NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATION OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS

1 IntroductionRecently there wele a lot of activity concerning the oscillatory behavior ofsolutions of delay differential equations. We refer to the monographs [1,2,3,4].In particular, the oscillations and nonoscillations of solutions of the first orderneutral linear differential equation[x(t) -- p(t)x(t -- T)]' Q(t)x(t -- a) = 0 (*)have been investigated by many authors.In this paper, we consider the following first order nonlinear neutral diffe-rential equationwhereT) a E (0, co), al(t) …     

非线性二阶中立型微分方程的振动性

考虑中立型微分方程我们建立了方程(1)的若干振动准则。这些结果适用于强超线性和强次线性中立型微分方程,且当p(t)≡0时仍然成立。     

一类四阶差分方程振动性的充分必要条件

本文给出了四阶差分方程△＾２（ｒｎ△＾２ｙｎ）＋ｆ（ｎ,ｙｎ）＝０,ｎ＝０,１,２,…的所有解（有界解）振动的一些充分条件和必要条件。     

Asymptotic properties of a new Sturm–Liouville problem with retarded argument

In this work, we study the asymptotic properties of a new Sturm–Liouville problem with retarded argument. Contrary to previous works, differential equation includes eigenparameter as a quadratic function. In the considered problem arise new difficulties. Copyright © 2013 John Wiley & Sons, Ltd.     

OSCILLATION CRITERIA OF SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS OF SECOND ORDER

Oscillatory behavior of solutions of second order nonlinear difference equation is studied. Oscillation criteria for its solutions are given. Examples are given in the text to illustrate the results.     

一类二阶微分方程周期解的存在性

研究二阶非线性滞后型微分方程x。(t)+P[x(。t)]+Q[x(。t)]R[x(t-r)]=f(t)通过Lyaponov方法给出了ω-周期解的存在性定理和时滞范围的简明表达式,推广了一些原有结果.     

关于运用Liapunov函数的滞后型系统部分稳定性的注释

Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.     

OSCILLATION OF CERTAIN SECOND ORDER DIFFERENCE EQUATIONS

1IntroductionConsiderasec0nd0rderdifferenceequationwherethef0rwarddifference0perat0rAisdefinedbyAx.=xn 1-x.,{ri}2',isarealp0sitivesequence,ffN(n0)xR-Risc0ntinuousinxERf0reverynEN(no),xf(n,x)>Oforx/0,nEN(n0 1),N(s)isdefinedbyN(s)={s,s l,'-},seN(n0).Byasolution0f(E)wemeanarealsequence{I.}7',satisfying(E).Asolutionof(E)iscal1edn0n0scil1at0ry,ifitiseventua1lypositive0rnegative,otherwiseitiscalledoscillat0ry.Eq.(E)isca1led0scil1atoryifa11so1utions-of(E)are0scillat0ry.Definiti0n1Thefuncti…     

On Differential Equations with Retarded Argument

We study the limit properties of solutions to one class of systems of differential equations as the number of equations and some parameters tend to infinity. We establish a close connection between solutions to these systems and differential equations with retarded argument. Our convergence theorems provide a new method for approximation of solutions to nonlinear differential equations with retarded argument.Original Russian Text Copyright © 2005 Demidenko G. V. and Likhoshvai V. A.The authors were supported by the Russian Foundation for Basic Research (Grants 03-01-00328, 03-04-48506, 03-04-48829, 03-07-96833, 04-01-00458, 05-04-49068, and 05-07-90274), the Integration Grant of the Siberian Branch of the Russian Academy of Sciences (No. 119 and 148), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8273), and NSF:FIBR (Grant EF-0330786).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 538–552, May–June, 2005.     

具分段常数变元的脉冲微分方程周期解的存在性

通过构造差分方程的周期数列解，研究了一类具有分段常数变元的脉冲微分方程周期解的存在性．     

Asymptotic formulations of the eigenvalues and eigenfunctions for a boundary value problem

In this work, a discontinuous boundary‐value problem with retarded argument that contains a spectral parameter in the transmission conditions at the point of discontinuity is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. Copyright © 2012 John Wiley & Sons, Ltd.     

On a Sturm–Liouville type problem with retarded argument

In this work, a Sturm–Liouville‐type problem with retarded argument, which contains spectral parameter in the boundary conditions and with transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. Copyright © 2012 John Wiley & Sons, Ltd.     

具偏差变元的Rayleigh方程周期解问题

利用Mawhin重合度拓展定理研究了一类具偏差变元的Rayleigh方程x”(t)+f(x'(t))+g(x(t-τ(t)))=p(t)周期解问题,得到了周期解存在性的若干新的结果,推广了已有的结果(见文[8]).     

OSCILLATIONS OF FIRST ORDER NONLINEAR RETARDED DIFFERENTIAL EQUATIONS

ＯＳＣＩＬＬＡＴＩＯＮＳＯＦＦＩＲＳＴＯＲＤＥＲＮＯＮＬＩＮＥＡＲＲＥＴＡＲＤＥＤＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＷａｎｇＱｉｒｕ（王其如）（ＬｕｏｙａｎｇＴｅａｃｈｅｒｓ＇Ｃｏｌｌｅｇｅ，洛阳师范高等专科学校，邮编：４７１０２２）Ａｂｓ...     

Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 62–68, January, 1998.     

一类强退化拟线性抛物方程的弱解

In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.     

一类具复杂偏差变元的微分方程的周期解存在性

应用重合度理论研究了一类具复杂偏差变元的二阶Liénard微分方程的周期解存在性问题,改进和推广了以往文献的相关结果.     

A regularized trace formula and oscillation of eigenfunctions of a Sturm‐Liouville operator with retarded argument at 2 points of discontinuity

In this study, first, a formula for regularized sums of eigenvalues of a Sturm‐Liouville problem with retarded argument at 2 points of discontinuity which contains a spectral parameter in the boundary conditions is obtained. After that, oscillation properties of the related problem is investigated. Finally, under the condition that a subset of nodal points is dense in definition set, the potential function is determined.