Abstract and Applied Analysis

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

Yang-Cong Qiu and Qi-Ru Wang

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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 Δ ( t ) ) ) Δ + r ( t ) k 2 ( x ( t ) , x Δ ( t ) ) x Δ ( t ) + f ( t , x ( σ ( 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

Permanent link to this document
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


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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