Properties Of Solution Set Of Generalized Polynomial Variational Inequality

In the field of variational inequalities and complementarity problems,the existence and uniqueness of solutions and the nonempty compactness of solution sets are two important research directions.It includes tensor complementarity problem,polynomial complementarity problem,tensor variational inequality,polynomial variational inequality,etc.The purpose of this paper is to establish the existence and uniqueness theorem of the solution of the generalized polynomial variational inequality,and to study the nonempty compactness of its solution set.For generalized variational inequalities,a famous theorem of the existence and uniqueness of solutions is established by Pang and Yao.We find that when the function involved is a polynomial,the conditions in this theorem are generally not true.Therefore,the existence and uniqueness theorem of Pang and Yao cannot be applied to generalized polynomial variational inequalities.In this paper,monotonicity and other conditions are used,Combined with the properties of structure tensor and polynomials,using the tools of degree theory and exception family,we establish the existence and uniqueness theorem of solutions of generalized polynomial variational inequality,and construct examples to verify the conclusions.Furthermore,the nonempty compactness of solution set of generalized polynomial variational inequality on closed convex set and extended rectangular set is discussed.