Fundamental Existence Theory of Inequality Problems

Abstract

The first purpose of this Chapter is to list and prove the fundamental existence theorems applicable to the study of inequality problems. Variational and hemivariational inequalities are studied for several important classes of operators among which monotone and hemicontinuous operators, semicoercive operators, nonlinear perturbations of semicoercive operators, maximal monotone operators and pseudomonotone perturbations of maximal monotone operators. The second purpose of this Chapter is to draw from the aforementioned abstract theorems the basic methods which can be used to study inequality problems. For instance, the monotonicity method (Sections 3.3, 3.11), the projection method (Section 3.2), the Fichera’s approach (Section 3.4), the recession approach (Section 3.5), the method of lower and upper solutions (Section 3.6), the method of maximal monotone operators and semigroup of contractions (Sections 3.7, 3.9), the Brézis approach (Section 3.8) are here discussed. The results of this chapter will be used later in Chapters 5, 6 and 7 so as to study various classes of elliptic, parabolic and hyperbolic unilateral problems.