具有分布时滞的二阶非线性中立型时标动力方程的振动定理

利用广义Riccati变换技术, 研究时标上具有分布时滞的二阶非线性中立型动力方程 \[\big(r(t)\big(\big(y(t)+p(t)y(\tau(t))\big)^{\it \Delta}\big)^\beta\big)^{\it \Delta}+\int_c^dF(t,\xi,y(\delta(t,\xi))){\it \Delta}\xi=0 \]的振动性, 其中$\beta>0$是两个正奇数之比, 获得了方程所有解振动的几个充分条件,推广和改进了一些已知的结果, 并给出了几个应用实例.     

OSCILLATION THEOREMS FOR DELAY AND NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Oscillation criteria for the delay differential equationx′(t)+p(t)x(t-т(t))=0,where p, т are non-negative real-valued continuous functions are investigated inthe case when the numbers k=integral from n=t-т(t) to t(p(s)ds),L=integral from n=t-т(t) to t(p(s)ds)satisfy 0≤k<1/e and 1/e≤L<1. The present result improves almost allresults of the literature concerning it. Furthermore, it is established that allsolutions of the odd-order neutral delay differential equation(x(t)-px(t-т))~(n)+Q(t)x(t-σ)=0,where 0≤p<1,т,σ∈(0,∞)and Q(t)≥0,are oscillatory ifintegral from n=t-σ to t ((t-s)~(n-1)Q(s)ds>(1/e)(1-p)(n-1))!.This result generalizes a theorem of Gopalsamy et. al (Czech. Math. J., 42(1992),313-323)and also extends a very well-known result of Ladas (ApplicableAnalysls, 9(1979),93-98).     

CONDITIONS FOR OSCILLATION OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper,we study one type of odd-order neutral delay differential equation. Suffcient conditions for the oscillation of every solution to this type of differential equation are given.     

测度链上一类三阶半线性中立型时滞动力方程的振动准则

利用广义Riccati代换及时标上一些分析技巧给出时标上一类三阶半线性动力方程的振动准则,并利用例子说明其结果的可用性,推广了一些已知的二阶或三阶动力方程振动性的结果.     

OSCILLATION CRITERIA FOR SECOND-ORDER NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES

In this paper, we consider a class of second-order neutral delay dynamic equations on a time scale T. By means of Riccati transformation technique, we establish some new oscillation criteria in two different conditions. The obtained results enrich the well-known oscillation results for some dynamic equations.     

OSCILLATION RESULTS FOR A SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Some new oscillation criteria are established for a second order neutral delay differential equations. These results improve oscillation results of Y.V. Rogo-vchenko for the retarded delay differential equations. The relevance of our theorems is illustrated with two carefully selected examples.     

二阶非线性中立型微分方程的振动性

利用广义Riccati变换和权函数积分平均技巧,建立了非线性时滞中立型方程:[r(t)(x(t)-p(t)x(t-τ))']'+q(t)f(x(t-δ))h(x'(t))=0的振动准则,其中τ和δ是非负常数,a,p,q∈C([to,∞),R),f和h∈C(R,R).这些结果补充了大量存在性结论和处理以前结果不能解决的问题.特别地,以实例说明本文结果是实质性推广.     

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.     

A NEW OSCILLATION CRITERION FOR FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION

In this paper, a new sufficient condition for the oscillation of all solutions of first order neutral delay differential equations is obtained. Secondly, the result can also be extended to a general neutral differential equation, and many known results in the literatures are improved.     

OSCILLATION OF SECOND-ORDER NEUTRAL DYNAMIC FUNCTIONAL EQUATIONS

In this paper we establish some oscillation criteria for the second-order neutral dynamic delay equations on time scales. By using the generalized Riccati technique, new oscillation criteria are obtained.     

ASYMPTOTIC BEHAVIOR OF DYNAMIC EQUATIONS ON A PERIODIC TIME SCALES

We first prove some basic results on the periodic time scales, then on a special but practical periodic time scales T =∪[k(a + b),k(a + b) + a], the asymptotic behavior of x△ = px is explored with specific emphasis given to how the graininess of the time scales affects stability. At last, we analyze the asymptotic properties of the solutions of planar linear dynamic system, and obtain some sufficient conditions for the stability of the equalibium (0,0).     

OSCILLATION FOR A CLASS OF SECOND ORDER DELAY DIFFERENTIAL EQUATIONS WITH NONLINEAR NEUTRAL TERM

By the averaging technique,we establish some oscillation theorems for a class of second order delay differential equations with nonlinear neutral term,the results obtained extend some known results in the previous literatures.     

中立型时滞差分方程解的渐近性

给出了中立型时滞差分方程△（ｘｎ＋ｐｘｎ－ｋ）＋ｑｎｆ（ｘｎ－ｍ）＝０一切非振动解趋于零（ｎ→∞）的充分条件，证明了方程（Ｅ）一切解振动的两个新的定理。     

AN OSCILLATION CRITERION FOR EVEN ORDER NEUTRAL DIFFERENCE EQUATIONS

An oscillation criterion is obtained for even order neutral type differenceequations of the following formΔ~m(x_n+α_nx_(n-т))+f(n,x_n,x_(n-σ))=0,n=n_0,n_0+1,…,where m≥2 is even, n_0 is a nonnegative integer, Δ is the forward differenceoperator defined by Δx_n=x_(n+1)-x_n, and for i≥1, Δ~i is the i~(th)-order forwarddifference operator defined by Δ~ix_n=Δ(Δ~(i-1)x_n),т and σ are positive integers.     

LINEARIZED OSCILLATION FOR ODD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

By an associate linear equation, we obtain a linearized oscillation result of certain odd-order nonlinear neutral delay differential equation. The result answers partially an open problem proposed by Gyori and Ladas.     

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.     

NEW OSCILLATION CRITERIA FOR FIRST ORDER NEUTRAL DIFFERENCE EQUATIONS

1 IntroductionIn recent yearss those who were interested in studying the neutral differenceequation were growing more and more. The oscillation and nonoscillation ofsolutions of neutral difference equation have been extensively investigated (see[l][2][3]).Our aim in this paper is to obtain some sufficient conditions for the oscillation of all solutions of neutral difference equationA(x(n) -- p(n)x(n -- m)) Q(n)x(n -- k) = 0, n 2 no, (1)wherep(n), Q(n) 6 C({no, no 1,' .}, R ), m > 0, and…     

OSCILLATION OF A NONLINEAR DELAY DIFFERENTIAL EQUATION WITH NONLINEAR NEUTRAL TERM

This paper is concerned with the following nonlinear delay differential equation with nonlinear neutral term[h(t)(y(t) +φ(t,y(τ(t)))']' + q(t)f(y(g(t))) = 0. An oscillation criterion for the above equation is obtained.     

ASYMPTOTICS AND OSCILLATION FOR FIRST ORDER MIXED NEUTRAL DIFFERENTIAL EQUATIONS

1IntroductionConsidertheneutraldifferentialequationswherepiERfori=l,2,...,m;o相似文献   

OSCILLATION IN FIRST ORDER NEUTRAL DIFFERENTIAL EQUATIONS

A new oscillation result is established for the first order neutral differential equationx(t) - px(t - (?)) + qx(t - σ) = 0.The results extends and improves some existing results in the literature.