A discrete equationΔy(n)=β(n)[y(n−j)−y(n−k)]with two integer delayskandj, k>j≥0is considered forn→∞. We assumeβ:ℤn0−k∞→(0,∞), whereℤn0∞={n0,n0+1,…},  n0∈ℕandn∈ℤn0∞. Criteria for the existence of strictly monotone and asymptotically convergent solutions forn→∞are presented in terms of inequalities for the functionβ. Results are sharp in the sense that the criteria are valid even for some functionsβwith 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.

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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.

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The asymptotic behavior of the solutions of the first-order differential equationẏ t n i 1 β i t y t − δ i − y t − τ i containing delays is studied with β i :The attention is focused on an analysis of the asymptotical convergence of solutions. A criterion for the asymptotical convergence of all solutions, characterized by the existence of a strictly increasing bounded solution, is proved. Relationships with the previous results are discussed, too.