Oscillation theorems for second order matrix differential systems

By employing the generalized Riccati technique and the integral averaging technique, new oscillation criteria are established for a class of second order matrix differential systems. These criteria extend, improve and unify a number of existing results and handle a number of cases not covered by known criteria. In particular, several interesting examples that illustrate the importance of our results are included. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Oscillation theorems for second order nonlinear differential equations

Some oscillation criteria are established for certain second order nonlinear differential equations of the form (a(t)ψ(x(t)) x^. (t))^. + p(t) x^. (t) + q(t)f(x(t)) = 0. These criteria improve upon some of the known results by Kura, Kamenev and Philos.     

Oscillation theorems for second order quasilinear perturbed differential equations

Abstract. New oscillation criteria for the second order perturbed differential equation are pre-sented. The special case of the results includes the corresponding results in previous papers,extends and unifies a number of known results.     

Oscillation theorems for certain second order self-adjoint matrix differential systems

By means of generalized averaging pair technique and Riccati transformation method, oscillation criteria for self-adjoint differential matrix system of the form

     

Oscillation of second order quasilinear neutral delay differential equations

In this paper, by employing Riccati transformation technique, some new sufficient conditions for the oscillation criteria are given for the second order quasilinear neutral delay differential equations with delayed argument in the form $$\bigl(r(t)\bigl|z'(t)\bigr|^{\alpha-1}z'(t)\bigr)'+q(t)f\bigl(x\bigl(\sigma(t)\bigr)\bigr)=0,\quad t\geq t_0,$$ where z(t)=x(t)?p(t)x(??(t)), 0??p(t)??p<1, lim _t???? p(t)=p ₁<1, q(t)>0, ??>0. Two examples are considered to illustrate the main results.     

二阶非线性椭圆型微分方程解的振动比较定理

建立几个微分不等式,讨论了一类二阶非线性椭圆型微分方程解的振动性,得到几个新的振动比较定理．     

Oscillation and wandering of solutions to a second order differential equation

The Lyapunov’s oscillation and wandering characteristics of solutions to a second order linear equation are defined, namely, the mean frequency of a solution, of its derivative or their various linear combinations, the mean angular velocity of the vector composed of a solution and its derivative, also wandering indices derived from that velocity. Nearly all of the values introduced for any equation are proved to be the same: just all for the autonomic equation (moreover, they coincide with the absolute values of the imaginary parts of the roots of the characteristic polynomial), but even for the periodic one, generally speaking, not all.     

Oscillation theorems for fourth‐order delay differential equations with a negative middle term

This paper deals with the oscillation of the fourth‐order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third‐order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained.     

Periodic solutions of a nonlinear second order differential equation with delay

It is proved that the autonomous difference-differential equation

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(0) has, for a broad class of functions f, a nonconstant periodic solution whenever the associated characteristic equation has a root with positive real part.     

Oscillation theorems for second order nonlinear dynamic equations

Some new criteria for the oscillation of nonlinear dynamic equations of the form $$\bigl(a(t)(x^{\Delta}(t))^{\alpha}\bigr)^{\Delta}+f(t,x^{\sigma}(t))=0$$ on a time scale $\mathbb{T}$ are established.     

Oscillation theorems for fourth order functional differential equations

The authors investigate the oscillatory behavior of all solutions of the fourth order functional differential equations $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})+q(t)f(x[g(t)])=0$ and $\frac{d^{3}}{dt^{3}}(a(t)(\frac{dx(t)}{dt})^{\alpha})=q(t)f(x[g(t)])+p(t)h(x[\sigma(t)])$ in the case where ∫ ^∞ a ^?1/α (s)ds<∞. The results are illustrated with examples.     

Oscillation of second order neutral delay differential equations of Emden-Fowler type

We present new oscillation criteria for the second order nonlinear neutral delay differential equation [y(t)-py(t-τ)]'+ q(t)y ^λ (g(t)) sgn y(g(t)) = 0, t ≧ t ₀. Our results solve an open problem posed by James S.W . Wong [24]. The relevance of our results becomes clear due to a carefully selected example. This revised version was published online in June 2006 with corrections to the Cover Date.