Weighted Asymptotically Periodic Solutions of Linear Volterra DifferenceEquations

Abstract

A linear Volterra difference equation of the form $x(n+1)=a\left(n\right)+b\left(n\right)x\left(n\right)+{{\displaystyle \sum }}_{i=0}^{n}K(n,i)x\left(i\right),$ where $x:{\mathbb{N}}_0\to ℝ$, $a:{\mathbb{N}}_0\to ℝ$, $K:{\mathbb{N}}_0\times {\mathbb{N}}_0\to ℝ$ and $b:{\mathbb{N}}_0\to ℝ\setminus \left\{0\right\}$ is $\omega$-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on ${{\displaystyle \prod }}_{j=0}^{\omega -1}b\left(j\right)$ is assumed. The results generalize some of the recent results.