Oscillatory Property of n-th Order Functional Differential Equations

The authors study oscillatory property of nonlinear functional differential equation $L_nx(t)+p(t)f(x(t),x(g(t)))=r(t)$(1) where L_nx(t) is an n-th order linear differential operator defined by $L_0x(t)=x(t)$, $L_kx(t)=\frac{d}{dt}(a_k-1(t)L_k-1x(t)),k=1,2,\cdots,n.$ Sufficient conditions are obtained which guarantee that all continuable solutions of (1) are oscillatory or tend to zero as t\rightarrow \infinity.     

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.     

高阶具非线性中立项不稳定型时滞微分方程的振动性和非振动性

在α>1,且0<β<α情性下研究了高阶具非线性中立项不稳定型时滞微分方程[x(t)-pxα(t-τ)](n)=q(t)xβ(t-σ),(t≥t0)的振动和非振动性.利用一些新的技巧,获得了上述方程有界解振动的振动准则和至少存在一个非振动解的非振动准则,所得结果补充和推广了已有文献部分结果.     

二阶线性矩阵微分系统的振动性

讨论了二阶线性矩阵微分系统(P(t)Y′(t))′+Q(t)Y(t)=0,t≥t0,其中P(t),Q(t) 和Y(t)是n×n实连续矩阵函数, P(t)和Q(t)是对称的且P(t)>0是正定矩阵．利用推广的Riccati变换,采用两种不同的方法,得到了该系统振动的若干判据．所得结果推广和改进了已知的相应结果．     

三阶非线性中立型微分方程的振动性

研究三阶中立型时滞微分方程(r(t)[x(t)+p(t)x(σ(t))]″)′+q(t)f(x(t),x[q(t)])h(x′(t))=0的振动性和渐进性.给出了方程一切解振动或者渐近趋向于零的若干充分条件.     

应用推广的比较法得到的概率估计的几个结果

我们推广了比较法, 提出一个新方法得到 概率$\pr(X\leq t)$的可达的半参数界, 这里$X\in[0,M]$有给定的\, $\ep X=m_1$和$\ep X^2=m_2$. 我们的证明是初等的.     

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。     

一类中立型时滞抛物偏微分方程的强迫振动性

研究了一类中立型时滞抛物偏微分方程:t(u(x,t)-pu(x,t-τ))-∑rk=1ak(t)Δu(x,t-ρk(t))+∑mj=1qj(t)u(x,t-σj(t))=e(x,t),的强迫振动性(其中(x,t)∈Ω×[0,∞)≡G,Ω是n维欧几里得空间Rn中带有逐段光滑边界Ω的有界区域,Δ是Rn中带有三类不同边值条件的拉普拉斯算子,强迫项e(x,t)是定义在G上的一个振荡函数),给出了一些新的振动性判据,这些结果推广了已知的一些结论.     

一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.     

单位方体上沿曲面的振荡积分在Sobolev空间上的有界性

研究了欧氏空间R~2中单位方体Q~2=[0,1]~2上沿曲面(t,s,γ(t,s))的振荡奇异积分算子T_(α,β)f(u,v,x)=∫_(Q~2)f(u-t,v-s,x-γ(t,s))e~(it~(-β_1)s~(-β_2))t~(-1-α_1)s~(-1-α_2)dtds从Sobolev空间L_τ~p(R~(2+n))到L~p(R~(2+n))中的有界性,其中x∈R~n,(u,v)∈R~2,(t,s,γ(t,s))=(t,s,t~(P_1)s~(q_1),t~(p_2)s~(q_2),…,t~(p_n)s~(q_n))为R~(2+n)上一个曲面,且β_1α_1≥0,β_2α_20.这些结果推广和改进了R~3上的某些已知的结果.作为应用,得到了乘积空间上粗糖核奇异积分算子的Sobolev有界性.     

Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales

This paper is concerned with the oscillatory properties of even order advanced type dynamic equation with mixed nonlinearities of the form $$\bigl[r(t)\varPhi_\alpha\bigl(x^{\Delta^{n-1}}(t) \bigr) \bigr]^\Delta+ p(t)\varPhi_\alpha\bigl(x\bigl(\delta(t)\bigr) \bigr) +\sum_{i=1}^kp_i(t) \varPhi_{\alpha_i} \bigl(x\bigl(\delta(t)\bigr) \bigr)=0 $$ on an arbitrary time scale $\mathbb{T}$ , where Φ _?(u)=|u|^??1 u. We present some new oscillation criteria for the equation by introducing parameter functions, establishing a new lemma, using a Hardy-Littlewood-Pólya inequality and an arithmetic-geometric mean inequality and developing a generalized Riccati technique. Our results extend and supplement some known results in the literature. Several examples are given to illustrate our main results.     

线性矩阵Hamilton系统振动性的区间判据

采用两种不同的方法,得到了线性矩阵Hamilton系统的振动性判据.这些振动性判据仅依赖于系数矩阵在[to,∞)的某些子区间上的性质,从而改进并推广了许多已知的Kamenev型振动准则.     

Oscillation Theorems for Neutral Differential Equations of Higher Order

In this paper we present some new oscillatory criteria for the n-th order neutral differential equations of the form The results obtained extend and improve a number of existing criteria.     

Asymptotic Behavior of Solutions to Second-Order Nonlinear Differential Equations

In this paper the authors study the oscillation and the asymptotic behavior of solutiosn to the second-order nonlinear differential equations $[\ddot x]\pmX(t,x,[\dot x])=0$(X_{\pm}) and give necessary and sufficient conditions for Equation (X_+) to have a bounded nonoscillatory solution or to be oscillatory. There are four classes of solutions to Equation (X_) with different asymptotic behavior. For each class of solution, the necessary or sufficient condition of the existence is obtained.     

Oscillatory and Asymptotic Behavior of First Order Functional Differential Equations

In this paper the author discusses the following first order functional differential equations: $x''(t)+[\int_a^b {p(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (1)$ $x''(t)+[\int_a^b {f(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (2)$ Some sufficient conditions of oscillation and nonoscillation are obtained, and two asymptotic properties and their criteria are given. These criteria are better than those in [1, 2], and can be used to the following equations: $x''(t)+[\sum\limits_{i = 1}^n {{p_i}(t)x[{g_i}(t)] = 0} \] (3)$ $x''(t)+[\sum\limits_{i = 1}^n {{f_i}(t)x[{g_i}(t)] = 0} \] (4)$     

Oscillating solutions of scalar delay-differential equations with state dependence

This paper is concerned with the oscillatory behavior of the delay-differential equation X'(t)=F(t,x_t) including the equations x'(t)=-a(t)x(t-r(t,x(t))), [display math001] as special cases.We give conditions for the existence of a nonoscillatory solution of (1) and criteria for the oscillation of all solutions of (1), aiming at extending or generalizing to (1) some of the recent oscillation and nonoscillation results for delay equations of the form x'(t)=-a(t)x(t-p)).     

���帺������β��������ͼ������ֵβ���ʵĽ�����

Let $X_1,X_2,\ldots,X_n$ be a sequence of extended negatively dependent random variables with distributions $F_1,F_2,\ldots,F_n$,respectively. Denote by $S_n=X_1+X_2+\cdots+X_n$. This paper establishes the asymptotic relationship for the quantities $\pr(S_n>x)$, $\pr(\max\{X_1,X_2, \ldots,X_n\}>x)$, $\pr(\max\{S_1,S_2$, $\ldots,S_n\}>x)$ and $\tsm_{k=1}^n\pr(X_k>x)$ in the three heavy-tailed cases. Based on this, this paper also investigates the asymptotics for the tail probability of the maximum of randomly weighted sums, and checks its accuracy via Monte Carlo simulations. Finally, as an application to the discrete-time risk model with insurance and financial risks, the asymptotic estimate for the finite-time ruin probability is derived.     

Oscillatory properties of second-order nonlinear neutral dynamic equations with positive and negative coefficients

This paper is to investigate the oscillatory properties of second-order nonlinear neutral dynamic equation with positive and negative coefficients of the form on an arbitrary time scale $\mathbb{T}$ . We establish several oscillation criteria for the equation by using a generalized Riccati technique. The obtained results extend and improve certain existing results in the literature. Examples are shown to illustrate our main results.     

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.     

中立型方程d/(dt)[x(t)+px(t-τ)-rx(t-ρ)]+qx(t-s)-hx(t-v)=0的振动性

本文研究了既具正系数又具负系数的一阶中立型微分方程ｄｄｔ［ｘ（ｔ）＋ｐｘ（ｔ－τ）－ｒｘ（ｔ－ρ）］＋ｑｘ（ｔ－ｓ）－ｈｘ（ｔ－ｖ）＝０其中ｐ，ｒ，τ，ρ，ｑ，ｈ，ｓ，ｖ都是正的常数，建立了一切解振动的带有若干个可调参数的一个充要条件和一系列充分性判据及某种形式的一个充要条件．有些充要条件和充分性判据包含或改进了前人的相应结果．