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 D f ((x) over bar) at an approximate solution (x) over bar. We present rigorous computer-assisted calculations showing that T is a contraction near (x) over bar, thus yielding the existence of a solution. Since D f ((x) over bar) 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.