Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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RICCATI TRANSFORMATION AND SUBLINEAR OSCILLATION FOR SECOND ORDER NEUTRAL DELAY DYNAMIC EQUATIONS

  • Received : 2011.11.20
  • Accepted : 2012.02.17
  • Published : 2012.09.30
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Abstract

This work is concerned with oscillation of the second order sublinear neutral delay dynamic equations of the form $$\(r(t)\;\((y(t)+p(t)y(a(t)))^{\Delta}\)^{\gamma}\)^{\Delta}+q(t)y^{\gamma}({\beta}(t))=0$$ on a time scale $\mathcal{T}$ by means of Riccati transformation technique, under the assumptions $\int^{\infty}_{t_0}\(\frac{1}{r(t)}\)^{\frac{1}{\gamma}}$ ${\Delta}t={\infty}$ and $\int^{\infty}_{t_0}\(\frac{1}{r(t)}\)^{\frac{1}{\gamma}}$ ${\Delta}t$ < ${\infty}$, where 0 < ${\gamma}{\leq}1$ is a quotient of odd positive integers.

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References

  1. R.P. Agarwal, M. Bohner, D.O'Regan, A. Peterson; Dynamic equations on time scales : a survey, J. Compu. Appl. Math. Vol. 141, No. 1-2(2002), pp. 1-26. https://doi.org/10.1016/S0377-0427(01)00432-0
  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), pp. 203-217. https://doi.org/10.1016/j.jmaa.2004.06.041
  3. M. Bohner, A. Peterson; Dynamic equations on Time scales : An Introduction with Applications, Birkhauser, Boston, 2001.
  4. M. Bohner, A. Peterson; Advances in Dynamic Equations on Time scales : Birkhauser, Boston, 2003.
  5. L. Erbe, T.S. Hassan, A. Peterson, S.H. Saker : Oscillation criteria for sub-linear half linear delay dynamic equations, Inter. J. Difference Equn. No. 2, 3 (2008), pp. 227-245.
  6. Z. Han, B. Shi, S. Sun : Oscillation criteria for second order delay dynamic equations on Time scales, Adv. Differene Equn. ID 70730 (2007), pp. 1-16.
  7. T.S. Hassan; Oscillation criteria for half-linear dynamic equations on time scales : J. Math. Anal. Appl. 345(2008), pp. 176-185. https://doi.org/10.1016/j.jmaa.2008.04.019
  8. S. Hilger; Analysis on measure chains - a unified approach to continuous and discrete calculus, Results in Mathematics, Vol.18, No. 1-2(1990), pp. 18-56. https://doi.org/10.1007/BF03323153
  9. Y. Sahiner; Oscillation of second order delay differential equations on time scales, Nonlinear Analysis : Theory, Methods and Applications, Vol.-63, No. 5-7 (2005), pp. e1073-e1080. https://doi.org/10.1016/j.na.2005.01.062
  10. S.H. Saker; Oscillation of second order nonlinear neutral delay dynamic equations on time scales, J. Comp. Appl. Math. Vol.-187, No. 2(2006), pp. 123-141. https://doi.org/10.1016/j.cam.2005.03.039
  11. S.H.Saker, D.O'Regan; New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution, Commun Nonlinear Sci Numer Simulat 16(2011), 423-434. https://doi.org/10.1016/j.cnsns.2009.11.032
  12. A.K. Tripathy; Some oscillation results for second order nonlinear dynamic equations of neutral type, Nonlinear Analysis : Theory, Methods and Applications, 71(2009), pp. e1727- e1735. https://doi.org/10.1016/j.na.2009.02.046
  13. A.K. Tripathy, G.B. Tenali; Oscillation results for second order neutral delay dynamic equations,J. Functional Differential Equations,17(2010),pp.329-344.
  14. B.G. Zhang, S. Zhu; Oscillation of second order nonlinear delay dynamic equations on time scales, Compu. Math. Appl. Vol. 49, No.4 (2005), pp. 599-609. https://doi.org/10.1016/j.camwa.2004.04.038