OSCILLATION FOR NONLINEAR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES

Through the use of generalized Riccati transformation techniques, we establish some oscillation criteria for one type of nonlinear dynamic equation on time scales. Several examples, including a semilinear dynamic equation and a nonlinear Emden-Fowler dynamic equation, are also given to illustrate these criteria and to improve the results obtained in some references.     

一类二阶非线性阻尼微分方程的振动性

研究了一类二阶非线性阻尼微分方程解的振动性,建立了三个新的振动性定理,推广了Cecchi M和Marini M(Rocky Mount J Math,1992,22:1259-1276)的结果.     

二阶非线性阻尼常微分方程的振动性定理

考虑二阶非线性阻尼微分方程(α(t)ψ(x(t))x′(t))′ p(t)x′(t) q(t)f(x(t))=0 (1)和二阶非线性微分不等式x(t){(α(t)ψ(x(t))x′(t))′ p(t)x′(t) q(t)f(x(t))}≤0,(2)其中α,p,q∈C([t_0,∞)→(-∞,∞)),ψ,f∈C(R→R),并且α(t)>0,xf(x)>0 (x≠0).此外,我们总假设方程(1)的每一个解 x(t)可以延拓于[t_0, ∞)上.在任何无穷区间[T,∞)上,x(t)不恒等于零,这样的解叫正则解.一个正则解,若它有任意大的零点,则称为振动的;否则就称为非振动的.若方程(1)的所有正则解是振动的,则称方程(1)是振动的.关于不等式(2)的振动性的定义,与方程(1)的振动性的定义完全类似,不再赘述.     

OSCILLATION CRITERIA FOR SECOND ORDER QUASILINEAR DIFFERENTIAL EQUATIONS

1 IntroductionThis paper is concerned with the problem of oscillation of second-orderquasilinear differential equationsFOr equation (1), the following conditions, which are referred to (FO), arealways assumed to be valid(a) a > 0 is a constant;(b) F: [to, co) x R x R x R x R - R is a continuous function;(c) sgn F(t, x) u? v3 w) = sgn x for each t 2 to and x, u? v? w E R,(d) r(t) E C([to, co); (0, co)) and jco ddt = co.' (e) T: [to, co) - R is continuous, T(t) 5 t for t 2 to and ill7T(t)…     

AN OSCILLATION CRITERIA FOR SECOND ORDERFUNCTIONAL EQUATIONS

1 IntroductionThe oscillation theory of functional dtherelltial equations and dmerence eqllations 'withdiscrete arguments lias been extensively developed during the last few decades. Recently, theoscillatory behavior of difference equations with continuous argUnlents has beell studied illsome paPers. We are refer to [1,8,11,1115J. lll the above melltioned paPers tlie equationsunder consideration are called difference eqllatiolls with continuous argument (or continuousvariable or colltinuous …     

OSCILLATION CRITERIA FOR STRONGLY SUPERLINEAR AND STRONGLY SUBLINEAR HIGHER ORDER DYNAMIC EQUATIONS ON TIME SCALES

We establish some new criteria for the oscillation of even order nonlinear dynamic equation. We study the case of strongly super-linear and the case of strongly sub-linear subject to various conditions.     

二阶非线性扰动微分方程的振动性准则

本文主要研究了二阶非线性扰动微方程的振动性,所得的振动性准则包含和推广了一些现有的结果。     

OSCILLATION CRITERIA FOR SECONDORDER NONLINEAR NEUTRALDIFFERENTIAL EQUATIONS

1IntroductionConsiderthesecondordernonlinearneutraldelaydifferentialequationwheret2to,rando4renonnegativeconstants,a3p,qEC([to,co);R),fEC(R;R).Throughoutthispaper,weassumethatItiswellknownthatthesecondorderneutraldifferentialequationshavemanyapplicationsintheproblemsdealingwithsuchasvibratingmassesattachedtoanelasticbarandsomevariationalproblems(seeHale[1]).ManyauthorshasextensivelyinvestigatedtheoscillationofEq.(E)anditsspecialcases.Forexample,Horng-JaanLiandWei-LingLiuL2'3]andRuan[41…     

OSCILLATION CRITERIA FOR NONLINEAR SECOND ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

1.IntroductionOscillationtheoryforellipticdifferentialequationswithvariablecoefficielltshasbeenextensivelydevelopedinrecentyearsbymanyauthors(seee-g-,Allegretto[1,2l,Bugir['],Fiedler[5]'KitamuraandKusano['],Kural4]KusanoandNaitol1o]'NaitoandYoshidal12]'NoussairandSwanson[l3'14]'Swansonl15'l6]andthereferencescitedtherein).Inthispaper,weareconcernedwiththeoscillatorybehaviorofsolutionsofsecondorderellipticdifferelltialequatiollswithalternqtingcoefficients.Asusual,pointsinn-dimensionalEucli…     

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.     

OSCILLATION CRITERIA FOR EVEN ORDER DELAY DIFFERENTIAL EQUATIONS WITH NONLINEAR TERM

By Riccati transformation, we establish some new oscillation criteria for a class of even order delay differential equations with nonlinear term. In some sense, the results obtained extend some known results in the literature.     

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR NEUTRAL PERTURBED DYNAMIC EQUATIONS ON TIME SCALES

In this paper,we discuss the oscillatory behavior of a class of nonlinear neutral perturbed dynamic equations on time scales.Some new oscillation criteria are obtained for such dynamic equations.Finally,an example that dwells upon the sharp conditions of our result is also included.     

二阶非线性阻尼微分方程解的振动性质

本文研究了一类二阶非线性阻尼微分方程解的振动性,利用分类讨论和分析的方法,建立了两个新的振动性定理,推广和改进了已有的结果.     

OSCILLATION CRITERIA FOR SECOND ORDER DIFFERENCE EQUATIONS OF NEUTRAL TYPE WITH MIXED ARGUMENTS

In this paper, we establish some new oscillation criteria for second order difference equations of neutral type with mixed arguments. We extend some known oscillation criteria for second order difference equations.     

OSCILLATION THEOREMS FOR SECOND ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

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OSCILLATION THEOREMS FOR SECOND ORDER NEUTRAL DIFFERENCE EQUATIONS

The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equation to that of the first order equation.The comparison principles obtained essentially simplify the examination of the equations.     

OSCILLATION CRITERIA FOR A FOURTH ORDER SUBLINEAR DYNAMIC EQUATION ON TIME SCALE

Some new criteria for the oscillation of a fourth order sublinear and/or linear dynamic equation on time scale are established. Our results are new for the corresponding fourth order differential equations as well as difference equations.     

具高阶Laplace算子的非线性脉冲时滞双曲型方程的振动判据

研究一类具高阶Laplace算子的非线性脉冲时滞双曲型偏微分方程的振动性,利用特征函数法和一阶脉冲时滞微分不等式,获得了该类方程在两类不同边值条件下所有解振动的若干充分性判据.所得结论充分反映了脉冲和时滞在振动中的影响作用.     

COMPARISON THEOREMS AND OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH SEVERAL DELAYS

In this paper, we first establish the equivalence of the oscillation of thedifference equations with several delays of the form: Δx_n+sum from i=1 to m(pi(n)x_(n-k_i))=0 for n≥0and the second-order difference equations without delay of the form:where{pj(n)}is a sequence of nonnegative real numbers and{k_i}_(i=1)~m is a setof positive integers. Then we get some "sharp" conditions for oscillation andnon-oscillation of the first equation.     

EXISTENCE OF POSITIVE SOLUTIONS TO A THREE-POINT BOUNDARY VALUE PROBLEM FOR SECOND ORDER DYNAMIC EQUATIONS WITH DERIVATIVE ON TIME SCALES

In this paper, we consider the existence of positive solutions to a three-point boundary value problem for second order dynamic equations with derivative on time scales.Applying Leggett-Williams fixed point theorem, we obtain at least three positive solutions to the problem. An example is also presented to illustrate the applications of the results obtained.