In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form ∆(pn−1∆yn−1) + qyn = 0, n ≥ 1, where q is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type ∆(pn−1∆yn−1) + qng(yn) = fn−1, n ≥ 1, where, unlike earlier works, fn > 0 or 6 0 (but 6≡ 0) for large n. Further, these results are used to obtain sufficient conditions for non-oscillation of all solutions of forced linear third order difference equations of the form yn+2 + anyn+1 + bnyn + cnyn−1 = gn−1, n ≥ 1.
We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations u ′′(x) +∑ i pi(x)u ′ (hi(x)) +∑ i qi(x)u(gi(x)) = 0 without the delay conditions hi(x), gi(x) ≤ x, i = 1, 2, . . ., and u ′′(x) + ∫ ∞ 0 u ′ (s)dsr1(x, s) + ∫ ∞ 0 u(s)dsr0(x, s) = 0.