Research On Some Hyperbolic Variational-hemivariational Inequalities

The study of variational-hemivariational inequalities plays an important role in qualitative analysis of various problems in mechanics,physics and engineering sciences.In this paper,the existence of solutions of some hyperbolic variational-hemivariational inequalities is studied by combining the theories of partial differential equations,nonlinear analysis,setvalued analysis and Rothe's method.Chapter 1 is the introduction,which introduces the research background and significance of variational-hemivariational inequalities,analyzes the research status and development trends at home and abroad in recent years,and gives the main results of this paper.In chapter 2,we introduce the important definitions and conclusions in the function space,and give some inequalities and important definition lemmas.In chapter 3,we mainly study a class of hyperbolic variational inequalities with integral terms corresponding to history-dependent operators in contact mechanics.We combine monotone operator theory and Rothe's method to obtain the existence of solutions of hyperbolic variational inequalities.Firstly,the original hyperbolic variational inequality problem is transformed into elliptic discrete problem,and then the corresponding priori estimation and approximate solution sequence are obtained.Finally,it is proved that the convergent function is the solution to variational inequality.In chapter 4,we study a class of hyperbolic variational-hemivariational inequality problem.This inequality not only contains the time differential convex function and at the same time Clarke's subdifferential,and similar to the second chapter,does not contain on the first order derivative term of time,this leads to lack of a derivative of a prior estimate and making the proof of the convergence calculation more difficult.The existence of solutions for hyperbolic inequalities is also proved by Rothe's method.In chapter 5,we study a class of hyperbolic quasi-variational hemivariational inequalities,in which Lipschitz continuous functions are binary functions and depend on the solution u(t).By using Rothe's method and Banach's fixed point theorem,we obtain the existence of solutions of hyperbolic quasi-variational hemivariational inequalities.In Chapter 6,compared with the previous chapters,we no longer study an inequality,but a class of coupled systems composed of hyperbolic variational-hemivariational inequalities and differential equations.We first prove the uniqueness of the solution of the differential equation,and then prove the existence of the solution of the coupled system according to Rothe's method.Chapter 7 summarizes the work of this paper and gives the future research idea.