Necessary and sufficient conditions for oscillation of first order neutral delay difference equations

  • Authors

    • A. Murgesan DEPARTMENT OF MATHEMATICS,GOVERNMENT ARTS COLLEGE (AUTONOMOUS), SALEM - 636 007, TAMIL NADU, INDIA.
    • P. Sowmiya DEPARTMENT OF MATHEMATICS, GOVERNMENT ARTS COLLGE (AUTONOMOUS), SALEM - 636 007, TAMIL NADU, INDIA.
    2015-04-25
    https://doi.org/10.14419/gjma.v3i2.4560
  • Oscillatory behavior, Neutral, Delay difference equation, Constant coefficients.

  • Abstract

    In this paper, we obtained some necessary and sufficient conditions for oscillation of all the solutions of the first order neutral delay difference equation with constant coefficients of the form
    \begin{equation*} \quad \quad \quad \quad \Delta[x(n)-px(n-\tau)]+qx(n-\sigma)=0, \quad \quad n\geq n_0 \quad \quad \quad \quad \quad \quad {(*)} \end{equation*}
    by constructing several suitable auxiliary functions. Some examples are also given to illustrate our results.

    Author Biography

    • A. Murgesan, DEPARTMENT OF MATHEMATICS,GOVERNMENT ARTS COLLEGE (AUTONOMOUS), SALEM - 636 007, TAMIL NADU, INDIA.
      ASSISTANT PROFESSOR, DEPARTMENT OF MATHEMATICS

  • References

    1. [1] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York, 1999.

      [2] R.P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer, Dodrecht, 1997.

      [3] M. P. Chen, B. S. Lalli and J. S. Yu, Oscillation in neutral delay difference equations with variable coefficients, Comput. Math. Appl. 29(3) (1995), 5-11.

      [4] S. N. Elaydi, An Introduction to Difference Equations, Springer Verlag, New York, 1996.

      [5] Ethiraju Thandapani, Ramalingam Arul and Palanisamy S. Raja, Oscillation of first order neutral delay difference equations, Appl. Math. E-Notes, 3(2003), 88-94.

      [6] D. A. Georgiou, E. A. Grove and G. Ladas, Oscillations of neutral difference equations, Appl. Anal. 33(1989), 243-253.

      [7] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.

      [8] W. G. Kelley and A. C. Peterson, Difference Equations: An introduction with Applications, Clarendon Press, Oxford 1991.

      [9] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations, Academic Press, New york, 1988.

      [10] B. S. Lalli and B. G. Zhang, On existence of positive solutions and bounded oscillations for neutral difference equations, J. Math. Anal. Appl. 166(1992), 272-287.

      [11] B. S. Lalli and B. G. Zhang, Oscillation and comparison theorems for certain neutral difference equations, J. Austral. Math. Soc. Ser. B34(1992), 245-256.

      [12] G. Ladas, Explicit conditions for the oscillation of difference equations, J. Math. Anal. Appl. 153(1990), 276-287.

      [13] zkan calan, Oscillation criteria for systems of difference equations with variable coefficients, Appl. Math. E-Notes, 6(2006), 119-125.

      [14] X. H. Tang and Xiaoyan Lin, Necessary and sufficient conditions for oscillation of first - order nonlinear neutral difference equations, Comput. Math. Appl. 55(2008), 1279-1292.

      [15] Xiaohui Gong, Xiaozhu zhong, Jianqiang Jia, Rui Ouyang and Hongqiang Han, Oscillation of first order Neutral Difference Equation, Modern Applied Science, 3(8) (2009), 90-94.

      [16] Ying Gao and Guang Zhang, Oscillation of nonlinear first order neutral difference equations, Appl. Math. E-Notes, 1(2001), 5-10.

      [17] Z. Zhou and J. S. Yu, Linearized oscillations for difference equations of neutral type, Math. Sci. Res. Hot-Line 1(11)(1997), 1-8.

  • Downloads

  • How to Cite

    Murgesan, A., & Sowmiya, P. (2015). Necessary and sufficient conditions for oscillation of first order neutral delay difference equations. Global Journal of Mathematical Analysis, 3(2), 61-72. https://doi.org/10.14419/gjma.v3i2.4560

    Received date: 2015-03-29

    Accepted date: 2015-04-20

    Published date: 2015-04-25