Stability criteria for linear second order delay differential system

Consider second order delay differential system where r is a positive constant and all coefficients are real constants.Our main results are as follows:(1) The maximal length of the delay for which the stability of system (*) is maintained is given in the case where the zero solution of system (*) is asymptotically stable in the absence of delay.(2) The necessary and sufficient criteria for judging that asymptetical stability of system (*) is preserved for an arbitrary large delay are obtained.     

THE HOLDER CONTINUITY OF A CLASS OF 3-DIMENSION ULTRAPARABOLIC EQUATIONS

We obtained the Cα continuity for weak solutions of a class of ultraparabolic equations with measurable coeffcients of the form
δt u = δx（a（x, y, t）δx u） ＋ b0（x, y, t）δxu ＋ b（x, y, t）δyu, which generalized our recent results on KFP equations.     

某些二阶系统正解的存在性

考察如下边值问题正解的存在性x″(t) λa(t) f (x(t) ,y(t) ) =0y″(t) λb(t) g(x(t) ,y(t) ) =0x(0 ) =x(1 ) =y(0 ) =y(1 ) =0其中 f ,g:R × R R ;a,b:[0 ,1 ] R .所有的函数都被假定是连续的 ,此外 f ,g满足某些增长性条件 .本文得到了一些正解的存在性结果 .     

周期系数的平面Hamilton系统的平衡解的稳定性

本文考虑周期系数的平面Hamilton系统H(x,y,t)=H2(x,y,t)+H4(x,y,t)+d(x,y,t)的平衡解的稳定性。其中H2(x,y,t)=1/2[a(t)x2+y2],H4(x,y,t)=b4(t)x4+b2(t)(xy)2+b0(t)y4以及a(t),b0(t),b2(t),b4(t)是连续的T-周期函数,d(x,y,t)关于时间也是T-周期,在原点附近其阶为(x2+y2)3.     

二阶拟线性微分方程组边值问题的三个对称正解

本文讨论二阶拟线性微分方程组边值问题( p(x'))'+a(t),(t,x,y)=0,( q(y'))'+b(t)g(t,x,y)=0,x(0)-B0(x'(0))=x(1)+Bo(x'(1))=0,y(0)-B1(y'(0))=y(1)+B1(y'(1))=0,其中f,g是非负连续的函数.利用五个泛函的不动点定理,赋予f和g一些增长条件保证至少三个对称正解的存在性.     

具p-Laplacian算子型奇异方程组边值问题正解的存在性

本文讨论了一类具p-Laplacian算子型奇导方程组边值问题(φp(x'))'+α1(t),f(x(t),y(t))=0,(φp(y'))'+α2(t)g(x(t),y(t))=0,x(0)-β1x'(0)=0,x(1)+δ1x'(1)=0,y(0)-β2Y'(0)=0,y(1)+δ2y'(1)=0正解的存在性,其中φp(x)=|x|p-2x,p>1.通过使用不动点指数定理,在适当的条件下,建立了这类奇异方程组边值问题存在一个或者多个正解的充分条件.这些结果能用来研究椭圆型方程组边值问题径向对称解的存在性.     

带有振动系数的一类高阶中立型非线性受迫微分方程的振动准则

在本文中,我们获得形如[y(t) l∑j=1pj(t)y(σj(t))](n) ∫0q(t,s)f(y(t s))dσ(s)=h(t)的带有振动系数的一类高阶中立型非线性受迫微分方程的振动准则.     

On the solutions of certain linear nonhomogeneous second-order differential equations

The system Nω=(N-α)ω+y, N= bN+aωω^T, N(t)?R^m×m, ω(t)?R^m which originally arose from a model for the pathological behavior of neural networks, is studied. Similar equations can arise in a variety of applications. It is shown that if N(0) is positive definite, then solutions exist for all time. Equilibrium points are determined. N is found to be singular at the equilibrium points, making the analysis of the asymptotic properties of the system non-trivial. The asymptotic behavior when y = 0 is completely described. Some results are proven on the asymptotic behavior of N and ω when y≠0     

具有拟周期系数的schrodinger方程的可约化与平衡点的线性稳定性

对Schrodinger方程(A):iut-uxx+c(t)u=0u(t,0)=u(t,2π)=0,u(t,x)=∑∞n=1qn(t)n(x)进行讨论.n(x)是特征方程y″+λy=0y(0)=y(2π)=0中特征值对应的特征函数,c(t)=a+εc1(t),其中a是常数,c1(t)是以ω为频率的拟周期函数.直接判断方程的稳定性十分困难,把方程中的c(t)约化为常数,然后利用约化后的结果来判断方程(A)的平衡点的线性稳定性,方法简单实用.     

极限点型 Sturm-Liouville 算子乘积的自伴性

假设微分算式l(y)=-(py') qy,t∈[a,∞),满足lk(y)(k=1,2,3)均为极限点型,作者研究了由l(y)生成的两个微分算子Li(i=1,2)的乘积L2L1的自伴性问题并获得其自伴的充分必要条件．同时研究了由l(y)=-y" qy,t∈[a,∞),生成的三个微分算子Li(i=1,2,3)的乘积L3L2L1的自伴性问题．     

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Let p_M(t,x,y) be the minimal heat kernel of a d-dimenional compact Riemannian manifold M for any time t\in(0,1] and x,y\in M. Using the horizontal Brown bridge on M, we prove that, for any nonnegative integers n and m, there is a constant C depending on n,m and the manifold M, such that |\nabla^n_x\nabla^m_y\ln p_M(t,x,y)|\leq C[d(x,y)/t+1/\sqrt{t}\,]^{n+m}$, which generalizes the conclusion of the higher derivatives of the logarithmic heat kernel \ln p_M(t,x,y) about single variable in \ncite{1}.     

On FBDF5 Method for Delay Differential Equations of Fractional Order with Periodic and Anti-Periodic Conditions

In this paper, we study the fractional backward differential formula (FBDF) for the numerical solution of fractional delay differential equations (FDDEs) of the following form: \(\lambda _n {}_0^C D_t^{\alpha _n } y(t - \tau ) + \lambda _{n - 1} {}_0^C D_t^{\alpha _{n - 1} } y(t - \tau ) + \cdots + \lambda _1 {}_0^C D_t^{\alpha _1 } y(t - \tau ) + \lambda _{n + 1} y(t) = f(t), t \in [0,T]\), where \( \lambda _i \in \) \(\mathbb {R}\,(i = 1,\ldots ,n + 1)\,,\,\lambda _{n + 1} \ne 0,\,\, 0 \leqslant \alpha _1< \alpha _2< \cdots< \alpha _n < 1,\,\,T > 0,\) in Caputo sense. We find the Green’s functions for this equation corresponding to periodic/anti-periodic conditions in term of the Mittag-Leffler type. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Finally, some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis     

脉冲中立型时滞微分方程解的振动性

本文讨论一阶脉冲中立型时滞微分方程［ｙ（ｔ）＋Ｐｙ（ｔ－σ）］′＋Ｑ（ｔ）ｙ（ｔ－σ）＝０，ｔ０，ｔ≠ｔｋ，ｋ＝１，２，…，ｙ（ｔ＋ｋ）－ｙ（ｔ－ｋ）＝ｂｋｙ（ｔｋ），ｋ＝１，２，…，｛（Ｅ）这里τ，σ，Ｐ均为常数，τ＞０，σ＞０，Ｑ（ｔ）∈Ｃ（［０，∞），Ｒ＋），ｂｋ＞－１，ｋ＝１，２，…．分三种情况，Ｐ－１；－１＜Ｐ＜０；Ｐ＞０给出了方程（Ｅ）所有解振动的充分条件．     

二阶非线性中立型微分方程的振动性

利用Ho lder不等式研究一类非线性项具时滞的二阶中立型时滞微分方程{r(t)[y(t)+p(t)y(t-τ)]′2m+1}′+q(t)f[y(t-σ)]=0(t>t0)的振动性.给出了该方程的解振动的若干充分条件,所得结果推广了已有的相应结论.     

二阶微分系统奇异正定超线性周期边值问题的多重正解

主要研究了二阶微分系统具有奇异正定超线性周期边值问题多重正解的存在性问题,利用Leray-Schauder抉择定理和锥不动点定理给出了奇异正定超线性周期边值问题-(p(t)x′)′+q1(t)x=f1(t,x,y),t∈I=[0,1]-(p(t)y′)′+q2(t)y=f2(t,x,y)x(0)=x(1),x[1](0)=x[1](1)y(0)=y(1),y[1](0)=y[1](1)(1.1)的多重正解的存在性,其中非线性项fi(t,x,y)(i=1,2)在x=∞,y=∞点处超线性,在(x,y)=(0,0)处具有奇性.这里定义x[1](t)=p(t)x′(t),y[1](t)=p(t)y′(t)为准导数,其中系数p(t),qi(t)(i=1,2)是定义在[0,1]上的可测函数,且p(t)>0,qi(t)>0(i=1,2),a.e[0,1],fi(t,x,y)∈C(I×R×R,R+),R+=(0,+∞).     

On second order functional differential equations and inequalities with deviating arguments

A necessary and sufficient condition is established in order that (i) the retarded differential equation $$y''(t) = p_0 y(t) + f(y(t - \tau _1 ),...,y(t - \tau _N ))$$ has no bounded nonoscillatory solution and (ii) the advanced differential equation $$y''(t) = p_0 y(t) + f(y(t + \tau _1 ),...,y(t + \tau _N ))$$ has no unbounded nonoscillatory solution, wherep ₀≥0 and τ _j > 0,1 ?i ?N, are constants. Differential inequalities related to (^*) and (^**) are also studied. Finally, an oscillation criterion is given for a class of differential equations containing both retarded and advanced arguments.     

On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients

In this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem.     

空间情形下旋转体体积的计算

运用定积分中的元素法,给出了空间曲线绕空间直线旋转一周所成的旋转曲面与垂直于旋转轴的两个平面所围成的旋转体体积的计算公式:V=π(m2+n2+p2)23∫tt12{[p(y(t)-b)-n(z(t)-c)]2+[m(z(t)-c)-p(x(t)-a)]2+[n(x(t)-a)-m(y(t)-b)]2}m.x′(t)+n.y′(t)+p.z′(t)dt从而将平面图形的旋转体体积推广到了空间情形.     

一个描述血吸虫病的数学模型的周期解

研究一个描述血吸虫病的周期微分方程模型dx/dt=-rx+A/S(t)y,dy/dt=-δ(t)y+B(S(t)-y) x2/1+ x.数值计算发现该系统同时具有渐近稳定的零解和一个正周期解.通过证明该系统解的有界性,并在一个函数空间上构造单调有界序列,进而证明了在一定条件下正周期解的存在性.