Oscillation of third-order quasilinear neutral dynamic equations on time scales with distributed deviating arguments

Volume 17, Issue 1, pp 41-52 http://dx.doi.org/10.22436/jmcs.017.01.04

Publication Date: March 15, 2017 Submission Date: April 20, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2049 Downloads
• 3537 Views

Authors

M. Tamer Senel - Department of Mathematics, Faculty of Sciences, Erciyes University, 38039, Kayseri, Turkey. Nadide Utku - Institute of Sciences, Erciyes University, 38039, Kayseri, Turkey.

Abstract

The aim of this paper is to give oscillation criteria for the third-order quasilinear neutral delay dynamic equation \begin{equation*} \bigg[r(t)\big([x(t)+p(t)x(\tau_{0}(t))]^{\Delta\Delta}\big)^{\gamma}\bigg]^{\Delta}+\int_{c}^{d}q_{1}(t)x^{\alpha}(\tau_{1}(t,\xi))\Delta\xi+\int_{c}^{d}q_{2}(t)x^{\beta}(\tau_{2}(t,\xi))\Delta\xi=0, \end{equation*} on a time scale \(\mathbb{T}\), where \(0<\alpha<\gamma<\beta\). By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.

Share and Cite

Keywords

• Oscillation
• third order quasilinear neutral dynamic equation with distributed deviating arguments
• time scales.

MSC

•  34K11
•  34C11

References

• [1] R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: a survey, Dynamic equations on time scales, J. Comput. Appl. Math., 141 (2002), 1–26.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] R. P. Agarwal, D. O’Regan, S. H. Saker, Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl., 300 (2004), 203–217.
  • View Article
  • Google Scholar


• [3] M. Bohner, A. Peterson, Dynamic equations on time scales, An introduction with applications, Birkhäuser Boston, Inc., Boston, MA (2001)


• [4] T. Candan, Oscillation of second-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments, Comput. Math. Appl., 62 (2011), 4118–4125.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] T. Candan, Asymptotic properties of solutions of third-order nonlinear neutral dynamic equations, Adv. Difference Equ., 2014 (2014 ), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] S. R. Grace, J. R. Graef, M. A. El-Beltagy, On the oscillation of third order neutral delay dynamic equations on time scales, Comput. Math. Appl., 63 (2012), 775–782.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] Z.-L. Han, T.-X. Li, S.-R. Sun, C.-H. Zhang, Oscillation behavior of third-order neutral Emden-Fowler delay dynamic equations on time scales, Adv. Difference Equ., 2010 (2010), 23 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge (1988)


• [9] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] T.-X. Li, Z.-L. Han, S.-R. Sun, Y. Zhao, Oscillation results for third order nonlinear delay dynamic equations on time scales, Bull. Malays. Math. Sci. Soc., 34 (2011), 639–641.
  • MathSciNet
  • Google Scholar


• [11] T.-X. Li, Z.-L. Han, C.-H. Zhang, Y. Sun, Oscillation criteria for third-order nonlinear delay dynamic equations on time scales, Bull. Math. Anal. Appl., 3 (2011), 52–60.
  • MathSciNet
  • Google Scholar


• [12] S. H. Saker, J. R. Graef, Oscillation of third-order nonlinear neutral functional dynamic equations on time scales, Dynam. Systems Appl., 21 (2012), 583–606.
  • MathSciNet
  • Google Scholar


• [13] M. T. Şenel, N. Utku, Oscillation behavior of third-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments, Filomat, 28 (2014), 1211–1223.
  • View Article
  • MathSciNet
  • MATH


• [14] N. Utku, M. T. Şenel, Oscillation behavior of third-order quasilinear neutral delay dynamic equations on time scales, Filomat, 28 (2014), 1425–1436.
  • View Article
  • MathSciNet
  • MATH


• [15] C.-H. Zhang, S. H. Saker, T.-X. Li , Oscillation of third-order neutral dynamic equations on time scales, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20 (2013), 333–358.
  • Google Scholar