Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

OSCILLATION THEOREMS FOR SECOND-ORDER MIXED-TYPE NEUTRAL DYNAMIC EQUATIONS ON SOME TIME SCALES

  • Sun, Jing (School of Information and Electronic Engineering, Shandong Institute of Business and Technology)
  • Received : 2011.05.26
  • Accepted : 2011.08.04
  • Published : 2012.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

Some oscillation results are presented for the second-order neutral dynamic equation of mixed type on a time scale unbounded above $$\(r(t)[x(t)+p_1(t)x(t-{\tau}_1)+p_2(t)x(t+{\tau}_2)]^{\Delta}\)^{\Delta}+q_1(t)x(t-{\tau}_3)+q_2(t)x(t+{\tau}_4)=0.$$ These criteria can be applied when $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=h{\mathbb{Z}}$ and $\mathbb{T}=\mathbb{P}_{a,b}$. Two examples are also provided to illustrate the main results.

Keywords

References

  1. R. P. Agarwal, S. R. Grace, and D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, Taylor and Francis, London, 2003.
  2. R. P. Agarwal, D. O'Regan, and S. H. Saker, Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl. 300 (2004), 203-217. https://doi.org/10.1016/j.jmaa.2004.06.041
  3. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, an introduction with applications, Birkhaauser, Boston, 2001.
  4. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
  5. E. Braverman and B. Karpuz, Nonoscillation of second-order dynamic equations with several delays, Abstr. Appl. Anal. 2011 (2011), 1-34.
  6. L. Erbe, T. S. Hassan, and A. C. Peterson, Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comput. 202 (2008), 343-357.
  7. J. K. Hale, Theory of Functional Differential Equations, Spring-Verlag, New York, 1977.
  8. S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56. https://doi.org/10.1007/BF03323153
  9. B. Karpuz, Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coeffcients, E. J. Qualitative Theory of Diff. Equ. 34 (2009), 1-14.
  10. T. Li, Comparison theorems for second-order neutral differential equations of mixed type, Electron. J. Diff. Equ. 167 (2010), 1-7.
  11. T. Li, B. Baculikova and J. Dzurina, Oscillation results for second-order neutral differential equations of mixed type, Tatra Mt. Math. Publ. 48 (2011), 101-116.
  12. T. Li, Z. Han, S. Sun and D. Yang, Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales, Adv. Differ. Equ. 2009 (2009), 1-10.
  13. T. Li, Z. Han, S. Sun and C. Zhang, Forced oscillation of second-order nonlinear dynamic equations on time scales, E. J. Qualitative Theory of Diff. Equ. 60 (2009), 1-8.
  14. E. Thandapani and V. Piramanantham, Oscillation criteria of second-order neutral delay dynamic equations with distributed deviating arguments, E. J. Qualitative Theory of Diff. Equ. 61 (2010), 1-15.