Research On The Solvability Of A System Of Hemivariational Inequalities

The theory of variational inequalities is an important method to study nonlinear boundary value problems.By the development in the past decades,variational inequality has been applied in many fields such as contact mechanics,management science and economic equilibrium.Thanks to the new demands proposed by the practical models,the mathematical literature about the theory becomes more and more extensive and impressive,and people proposed the so-called hemivariational inequality in this progress.As a special case of inclusion problem,the notion of hemivariational inequality is also to be a development of variational inequality by combining with the notion of subdifferential of Clarke.Generally speaking,nonconvex and nonsmooth energy functions can lead to hemivaritional inequalities,while a convex energy function leads to a variational inequality.Both variational inequalities and hemivariational inequalities are inequalities of variational form.One can easily obtain the relation of equivalence between a variational inequality and the corresponding variational equation.This work aims at investigating the solvability of a system of hemivariational inequalities,which consists of two coupled evolution hemivarational inequalities,related to a class of dynamic problems in contact mechanic analysis.In this thesis,we study the existence and uniqueness of the solution to the system within the frame of theory of nonlinear analysis and nonsmooth analysis.The thesis consists of four chapters.In chapter one,we present the background and study status of hemivariationalites.After then,some useful results and tools are introduced in chapter two,which are derived from functional analysis,set-valued analysis,nonsmooth analysis and theory of nonlinear operators.In chapter three,we apply a surjective theorem for L-pseudomonotone operators,in which operator L is a linear densely defined maximal monotone operator,to study the existence of the system when the initial conditions are regular.Then we give a prior estimate of the solutions and show existence in the general case by convergence method.With extra conditions including relaxed monotonicity,we prove the uniqueness of the solution at last.