On the Stability of Volterra Difference Equations of Convolution Type

ABSTRACT In 4, S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equation x n = ∑ i = 0 n - 1 a n - i x i , n ≥ 1 , by localizing the roots of its characteristic equation 1 - ∑ n = 1 ∞ a n z n = 0 . The assumption that (an ) ∈ ℓ1 was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if the ratio R of convergence of the power series of the previous equation equals one. In fact, when R = 1, this characterization conflicts with a result obtained by Erdo¨ s et al. in 8. Here, we analyze the R = 1 case and show that some parts of that characterization still hold. Furthermore, studies on stability for the R < 1 case are presented. Finally, we study some results related to stability via finite approximation.