Solution
= Solution
The <Leray-Schauder fixed point theorem> says that if $X$ is a Banach space and $T:X\to X$ is continuous and compact, and the set
$$
\{u\in X:u=tT(u)\text{ for some }0\leq t\leq1\}
$$
is bounded, then $T$ has a fixed point.
= Solution
The <Leray-Schauder fixed point theorem> says that if $X$ is a Banach space and $T:X\to X$ is continuous and compact, and the set
$$
\{u\in X:u=tT(u)\text{ for some }0\leq t\leq1\}
$$
is bounded, then $T$ has a fixed point.