Oscillation theorems for second-order nonlinear difference equations

In this paper, discrete inequalities are used to offer sufficient conditions for oscillation of all solutions of second-order nonlinear difference equations with alternating coefficients.     

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Oscillation theorems for certain delay partial difference equations

In this work we obtain some new oscillation criteria for certain delay partial difference equations with continuous arguments by some new techniques.     

Oscillation criteria for second-order nonlinear delay dynamic equations

In this paper, we consider the second-order nonlinear delay dynamic equation

(r(t)xΔ(t)Δ⁾+p(t)f(x(τ(t)))=0,     

Oscillation of second-order quasilinear neutral delay difference equations

By using the Riccati transformation and mathematical analytic methods,some sufficient conditions are obtained for oscillation of the second-order quasilinear neutral delay difference equations Δ[r n |Δz n | α-1 Δ z n ] + q n f (x n-σ)=0,where z n=x n + p n x n τ and ∞ Σ n=0 1 /r n 1/α < ∞.     

Oscillation criteria for second-order nonlinear neutral delay dynamic equations

In this paper we will establish some oscillation criteria for the second-order nonlinear neutral delay dynamic equation

(r(t)((y(t)+p(t)y(t−τ)Δ⁾γ⁾Δ⁾+f(t,y(t−δ))=0     

Oscillation theorems for second-order nonlinear differential equations with damped term

Several oscillation criteria are given for the second-order damped nonlinear differential equation (a(t)[y′(t)]^σ′i +p(t)[y′(t)]^σ +q(t)f(y(t)) = 0, where σ > 0 is any quotient of odd integers, a ϵ C(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [t_o, ∞), and f ϵ Cl (R, R) such that xf (x) > 0 for x≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.     

Oscillation theorems for second-order half-linear differential equations

Oscillation criteria for the second-order half-linear differential equation

[r(t)|ξ′(t)|⁻¹ ξ′(t)]′ + p(t)|ξ(t)|⁻¹ξ(t)=0, t t₀

are established, where > 0 is a constant and exists for t [t₀, ∞). We apply these results to the following equation:

where , D = (D₁,…, D_N), Ω_a = x ^N : |x| ≥ a} is an exterior domain, and c C([a, ∞), ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.     

Oscillation theorems for second-order functional differential equations

This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ₁, τ₂ be nonnegative continuous functions on (0, ∞) such that τ₁ ? τ₂ is a bounded function on [1, ∞) and t ? τ₁(t) → ∞ if t → ∞. Then $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_1\text{(t)) = 0}$ is oscillatory if and only if $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_2\text{(t)) = 0}$ is oscillatory.     

A remark for “Oscillation theorems for second-order nonlinear partial difference equations”

In this remark, we shall show a counter example for the main result of the paper [S.T. Liu, Y.Q. Liu, Oscillation theorems for second-order nonlinear partial difference equations, J. Comput. Appl. Math. 132 (2001) 479-482].     

Oscillation theorems for second-order superlinear neutral differential equations

The purpose of this paper is to investigate the oscillation of the second-order neutral differential equations of the form (E) $$ (r(t)|z'(t)|^{\alpha - 1} z'(t))' + q(t)|x(\sigma (t))|^{\alpha - 1} x(\sigma (t)) = 0, $$ where z(t) = x(t) + p(t)x(τ(t)). The obtained comparison principles essentially simplify the examination of the studied equations. Further, our results extend and improve the results in the literature.