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 this note we consider the third order linear difference equations of neutral type (E) ∆ 3 [x(n) − p(n)x(σ(n))] + δq(n)x(τ (n)) = 0, n ∈ N(n0), where δ = ±1, p, q : N(n0) → ℝ+; σ, τ : N(n0) → ℕ, lim n→∞ σ(n) = lim n→∞ τ (n) = ∞. We examine the following two cases: {0 < p(n) ≤ 1, σ(n) = n + k, τ (n) = n + l}, {p(n) > 1, σ(n) = n − k, τ (n) = n − l}, where k, l are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.