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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 $\left(k+1\right)$st order

${\Delta }x\left(n\right)=-p\left(n\right)x\left(n-k\right),$

where $p$ is a positive function defined on $ℤ\cap \left[a,\infty \right)$ and $a$ is an integer, with initial conditions $x\left(n\right)=\varphi \left(n\right)$ for $a-k\le n\le a$ and prescribed constants $\varphi \left(n\right)\in ℝ$. Using a classical comparison result the authors show that a positive solution $x\left(n\right)$ exists if $p\left(n\right)$ is dominated for large $n$ by an explicitly given auxiliary function. Moreover, the size of $x\left(n\right)$ is controlled. A comparison with known results is included.

##### MSC:
 39A22 Growth, boundedness, comparison of solutions (difference equations) 39A06 Linear equations (difference equations) 39A12 Discrete version of topics in analysis 39A21 Oscillation theory (difference equations) 34K11 Oscillation theory of functional-differential equations