Accurate solution estimates for nonlinear nonautonomous vector difference equations

Rigoberto Medina; M. I. Gil

doi:10.1155/S1085337504306184

Accurate solution estimates for nonlinear nonautonomous vector difference equations

Rigoberto Medina, M. I. Gil

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The paper deals with the vector discrete dynamical system X_k+1 = A_kX_k + f_k (x_k). The well-known result by Perron states that this system is asymptotically stable if A _k = A = const is stable and f_k(x) f̃(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.

Original language	English
Pages (from-to)	603-611
Number of pages	9
Journal	Abstract and Applied Analysis
Volume	2004
Issue number	7
DOIs	https://doi.org/10.1155/S1085337504306184
State	Published - 29 Jun 2004

Access to Document

10.1155/S1085337504306184

Cite this

@article{f9c4b44fca9b4e64aa479a725b730e2e,

title = "Accurate solution estimates for nonlinear nonautonomous vector difference equations",

abstract = "The paper deals with the vector discrete dynamical system Xk+1 = AkXk + fk (xk). The well-known result by Perron states that this system is asymptotically stable if A k = 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.",

author = "Rigoberto Medina and Gil, {M. I.}",

year = "2004",

month = jun,

day = "29",

doi = "10.1155/S1085337504306184",

language = "English",

volume = "2004",

pages = "603--611",

journal = "Abstract and Applied Analysis",

issn = "1085-3375",

publisher = "Hindawi Publishing Corporation",

number = "7",

}

TY - JOUR

T1 - Accurate solution estimates for nonlinear nonautonomous vector difference equations

AU - Medina, Rigoberto

AU - Gil, M. I.

PY - 2004/6/29

Y1 - 2004/6/29

N2 - The paper deals with the vector discrete dynamical system Xk+1 = AkXk + fk (xk). The well-known result by Perron states that this system is asymptotically stable if A k = A = const is stable and fk(x) f̃(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.

AB - The paper deals with the vector discrete dynamical system Xk+1 = AkXk + fk (xk). The well-known result by Perron states that this system is asymptotically stable if A k = A = const is stable and fk(x) f̃(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.

UR - http://www.scopus.com/inward/record.url?scp=4944250791&partnerID=8YFLogxK

U2 - 10.1155/S1085337504306184

DO - 10.1155/S1085337504306184

M3 - Article

AN - SCOPUS:4944250791

VL - 2004

SP - 603

EP - 611

JO - Abstract and Applied Analysis

JF - Abstract and Applied Analysis

SN - 1085-3375

IS - 7

ER -

Accurate solution estimates for nonlinear nonautonomous vector difference equations

Abstract

Access to Document

Other files and links

Fingerprint

Cite this