  We present a method designed for computing solutions of infinite dimensional
non linear operators $f(x) = 0$ with a tridiagonal dominant linear part. We
recast the operator equation into an equivalent Newton-like equation $x = T(x)
= x - Af(x)$, where $A$ is an approximate inverse of the derivative
$Df(\overline x)$ at an approximate solution $\overline x$. We present rigorous
computer-assisted calculations showing that $T$ is a contraction near
$\overline x$, thus yielding the existence of a solution. Since $Df(\overline
x)$ does not have an asymptotically diagonal dominant structure, the
computation of $A$ is not straightforward. This paper provides ideas for
computing $A$, and proposes a new rigorous method for proving existence of
solutions of nonlinear operators with tridiagonal dominant linear part.
