## Abstract and Applied Analysis

### Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

#### Abstract

We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: $(p(t){|{x}^{{\mathrm{\Delta }}^{n-1}}(t)|}^{\alpha -1}{x}^{{\mathrm{\Delta }}^{n-1}}(t){)}^{\mathrm{\Delta }}+f(t,x(\delta (t))) + {\int }_{a}^{\sigma (b)}k(t,s){|x(g(t,s))|}^{\theta (s)}\text{s}\text{g}\text{n}(x(g(t,s)))\mathrm{\Delta }\xi (s)=0,$ where $t\in [{t}_{0},\mathrm{\infty }{)}_{\mathrm{\Bbb T}}:=[{t}_{0},\mathrm{\infty })\cap \mathrm{\Bbb T}$, $\mathrm{\Bbb T}$ a time scale which is unbounded above, $n{\geqslant}2$ is even, $|f(t,u)|{\geqslant}q(t)|{u}^{\alpha }|$, $\alpha >0$ is a constant, and $\theta :[a,b{]}_{{\mathrm{\Bbb T}}_{1}}\to \Bbb R$ is a strictly increasing right-dense continuous function; $p,q:[{t}_{0},\mathrm{\infty }{)}_{\mathrm{\Bbb T}}\to \Bbb R$, $k:[{\mathrm{t}}_{0},\mathrm{\infty }{)}_{\mathrm{\Bbb T}}{\times}[a,b{]}_{{\mathrm{\Bbb T}}_{1}}\to \Bbb R$, $\delta :[{t}_{0},\mathrm{\infty }{)}_{\mathrm{\Bbb T}}\to [{t}_{0},\mathrm{\infty }{)}_{\mathrm{\Bbb T}}$, and $g:[{t}_{0},\mathrm{\infty }{)}_{\mathrm{\Bbb T}}{\times}[a,b{]}_{{\mathrm{\Bbb T}}_{1}}\to [{t}_{0},\mathrm{\infty }{)}_{\mathrm{\Bbb T}}$ are right-dense continuous functions; $\xi :[a,b{]}_{{\mathrm{\Bbb T}}_{1}}\to \Bbb R$ is strictly increasing. Our results extend and supplement some known results in the literature.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 395381, 8 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273243

Digital Object Identifier
doi:10.1155/2014/395381

Mathematical Reviews number (MathSciNet)
MR3186963

Zentralblatt MATH identifier
07022304

#### Citation

Liu, Haidong; Ma, Cuiqin. Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals. Abstr. Appl. Anal. 2014 (2014), Article ID 395381, 8 pages. doi:10.1155/2014/395381. https://projecteuclid.org/euclid.aaa/1412273243