  Building up on classical linear formulations, we posit that a broad class of
problems in signal synthesis and in signal recovery are reducible to the basic
task of finding a point in a closed convex subset of a Hilbert space that
satisfies a number of nonlinear equations involving firmly nonexpansive
operators. We investigate this formalism in the case when, due to inaccurate
modeling or perturbations, the nonlinear equations are inconsistent. A relaxed
formulation of the original problem is proposed in the form of a variational
inequality. The properties of the relaxed problem are investigated and a
provenly convergent block-iterative algorithm, whereby only blocks of the
underlying firmly nonexpansive operators are activated at a given iteration, is
devised to solve it. Numerical experiments illustrate robust recoveries in
several signal and image processing applications.
