The Existence Of Solution To Set-valued Variational Inequalities

In this dissertation, we focus on the existence of solution to variational inequality and optimization problems. The dissertation is trying to substitute exceptional family for coercivity condition to study the existence of solution to variational inequality and optimization problems. It is organized as follows:In Chapter 1, we introduce the background of variational inequality and nonlinear complementarity problems and the development of exceptional family, and some basic conceptions and lemmas which are used in this dissertation.In Chapter 2, firstly we define a new conception of exceptional family, then we prove a Leary — Schauder type fixed point theorem for acyclic set-valued mapping in Banach spaces.Theorem 2.2.1 (heray — Schauder type fixed point theorem) Let K be a nonempty unbounded closed convex set of Banach space E. F ∈ V_C(K,H) and F is a compact mapping, U is a relatively open set in K, 0 ∈ U is bounded, satisfying Leray — Schauder conditionx (?) λ(F(x) + a) - a, (?)_x∈(?)_KU, (?)λ∈[0,1].Then F has a fixed point in U.We present a sufficient condition of the existence of solution to variational inequality over a general unbounded closed convex set in Hilbert spaces by the Leary — Schauder type fixed point theorem and exceptional family, this condition is weaker than some well-known conditions. We obtain some existence theorems of the solution.Theorem 2.3.1 Let ifbea nonempty unbounded closed convex set of H, I — F ∈ V_C(K,H) and I— F is a compact mapping, then SVI(K,F) has either a solution or an exceptional family for any x|^ ∈ K.In Chapter 3, we present a new conception of exceptional family in reflexive and...