Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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OSCILLATORY BEHAVIOR OF A CERTAIN CLASS OF SECOND-ORDER NONLINEAR PERTURBED DYNAMIC EQUATIONS ON TIME SCALES

  • Saker, Samir H. (DEPARTMENT OF MATHEMATICS COLLEGE OF SCIENCE KING SAUD UNIVERSITY, DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE MANSOURA UNIVERSITY)
  • Received : 2005.06.24
  • Published : 2010.07.01
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Abstract

This paper is concerned with the asymptotic behavior of solutions of the second-order nonlinear perturbed dynamic equation $$(r(t)x^{\Delta}(t))^{\Delta}\;+\;F(t,\;x^{\sigma}))=G(t,\;x^{\sigma},\;(x^{\Delta})^{\sigma})$$ on a time scale $\mathbb{T}$. By using a new technique we establish some sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the known oscillation results on the literature for the perturbed dynamic equations on time scales. Some examples illustrating our main results are given.

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References

  1. R. P. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, Dynamic equations on time scales: a survey, Dynamic equations on time scales. J. Comput. Appl. Math. 141 (2002), no. 1-2, 1-26.
  2. R. P. Agarwal, M. Bohner, and S. H. Saker, Oscillation of second order delay dynamic equations, Can. Appl. Math. Qurt 13 (2005), no. 1, 1-17.
  3. 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), no. 1, 203-217.
  4. E. A. Bohner, M. Bohner, and S. H. Saker, Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations, Electron. Trans. Numer. Anal. 27 (2007), 1-12.
  5. E. A. Bohner and J. Hoffacker, Oscillation properties of an Emden-Fowler type equation on discrete time scales, J. Difference Eqns. Appl. 9 (2003), no. 6, 603-612.
  6. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.
  7. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
  8. M. Bohner and S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math. 34 (2004), no. 4, 1239-1254.
  9. M. Bohner and S. H. Saker, Oscillation criteria for perturbed nonlinear dynamic equations, Math. Comput. Modelling 40 (2004), no. 3-4, 249-260. https://doi.org/10.1016/j.mcm.2004.03.002
  10. L. Erbe, Oscillation criteria for second order linear equations on a time scale, Canad. Appl. Math. Quart. 9 (2001), no. 4, 345-375.
  11. L. Erbe and A. Peterson, Riccati equations on a measure chain, Dynamic systems and applications, Vol. 3 (Atlanta, GA, 1999), 193-199, Dynamic, Atlanta, GA, 2001.
  12. L. Erbe and A. Peterson, Oscillation criteria for second-order matrix dynamic equations on a time scale, J. Comput. Appl. Math. 141 (2002), no. 1-2, 169-185.
  13. L. Erbe and A. Peterson, Boundedness and oscillation for nonlinear dynamic equations on a time scale, Proc. Amer. Math. Soc. 132 (2004), no. 3, 735-744.
  14. L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London Math. Soc. (2) 67 (2003), no. 3, 701-714.
  15. L. Erbe, A. Peterson, and S. H. Saker, Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. Comput. Appl. Math. 181 (2005), no. 1, 92-102.
  16. 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.
  17. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, 2nd Ed. Cambridge Univ. Press 1952.
  18. 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
  19. S. H. Saker, Oscillation of nonlinear dynamic equations on time scales, Appl. Math. Comput. 148 (2004), no. 1, 81-91. https://doi.org/10.1016/S0096-3003(02)00829-9
  20. S. H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005), no. 2, 375-387. https://doi.org/10.1016/j.cam.2004.09.028
  21. S. H. Saker, Boundedness of solutions of second-order forced nonlinear dynamic equations, Rocky Mountain J. Math. 36 (2006), no. 6, 2027-2039. https://doi.org/10.1216/rmjm/1181069359
  22. S. H. Saker, New oscillation criteria for second-order nonlinear dynamic equations on time scales, Nonlinear Funct. Anal. Appl. 11 (2006), no. 3, 351-370.
  23. 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

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