ABSTRACT
In 44. S. Elaydi. Stability of Volterra difference equations of convolution type, Dynamical Systems. Nankai Ser. Pure Appl. Math. Theoret. Phys., 4 (1993), 66-72., S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equation
by localizing the roots of its characteristic equation
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 88. P. Erdös, W. Feller & H. Pollard. A property of power series with positive coefficients. Bull. Amer. Math. Soc., 55 (1949), 201-204.. 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.
Keywords:
difference equation; stability; convolution