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.
  • 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.

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  • 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