Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

We present a method designed for computing solutions of infinite dimensional nonlinear 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(x¯) at an approximate solution x¯. We present rigorous computer-assisted calculations showing that T is a contraction near x¯, thus yielding the existence of a solution. Since Df(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.