Leray—Schauder Topological Degree

Abstract

To study the number of solutions of equations like

$$ \Phi (u) = b, $$

where Ω is an open set in a Banach space X, $\Phi :\overline \Omega \to X$ and $b \in X$, and based on a similar idea of Brouwer for continuous maps defined in finite-dimensional spaces, Leray and Schauder [64] introduced a topological tool, called the degree.