Nash-Moser iteration argument
The Nash-Moser iteration argument is an iteration scheme based upon Newton's method for finding roots of a nonlinear equation. In order to make the scheme converge, one needs to apply regularizing operators (such as Littlewood-Paley multipliers) between each iteration of Newton's method.
This iteration argument is mostly employed in very nonlinear situations, such as quasilinear equations, and tends to require a large amount of regularity. For semilinear equations the Duhamel iteration argument is usually more efficient and gives stronger results.