Abstract and Applied Analysis
Volume 2004 (2004), Issue 7, Pages 603-611
doi:10.1155/S1085337504306184
Accurate solution estimates for nonlinear
nonautonomous vector difference equations
1Departmento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933, Osorno, Chile
2Department of Mathematics, Ben–Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
Received 27 August 2002
Copyright © 2004 Rigoberto Medina and M. I. Gil'. 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.
Abstract
The paper deals with the vector discrete dynamical system
xk+1=Akxk+fk(xk). Thewell-known result by
Perron states that this system is asymptotically stable if
Ak≡A=const
is stable and fk(x)≡f˜(x)=o(‖x‖). Perron's result gives no information about the
size of the region of asymptotic stability and norms of
solutions. In this paper, accurate estimates for the norms of
solutions are derived. They give us stability conditions for
(1.1) and bounds for the region of attraction of the
stationary solution. Our approach is based on the “freezing”
method for difference equations and on recent estimates for the
powers of a constant matrix. We also discuss applications of our
main result to partial reaction-diffusion difference equations.