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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 X A * X A = C , where A and C are given operators in a Hilbert space H 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 H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . 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 discrete Lyapunov equation; difference equations; Hilbert space; dichotomy; exponential stability
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 Style

Gil’, 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

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