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In this paper we introduce a new algorithm for solving perturbed nonlinearfunctional equations which admit a right-invertible linearization, but with aninverse that loses derivatives and may blow up when the perturbation parameter$\epsilon$ goes to zero. These equations are of the form $F_\epsilon(u)=v$ with$F_\epsilon(0)=0$, $v$ small and given, $u$ small and unknown. The maindifference with the by now classical Nash-Moser algorithm is that, instead ofusing a regularized Newton scheme, we solve a sequence of Galerkin problemsthanks to a topological argument. As a consequence, in our estimates there areno quadratic terms. For problems without perturbation parameter, our resultsrequire weaker regularity assumptions on $F$ and $v$ than earlier ones, such asthose of Hormander. For singularly perturbed functionals, we allow $v$ to belarger than in previous works. To illustrate this, we apply our method to anonlinear Schrodinger Cauchy problem with concentrated initial data studied byTexier-Zumbrun, and we show that our result improves significantly on theirs.

A surjection theorem for maps with singular perturbation and loss of derivatives