# 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 - A, 2015, 35 (10) : 4765-4789. doi: 10.3934/dcds.2015.35.4765
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##### References:
 [1] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [2] Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511 [3] Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 [4] Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017 [5] Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

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