## 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.

### MSC:

 39A11 Stability of difference equations (MSC2000)
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### References:

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