In the paper we consider the difference equation of neutral type (E) ∆3 [x(n) − p(n)x(σ(n))] + q(n)f(x(τ (n))) = 0, n ∈ N(n0 ), where p, q : N(n0 ) → R+; σ, τ : N → Z, σ is strictly increasing and lim n→∞ σ(n) = ∞; τ is nondecreasing and lim n→∞ τ (n) = ∞, f : R → R, xf(x) > 0. We examine the following two cases: 0 < p(n) ≤ λ ∗ < 1, σ(n) = n − k, τ (n) = n − l, and 1 < λ∗ ≤ p(n), σ(n) = n + k, τ (n) = n + l, where k, l are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as n → ∞ with a weaker assumption on q than the usual assumption ∑∞ i=n0 q(i) = ∞ that is used in literature.
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.
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.