Leray-Schauder Degree

Abstract

The objective of Leray-Schauder degree theory is the same as that of the fixed point theory of the first part of the book. We want to demonstrate that if certain hypotheses are satisfied, then we can conclude that a map f has a fixed point, that is, that f(x) = x. If the hypotheses are of the right type, we can hope to verify them in settings that arise in analysis and conclude that an analytic problem has a solution because we’ve managed to describe its solutions as fixed points. A major difference between Leray-Schauder theory and what we studied previously is the local nature of our new theory. A fixed point theorem generally states the existence of a fixed point somewhere in the domain of a map defined on an entire space. Degree theory, as in the last chapter, is concerned with a map defined on $\bar U$, the closure of a specified open set U. Leray-Schauder theory seeks conditions that imply the map has a fixed point specifically on U.