In this paper we consider the third-order nonlinear delay differential equation (∗) (a(t) x ′′(t) ) γ ) ′ + q(t)x γ (τ (t)) = 0, t ≥ t0, where a(t), q(t) are positive functions, γ > 0 is a quotient of odd positive integers and the delay function τ (t) 6 t satisfies lim t→∞ τ (t) = ∞. We establish some sufficient conditions which ensure that (∗) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.
In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form yn+3 + rnyn+2 + qnyn+1 + pnyn = 0, n ≥ 0. These results are generalization of the results concerning difference equations with constant coefficients yn+3 + ryn+2 + qyn+1 + pyn = 0, n ≥ 0. Oscillation, nonoscillation and disconjugacy of a certain class of linear third order difference equations are discussed with help of a class of linear second order difference equations.