Section 7.3 The Leray-Schauder Principle
The Leray-Schauder principle shows how a-priori estimates can be used to find solutions to equations and fixed-points of operators.Theorem 7.3.1.
Let \(X\) be a Banach space and \(A:X\to X\) a compact linear map. Suppose each solution of \(u=\gamma\, A u\) satisfies an a-priori estimate
\begin{equation*}
\|u\|\leq c \qquad\mbox{ for all }\quad \gamma\in[0,1].
\end{equation*}
Then \(u=Au \) has a solution.Proof.
\begin{equation}
L u :=\left\{\begin{array}{ll} Au \amp \qquad \mbox{ if } \|Au\|\leq 2 c,\\
2c \frac{Au}{\|Au\|} \amp \qquad \mbox{ if } \|Au\|> 2c . \end{array}\right. \tag{7.3.1}
\end{equation}
Then \(\|Lu\|\leq 2c\) for all \(u\in X\) and \(L:M\to M\text{.}\) The map \(L\) is continuous and compact, since \(A\) is continuous and compact. By Schauder's fixed point theorem, \(L\) has a fixed point \(Lu = u,\)\(u\in M\text{.}\) If \(\|Au\|\leq 2c\text{,}\) then \(Au=Lu=u\) and we are done. If \(\|Au\|> 2c\text{,}\) then
\begin{equation*}
\|u\|=\|Lu\|=2c\frac{\|Au\|}{\|Au\|} = 2c.
\end{equation*}
On the other hand we find that
\begin{equation*}
u = Lu = \frac{2c}{\|Au\|} \; Au = \gamma\, Au, \qquad\mbox{ for some }\gamma\in (0,1).
\end{equation*}
Hence the a-priori estimate applies to \(u\) and we find \(\| u \|\leq c\text{,}\) which is a contradiction. The fixed point of \(L\) satisfies \(\|Au\|\leq 2 c\) and it is also a fixed point of \(A\text{.}\)