Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

Maxime Breden; Laurent Desvillettes; Jean-Philippe Lessard

doi:10.3934/dcds.2015.35.4765

Article Contents

2015, Volume 35, Issue 10: 4765-4789. Doi: 10.3934/dcds.2015.35.4765

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Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

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

Received: May 31, 2014

Revised: December 31, 2014

Published: September 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 47H10, 97N20; Secondary: 42A10, 34B08, 65L10.

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