  Let $1\le p\le \infty$. In this paper, we consider solving a nonlinear
functional equation $$f(x)=y,$$ where $x, y$ belong to $\ell^p$ and $f$ has
continuous bounded gradient in an inverse-closed subalgebra of ${\mathcal
B}(\ell^2)$, the Banach algebra of all bounded linear operators on the Hilbert
space $\ell^2$. We introduce strict monotonicity property for functions $f$ on
Banach spaces $\ell^p$ so that the above nonlinear functional equation is
solvable and the solution $x$ depends continuously on the given data $y$ in
$\ell^p$. We show that the Van-Cittert iteration converges in $\ell^p$ with
exponential rate and hence it could be used to locate the true solution of the
above nonlinear functional equation. We apply the above theory to handle two
problems in signal processing: nonlinear sampling termed with instantaneous
companding and subsequently average sampling; and local identification of
innovation positions and qualification of amplitudes of signals with finite
rate of innovation.
