# American Institute of Mathematical Sciences

October  2015, 35(10): 4765-4789. doi: 10.3934/dcds.2015.35.4765

## Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

 1 CMLA, ENS Cachan & CNRS, 61 avenue du Président Wilson, 94230 Cachan, France, France 2 Département de Mathématiques et de Statistique, Université Laval, 1045 avenue de la Médecine, Québec, QC, G1V0A6, Canada

Received  June 2014 Revised  January 2015 Published  April 2015

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 $Df(\overline{x})$ 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.
Citation: Maxime Breden, Laurent Desvillettes, Jean-Philippe Lessard. Rigorous numerics for nonlinear operators with tridiagonal dominant linear part. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4765-4789. doi: 10.3934/dcds.2015.35.4765
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