## Abstract and Applied Analysis

### Oscillation Results for Second-Order Nonlinear Damped Dynamic Equations on Time Scales

#### Abstract

This paper is concerned with second-order nonlinear damped dynamic equations on time scales of the following more general form ${(p(t){k}_{1}(x(t),{x}^{\Delta }(t)))}^{\Delta }+r(t){k}_{2}(x(t),{x}^{\Delta }(t)){x}^{\Delta }(t)+f(t,x(\sigma (t)))=0$. New oscillation results are given to handle some cases not covered by known criteria. An illustrative example is also presented.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 351256, 7 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425049098

Digital Object Identifier
doi:10.1155/2014/351256

Mathematical Reviews number (MathSciNet)
MR3198180

#### Citation

Qiu, Yang-Cong; Wang, Qi-Ru. Oscillation Results for Second-Order Nonlinear Damped Dynamic Equations on Time Scales. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 351256, 7 pages. doi:10.1155/2014/351256. https://projecteuclid.org/euclid.aaa/1425049098

#### References

• S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten [Ph.D. thesis], Universität Würzburg, 1988.
• 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.
• R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics. Resultate der Mathematik, vol. 35, no. 1-2, pp. 3–22, 1999.
• R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
• M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.
• M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.
• A. Del Medico and Q. Kong, “Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain,” Journal of Mathematical Analysis and Applications, vol. 294, no. 2, pp. 621–643, 2004.
• A. Del Medico and Q. Kong, “New Kamenev-type oscillation criteria for second-order differential equations on a measure chain,” Computers & Mathematics with Applications, vol. 50, no. 8-9, pp. 1211–1230, 2005.
• O. Došlý and S. Hilger, “A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 147–158, 2002, Dynamic equations on time scales.
• H. Huang and Q.-R. Wang, “Oscillation of second-order nonlinear dynamic equations on time scales,” Dynamic Systems and Applications, vol. 17, no. 3-4, pp. 551–570, 2008.
• Y.-C. Qiu and Q.-R. Wang, “Interval oscillation criteria of second-order nonlinear dynamic equations on time scales,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 952932, 16 pages, 2012.
• Y.-C. Qiu and Q.-R. Wang, “Kamenev-type oscillation criteria of second-order nonlinear dynamic equations on time scales,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 315158, 12 pages, 2013.
• A. Tiryaki and A. Zafer, “Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 1, pp. 49–63, 2005.
• Q.-R. Wang, “Oscillation criteria for nonlinear second order damped differential equations,” Acta Mathematica Hungarica, vol. 102, no. 1-2, pp. 117–139, 2004.
• Q.-R. Wang, “Interval criteria for oscillation of certain second order nonlinear differential equations,” Dynamics of Continuous, Discrete & Impulsive Systems, vol. 12, no. 6, pp. 769–781, 2005. \endinput