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Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales

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Abstract

This paper is concerned with the oscillatory properties of even order advanced type dynamic equation with mixed nonlinearities of the form

$$\bigl[r(t)\varPhi_\alpha\bigl(x^{\Delta^{n-1}}(t) \bigr) \bigr]^\Delta+ p(t)\varPhi_\alpha\bigl(x\bigl(\delta(t)\bigr) \bigr) +\sum_{i=1}^kp_i(t) \varPhi_{\alpha_i} \bigl(x\bigl(\delta(t)\bigr) \bigr)=0 $$

on an arbitrary time scale \(\mathbb{T}\), where Φ ∗(u)=|u|∗−1 u. We present some new oscillation criteria for the equation by introducing parameter functions, establishing a new lemma, using a Hardy-Littlewood-Pólya inequality and an arithmetic-geometric mean inequality and developing a generalized Riccati technique. Our results extend and supplement some known results in the literature. Several examples are given to illustrate our main results.

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References

  1. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  2. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

  3. Agarwal, R.P., Zafer, A.: Oscillation criteria for second order forced dynamic equations with mixed nonlinearities. Adv. Differ. Equ. 2009, 1–20 (2009)

    MathSciNet  Google Scholar 

  4. Agarwal, R.P., Anderson, D.R., Zafer, A.: Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities. Comput. Math. Appl. 59, 977–993 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Grace, S.R., Agarwal, R.P., Zafer, A.: Oscillation of higher order nonlinear dynamic equations on time scales. Adv. Differ. Equ. 2012, 1–18 (2012)

    Article  MathSciNet  Google Scholar 

  6. Grace, S.R.: On the oscillation of nth-order dynamic equations on time scales. Mediterr. J. Math. (2013). doi:10.1007/s00009-012-0201-9

    MathSciNet  Google Scholar 

  7. Sun, Y.G., Wong, J.S.W.: Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities. J. Math. Anal. Appl. 337, 549–560 (2007)

    Article  MathSciNet  Google Scholar 

  8. Agarwal, R.P., Bohner, M.: Basic calculus on time scales and some of its applications. Results Math. 35, 3–22 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hardy, G.H., Littlewood, J.E., Pólya, G.P.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  10. Beckenbach, E.F., Bellman, R.: Inequalities. Springer, Berlin (1961)

    Book  Google Scholar 

  11. Erbe, L., Hassan, T.S., Peterson, A.: Oscillation of third order nonlinear functional dynamic equations on time scales. Differ. Equ. Dyn. Syst. 18, 199–227 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zafer, A.: On oscillation and nonoscillation of second-order dynamic equations. Appl. Math. Lett. 22, 136–141 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Karpuz, B.: Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients. Electron. J. Qual. Theory Differ. Equ. 2009, 1–14 (2009)

    MathSciNet  Google Scholar 

  14. Braverman, E., Karpuz, B.: Nonoscillation of first-order dynamic equations with several delays. Adv. Differ. Equ. 2010, 1–22 (2010)

    Article  MathSciNet  Google Scholar 

  15. Şenel, M.T.: Kamenev-type oscillation criteria for the second-order nonlinear dynamic equations with damping on time scales. Abstr. Appl. Anal. 2012, 1–18 (2012)

    Google Scholar 

  16. Mert, R.: Oscillation of higher-order neutral dynamic equations on time scales. Adv. Differ. Equ. 2012, 1–16 (2012)

    Article  MathSciNet  Google Scholar 

  17. Zhang, C., Li, T., Agarwal, R.P., Bohner, M.: Oscillation results for fourth-order nonlinear dynamic equations. Appl. Math. Lett. (2012). doi:10.1016/j.aml.2012.04.018

    MathSciNet  Google Scholar 

  18. Saker, S.H., Grace, S.R.: Oscillation criteria for quasi-linear functional dynamic equations on time scales. Math. Slovaca 62, 501–524 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Candan, T.: Oscillation of second-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments. Comput. Math. Appl. 62, 4118–4125 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hassan, T.S.: Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales. Appl. Math. Comput. 217, 5285–5297 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, T., Han, Z., Sun, S., Zhao, Y.: Oscillation results for third order nonlinear delay dynamic equations on time scales. Bull. Malays. Math. Soc. 34(2), 639–648 (2011)

    MATH  MathSciNet  Google Scholar 

  22. Thandapani, E., Piramanantham, V., Pinelas, S.: Oscillation criteria for second-order neutral delay dynamic equations with mixed nonlinearities. Adv. Differ. Equ. 2011, 1–14 (2011)

    Article  MathSciNet  Google Scholar 

  23. Han, Z., Li, T., Sun, S., Zhang, C., Han, B.: Oscillation criteria for a class of second-order neutral delay dynamic equations of Emden-Fowler type. Abstr. Appl. Anal. 2011, 1–26 (2011)

    Google Scholar 

  24. Ünal, M., Zafer, A.: Oscillation of second-order mixed-nonlinear delay dynamic equations. Adv. Differ. Equ. 2010, 1–21 (2010)

    Article  Google Scholar 

  25. Lin, Q., Jia, B., Wang, Q.: Forced oscillation of second-order half-linear dynamic equations on time scales. Abstr. Appl. Anal. 2010, 1–10 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of P.R. China (Grant No. 11271311) and the Natural Science Foundation of Hunan of P.R. China (Grant No. 11JJ3010).

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Authors and Affiliations

  1. College of Science, Hunan Institute of Engineering, 88 East Fuxing Road, Xiangtan, 411104, P.R. China

    Da-Xue Chen

  2. School of Information & Engineering, Henan Institute of Science and Technology, Xinxiang, 453003, P.R. China

    Pei-Xin Qu

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  1. Da-Xue Chen
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Correspondence to Da-Xue Chen.

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Chen, DX., Qu, PX. Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales. J. Appl. Math. Comput. 44, 357–377 (2014). https://doi.org/10.1007/s12190-013-0697-6

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  • DOI: https://doi.org/10.1007/s12190-013-0697-6

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