Non-oscillation of solutions of difference equations of third order

Abstract

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form y(n + 3) + α(n)y(n + 2) + β(n)y(n + 1) + γ (n)y(n) = 0, n ≥ 0, 13y(n − 1) + a(n)12y(n − 1) + b(n)1y(n) + c(n)y(n) = 0, n ≥ 1, and 1(p(n − 1)12y(n − 1)) + q(n)1y(n) + r(n)y(n) = 0, n ≥ 1, where γ (n) ̸ = 0 and p(n) > 0. The technique developed depends on non-oscillation of certain linear homogeneous second order difference equations.