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Oscillation of Second-Order Nonlinear Neutral Dynamic Equations with Noncanonical Operators

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Abstract

The study of half-linear differential equations has become an important area of research due to the fact that such equations occur in a variety of real-world problems such as in the study of \(p\)-Laplace equations, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. On the basis of these background details, we study oscillatory behavior of a class of second-order neutral functional dynamic equations on a time scale. New criteria improve and complement related results reported in the literature. Some examples are included to illustrate the results obtained. In particular, an example regarding the second-order neutral differential equation is also provided to show that these theorems improve those in the continuous case.

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References

  1. Agarwal, R.P., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141, 1–26 (2002)

  2. Agarwal, R.P., Bohner, M., Tang, S., Li, T., Zhang, C.: Oscillation and asymptotic behavior of third-order nonlinear retarded dynamic equations. Appl. Math. Comput. 219, 3600–3609 (2012)

    Article  MathSciNet  Google Scholar 

  3. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  4. Agarwal, R.P., O’Regan, D., Saker, S.H.: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J. Math. Anal. Appl. 300, 203–217 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Akın-Bohner, E., Bohner, M., Saker, S.H.: Oscillation criteria for a certain class of second order Emden–Fowler dynamic equations. Electron. Trans. Numer. Anal. 27, 1–12 (2007)

    MATH  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  8. Bohner, M., Saker, S.H.: Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mt. J. Math. 34, 1239–1254 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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 

  10. Chen, D.: Oscillation of second-order Emden–Fowler neutral delay dynamic equations on time scales. Math. Comput. Model. 51, 1221–1229 (2010)

  11. Chen, D.: Bounded oscillation of second-order half-linear neutral delay dynamic equations. Bull. Malays. Math. Sci. Soc. 36, 807–823 (2013)

  12. Erbe, L., Hassan, T.S., Peterson, A.: Oscillation criteria for nonlinear functional neutral dynamic equations on time scales. J. Differ. Equ. Appl. 15, 1097–1116 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grace, S.R., Agarwal, R.P., Bohner, M., O’Regan, D.: Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3463–3471 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grace, S.R., Bohner, M., Agarwal, R.P.: On the oscillation of second-order half-linear dynamic equations. J. Differ. Equ. Appl. 15, 451–460 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Han, Z., Li, T., Sun, S., Sun, Y.: Remarks on the paper [Appl. Math. Comput. 207 (2009), 388–396]. Appl. Math. Comput. 215, 3998–4007 (2010)

  16. Han, Z., Li, T., Sun, S., Zhang, C.: On the oscillation of second-order neutral delay dynamic equations on time scales. Afr. Dias. J. Math. 9, 76–86 (2010)

    MATH  MathSciNet  Google Scholar 

  17. 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 

  18. 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 

  19. Hilger, S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Karpuz, B.: Asymptotic behavior of bounded solutions of a class of higher-order neutral dynamic equations. Appl. Math. Comput. 215, 2174–2183 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, T., Agarwal, R.P., Bohner, M.: Some oscillation results for second-order neutral dynamic equations. Hacet. J. Math. Stat. 41, 715–721 (2012)

    MATH  MathSciNet  Google Scholar 

  22. Li, T., Han, Z., Sun, S., Yang, D.: Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales. Adv. Differ. Equ. 2009, 1–10 (2009)

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  24. Şahiner, Y.: Oscillation of second order neutral delay and mixed type dynamic equations on time scales. Adv. Differ. Equ. 2006, 1–9 (2006)

    Google Scholar 

  25. Saker, S.H.: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J. Comput. Appl. Math. 187, 123–141 (2006)

  26. Saker, S.H.: Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales. Nonlinear Oscil. 13, 407–428 (2011)

    Article  MathSciNet  Google Scholar 

  27. Saker, S.H., Agarwal, R.P., O’Regan, D.: Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales. Appl. Anal. 86, 1–17 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Saker, S.H., Agarwal, R.P., O’Regan, D.: Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales. Acta Math. Sin. 24, 1409–1432 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Saker, S.H., O’Regan, D.: New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution. Commun. Nonlinear Sci. Numer. Simul. 16, 423–434 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Spedding, V.: Taming nature’s numbers. New Scientist 179, 28–31 (2003)

    Google Scholar 

  31. Sun, S., Li, T., Han, Z., Li, H.: Oscillation theorems for second-order quasilinear neutral functional differential equations. Abstr. Appl. Anal. 2012, 1–17 (2012)

    Google Scholar 

  32. Sun, S., Li, T., Han, Z., Zhang, C.: On oscillation of second-order nonlinear neutral functional differential equations. Bull. Malays. Math. Sci. Soc. 36, 541–554 (2013)

  33. Tang, S., Gao, C., Li, T.: Oscillation theorems for second-order quasi-linear delay dynamic equations. Bull. Malays. Math. Sci. Soc. 36, 907–916 (2013)

  34. Tripathy, A.K.: Some oscillation results for second order nonlinear dynamic equations of neutral type. Nonlinear Anal. 71, 1727–1735 (2009)

    Article  MathSciNet  Google Scholar 

  35. Tripathy, A.K.: Riccati transformation and sublinear oscillation for second order neutral delay dynamic equations. J. Appl. Math. Inf. 30, 1005–1021 (2012)

    MATH  MathSciNet  Google Scholar 

  36. Wu, H., Zhuang, R., Mathsen, R.M.: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Appl. Math. Comput. 178, 321–331 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Xu, R., Meng, F.: Some new oscillation criteria for second order quasi-linear neutral delay differential equations. Appl. Math. Comput. 182, 797–803 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yang, Q., Xu, Z.: Oscillation criteria for second order quasilinear neutral delay differential equations on time scales. Comput. Math. Appl. 62, 3682–3691 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zhang, C., Li, T., Agarwal, R.P., Bohner, M.: Oscillation results for fourth-order nonlinear dynamic equations. Appl. Math. Lett. 25, 2058–2065 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhang, S., Wang, Q.: Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl. Math. Comput. 216, 2837–2848 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research is supported by NNSF of P. R. China (Grant Nos. 61034007, 51277116, 50977054). The authors express their sincere gratitude to the Editors for useful comments that helped to accentuate important details.

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Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, Shandong, People’s Republic of China

    Chenghui Zhang & Tongxing Li

  2. Department of Mathematics, Texas A&M University–Kingsville, 700 University Blvd., Kingsville, TX, 78363-8202, USA

    Ravi P. Agarwal

  3. Department of Mathematics and Statistics, Missouri S&T, Rolla, MO, 65409-0020, USA

    Martin Bohner

Authors
  1. Chenghui Zhang
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  2. Ravi P. Agarwal
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  3. Martin Bohner
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  4. Tongxing Li
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Correspondence to Tongxing Li.

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Communicated by Shangjiang Guo.

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Zhang, C., Agarwal, R.P., Bohner, M. et al. Oscillation of Second-Order Nonlinear Neutral Dynamic Equations with Noncanonical Operators. Bull. Malays. Math. Sci. Soc. 38, 761–778 (2015). https://doi.org/10.1007/s40840-014-0048-2

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  • DOI: https://doi.org/10.1007/s40840-014-0048-2

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