Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

Abstract

The paper investigates a dynamic equation $\Delta y\left(t_n\right)=\beta \left(t_n\right)\left[y\left(t_{n-j}\right)-y\left(t_{n-k}\right)\right]$ for $n\to \infty$, where $k$ and $j$ are integers such that $k>j\ge 0$, on an arbitrary discrete time scale $\mathbb{T}:=\left\{t_n\right\}$ with $t_n\in ℝ$, $n\in {\mathbb{Z}}_{n_0-k}^{\infty }=\{n_0-k,n_0-k+1,\dots \}$, $n_0\in \mathbb{N}$, , $\Delta y\left(t_n\right)=y\left(t_{n+1}\right)-y\left(t_n\right)$, and ${\text{lim}}_{n\to \infty }{t}_n=\infty$. We assume $\beta :\mathbb{T}\to \left(0,\infty \right)$. It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for $n\to \infty$. The results are presented as inequalities for the function $\beta$. Examples demonstrate that the criteria obtained are sharp in a sense.