The Methods Of Exceptional Family And Two Classes Of Variational Inequalities

Variational inequality theory is an important branch of nonlinear analysis with a wide range of applications in nonlinear optimization theory, differential equation, control problem, game theory and social economic equilibrium theory. A basic problem in variational inequalities is the existence of solutions. In this paper, we focus on the existence of solution for two classed of variational inequalities by using the techniques of exceptional family. This paper is organized as follows:In chapter1, we introduce some backgrounds and developments of variational inequality and the exceptional family. Moreover, we recall some basic conceptions and notations which used in this dissertation.In chapter2, we study the following variational inequality (VI(C,F)):find x*Ε C～ac1C and u Ε F(x*) such that <u,x-x*>>0,(?)x Ε C, where C1is a bounded, closed and convex subset in Banach space E, V is a relative open subset of C1, V is convex, C=V, F:C～ac1C→2E*is a admissible mapping.The existing literature always need to assume that F is well defined on C. However, the above variational inequality does not require F is well defined on the relative boundary ac1C. Firstly, we introduce the generalized projection operator in reflexive Banach spaces and obtain the norm-weak upper semicontinuity of generalized projection operator without the strictly convex assumption. Secondly, we prove a Leray-Schauder fixed point theorem and convert the above variational inequality to a fixed point problem. Furthermore, we propose an exceptional family of elements for the above variational inequality and show that the nonexistence of exceptional family of elements is a sufficient condition for the solvability of variational inequality. We present some sufficient conditions for the nonexistence of an exceptional family and obtain some existence results of solutions to the above variational inequality. Finally, as an application, we give a new proof to Walrasian equilibrium Existence Theorem under some continuity, monotonicity and convex assumptions in finite pure exchange economics.In chapter3, we study the existence of solution for a class of implicit set-valued variational inequality in reflexive, strictly convex and smooth Banach space. Firstly, we introduce the notion of generalized projection operator (?) for the implicit set-valued variational inequality and prove some properties of this operator. Secondly, we prove a Leray-Schauder fixed point theorem similar to the one in Chapter2and convert the implicit set-valued variational inequality to a fixed point problem. Finally, by applying the fixed point theory and the notion of exceptional family, we discuss the existence of solution for implicit set-valued variational inequality with compact continuous mapping.