Unilateral Analysis

Abstract

The purpose of this chapter is to provide some notions and fundamental results of convex analysis which will be used throughout this book. Starting with the notion of convexity, some basic results on convex and lower semi-continuous functionals are given. Particular attention is paid to the separation theorems of convex sets. There follow some results on lower estimate of lower semi-continuous convex functions. In particular we deal with a significant result of Szulkin. The famous Ekeland’s variational principle is also presented. Projection operators on closed convex sets are discussed. The chapter ends with four mathematical principles of particular interest in the study of inequality problems: The KKM principle, Minty’s principle, the complementarity principle and the variational principle.KeywordsBanach SpaceVariational InequalityLipschitz FunctionGeneralize GradientReal Banach SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.