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
Rigoberto Medina1
and M.I. Gil'2
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
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.