Forced oscillation of second-order nonlinear differential equations with positive and negative coefficients

In this paper we give new oscillation criteria for forced super- and sub-linear differential equations by means of nonprincipal solutions.     

Existence of positive solutions of second-order nonlinear neutral differential equations with positive and negative terms

In this paper, the existence of bounded positive solution of the second-order nonlinear neutral differential equations are studied. Some new sufficient conditions are given. Furthermore, examples given show that the results here are almost sharp.     

具有正负系数的中立型微分方程的振动性

研究具有正负系数的中立型微分方程（x(t)-Cx(t-r)‘ px(t-τ)-qx(t-σ)=0,在允许C q(τ-σ)≤1不成立的条件下,建立了方程（*）的振动性准则。     

Oscillatory behavior of the second-order nonlinear neutral difference equations

In this paper, we consider the oscillation of the second-order neutral difference equation $$\Delta ^2 \left( {x_n - px_{n - \tau } } \right) + q_n f\left( {x_{n - \sigma _n } } \right) = 0$$ as well as the oscillatory behavior of the corresponding ordinary difference equation $$\Delta ^2 z_n + q_n f\left( {R\left( {n,\lambda } \right)z_n } \right) = 0$$ .     

On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients

In this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem.     

正负系数中立型差分方程的全局吸引性

A neutral difference equation with positive and negative coefficients△(xn-cnxn-k) pnxn-l-qnxn-r=0,n=0,1,2.…，is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.     

正负系数的扰动的中立型微分方程的稳定性

In this paper the perturbed neutral differential equation with positive and negative coefficients d/dt[x(t)-C(t)x(t-r)] p(t)x(t-x)-Q(t)x(t-δ)=f(t,x(t)),t≥t0is considered. Sufficient conditions for the zero solution of this equation to be uniformly stable as well as asymptotically stable are obtained.     

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

正负系数中立型差分方程的正解的存在性

In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions in the literature     

Oscillations of solutions of neutral differential equations with positive and negative coefficients

We obtain some new results for oscillation of all solutions of the neutral differential equation with positive and negative coefficients

where P, Q, R ϵ C([t₀, ∞), R), r ϵ (0, ∞, and τ, σ ϵ (0, ∞). These new results are obtained by establishing and using some new lemmas which are interesting in their own right and which may have further applications in analysis.     

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     

Existence for positive solutions of second-order neutral nonlinear differential equations

We consider the nonlinear neutral differential equations. This work contains some sufficient conditions for the existence of a positive solution which is bounded with exponential functions. The case when the solution converges to zero is also treated.     

Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients

Sufficient conditions are established for the asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients

     

Oscillation of second-order nonlinear neutral dynamic equations on time scales

This paper is concerned with the oscillation of second-order nonlinear neutral dynamic equations of the form

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

Oscillatory of unstable type second-order neutral difference equations

We consider the problem of oscillation and nonoscillation solutions for unstable type second-order neutral difference equation: Δ²(x(n) − p(n)x(n −τ)) =q(n)x(g(n)). (1) In this paper, we obtain some conditions for the bounded solutions of Eq(1) to be oscillatory and for the existence of the nonoscillatory solutions.     

Oscillation of higher order neutral functional difference equations with positive and negative coefficients

Sufficient conditions are obtained so that every solution of the neutral functional difference equation

Oscillation of neutral delay difference equations of second order with positive and negative coefficients

This paper is concerned with a class of neutral difference equations of second order with positive and negative coefficients of the forms

Oscillation criteria for second order neutral dynamic equations of Emden-Fowler type with positive and negative coefficients on time scales

We investigate oscillation of certain second order neutral dynamic equations of Emden-Fowler type with positive and negative coefficients. We use some different techniques and apply Riccati transformation to establish new oscillatory criteria which include two necessary and sufficient conditions. Moreover, we point out that how the power γ plays its role. Some interesting examples are given to illustrate the versatility of our results.