Mixed Quasi-Variational Inequality And Its Application

Variational inequality has been widely used in the fields of finance,economics,trans-portation,optimization,operations research,and the engineering sciences.The mixed quasi-variational inequality problem is a complex variational inequality problem.It contains quasi-optimization problem,variational inequality problem,quasi-variational inequality problem,and mixed variational inequality problem.The existence of solution is the basic problem of the mixed quasi-variational inequality.There is no literature to use the exceptional family of elements method to study the existence of solution for mixed quasi-variational inequality problem.In this paper,we study the existence of the solution for mixed quasi-variational in-equality.We mainly use the fixed point theory,the exceptional family of elements,and the iterative algorithm of generalized f-projection operator to study the existence of the solution of the corresponding problem.The details of this article are as follows:In chapter 1,we briefly introducue the development backgrounds and research status of mixed quasi-variational inequality,quasi-variational inequality,and exceptional family of elements.In addition,the basic conceptions and common symbols are also given.In chapter 2,we study the existence of solution for mixed quasi-variational inequality.Firstly,the definition of generalized f-projection operator is introduced and the continuity of generalized f-projection operator is obtained.The fixed point,problem is here reformulated by means of mixed quasi-variational inequality problem.Secondly,we introduce a conception of exceptional family of elements for mixed quasi-variational inequality.Then,we prove that nonexistence exception family of elements for mixed quasi-variational inequality is a.sufficient condition for existence of solution for mixed quasi-variational inequality by a Leray-Schauder type fixed point theorem.Finally,some sufficient conditions for the existence of exception family of elements for mixed quasi-variational inequality are given.We prove that when S= {x ? X:x ? K(x)} is a bounded sets,then solution of mixed quasi-variational inequality exist.When S = {x ? X:x ? K(x)} is a unbounded sets,some sufficient conditions for the existence of solution for mixed quasi-variational inequality and the nonemptiness and boundedness of solution set for mixed quasi-variational inequality are given.In chapter 3,when the constraint set function K satisfies certain conditions,the gen-eralized f-projection operator is locally Lipschitz mapping by using inequality in Banach space.We give an iterative algorithm to solve the mixed quasi-variational inequality MQVI(K,F,f)and prove that the sequence defined by the iterative algorithm exists a subsequence strongly converging to the solution of mixed quasi-variational inequality.In chapter 4,we study the existence of a class of generalized quasi-variational inequality and its application in economic equilibrium.Firstly.we give the inverse example of the main theorem in Milasi[M.Milasi,Existence theorem for a class of generalized quasi-variational inequalities,Journal of Global Optimization,2014,60(4):679-688]to show that the theorem is not true.As we all known,the composites of finite acyclic mapping exists a fixed point on compact convex set.Then we prove the existence of the solution for the generalized quasi-variational inequality.We apply the existence result of generalized quasi-variational inequality to economic equilibrium problem and obtain a existence result of the competitive equilibrium.