= Solution
For <Banach spaces> $X,Y$, a <nonlinear compact operator> $T:X\to Y$ is a <continuous map> that sends every <bounded set> to a <relatively compact subset> of $Y$: the closure of its image is a <compact set>. For a <continuous map>, the image condition is equivalent to requiring that every bounded <sequence> $(x_j)$ has a <subsequence> along which $T(x_j)$ converges in the <norm topology> of $Y$. For a <linear operator>, this is the usual <compact operator> definition; the continuity requirement is included explicitly here because the subsequent applications are nonlinear.
The compact-domain version of the <Schauder fixed-point theorem> says
$$
\boxed{K\ne\varnothing\text{ compact and convex in a Banach space},\quad
T:K\to K\text{ continuous}\quad\Longrightarrow\quad Tu=u\text{ for some }u\in K.}
$$
A useful equivalent version replaces $K$ by a nonempty closed bounded <convex set> $C$ and assumes $T:C\to C$ is a <continuous map> with <relatively compact> image. This latter form will be used for the <partial differential equation>. Both forms assert existence of a <fixed point>, without asserting uniqueness.
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