Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case
Summary: A discrete equation \(\Delta y(n) = \beta(n)[y(n - j) - y(n - k)]\) with two integer delays \(k\) and \(j\), \(k > j \geq 0\) is considered for \(n \rightarrow \infty\). We assume \(\beta : \mathbb Z^{\infty}_{n_{0}-k} \rightarrow (0, \infty)\), where \(\mathbb Z^{\infty}_{n_0} = \{ n_0, n_0 + 1, \dots \}\), \(n_0 \in \mathbb N\) and \(n \in \mathbb Z^{\infty}_{n_0}\). Criteria for the existence of strictly monotone and asymptotically convergent solutions for \(n \rightarrow \infty\) are presented in terms of inequalities for the function \(\beta\). Results are sharp in the sense that the criteria are valid even for some functions \(\beta\) with a behavior near the so-called critical value, defined by the constant \((k - j)^{-1}\). Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.
scientific article; zbMATH DE number 5930775