## 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

#### References

• S. R. Grace, “On the oscillation of $n$th order dynamic equations on time-scales,” Mediterranean Journal of Mathematics, vol. 10, no. 1, pp. 147–156, 2013.
• S. R. Grace, R. P. Agarwal, and A. Zafer, “Oscillation of higher order nonlinear dynamic equations on time scales,” Advances in Difference Equations, vol. 2012, article 67, 2012.
• D. X. Chen, “Oscillation and asymptotic behavior for $n$th-order nonlinear neutral delay dynamic equations on time scales,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 703–719, 2010.
• R. Mert, “Oscillation of higher-order neutral dynamic equations on time scales,” Advances in Difference Equations, vol. 2012, article 68, 2012.
• L. Erbe, B. Jia, and A. Peterson, “Oscillation of $n$th order superlinear dynamic equations on time scales,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 2, pp. 471–491, 2011.
• B. Karpuz, “Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2174–2183, 2009.
• B. Karpuz, “Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 9, pp. 1–14, 2009.
• D. X. Chen and P. X. Qu, “Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales,” Journal of Applied Mathematics and Computing, vol. 44, no. 1-2, pp. 357–377, 2014.
• S. R. Grace, “On the oscillation of higher order dynamic equations,” Journal of Advanced Research, vol. 4, pp. 201–204, 2013.
• L. Erbe, B. Karpuz, and A. C. Peterson, “Kamenev-type oscillation criteria for higher-order neutral delay dynamic equations,” International Journal of Difference Equations, vol. 6, no. 1, pp. 1–16, 2011.
• T. Sun, H. Xi, and W. Yu, “Asymptotic behaviors of higher order nonlinear dynamic equations on time scales,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 177–192, 2011.
• T. X. Sun, H. J. Xi, and X. F. Peng, “Asymptotic behavior of solutions of higher-order dynamic equations on time scales,” Advances in Difference Equations, vol. 2011, Article ID 237219, 14 pages, 2011.
• Z. G. Zhang, W. L. Dong, Q. L. Li, and H. Y. Liang, “Existence of nonoscillatory solutions for higher order neutral dynamic equations on time scales,” Journal of Applied Mathematics and Computing, vol. 28, no. 1-2, pp. 29–38, 2008.
• S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
• M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001.
• R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics, vol. 35, no. 1-2, pp. 3–22, 1999.
• D. X. Chen, “Oscillation of second-order Emden-Fowler neutral delay dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1221–1229, 2010.
• G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1952.
• Y. Sun, “Interval oscillation criteria for second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals,” Abstract and Applied Analysis, vol. 2011, Article ID 719628, 14 pages, 2011. \endinput