On oscillation of difference equations with continuous time and variable delays

Abstract

We consider existence of positive solutions for a difference equation with continuous time, variable coefficients and delays $$x(t+1)-x(t)+\sum _{k=1}^m{a}_k(t)x(h_k(t))=0,a_k(t)\ge 0,h_k(t)\le t,t\ge 0,k=1,\dots ,m.$$ We prove that for a fixed $h(t)\not\equiv t$, a positive solution may exist for $a_k$ exceeding any prescribed $M>0$, as well as for constant positive $a_k$ with $h_k(t)\le t-n$, where $n\in \mathbb{N}$ is arbitrary and fixed. The point is that for equations with continuous time, non-existence of positive solutions with $\inf x(t)>0$ on any bounded interval should be considered rather than oscillation. Sufficient conditions when such solutions exist or do not exist are obtained. We also present an analogue of the Grönwall-Bellman inequality for equations with continuous time, and examine the question when the equation has no positive non-increasing solutions. Counterexamples illustrate the role of variable delays.

12 pages