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.     

Interval criteria for oscillation of second-order nonlinear differential equations

By employing the generalized Riccati technique and the integral averaging technique, new interval oscillation criteria are established for second-order nonlinear differential equations.     

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.     

New oscillation criteria for second-order nonlinear neutral delay differential equations

In this paper, several new oscillation criteria for the second-order nonlinear neutral delay differential equation

     

New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping

Some new oscillation criteria are established for the nonlinear damped differential equation (r(t)y′)′+p(t)y′+q(t)f(y)=0 that are different from most known ones in the sense that they are based on a class of new functions Φ(t,s,r) defined in the sequel. Our results are sharper than some previous results which can be seen by the examples at the end of this paper.     

Interval oscillation criteria for a forced second-order nonlinear differential equations with damping

In this paper, we are concerned with the oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using classical variational principle and averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. Example is also given to illustrate the results.     

New oscillation criteria for a class of second-order nonlinear forced differential equations

By using classical variational principle and averaging technique, new oscillation criteria are established for second-order nonlinear forced differential equations

     

Sharp oscillation criteria for second-order neutral delay differential equations

This paper is a continuation of a recent work by the authors on the oscillatory properties of second-order half-linear neutral delay differential equations. Providing a new apriori bound for a nonoscillatory solution, we present a new oscillation criterion, which essentially improves the existing ones. In a particular nonneutral case, the obtained oscillation constant is unimprovable.     

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.     

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 criteria for second-order nonlinear differential equations with damping

For a second-order nonlinear differential equation with a damping term, we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades. Two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations, and both sets of oscillation criteria established in the paper are of independent interest. Several examples are provided to illustrate the relevance of the new theorems.     

Oscillation criteria for nonlinear second-order differential equations with damping

Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x″+ p ( t ) x′+ q ( t ) f ( x ) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 694–700, May, 2008.     

On the oscillation of certain second-order nonlinear differential equations

In this paper, oscillation criteria for the nonlinear second-order ordinary differential equation

     

Oscillation criteria for second-order nonlinear differential equations with integrable coefficient

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

[a(t)|y′(t)|^σ−1y′(t)|′+q(t)f(y(t))=r(t)

where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.     

Oscillation criteria for forced second-order nonlinear differential equations with damping

In this paper, we are concerned with oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using a classical variational principle and an averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. An example is also given to illustrate the results.     

A remark on oscillation criteria for the solutions of nonlinear differential equations

Oscillation conditions are obtained for all the solutions of the equation L^ku + c(x, u)=f(x), where L is the Laplace-Beltrami operator in Euclidean and hyperbolic spaces.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 458–464, April, 1990.