## Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case.(English)Zbl 1220.39004

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.

### MSC:

 39A12 Discrete version of topics in analysis 39A06 Linear difference equations 34K06 Linear functional-differential equations
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### References:

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