the weak convergence of the approxi-mating sequence,i.e.the approximation sequence converges weakly to the solution of the generalized hemivariational inequality problem.In addition,in order to clarify the significance of the research in this thesis,some special cases of the generalized hemivari-ational inequality problem are considered under different conditions,and the degenerated versions of the results obtained are also stated accordingly.