Oscillation of bounded solutions of higher order nonlinear neutral difference equations (Q2910166)

This paper is concerned with the oscillation properties of \(m^{th}\) order neutral difference equations of the form NEWLINE\[NEWLINE\Delta^{m}[x(n)+cx(\tau(n))]+q(n)f(x(\sigma(n))=0, n\geq n_0.NEWLINE\]NEWLINE Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Especially, the oscillation criteria obtained in this paper improves the corresponding results due to \textit{R. P. Agarwal} et al. [Appl. Math. Lett. 10, No. 1, 71--78 (1997; Zbl 0890.39019)]. Finally, the author studied the equation when a certain type of forcing term is present and \(c\) is a sequence \(c(n)\). Several examples are inserted to illustrate the results of the paper.