Leray–Schauder method

Summary

The Leray–Schauder method proves that a nonlinear equation $u = T(u)$ with a compact map $T$ has a solution. It deforms the equation into one that is easy to solve. One embeds the problem in a family $u = T(\lambda, u)$, $0 \le \lambda \le 1$, and proves an a priori bound: all solutions for all $\lambda$ lie in a fixed ball. The Leray–Schauder degree is then the same for every $\lambda$, and a nonzero degree at $\lambda = 0$ forces a solution at $\lambda = 1$. It is a main tool for nonlinear integral equations and for quasilinear elliptic boundary value problems.