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Zbl 1207.39014
Baštinec, J.; Dibl{\'\i}k, J.; Šmarda, Z.
Existence of positive solutions of discrete linear equations with a single delay.
(English)
[J] J. Difference Equ. Appl. 16, No. 9, 1047-1056 (2010). ISSN 1023-6198

Consider the linear scalar discrete equation of $(k+1)$st order $$\Delta x(n)=-p(n)x(n-k),$$ where $p$ is a positive function defined on $\mathbb{Z} \cap [a, \infty)$ and $a$ is an integer, with initial conditions $x(n)=\varphi(n)$ for $a-k \leq n \leq a$ and prescribed constants $\varphi(n) \in \mathbb{R}$. Using a classical comparison result the authors show that a positive solution $x(n)$ exists if $p(n)$ is dominated for large $n$ by an explicitly given auxiliary function. Moreover, the size of $x(n)$ is controlled. A comparison with known results is included.
[Johanna Michor (Wien)]
MSC 2000:
*39A22
39A06
39A12 Discrete version of topics in analysis
39A21
34K11 Oscillation theory of functional-differential equations

Keywords: discrete delayed equation; oscillating solution; positive solution; asymptotic behaviour

Cited in: Zbl 1200.39002

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