= Solution
The <Fredholm alternative for an elliptic Dirichlet problem> and part (b) give an inverse $S:Y\to X$ of norm at most $K\geq1$. Choose
$$
\varepsilon=\min\left(1,\frac1{8KC}\right),\qquad
\delta=\min\left(\frac{\varepsilon}{4K},\frac1{4K}\right),\qquad
\varepsilon_0=\frac{\varepsilon}{4K}.
$$
On the closed radius-$\varepsilon$ ball $\mathcal B\subset X$, set $T(v)=S(\mathcal Q(v)+f)$. The stated inequalities give
$$
\|T(v)\|_{C^{2,\alpha}}\leq K(C\varepsilon^2+\delta+\varepsilon_0)
\leq\tfrac58\varepsilon,
$$
and
$$
\|T(v)-T(w)\|_{C^{2,\alpha}}
\leq K(2C\varepsilon+\delta)\|v-w\|_{C^{2,\alpha}}
\leq\tfrac12\|v-w\|_{C^{2,\alpha}}.
$$
Thus $T$ is a <contraction mapping> of the complete metric space $\mathcal B$ into itself. The <Banach fixed-point theorem> gives $u=T(u)$, solving the required <nonlinear elliptic boundary value problem>. This <small-data existence for a nonlinear elliptic Dirichlet problem> also gives uniqueness within this small ball; it does not assert global uniqueness.
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