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Existence of positive solutions of discrete linear equations with a single delay. (English) Zbl 1207.39014
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.

39A22Growth, boundedness, comparison of solutions (difference equations)
39A06Linear equations (difference equations)
39A12Discrete version of topics in analysis
39A21Oscillation theory (difference equations)
34K11Oscillation theory of functional-differential equations
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