Interval criteria for oscillation of second-order half-linear differential equations

By employing a generalized Riccati technique and an integral averaging method, interval oscillation criteria are established for the second-order half-linear differential equation [r(t)|x′(t)|^α−1x′(t)]′+q(t)|x(t)|^α−1x(t)=0. These criteria are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line. They also extend, improve, and complement a number of existing results, and can be applied to extreme cases such as . In particular, several interesting examples that illustrate the importance of our results are included.     

Half-linear differential equations: Linearization technique and its application

Oscillatory properties of the half-linear second-order differential equation

(*)     

Adapted Riccati technique and non-oscillation of linear and half-linear equations

The aim of this paper is to mention a generalization of the adapted Riccati equation and, using this method, to prove a non-oscillatory result concerning half-linear differential equations with coefficients having mean values. Note that this result is new even for linear equations.     

Methods of oscillation theory of half-linear second order differential equations

In this paper we investigate oscillatory properties of the second order half-linear equation

Perturbations of the half-linear Euler-Weber type differential equation

We investigate oscillatory properties of the half-linear second order differential equation

     

Integral condition for oscillation of half-linear differential equations with damping

The purpose of this paper is to provide an oscillation theorem that can be applied to half-linear differential equations with time-varying coefficients. A parametric curve by the coefficients is focused in order to obtain our theorem. This parametric curve is a generalization of the curve given by the characteristic equation of the second-order linear differential equation with constant coefficients. The obtained theorem is proved by transforming the half-linear differential equation to a standard polar coordinates system and using phase plane analysis carefully.     

二阶非线性矩阵微分方程的振动性

Some new oscillation criteria are established for the second-order matrix differential system(r(t)Z′(t))′ p(t)Z′(t) Q(t)F(Z′(t))G(Z(t)) = 0, t ≥ to ＞ 0,are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.     

Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation ^′[p(t)Φ(x^′)]+q(t)Φ(x)=0, where Φ(u)=|u|^r−1u with r>0, 1/p and q are locally integrable on R₊; p>0, q?0 a.e. on R₊, and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p≡1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 0<r<1 and on oscillation when r>1. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373].     

New oscillation criteria of second-order nonlinear differential equations

By employing a class of new functions Φ=Φ(t,s,l) and a generalized Riccati technique, some new oscillation and interval oscillation criteria are established for the second-order nonlinear differential equation

(r(t)y^′(t))^′+Q(t,y(t),y^′(t))=0.     

具有散度形式的拟线性椭圆型微分方程的区域振动准则

对一类具有散度形式的拟线性椭圆型微分方程建立了若干新的振动准则,所得结果仅依赖于方程在外区域Ω(?)R~n的一个区域序列的信息而有别于已知的大多数结论.     

Ordinary differential equations in the oscillation theory of partial half-linear differential equation

In the paper we study the damped half-linear partial differential equation

     

Some new oscillation theorems for second order nonlinear elliptic equations with damping

In this paper, some new oscillation criteria are obtained for second order elliptic differential equations with damping of the form

     

Integral averaging technique and oscillation of certain even order delay differential equations

Using the generalized Riccati technique and the averaging technique, new oscillation criteria for certain even order delay differential equation of the form

     

Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations

We present new interval oscillation criteria related to integral averaging technique for certain classes of second-order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line. They generalize and improve some known results. Examples are also given to illustrate the importance of our results.     

Oscillation and Nonoscillation of Higher Order Self-Adjoint Differential Equations

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾+q(t)y = 0 (*) are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾ - µ,t^2n-y = 0, where _n, is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.     

Oscillation and Nonoscillation in Delay or Advanced Differential Equations and in Integrodifferential Equations

Some new oscillation and nonoscillation criteria are given for linear delay or advanced differential equations with variable coefficients and not (necessarily) constant delays or advanced arguments. Moreover, some new results on the existence and the nonexistence of positive solutions for linear integrodifferential equations are obtained.     

Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping

In this paper, we are concerned with the oscillation of third order nonlinear delay differential equations of the form

(r₂(t)(r₁(t)y^′)′)′+p(t)y^′+q(t)f(y(g(t)))=0.