Subdominant positive solutions of the discrete equation $\Delta u(k+n)=-p(k)u(k)$

Abstract

A delayed discrete equation $\Delta u\left(k+n\right)=-p\left(k\right)u\left(k\right)$ with positive coefficient $p$ is considered. Sufficient conditions with respect to $p$ are formulated in order to guarantee the existence of positive solutions if $k\to \infty$. As a tool of the proof of corresponding result, the method described in the author's previous papers is used. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (if $k\to \infty$) with the speed not smaller than the speed characterized by the function $\sqrt{k}·{\left(n/\left(n+1\right)\right)}^k$. A comparison with the known results is given and some open questions are discussed.