# On oscillation of difference equations with continuous time and variable delays

## TL;DR

This paper investigates the existence and non-existence of positive solutions for a class of continuous-time difference equations with variable delays, providing conditions and inequalities that characterize their oscillatory behavior.

## Contribution

It introduces new criteria for the existence of positive solutions in difference equations with continuous time and variable delays, and extends the Grönwall-Bellman inequality to this context.

## Key findings

- Positive solutions can exist even with large coefficients and certain delay conditions.
- Conditions are established for the non-existence of positive non-increasing solutions.
- An analogue of the Grönwall-Bellman inequality is developed for these equations.

## Abstract

We consider existence of positive solutions for a difference equation with continuous time, variable coefficients and delays $$ x(t+1)-x(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))=0, \quad a_k(t) \geq 0, ~~h_k(t) \leq t, \quad t \geq 0, \quad k=1, \dots, m. $$ 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 $h_k(t) \leq t-n$, 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 Gr\"{o}nwall-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.