Nonlinear Operators and Fixed Points

Abstract

*In this chapter, first we study certain broad classes of nonlinear operators that arise naturally in applications and then we develop some related degree theories and fixed point theorems. We start with compact operators, we conduct a detailed study of both linear and nonlinear operators. Then we pass to a broader class of nonlinear operators of monotone-type. We deal with operators from a reflexive Banach space X to its dual X * (monotone operators and their generalizations) and with operators from X into itself (accretive operators). The latter, are related to the generation theory of linear and nonlinear semigroups. We deal with both. Then, we present the degree theories of Brouwer, (finite-dimensional) and Leray–Schauder and Browder–Skrypnik (infinite-dimensional) and several interesting topological applications of them. Finally we present the main aspects of fixed point theory. We consider metric fixed point theory, topologucal fixed point theory and the interplay between fixed point theory and partial order. In this direction we introduce and use the so-called “fixed point index”