Oscillation condition for first order linear dynamic equations on time scales

Abstract

In this paper, we deal with the first-order dynamic equations withnonmonotone arguments\begin{equation*}y^{\Delta }(t)+\underset{i=1}{\overset{m}{\sum }}p_{i}(t)y\left( \tau_{i}(t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}}\end{equation*}where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},%\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{%T}},\mathbb{T}\right) $ and $\tau _{i}(t)\leq t,\ \lim_{t\rightarrow \infty}\tau _{i}(t)=\infty $ for $1\leq i\leq m$. Also, we present a new sufficient condition for theoscillation of delay dynamic equations on time scales. Finally, we give anexample illustrating the result.