Convergence Analysis On The Solutions For A Class Of Evolution Variational-Hemivariational Inequalities With Constraint Sets

As one of the important theories for solving nonlinear problems,variational inequal-ity is a kind of important mathematical problem.With the development of its morden mathematical theory in the sixties of last century and the untiring work of many math-ematical scholars and researchers in this filed,the theory of variational inequalties has been broadly applied in the theoretical and applied discipliens including contact mechan-ics,differential equations,optimal control,engineering management,nonlinear program-ming and so on.With the deepening and development of variational inequality prob-lems in their application field,a class of extended variational inequalities are proposed when the classical variational inequality problem involves both convex functions and non-convex and nonsmooth functions,the corresponding inequality problem is called variational-hemivariational inequality problem.Over the past three decades,the variational-hemivariational inequality problems and their theories have also developed rapidly,and have been widely applied in the filed of contact mechanics,fluid mechanics,control theory and other subjects.In general,when solving practial problems,the data of the obtained variational-hemivariational inequality models for practical problems are always perturbed since the accurate data are affected by errors and other reasons,which may induce the ill-posedness of the corresponding problem,i.e.either no solution or multiple solutions for the problem.Therefore,it does make sense for us to study the solvability of the perturbed variational-hemivariational inequlities and the relationship of solutions between the perturbed prob-lems and the original problem.In the present thesis,we focus on a class of evolution variational-hemivariational inequalities with constraint sets as follows.(?)where Xis a reflexive Banach space with X*being its dual space,u=du/dt stands for the generaliezd derivative,j°(u,v-u)represents the generalized directional derivative of the locally Lipschitz function j:X?R at u in the direction v-u,K is a nonempty,closed and convex subset of X,A:X?X*is a single-valued operator,and f is an element of X*.Firstly,we transfer the considered variational-hemivariational inequality problem to an inclusion problem,and we get the existence and uniqueness of solution of the original problem based on the surjective theorem in reflexive Banach space.Then,according to the condition for the perturbed data,we consider different problems to get the convergence results for the variational-hemivariational inequlities.When the smallness condition is unsatisfied,we use the regularization method to study the solvability of the regularized problems obtained from the perturbed variational-hemivariational inequlities,we get the existence and uniqueness of solution.Using the relationship of the original data and its perturbed data,and combined with the set convergence in the sense of Mosco,we obtain that the sequence of regularized solutions converges strongly to the orginal solution.When the smallness condition of the perturbed data is satisfied,we directly study the perturbed problems,we have the unique solvability of their solutions and the similar result of strong convergence of the sequence of the perturbed solutions.