On existence of positive solutions for linear difference equations with several delays.

*(English)*Zbl 1124.39002The authors consider a scalar linear difference equation with several delays

$$x(n+1)-x\left(n\right)=-\sum _{i=1}^{m}{a}_{i}\left(n\right)x\left({h}_{i}\left(n\right)\right),\phantom{\rule{1.em}{0ex}}{h}_{i}\left(n\right)\le n,\phantom{\rule{1.em}{0ex}}n>{n}_{0}\xb7$$

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.

Reviewer: Miloš Čanak (Beograd)

##### MSC:

39A11 | Stability of difference equations (MSC2000) |

39A12 | Discrete version of topics in analysis |