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Zbl 1124.39002
Berezansky, Leonid; Braverman, Elena
On existence of positive solutions for linear difference equations with several delays.
(English)
[J] Adv. Dyn. Syst. Appl. 1, No. 1, 29-47 (2006). ISSN 0973-5321; ISSN 0974-021X/e

The authors consider a scalar linear difference equation with several delays $$x(n+ 1)- x(n)= -\sum^m_{i=1} a_i(n) x(h_i(n)),\quad h_i(n)\le n,\quad n> n_0.$$ After some preliminaries (Section 2) they present nonoscillation criteria (Section 3), comparison theorems (Section 4) and explicit nonoscillation and oscillation results (Section 5). Section 6 contains some numerical examples. For difference equations with variable delay they extend several results, which are well known for delay differential equations. The main tool in this investigation is the solution representation formula and properties of the fundamental function.
MSC 2000:
*39A11 Stability of difference equations
39A12 Discrete version of topics in analysis

Keywords: oscillation; nonoscillation; comparison theorems; numerical examples; variable delays

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