= Small-data existence for a nonlinear elliptic Dirichlet problem
Suppose the inverse $S:C^{0,\alpha}\to C^{2,\alpha}_0$ of a linear <elliptic boundary value problem> is bounded by $K$, and a nonlinear map satisfies
$$
\|Q(v)\|_{C^{0,\alpha}}\leq C\|v\|_{C^{2,\alpha}}^2+\delta,\qquad
\|Q(v)-Q(w)\|_{C^{0,\alpha}}\leq[C(\|v\|+\|w\|)+\delta]\|v-w\|.
$$
Then $T(v)=S(Q(v)+f)$ maps a sufficiently small closed ball into itself and is a <contraction mapping> when $\delta$ and $\|f\|_{C^{0,\alpha}}$ are sufficiently small. The <Banach fixed-point theorem> gives a solution of $Lu=Q(u)+f$ with zero <Dirichlet boundary condition>, unique within that ball. Invertibility follows from the <Fredholm alternative for an elliptic Dirichlet problem> when the homogeneous <null space> is zero.
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