Iteration scheme

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An iteration scheme seeks to construct an exact solution to an equation (typically nonlinear) by recursive constructing a sequence of approximate solutions, and then showing that these approximate solutions converge in a suitable sense to an exact solution. It is related to the penalization method and variational method for constructing solutions, except that the approximating solutions are defined rather explicitly (albeit recursively); also, one typically does not rely on compactness and so there is no need to pass to a subsequence in order to extract a limit.

By far the most common iteration scheme employed in nonlinear dispersive and wave equations is the Duhamel iteration argument. For particularly nonlinear equations one may also use the Nash-Moser iteration argument.