Uniform stability of periodic discrete systems in Banach spaces

Abstract

Let q>1 be a fixed integer number. We prove that a discrete q-periodic evolution family

on a complex Banach space is uniformly asymptotically stable, that is, U(m, n) → 0 in the norm of when (m − n) → ∞, if and only if for each and each x ∈ X one has

In particular, we obtain the following result of Datko type. The family is uniformly asymptotically stable if and only if for each one has

Keywords:

• Discrete semigroups
• Discrete evolution families
• Stability
• Spectral radius of bounded linear operators

Keywords:

• Primary 47D03

Acknowledgements

The first author would like to acknowledge the support of Victoria University Discovery Research Grant D03/02 and the Faculty of Mathematics, West University of Timişoara research funding.

Notes

^∥ Email: [email protected]. URL: http://rgmia.vu.edu.au/sofo

^§ Email: [email protected]. URL: http://rgmia.vu.edu.au/ssdragomirweb.html

^¶ Email: [email protected]. URL: http://rgmia.vu.edu.au/cerone