Small-data existence for a nonlinear elliptic Dirichlet problem

Suppose the inverse $S:C^{0,\alpha }\to C_0^{2,\alpha }$ of a linear elliptic boundary value problem is bounded by $K$, and a nonlinear map satisfiesThen $T(v)=S(Q(v)+f)$ maps a sufficiently small closed ball into itself and is a contraction mapping when $\delta$ and $\Vert f{\Vert }_{C^{0,\alpha }}$ 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.