Oscillatory and asymptotic behavior of solutions of third-order quasi-linear neutral difference equations

Abstract

Abstract By using comparison theorems, the authors investigate the oscillatory and asymptotic behavior of solutions to the third-order quasi-linear neutral difference equation $$\begin{array}{} \displaystyle \Delta\left(a(n)\left(\Delta^2 z(n) \right)^\alpha \right)+q(n)x^\beta(\sigma(n))=0, \end{array}$$ where z(n) = x(n) + px(n − k). Under less restrictive conditions on the coefficient functions and on the delay argument σ(n) than currently found in the literature, their criteria improve a number of known related results. The results are illustrated with examples.