A new view on one problem of asymptotic behavior of solutions of delay difference equations
Consider the linear Volterra difference equation with real coefficients \[ x_{n+1}-x_n=ax_n+\sum_{j=1}^{\infty}K_jx_{n-j},\quad n\geq 0. \] In the first section, the author slightly generalizes an asymptotic result obtained by \textit{I.-G. E. Kordonis} and \textit{C. G. Philos} [J. Differ. Equations Appl. 5, No. 3, 219--233 (1999; Zbl 0973.39005)], where a stability criterion is derived via a positive root of the corresponding characteristic equation. In addition, a new generalization is developed for the second-order linear difference equation \[ x_{n+1}-x_n=ax_n+b x_{n-1}. \] The author suggests that similar ideas might be applied for a general Volterra difference equation and also for functional-differential equations.
scientific article; zbMATH DE number 5134609