Solution
= Solution
For $w\in X_\delta$, hypothesis (2) gives
$$
\|F(w)\|_{C^{0,\alpha}}\leq\|w\|_{C^0}^2\leq\delta^2.
$$
Combining the two estimates in part 4(i) with (5) yields
$$
\|T(w)\|_{C^{2,\alpha}}\leq C(\delta^2+\varepsilon).
$$
Choose $0<\delta<\min\{1,(2C)^{-1}\}$, so that $C\delta^2\leq\delta/2$, and then choose $\varepsilon\leq\delta/(2C)$. Uniformly for $w\in X_\delta$ we obtain $\|T(w)\|_{C^{2,\alpha}}\leq\delta$. Hence
$$
\boxed{T(X_\delta)\subseteq X_\delta.}
$$