Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations
Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva 84105, Israel
Axioms 2019, 8(1), 20; https://doi.org/10.3390/axioms8010020
Received: 15 January 2019 / Revised: 1 February 2019 / Accepted: 2 February 2019 / Published: 6 February 2019
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
The paper is devoted to the discrete Lyapunov equation , where A and C are given operators in a Hilbert space and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.
View Full-Text
Keywords:
discrete Lyapunov equation; difference equations; Hilbert space; dichotomy; exponential stability
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
MDPI and ACS Style
Gil’, M. Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations. Axioms 2019, 8, 20. https://doi.org/10.3390/axioms8010020
AMA Style
Gil’ M. Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations. Axioms. 2019; 8(1):20. https://doi.org/10.3390/axioms8010020
Chicago/Turabian StyleGil’, Michael. 2019. "Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations" Axioms 8, no. 1: 20. https://doi.org/10.3390/axioms8010020
Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.
Search more from Scilit