Study On Several Kinds Of Evolution Variational Inequalities

Variational inequality theory,which evolved and developed from classic variational problems,is an important part of nonlinear functional analysis.It has a wide range of applications in many disciplines,such as optimization theory,cybernetics,engineering applications,and economic models.This paper will study several types of problems on nonlinear evolution variational inequalities,including the regularization for quasi-variational inequalities,the regularization for quasi-mixed equilibrium problems,the "bang-bang" principle for implicit sweeping process,and the existence of solutions for nonlinear differential variational inequalities.(1)The regularization of solutions for elliptic quasi-variational inequalities in Banach spaces is studied.First,the existence result of the solutions of quasi-variational inequalities is obtained by using the fixed point theorem.Under the condition of mild coercivity,it is verified that the sequence of bounded regularized solutions will strongly converge to the solution of the initial quasi-variational inequality problem.Secondly,some conditions are given to ensure the boundedness of the regularization of quasi-variational inequalities.Finally,an example is presented to illustrate our main results.(2)The regularization of solutions for a class of quasi-mixed equilibrium problems is discussed.The quasi-mixed equilibrium problem is the generalization and extension of the quasivariational inequality.First,the existence result of Minty inequality for mixed equilibrium problem is obtained by using Tychonov’s fixed point theorem.Next,the existence of the solution for the quasi-mixed equilibrium problem is verified.Finally,some assumptions are put forward to ensure the convergence and the boundedness of the regularization solutions.(3)A kind of implicit sweeping process with perturbation in Hilbert space is discussed.The sweeping process is a special kind of variational inequality.Under the assumption of weakly constrained sets,combined with the catch-up algorithm,the existence and the uniqueness of the solutions for implicit sweeping process are established.The existence of the extremal point trajectory is obtained by applying the Schauder fixed point theorem,and it is further verified that the extremal point trajectory is dense in the solution set of the initial system.(4)A type of second-order nonlinear evolutionary differential equation systems restricted by generalized mixed variational inequalities are explored.The solution set of the variational inequalities is bounded,closed and convex by using Ky Fan’s inequality,and the corresponding relationship between the solutions of the variational inequality and the differential equation is established.Combining the non-compactness measure and the semigroup theory of operators,the existence of the solutions for second-order differential equation is obtained.