On the Stability of Volterra Difference Equations of Convolution Type


  • Higidio Portillo Oquendo Universidade Federal do Paraná
  • Jose Renato Ramos Barbosa Universidade Federal do Parana
  • Patricia Sánez Pacheco Universidade Tecnológica Federal do Paraná




difference equation, stability, convolution.


In \cite{Elaydi-10}, S.\ Elaydi obtained a characterization of the stability of
the null solution of the Volterra difference equation
x_n=\sum_{i=0}^{n-1} a_{n-i} x_i\textrm{,}\quad n\geq 1\textrm{,}
by localizing the roots of its characteristic equation
The assumption that $(a_n)\in\ell^1$ 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 Erd\"os et al in \cite{Erdos}. 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 state some new results
related to stability via finite approximation.

Author Biographies

Higidio Portillo Oquendo, Universidade Federal do Paraná

Professor Associado

Jose Renato Ramos Barbosa, Universidade Federal do Parana

Professor Associado

Patricia Sánez Pacheco, Universidade Tecnológica Federal do Paraná

Professor Adjunto




How to Cite

Oquendo, H. P., Barbosa, J. R. R., & Pacheco, P. S. (2018). On the Stability of Volterra Difference Equations of Convolution Type. Trends in Computational and Applied Mathematics, 18(3), 337. https://doi.org/10.5540/tema.2017.018.03.337



Original Article