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An Improved Oscillation Criteria For First Order Dynamic Equations, Özkan Öcalan

Turkish Journal of Mathematics

In this work, we consider the first-order dynamic equations \begin{equation*} x^{\Delta }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}} \end{equation*} 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) $ and $\tau (t)\leq t,\ \lim_{t\rightarrow \infty }\tau (t)=\infty $. When the delay term $\tau (t)$ is not necessarily monotone, we present a new sufficient condition for the oscillation of first-order delay dynamic equations on time scales.