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Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients

Received: 19 November 2015     Accepted: 7 December 2015     Published: 29 January 2016
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Abstract

In this paper some necessary and sufficient conditions are obtained to guarantee the oscillation for bounded and all solutions of second order nonlinear neutral delay difference equations. In Theorem 5 and Theorem 8, We have studied the oscillation criteria as well as the asymptotic behavior, where was established some sufficient conditions to ensure that every solution are either oscillates or |yn |→∞ as n→∞. Examples are given to illustrate the obtained results.

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 1)
DOI 10.11648/j.pamj.20160501.12
Page(s) 9-14
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Oscillation, Neutral Difference Equations, Second Order Difference Equations

References
[1] S. N. Elaydi: An introduction to difference equations. Springer-Verlag New York, Inc., 2005.
[2] E. George Chatzarakis, G. Christos Philos, P. Ioannis Stavroulakis: On the oscillation of solutions to linear difference equation with variable delay, Electronic Journal of Differential Equations, Vol. (2008), No. 50, 1–15.
[3] I. Gyori, G. Ladas: Oscillation theory of delay differential equations with applications. Oxford Clarendon Press, 1991.
[4] Hasan ögunmez, Özkan Öcalan: Oscillations of difference equations with several positive and negative coefficients. Fasciculi mathematic, 51(2013), 115–122.
[5] G. Ladas: Oscillations of difference equations with positive and negative coefficients, Rocky Mountain J. of Math., 20(4) (1990), 1051–1061.
[6] R. N. Rath, L. N. Padhy: Necessary and sufficient conditions for oscillations of solutions of a first order forced nonlinear difference equation with several delays. Fasc. Math., 35(2005), 99–113.
[7] R. N. Rath, L. N. Padhy: Oscillations and non-oscillations of neutral difference equations of first order with positive and negative coefficients. Fasc. Math., 37(2007), 57–65.
[8] X. H. Tang, J. S. Yu, D. H. Peng: Oscillations and non- oscillations of neutral difference equations with positive and negative coefficients. Computers and mathematics application, 39 (2000), 169–181.
[9] A. K. Tripathy, S. Panigrahi: Oscillations in nonlinear neutral difference equation with positive and negative coefficients: International Journal of Difference Equations. Vol. 5, No. 2, (2010), 251–265.
[10] Xiaohui Gong, Xiaozhu Zhong, Jianqiang Jia, Rui Ouyang, Hongqiang Han: Oscillations of first order neutral difference equations. Modern Applied science, Vol. 3, No. 8, (2009), 90–94.
[11] Zhiguo Luo, Jianhua Shen: New Oscillation Criteria for Delay Difference Equations. Journal of Mathematical Analysis and Applications, 264 (2001), 85–95.
[12] W. R. Wade: An Introduction to Analysis. Third edition, Pearson Prentice Hall, 2004.
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  • APA Style

    Hussain Ali Mohamad, Hala Majid Mohi. (2016). Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients. Pure and Applied Mathematics Journal, 5(1), 9-14. https://doi.org/10.11648/j.pamj.20160501.12

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    ACS Style

    Hussain Ali Mohamad; Hala Majid Mohi. Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients. Pure Appl. Math. J. 2016, 5(1), 9-14. doi: 10.11648/j.pamj.20160501.12

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    AMA Style

    Hussain Ali Mohamad, Hala Majid Mohi. Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients. Pure Appl Math J. 2016;5(1):9-14. doi: 10.11648/j.pamj.20160501.12

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  • @article{10.11648/j.pamj.20160501.12,
      author = {Hussain Ali Mohamad and Hala Majid Mohi},
      title = {Oscillations of Neutral Difference Equations of Second Order with Positive and Negative Coefficients},
      journal = {Pure and Applied Mathematics Journal},
      volume = {5},
      number = {1},
      pages = {9-14},
      doi = {10.11648/j.pamj.20160501.12},
      url = {https://doi.org/10.11648/j.pamj.20160501.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160501.12},
      abstract = {In this paper some necessary and sufficient conditions are obtained to guarantee the oscillation for bounded and all solutions of second order nonlinear neutral delay difference equations. In Theorem 5 and Theorem 8, We have studied the oscillation criteria as well as the asymptotic behavior, where was established some sufficient conditions to ensure that every solution are either oscillates or |yn |→∞ as n→∞. Examples are given to illustrate the obtained results.},
     year = {2016}
    }
    

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    AB  - In this paper some necessary and sufficient conditions are obtained to guarantee the oscillation for bounded and all solutions of second order nonlinear neutral delay difference equations. In Theorem 5 and Theorem 8, We have studied the oscillation criteria as well as the asymptotic behavior, where was established some sufficient conditions to ensure that every solution are either oscillates or |yn |→∞ as n→∞. Examples are given to illustrate the obtained results.
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Author Information
  • Dept. of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq

  • Dept. of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq

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