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

Abstract

The paper investigates a dynamic equation Δy t n β t n y t n−j − y t n−k for n → ∞, where k and j are integers such that k > j ≥ 0, on an arbitrary discrete time scale T : {t n } with t n ∈ R, n ∈ Z ∞ n0−k {n 0 − k, n 0 − k 1, . . .}, n 0 ∈ N, t n < t n 1 , Δy t n y t n 1 − y t n , and lim n → ∞ t n ∞. We assume β : T → 0, ∞ . 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 → ∞. The results are presented as inequalities for the function β. Examples demonstrate that the criteria obtained are sharp in a sense.

Permanence and Almost Periodic Solutions of a Discrete Ratio‐Dependent Leslie System with Time Delays and Feedback Controls

“…Already, many scholars have paid attention to the nonautonomous discrete population models, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world see [12][13][14][15][16][17][18][19][20][21][22][23] . Furthermore, the asymptotic convergence of solutions of difference equations with delay is one of the most important topics in the study of population dynamics, and many excellent results have already been obtained and seen great progress see [24][25][26] and the references cited therein .…”

“…Since time delays occur so often in nature, a number of ecological systems can be described as systems with time delays see 14,15,19,[21][22][23][24][25][26][27] . One of the most important problems for this type of system is to analyze the effect of time delays on the stability of the system.…”

Convergence of the solution of an impulsive differential equation with piecewise constant argument

“…They showed that some nonimpulsive systems can be stabilized by imposition of impulsive controls. Lately, the problem of the asymptotic constancy of solutions was studied for some functional differential equations [6,8,11,12,[15][16][17][18][19][20][21][22][23]41] and as well the same problem has been considered for some impulsive delay differential equations [10,30]. So, due to the practical reasons and the papers mentioned above one can be motivated to deal with the problem of asymptotic constancy of solutions of an impulsive differential equation with piecewise constant arguments.…”