Leigh Lunsford

Existence Results for Generalized Variational Inequalities

The study of variational inequalities has gained importance in the last twenty five years due to the wide variety of problems for which variational inequality techniques can be applied. In this dissertation we will study the question of existence of solutions to generalized variational inequalities, which are variational inequalities with set-valued mappings. We will obtain existence results for several types of generalized variational inequalities: the generalized variational inequality, the generalized quasi-variational inequality, the generalized quasi-variational-like inequality. By relaxing conditions of convexity, continuity, and compactness in our results, we generalize and extend previously known existence theorems. We then apply our results to generalized variational inequalities which contain monotone type operators. In particular, we will obtain existence results for classes of pseudo-monotone and (S)+ operators.