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

  • Authors

    • A. Murugesan DEPARTMENT OF MATHEMATICS,GOVERNMENT ARTS COLLEGE (AUTONOMOUS), SALEM - 636 007, TAMIL NADU, INDIA.
    • P. Sowmiya DEPARTMENT OF MATHEMATICS, GOVERNMENT ARTS COLLGE ( AUTONOMOUS), SALEM - 636007, TAMIL NADU, INDIA.
    2015-04-22
    https://doi.org/10.14419/ijams.v3i1.4462
  • Neutral, delay difference equation, oscillatory properties.
  • 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.

    Author Biography

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

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  • How to Cite

    Murugesan, A., & Sowmiya, P. (2015). Necessary and sufficient conditions for oscillations of first order neutral delay difference equations with constant coefficients. International Journal of Advanced Mathematical Sciences, 3(1), 12-24. https://doi.org/10.14419/ijams.v3i1.4462