On exact convergence rates for solutions of linear systems of Volterra difference equations. (English) Zbl 1119.39003

Appleby, John A. D.; Győri, István; Reynolds, David W.

J. Difference Equ. Appl. 12, No. 12, 1257-1275 (2006).

This paper is concerned with the asymptotic behavior of solutions of difference equations of the form \( z(n+1)=h(n)+\sum_{i=0}^n H(n,i)z(i),\quad n\in \mathbb Z^+, z(0)=z_0, \) where \(h: \mathbb Z^+\to \mathbb R^d\), \(H: \mathbb Z^+\times \mathbb Z^+\to \mathbb R^{d\times d}\), \(H(n,i)=0\) for \(i>n\), and \(z_0\in \mathbb R^d\). Sufficient conditions are given for the asymptotic constancy of solutions to the initial value problem associated with the above equations with a formula for the rate of convergence. Moreover, an explicit expression is obtained in the particular case of when the above equations are linear Volterra convolution equations of the form \[ x(n+1)=f(n)+\sum_{i=0}^nF(n-i)x(i),\quad n\in \mathbb Z^+. \] Several applications and examples are given.

Reviewer: Nguyen Van Minh (Carrollton)

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

asymptotic behavior; Volterra difference equation; asymptotic constancy; initial value problem; linear Volterra convolution equations

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