## Abstract

We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: $(p(t){\left|{x}^{{\mathrm{\Delta}}^{n-1}}(t)\right|}^{\alpha -1}{x}^{{\mathrm{\Delta}}^{n-1}}(t){)}^{\mathrm{\Delta}}+f(t,x(\delta (t)))\hspace{0.17em}+\hspace{0.17em}{\int}_{a}^{\sigma (b)}k(t,s){\left|x(g(t,s))\right|}^{\theta (s)}\text{s}\text{g}\text{n}(x(g(t,s)))\mathrm{\Delta}\xi (s)=0,$ where $t\in [{t}_{0},\mathrm{\infty}{)}_{\mathbb{T}}:=[{t}_{0},\mathrm{\infty})\cap \mathbb{T}$, $\mathbb{T}$ a time scale which is unbounded above, $n\u2a7e2$ is even, $\left|f(t,u)\left|\u2a7eq(t)\right|{u}^{\alpha}\right|$, $\alpha >0$ is a constant, and $\theta :[a,b{]}_{{\mathbb{T}}_{1}}\to \mathbb{R}$ is a strictly increasing right-dense continuous function; $p,q:[{t}_{0},\mathrm{\infty}{)}_{\mathbb{T}}\to \mathbb{R}$, $k:[{\mathrm{t}}_{0},\mathrm{\infty}{)}_{\mathbb{T}}\times [a,b{]}_{{\mathbb{T}}_{1}}\to \mathbb{R}$, $\delta :[{t}_{0},\mathrm{\infty}{)}_{\mathbb{T}}\to [{t}_{0},\mathrm{\infty}{)}_{\mathbb{T}}$, and $g:[{t}_{0},\mathrm{\infty}{)}_{\mathbb{T}}\times [a,b{]}_{{\mathbb{T}}_{1}}\to [{t}_{0},\mathrm{\infty}{)}_{\mathbb{T}}$ are right-dense continuous functions; $\xi :[a,b{]}_{{\mathbb{T}}_{1}}\to \mathbb{R}$ is strictly increasing. Our results extend and supplement some known results in the literature.

## Citation

Haidong Liu. Cuiqin Ma. "Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals." Abstr. Appl. Anal. 2014 1 - 8, 2014. https://doi.org/10.1155/2014/395381