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

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.


39A11 Stability of difference equations (MSC2000)
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