Schaefer's fixed point theorem

Remarks

The proof uses the Schauder fixed-point theorem. Choose $M$ larger than the norm of every $u$ in the bounded set of the hypothesis, and let $B$ be the closed ball of radius $M$. On $B$, define $\tilde A(u) = A(u)$ if $\norm{A(u)} \le M$ and $\tilde A(u) = M A(u)/\norm{A(u)}$ otherwise. Then $\tilde A$ maps $B$ continuously into a compact subset of $B$, so it has a fixed point $u$. If $\norm{A(u)} > M$, then $u = \lambda A(u)$ with $\lambda = M/\norm{A(u)} < 1$ and $\norm{u} = M$, which contradicts the choice of $M$; so $A(u) = u$.

The hypothesis is an a priori bound: one shows that all solutions of the family of problems $u = \lambda A(u)$ lie in a fixed ball, without knowing in advance that any exists.

Example: on $X = \R$, let $A(u) = 1 + \frac{1}{2}\sin u$. If $u = \lambda A(u)$ with $\lambda \in [0, 1]$, then $\abs{u} \le 3/2$, so the set is bounded, and $A$ has a fixed point.

The a priori bound is necessary: $A(u) = u + 1$ on $\R$ is continuous and compact, but has no fixed point, and the solutions $u = \lambda/(1 - \lambda)$ of $u = \lambda A(u)$ are unbounded as $\lambda \to 1$.

Going deeper

For a student

The theorem is a standard tool for nonlinear elliptic and parabolic equations, as in Evans's book on partial differential equations. One writes the equation as $u = A(u)$, where $A$ solves a linear problem with the nonlinearity frozen, and gets compactness of $A$ from elliptic regularity and a compact Sobolev embedding. The a priori bound is then proved by energy estimates or the maximum principle. It is a special case of the Leray–Schauder continuation principle, which is based on degree theory.