Research On Nonconvex And Nonsmooth Variational Inequalities And Their Applications

As a generalization of variational inequality problems,nonconvex and nonsmooth variational inequality problems are widely applied in Mechanics,Economics,Optimal Control and Engineering.Since nonconvex and nonsmooth variational inequality problems are characterized by different forms of nonconvex and nonsmooth energy functionals or nonconvex constraint sets,many methods and research results on classical variational inequalities are no longer applicable in the study of the nonconvex and nonsmooth variational inequality problems.In addition,the study of the nonconvex and nonsmooth variational inequalities began late,and many problems remain to be solved.Therefore,it is of great significance to study the nonconvex and nonsmooth variational inequality problems and their applications.The purpose of this dissertation is to study the solvability,well-posedness,convergence behavior of solutions and applications of the obtained related theoretical results in contact problems for the nonconvex and nonsmooth variational inequality problems.Chapter three of this dissertation proves a unique solvability result of a class of nonconvex and nonsmooth variational inequality problems,based on the theory of maximal monotone operators and the Banach fixed point theorem.First,the nonconvex and nonsmooth variational inequality problem considered is transformed into a fixed point problem governed by the resolvent operator of a maximal monotone operator.Then,the existence and uniqueness of solution to the nonconvex and nonsmooth variational inequality problem is obtained by proving the unique solvability of the fixed point problem.Finally,by using Picard,Krasnoselskij and Mann iterative algorithms for fixed point problems,corresponding numerical approximation methods for the nonconvex and nonsmooth variational inequality problems are obtained.Next,chapter four is committed to study gap functions for a kind of nonconvex and nonsmooth variational inequality problem.First,by constructing two kinds of gap functions with better properties,i.e.,the generalized difference gap function and the MoreauYosida regularized gap function,the nonconvex and nonsmooth variational inequality problem is transformed into an unconstrained minimization problem with zero as its optimal value.Then,some error bound results for the generalized difference gap function and the Moreau-Yosida regularized gap function are established,respectively.After that,by designing a descent iterative algorithm for the corresponding unconstrained optimization problem,a numerical algorithm for solving the nonconvex and nonsmooth variational inequality problem is obtained.Finally,the convergence of the algorithm is proved and a simple numerical example is presented.Then,fifth chapter of this dissertation studies well-posedness for one kind of nonconvex and nonsmooth variational inequality problem,including Tykhonov well-posedness,Levitin-Polyak well-posedness and well-posedness in the sense of Tykhonov triple.First,by introducing the concept of Tykhonov well-posedness,the equivalence of the Tykhonov well-posedness for the nonconvex and nonsmooth variational inequality problem and the corresponding fixed point problem is proved.Secondly,after giving the concept of LevitinPolyak well-posedness,the equivalent relationships among the Levitin-Polyak well-posedness for the nonconvex and nonsmooth variational inequality problem,its corresponding inclusion problem,and the existence and uniqueness of its solution are obtained.Finally,by defining well-posedness in the sense of Tykhonov triple,this dissertation unifies the concepts of the Tykhonov well-posedness and the Levitin-Polyak well-posedness for the nonconvex and nonsmooth variational inequality problem.Additionally,some metric characterizations and sufficient conditions of the well-posedness in the sense of Tykhonov triple for the nonconvex and nonsmooth variational inequality problem are established,and a continuous dependence result of its solution is provided.In the end,chapter six studies applications of the obtained related theoretical results in contact problems for nonconvex and nonsmooth variational inequality problems.First,a spring-rod system problem and a static friction contact problem with elastic body are introduced,and the corresponding nonconvex and nonsmooth variational inequality models are derived.Then,using the solvability and well-posedness results of the nonconvex and nonsmooth variational inequality problems,the solvability conditions and convergence of weak solutions for the two kinds of contact problems are obtained,and the corresponding mechanical interpretations are provided.