### 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.**How to Cite this Article:**Özkan Öcalan, Umut Mutlu Ozkan, Mustafa Kemal Yildiz, Oscillatory solutions for dynamic equations with non-monotone arguments, Journal of Mathematical and Computational Science, Vol 7, No 4 (2017), 725-738 Copyright © 2017 Özkan Öcalan, Umut Mutlu Ozkan, Mustafa Kemal Yildiz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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