New oscillation criteria for second order linear matrix differential equations with damping

By using the positive linear functional and the monotone subhomogeneous functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order linear matrix differential system
（P（t）X＇（t））＇ ＋ R（t）X＇（t） ＋ Q（t）X（t） =0, t ≥ to 〉 0
where P（t）, R（t）, Q（t） are n × n real continuous matrix functions, P（t） and R（t） are commutative. Theresults improve and generalize those given in some previous papers, which can be seen by the examples given at the end of this paper.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

Interval oscillation criteria for second order partial differential equations with delays

In this paper, we present new interval oscillation criteria related to integral averaging technique for second order partial differential equations with delays that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [_t0,∞)$[t_0,\infty )$, rather than on whole half-line. Our results are sharper than some previous results and handles the cases which are not covered by known criteria.     

An integral criterion for oscillation of linear differential equations of second order

It is proved that if for some n>2 the function x^1–nA_n(x), where A_n(x) is the n-th primitive ofa(x), is not bounded above, then the equation y +a(x)y = 0 oscillates.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 249–251, February, 1978.In conclusion, I thank R. S. Ismagilov for useful discussions about the problem of osillation.     

Necessary and sufficient conditions for oscillation of second order linear differential equations

Necessary and sufficient conditions are given for oscillations of second order linear differential equationx″+p(t)x′+q(t)x=0.     

一类二阶非齐次线性微分方程解的复振荡

研究了一类二阶非齐次线性微分方程f″+Ae~(az~n)f′+(B_1e~(bz~n)+B_0e~(dz~n))f=F(z)解的增长性和零点分布,其中F为级小于n的非零整函数,A,B1,B0为非零多项式.在复数a,b,d满足一定条件下,得到该方程的每一个解的超级和二级零点收敛指数的精确估计.     

Oscillation criteria for differential equations of second order

In this article, we give sufficient condition in the form of integral inequalities to establish the oscillatory nature of non linear homogeneous differential equations of the form

Oscillation criteria for second order nonlinear differential equations

Summary Various sufficient conditions are obtained which guarantee that all continuable solutions of (1.1) y″+q(t)y′+p(t)f(y)=0 are oscillatory. No explicit sign assumptions are made on p(t) although certain integral conditions are assumed to hold with regard to f(y), p(t) and q(t). Examples are given of the form p(t) = λ/t^μ + (βsint)/t^α, λ, β, μ, α>0. This research was supported by NRC Grant A-7673 and CMC Summer Research Grant. Entrata in Redazione il 4 settembre 1971.     

Nonoscillation criteria for quasilinear second order differential equations

Some nonoscillation criteria for quasilinear second order differential equations are obtained. These results generalize the classical results of Hille, Wintner and Opial and recent results of Elbert, Yan, Del Pino as well as Takasi and Yoshida.     

Nonoscillation criteria for second order nonlinear differential equations

A number of known nonoscillation criteria for the second order nonlinear differential equation y″ + q(x) y^γ = 0, γ > 0, where q is positive and locally of bounded variation, are improved by energy function techniques.     

Boundedness and stability criteria for linear ordinary differential equations of the second order

We establish some correlations for solutions of ordinary differential equations and the imaginary part of the complex solution of the corresponding Riccati equation. On the basis of these correlations and the I. M. Sobol’ theorem we prove some new stability and boundedness criteria for linear equations of the second order.     

Interval oscillation criteria for second order super‐half linear functional differential equations with delay and advanced arguments

Sufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the form where ?_δ (u) = |u |^{δ –1}u; α > 0, β ≥ α, and γ ≥ α are real numbers; k, p, q, e, τ, σ are continuous real‐valued functions; τ (t) ≤ t and σ (t) ≥ t with lim_{t →∞} τ (t) = ∞. The functions p (t), q (t), and e (t) are allowed to change sign, provided that p (t) and q (t) are nonnegative on a sequence of intervals on which e (t) alternates sign. As an illustrative example we show that every solution of is oscillatory provided that either m₁ or m₂ or r₀ is sufficiently large (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Oscillation criteria for second order nonlinear neutral differential equations

By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form

(r(t)(x(t)+p(t)x(t-τ))^′)^′+q(t)f(x(t),x(σ(t)))=0.     

Oscillation criteria for nonlinear second order damped differential equations

For all solutions of a class of second order nonlinear damped differential equations, new oscillation criteria are established. Asymptotic behavior for forced equations is also discussed.     

二阶中立型微分方程的振动准则

研究了一类具有连续偏差变元的二阶非线性中立型方程,得到了该类方程的振动准则     

Stability criteria for linear second order delay differential system

Consider second order delay differential system where r is a positive constant and all coefficients are real constants.Our main results are as follows:(1) The maximal length of the delay for which the stability of system (*) is maintained is given in the case where the zero solution of system (*) is asymptotically stable in the absence of delay.(2) The necessary and sufficient criteria for judging that asymptetical stability of system (*) is preserved for an arbitrary large delay are obtained.     

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].