On the Asymptotic Behavior of Volterra Difference Equations

Details On the Asymptotic Behavior of Volterra Difference Equations On the Asymptotic Behavior of Volterra Difference Equations

2010-10-26

arxiv.org/abs/1010.5454v2

Mathematics

Dynamical Systems

12 pages

Scientific paper

We consider the asymptotic behavior of solutions of the difference equations of the form $x(n+1)=Ax(n) + \sum_{k=0}^n B(n-k)x(k) + y(n)$ in a Banach space $\X$, where $n=0,1,2,...$; $A,B(n)$ are linear bounded operator in $\X$. Our method of study is based on the concept of spectrum of a unilateral sequence. The obtained results on asymptotic stability and almost periodicity are stated in terms of spectral properties of the equation and its solutions. To this end, a relation between the Z-transform and spectrum of a unilateral sequence is established. The main results extend previous ones.

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