Oscillatory solutions for dynamic equations with non-monotone arguments

Abstract

Consider the first-order delay dynamic equation

$$

x^{\Delta }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\in [t_{0},\infty )_{\mathbb{T}}

$$

where $p\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) $, $\tau \in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ is non-monotone, and $\tau (t)\leq t$ $,\ \lim_{t\rightarrow \infty }\tau(t)=\infty $. Under the assumption that the $\tau $ is non-monotone, we present sufficient conditions for the oscillation of first-order delay dynamic equations on time scales. An example illustrating the result is also given.