Research On The Existence Of Solutions To Some Nonlinear Differential Variational Inequalities Problems

The research of differential variational inequality problems,which has significant values of both theory and application,provides a unified framework to study the problems of differen-tial equations with parameters and dynamic variational inequalities.The differential variational inequality provides a powerful modeling paradigm for many applied problems,such as,electri-cal circuits with ideal diodes,differential Nash games,Coulomb friction for contacting bodies,dynamic traffic networks and hybrid engineering systems with variable structures.The research of nonlinear differential variational inequalities is a brand-new field with intercross and pen-etration of control theory and branch of mathematics such as nonlinear functional analysis,differential equations and variational inequalities.Also,there are immense potentialities and good perspectives to investigate the nonlinear differential variational inequalities.As a con-sequence,we study several kinds of nonlinear differential variational inequality problems in infinite dimensional spaces.(1)We study a class of anti-periodic problems for nonlinear second order differential varia-tional inequalities in abstract spaces.Firstly,we provide and prove the properties of solution set for variational inequality.Then,without the Lipschitz continuity of the nonlinear term,we first prove the existence of solutions for the differential variational inequality anti-periodic problems mainly by means of the Scorza-Dragoni property,topological degree theory and the Filippov implicit function lemma.(2)We consider a class of differential variational inequality problems with nonlocal bound-ary conditions,which consist of nonlinear evolution equations and generalized mixed variation-al inequalities.Firstly,we show and prove the properties of the solution set for generalized mixed variational inequalities.Then,the existence results for the differential variational in-equality problem are established and proved mainly by using a new method of combining the Yosida approximations of the infinitesimal generator of C0-semigroup and topological degree theory.Finally,we also obtain the weak compactness of mild solution set of the differential variational inequality problem.(3)We study a class of differential variational inequality problems with nonlocal boundary conditions,which are obtained by mixing nonlinear evolution inclusion equations with time depending operator and generalized variational inequalities.Firstly,we show the properties of the solution set for generalized variational inequalities.Then,without the compactness on the constraint set,on the evolution operator and on the nonlinear term,we establish and prove the existence results for the differential variational inequality problem mainly by applying the Kakutani-Fan-Glicksberg fixed point theorem.(4)We consider a class of fractional differential variational inequality problems with non-local boundary conditions,which are composed of nonlinear fractional evolution equations with delay and variational inequalities of elliptic type.Firstly,combining with the properties of the solution set for elliptic variational inequalities,we prove the existence results for the differential variational inequality problem by mainly applying the measure of noncompactness technique and the fixed point theorem with respect to k-set-contractive.Then,the uniqueness of mild solution is proved by mean of the Banach contraction mapping principle.(5)We study a class of nonlinear fractional evolution hemivariational inequalities control problem with with random disturbing.Firstly,a new set of sufficient conditions is established for the existence of mild solutions for the system.Then,the existence results are proved by using stochastic analysis techniques,fractional calculation,semigroup of operators theory,a fixed point theorem of multivalued maps and properties of generalized Clarke subdifferential.Next,the controllability of the control system is formulated and proved mainly by applying the fixed point technique.Finally,an example is given to illustrate the main results.