= Solution
Here $T$ is a <nonlinear compact operator>: it is continuous and sends bounded subsets of $M$ to <relatively compact subsets> of $X$. Equivalently, the images of any bounded input <sequence> have a norm-convergent <subsequence>.
The <Schauder fixed-point theorem> states that if $M$ is a nonempty closed bounded <convex set> in a <Banach space> and $T:M\to M$ is continuous and compact, then $\boxed{T\text{ has a fixed point in }M}$. Linearity, a contraction estimate, and uniqueness are not assumptions or conclusions of this theorem.
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