An oscillation criterion for first order delay dynamic equations. (English) Zbl 1254.34126

Agarwal, Ravi; Bohner, Martin

Funct. Differ. Equ. 16, No. 1, 11-17 (2009).

Summary: The authors consider a first-order delay dynamic equation of the form \[ x^\Delta(t)+ p(t) x(\tau(t))= 0 \] on a time scale \(\mathbb{T}\), where \(\mathbb{T}\) is unbounded above, \(p: \mathbb{T}\to\mathbb{R}\) is rd-continuous and nonnegative, \(\tau:\mathbb{T}\to \mathbb{T}\) is nondecreasing and satisfies \(\tau(t)< t\) for all \(t\in\mathbb{T}\) and \(\lim_{t\to\infty}\tau(t)= \infty\). They show that the above equation is oscillatory provided there exists a constant \(M\in (0,1)\) such that \[ \begin{gathered}\liminf_{t\to\infty}\, \int^t_{\tau(t)} p(s)\Delta s> M\qquad\text{and}\\ \limsup_{\tau\to\infty}\, \int^t_{\tau(t)} p(s)\Delta s> 1-(1-\sqrt{1-M})^2.\end{gathered} \] This result improves some known results.

MSC:

34N05	Dynamic equations on time scales or measure chains
34K11	Oscillation theory of functional-differential equations