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OSCILLATION OF NONLINEAR SECOND ORDER NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES

  • Agwo, Hassan A. (Department of Mathematics Ain Shams University Aculty of Education)
  • Published : 2008.05.31

Abstract

In this paper, we establish some oscillation criteria for nonautonomous second order neutral delay dynamic equations $(x(t){\pm}r(t)x({\tau}(t)))^{{\Delta}{\Delta}}+H(t,\;x(h_1(t)),\;x^{\Delta}(h_2(t)))=0$ on a time scale ${\mathbb{T}}$. Oscillatory behavior of such equations is not studied before. This is a first paper concerning these equations. The results are not only can be applied on neutral differential equations when ${\mathbb{T}}={\mathbb{R}}$, neutral delay difference equations when ${\mathbb{T}}={\mathbb{N}}$ and for neutral delay q-difference equations when ${\mathbb{T}}=q^{\mathbb{N}}$ for q>1, but also improved most previous results. Finally, we give some examples to illustrate our main results. These examples arc [lot discussed before and there is no previous theorems determine the oscillatory behavior of such equations.

References

  1. R. P. Agarwal, D. O'Regan, and S. H. Saker, Oscillation criteria for second-order non-linear neutral delay dynamic equations, J. Math. Anal. Appl. 300 (2004), no. 1, 203-217 https://doi.org/10.1016/j.jmaa.2004.06.041
  2. H. A. Agwo, On the oscillation of second order neutral delay dynamic equations with several delays and variable coefficients, Int. J. Appl. Math. Stat. 5 (2006), No. S06, 65-73
  3. E. Akin-Bohner, M. Bohner, and S. H. Saker, Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations, Electronic Transaction of Numerical Anal. (to appear)
  4. M. Bohner, L. Erbe, and A. Peterson, Oscillation for nonlinear second-order dynamic equations on a time scale, J. Math. Anal. Appl. 301 (2005), no. 2, 491-507 https://doi.org/10.1016/j.jmaa.2004.07.038
  5. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2003
  6. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkhauser, Boston, MA, 2001
  7. L. Erbe, A. Peterson, and S. H. Saker, Kamenev-type oscillation criteria for second-order linear delay dynamic equations, Dynam. Systems Appl. 15 (2006), no. 1, 65-78
  8. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations, With applications. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991
  9. J. Jiang and X. Li, Oscillation of second order nonlinear neutral differential equations, Appl. Math. Comput. 135 (2003), no. 2-3, 531-540 https://doi.org/10.1016/S0096-3003(02)00066-8
  10. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, 110. Marcel Dekker, Inc., New York, 1987
  11. X. Lin, Oscillation of second-order nonlinear neutral differential equations, J. Math. Anal. Appl. 309 (2005), no. 2, 442-452 https://doi.org/10.1016/j.jmaa.2004.08.023
  12. S. H. Saker, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, J. Comput. Appl. Math. 187 (2006), no. 2, 123-141 https://doi.org/10.1016/j.cam.2005.03.039
  13. J. S. W. Wong, Necessary and sufficient conditions for oscillation of second-order neutral differential equations, J. Math. Anal. Appl. 252 (2000), no. 1, 342-352 https://doi.org/10.1006/jmaa.2000.7063
  14. B. G. Zhang and Z. Shanliang, Oscillation of second-order nonlinear delay dynamic equations on time scales, Comput. Math. Appl. 49 (2005), no. 4, 599-609 https://doi.org/10.1016/j.camwa.2004.04.038
  15. H. A. Agwo, On the oscillation of second order nonlinear neutral delay dynamic equations, (to appear)
  16. S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18-56 https://doi.org/10.1007/BF03323153
  17. Y. Sahiner, On oscillation of second order neutral type delay differential equations, Appl. Math. Comput. 150 (2004), no. 3, 697-706 https://doi.org/10.1016/S0096-3003(03)00300-X

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  3. Necessary and Sufficient Conditions for Positive Solutions of Second-Order Nonlinear Dynamic Equations on Time Scales vol.2011, 2011, https://doi.org/10.1155/2011/769893
  4. Nonlinear oscillation of second-order neutral dynamic equations with distributed delay vol.39, pp.2, 2016, https://doi.org/10.1002/mma.3469
  5. Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales vol.2010, pp.1, 2010, https://doi.org/10.1186/1687-1847-2010-450264
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