Inverse Function for Banach Spaces

Question

I am learning about Calabi-Yau metrics on Kummer surfaces obtained by gluing Eguchi-Hanson spaces with a flat metric. In 'Calabi-Yau metrics on Kummer surfaces as a model glueing problem' Donaldson uses some inverse function theorem at the end of this paper to show that there is a solution in $L^2_5$ to the equation \begin{equation} \square f +h^{-3}Q(f)^2=h^3(\lambda(1+\eta)-1), \end{equation} whith $R,\lambda$ being a constant, $h,\eta$ functions and the non-linear differential operator $Q(f)$ has the property \begin{equation} \Vert (Rh)^{-3}(Q(g_1)^2-Q(g-2)^2)\Vert_{L^2_3}\leq C \Vert g_1-g_2\Vert_{L^2_5} (\Vert g_1 \Vert_{L^2_5}+\Vert g_2 \Vert_{L^2_5}). \end{equation} However i am not sure to 'which' inverse function theorem he is referring to. Is there any good reference for an inverse function thereom where the map is of the form \begin{equation} \Phi(x)=\Phi(0)+L(x)+N(x), \end{equation} whith $N$ non-linear, and $L$ having a right inverse?