Sufficient Conditions for the Oscillation of Odd-Order Neutral Dynamic Equations

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Basak Karpuz, Sufficient Conditions for the Oscillation of Odd-Order Neutral Dynamic Equations, Int. J. Anal. Appl., 14 (1) (2017), 69-76.

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Basak Karpuz

Abstract

In this paper, we study oscillation and asymptotic behaviour of odd-order delay dynamic equations. We first state an oscillation test for odd-order nonneutral equations, then by comparison we provide sufficient conditions for all solutions of neutral equations to be oscillatory or tend to zero depending on two main ranges of the neutral coefficient.

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References

  1. R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999), no. 1-2, 3-22.
  2. M. Bohner, Some oscillation criteria for first order delay dynamic equations, Far East J. Appl. Math. 18 (2005), no. 3, 289-304.
  3. M. Bohner, B. Karpuz and O. Ocalan, Iterated oscillation criteria for delay dynamic equations of first order, Adv. Difference Equ. 2008 (2008), Art.ID. 458687.
  4. M. Bohner and A.C. Peterson, Dynamic Equations on Time Scales An Introduction with Applications, Birkhäuser Boston, Inc., Boston, MA, 2001.
  5. E. Braverman and B. Karpuz, Nonoscillation of first-order dynamic equations with several delays, Adv. Difference Equ. 2010 (2010), Art.ID. 873459.
  6. Y. S ¸ahiner and I.P. Stavroulakis, Oscillations of first order delay dynamic equations, Dynam. Systems Appl. 15 (2006), no. 3-4, 645-655.
  7. P. Das, Oscillation criteria for odd order neutral equations, J. Math. Anal. Appl.,188 (1994), no. 1, 245-257.
  8. L. Erbe, R. Mert, A.C. Peterson, and A. Zafer, Oscillation of even order nonlinear delay dynamic equations on time scales, Czechoslovak Math. J. 63 (2013), no. 138(1), 265-279.
  9. L.H. Erbe, G. Hovhannisyan, and A.C. Peterson, Asymptotic behavior of n-th order dynamic equations, Nonlinear Dyn. Syst. Theory 12 (2012), no. 1, 63-80.
  10. L.H. Erbe, B. Karpuz, and A.C. Peterson, Kamenev-type oscillation criteria for higher-order neutral delay dynamic equations, Int. J. Difference Equ. 6 (2011), no. 1, 1-16.
  11. S.R. Grace, R.P. Agarwal, and A. Zafer, Oscillation of higher order nonlinear dynamic equations on time scales, Adv. Difference Equ. 2012 (2012), Art. ID. 67.
  12. S.R. Grace, R. Mert, and A. Zafer, Oscillatory behavior of higher-order neutral type dynamic equations, Electron. J. Qual. Theory Differ. Equ. 2013 (2013), Art. ID. 29.
  13. B. Karpuz, Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations, Appl. Math. Comput. 215 (2009), no. 6, 2174-2183.
  14. B. Karpuz, Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients, Electron. J. Qual. Theory Differ. Equ. 2009 (2009), Art.ID. 34.
  15. B. Karpuz, Existence and uniqueness of solutions to systems of delay dynamic equations on time scales, Int. J.Math. Comput. 10 (2011), no. M11, 48-58.
  16. B. Karpuz, Li type oscillation theorem for delay dynamic equations, Math. Methods Appl. Sci. 36 (2013), no. 9, 993-1002.
  17. B. Karpuz, Sufficient conditions for the oscillation and asymptotic behaviour of higher-order dynamic equations of neutral type, Appl. Math. Comput. 221 (2013), 453-462.
  18. B. Karpuz, Comparison tests for the asymptotic behaviour of higher-order dynamic equations of neutral type, Forum Math. 27(2015), no. 5, 2759-2773.
  19. B. Karpuz and O. Ocalan, New oscillation tests and some refinements for first-order delay dynamic equations, Turkish J. Math. 40 (2016), no. 4, 850-863.
  20. R. Mert, Oscillation of higher-order neutral dynamic equations on time scales, Adv. Difference Equ. 2012 (2012), Art. ID. 68.
  21. A.D. Myˇ skis, Linear homogeneous differential equations of the first order with retarded argument, Uspehi Matem. Nauk (N.S.) 5 (1950), no. 2(36), 160-162.
  22. X. Wu, T.X. Sun, H.J. Xi, and C.H. Chen, Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales, Adv. Difference Equ. 2013 (2013), Art.ID. 248.
  23. J. Yang, S. Liu, and X.K. Hou, Oscillation and existence of nonoscillatory solutions of forced higher-order neutral dynamic systems on time scales, Pure Appl. Math. (Xi'an) 25 (2009), no. 4, 665-670.
  24. B.G. Zhang and X.H. Deng, Oscillation of delay differential equations on time scales, Math. Comput. Modelling 36 (2002), no. 11-13, 1307-1318.