Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δ m

Abstract

Abstract In this article we obtain sufficient conditions for the oscillation of all solutions of the higher-order delay difference equation Δ m ( y n − ∑ j = 1 k p n j y n − m j ) + v n G ( y σ ( n ) ) − u n H ( y α ( n ) ) = f n , $$\begin{array}{} \displaystyle \Delta^{m}\big(y_n-\sum_{j=1}^k p_n^j y_{n-m_j}\big) + v_nG(y_{\sigma(n)})-u_nH(y_{\alpha(n)})=f_n\,, \end{array}$$ where m is a positive integer and Δ xn = x n+1 − xn . Also we obtain necessary conditions for a particular case of the above equation. We illustrate our results with examples for which it seems no result in the literature can be applied.