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Zbl 1119.39003
Applelby, John A.D.; Györi, István; Reynolds, David W.
On exact convergence rates for solutions of linear systems of Volterra difference equations. (English)
[J] J. Difference Equ. Appl. 12, No. 12, 1257-1275 (2006). ISSN 1023-6198

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 \Bbb Z^+, z(0)=z_0, $ where $h: \Bbb Z^+\to \Bbb R^d$, $H: \Bbb Z^+\times \Bbb Z^+\to \Bbb R^{d\times d}$, $H(n,i)=0$ for $i>n$, and $z_0\in \Bbb R^d$. 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 $$ x(n+1)=f(n)+\sum_{i=0}^nF(n-i)x(i),\quad n\in \Bbb Z^+. $$ Several applications and examples are given.
[Nguyen Van Minh (Carrollton)]

MSC 2000:: *39A11 Stability of difference equations

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

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