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 delaysx(t+1)−x(t)+∑k=1mak(t)x(hk(t))=0,ak(t)≥0,hk(t)≤t,t≥0,k=1,…,m.We prove that for a fixed h(t)≢t, a positive solution may exist for ak exceeding any prescribed M > 0, as well as for constant positive ak with hk(t)≤t−n, where n∈N is arbitrary and fixed. The point is that for equations with continuous time, non-existence of positive solutions with infx(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.