On conditions for exponential dichotomy for systems of difference equations under perturbation of coefficients

  • Bondar Anna A., anna.alex.bondar@gmail.com Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia
Keywords: systems of difference equations, exponential dichotomy, discrete Lyapunov equations, theorem on continuous dependence

Abstract

The problem of the exponential dichotomy for systems of linear difference equations with constant coefficients is considered. We investigate the question of admissible perturbations for the coefficient matrix under which the exponential dichotomy is preserved. Assuming the initial system of linear difference equations is exponentially dichotomous, we establish conditions for perturbations under which the perturbed system is also exponentially dichotomous. The conditions are written in the form of estimates on the norm of perturbation matrices and are of constructive character. Any spectral information was not used to obtain them, since the problem of finding the spectrum for non-self-adjoint matrices is ill-conditioned from the perturbation theory point of view. In the present paper, we apply an approach based on the solvability of the discrete Lyapunov matrix equations. Therefore, the established results can be used for the numerical study of the dichotomy problem.

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How to Cite
Bondar, A. (2020) “On conditions for exponential dichotomy for systems of difference equations under perturbation of coefficients”, Mathematical notes of NEFU, 26(4), pp. 3-13. doi: https://doi.org/10.25587/SVFU.2019.84.91.001.
Section
Mathematics