On the Stability of Volterra Difference Equations of Convolution Type

Abstract

ABSTRACT. In [4], S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equationby localizing the roots of its characteristic equationThe assumption that (a n ) ∈ 1 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 Erdös 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.

Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 &lt; α &lt; 1

“…The problem of the linear asymptotic stability of the fixed points of the discrete convolution equations with arbitrary kernels was investigated in [15] (for more reviews and publications on the subject see [14,[16][17][18][19]). Semi-analytic and numerical results for the case of fractional (power-law kernel) standard and logistic maps were obtained in [20,21] (see also reviews [4,22]).…”