Nash--Moser iteration and singular perturbations

We present a simple and easy-to-use Nash--Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter $\eps\to 0.$ The novel feature is to allow loss of powers of $\eps$ as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schr\"odinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems.