Necessary and sufficient conditions for oscillations of first order neutral delay difference equations with constant coefficients

  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract

    In this paper, we establish the necessary and sufficient conditions for oscillation of the following first order neutral delay difference equation 
    \begin{equation*} \quad \quad \quad \quad \quad \quad \quad \quad \quad\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*}
    where \(\tau\) and \(\sigma\) are positive integers, \(p\neq 0\) is a real number and \(q\) is a positive real number. We proved that every solution of (*) oscillates if and only if its characteristic equation
    \begin{equation*}\quad \quad \quad \quad\quad \quad \quad \quad\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\lambda-1)(1+p\lambda^{-\tau})+q\lambda^{-\sigma}=0\quad \quad \quad \quad \quad \quad \quad \quad {(**)} \end{equation*}
    has no positive roots.

  • Keywords

    Neutral; delay difference equation; oscillatory properties.

  • References

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

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

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

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

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

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

      [9] 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.

      [10] 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.

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

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




Article ID: 4462
DOI: 10.14419/ijams.v3i1.4462

Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.