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 we study the oscillation of the difference equations of the form Δ 2 xn+PnΔxn + f(n, Xn-ff, Δx n-h) = 0, in comparison with certain difference equations of order one whose oscillatory character is known. The results can be applied to the difference equation Δ 2 xn+pnΔxn +q n |x-_g|λ|Δxn -h |μ sgnx„-9 = 0, where A and \i are real constants, λ > 0 and μ ≥ 0.
Various new criteria for the oscillation of nonlinear neutral difference equations of the form Δi (xn — x n - h) + qn\xn~g\c sgns n -9 =0 , i = 1,2,3 and c > 0, are established.
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.
The asymptotic and oscillatory behavior of solutions of Volterra summation equations yn = pn ± n−1 ∑ s=0 K(n, s)f(s, ys), n ∈ where = {0, 1, 2,...}, are studied. Examples are included to illustrate the results.