Leray–Schauder degree

Summary

The Leray–Schauder degree is an integer that counts, with signs, the solutions of $u - K(u) = y$ in a bounded open set of a Banach space. Here $K$ is a compact map. It extends the Brouwer degree of maps between finite-dimensional spaces to this infinite-dimensional class. It does not change under compact deformations that keep solutions away from the boundary, and a nonzero degree guarantees a solution. This makes it a main tool to prove existence for nonlinear elliptic equations by continuation from a simpler problem.