n阶泛函微分方程边值问题

本文研究下列n阶RFDE边值问题:x⁽ⁿ⁾(t)=f(t,x_t,x(t),x′(t),…,x^(n-1)(t)),　t∈［0,T ］,x(t)=φ(t),t∈［-r,0］;x′(0)=η,x″(0)=η₂,…,x^(n-2) (0)=η_n-2,x^(j)(T)=A,其中j∈I={0,1,2,…,n-1},得到了解的存在性和唯一性新的结果.     

赋范线性空间的第二对偶空间

Theorem. A mapping x^** of X^* into sealar field Φ is a linear functional if and only if there exists a μ∈ba(S) such that X^**(X^*)=∫_sx^*(s)μ(ds),x^*∈X^* Theorem. F∈ba(S.Σ)^* if and only if there exists a λ∈ba (T) (T is the closed unit sphere of the space B(S,Σ))such that F(μ)=∫(∫_st(s)μ(ds))λ(dt),μ∈ba(S,Σ).     

一个与Wiener过程增量连续模有关的下极限收敛速度问题

设W(t),t≥0为标准Wiener过程,α_T为T的函数且0<α_T≤T,lim_T→∞ log(T/α_T)/loglogT=r,本文证明了 c₁(r/(1+r))^1/2≤liminf_T→∞(loglogT)^1/2max_αT≤t≤T|W(T)-W(T-t)|/{2t[log(T/t)+loglogt]}^1/2≤c₂(r/(1+r))^1/2,a.s,这儿c₁和c₂为正常数。     

Long-time Asymptotic Behavior for the Derivative Schr¨odinger Equation with Finite Density Type Initial Data*

In this paper, the authors apply ? steepest descent method to study the Cauchy problem for the derivative nonlinear Schr¨odinger equation with finite density type initial data iq_t + q_xx + i(|q|²q)x = 0,q(x, 0) = q0(x),where lim/x→±∞ q0(x) = q± and |q±| = 1. Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schr¨odinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region ξ =x/t with |ξ + 2| < 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) + O(t^?3/4 ),in which the leading term is N(I) solitons, the second term is a residual error from a ? equation. For the region |ξ + 2| > 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) ? t^?1/2 if¹¹ + O(t^?3/4 ),in which the leading term is N(I) solitons, the second t^?1/2 order term is soliton-radiation interactions and the third term is a residual error from a ? equation. These results are verification of the soliton resolution conjecture for the derivative Schr¨odinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively.     

无限长信号中值滤波列的收敛性

若x ={x(n) }_n∈Z是一实数列 ,对每一正整数 p ,x^(p) ={x^(p) (n) }_n∈Z表示x通过p次窗为 2k + 1的中值滤波后的序列 .证明了{x^{( 2p)} }p≥ 1和 {x^{( 2p -1)} }p≥ 1皆收敛 ,从而完全解决了无限长信号中值滤波的收敛性问题 .     

拟线性抛物方程组第一边界问题的有限差分法 ——Ⅲ.稳定性

讨论如下拟线性抛物组第一边值问题的显式、弱隐式和强隐式差分解u_t=(-1)^M+1A(x,t,u,…,u_x^M-1)u_x^2M+f(x,t,u,…,u_x^2M-1(x,t)∈Q_T={O<x<l,0<t≤T.}，u_x^k(0，t)=u_x^k(l，t)=0 (k=0，1，…，M -1)，0<t≤T，u(x，0)=φ(x)，0≤x≤l，其中u，φ和f是m维向量值函数，A是m×m正定矩阵，u_t=∂u/∂t，u_x^k=∂^ku/∂x^k.在以下意义下证明了该问题的一般有限差分格式的稳定性：即离散向量解在W₂^(2M，M)(Q_T)中的离散范数是连续地依赖于初始数据的H^M离散范数，以及矩阵A与自由项f的相应的离散范数.     

Duffing型方程解的有界性

证明了下列Duffing型方程的所有解的有界性 :d2x / dt2 +x²ⁿ⁺¹+Σ²ⁿ_j=0 x^jp^j(t) =0 ,n≥1,其中,p1,p2 ,… ,p2n是 1周期的有Lipschitz连续性的函数,p_n+1,… ,p2n是Zygmund连续的 .这表明Duffing型方程的解的有界性不必要求pj(t)的光滑性.     

一个非线性色散波方程的惟一连续性

该文证明:和如下耦合色散系统相联系的初值问题的充分光滑的解$(u,v)=(u(x,t),v(x,t))$, 如果在两个时刻有半线支集那么它们全为零. {∂ _tu+∂³_x u+∂ _x(u^p v^p+1)=0, ∂ _tv+∂³_x v+∂_x(u^p+1v^p)=0,x∈R,t≥ 0     

一类非线性波方程初边值问题解的爆破

该文研究如下具有非线性阻尼项和非线性源项的波方程的初边值问题 u_tt -u_xxt -u_xx -(σ(u²_x)u_x)_x+δ|u_t|^p-1u_t=μ|u|^q-1u, 0 < x <1, 0≤ t ≤T, (0.1) u(0, t)=u(1, t)=0, 0≤t≤ T, (0.2) u(x, 0)=u₀(x), u_t(x, 0)=u₁(x),0≤x≤1.(0.3) 文章将给出问题(0.1)--(0.3)的解在有限时刻爆破的充分条件, 同时将证明问题的局部广义解和局部古典解的存在性和唯一性.     

强不变原理中的最佳速度

设{X_n}为i.i.d.r.v.s.,EX₁=0,EX₁~2=1,S_n=sum from i=1 to n(X_i),H(x)>0 (x≥0)为非降连续函数,对某γ>0和x₀>0,当x≥x₀时,x^-2-γH(x)非降,x^-1logH(x)非增,且x^-1logH(x)→0(x→∞),则有一标准Wiener过程{W(t),t≥0},使得 S_n-W(n)=O(invH(n))a.s.(n→∞)的充分必要条件是:对任何t>0有EH(t|X₁|)<∞.     

二阶非线性矩阵微分方程的振动性

Some new oscillation criteria are established for the second-order matrix differential system(r(t)Z′(t))′ p(t)Z′(t) Q(t)F(Z′(t))G(Z(t)) = 0, t ≥ to ＞ 0,are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.     

具有振动系数的二阶非线性中立型时滞动力方程的有界振动性

研究时标T上具有振动系数的二阶非线性中立型时滞动力方程(r(t)(y(t)+p(t)y(r(t))]△)α)△+f(t,y(δ(t)))=0的有界振动性,其中p是一个定义于T上的振动函数,α＞0是两个正奇数之比.利用一种Riccati变换技术,获得了该方程所有有界解振动的几个充分条件,推广和补充了文献中要求p(t)≥ 0的一些结果,并举例说明了该文主要结果的应用.     

Oscillation criteria for quasi-linear functional dynamic equations on time scales

This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation

$$(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$$

on a time scale \(\mathbb{T}\) where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on \(\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}\) and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty \). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.

     

Oscillation criteria for sublinear differential equations with damping

We present some criteria for the oscillation of the second order nonlinear differential equation [a(t)ψ(x(t))x'(t)]' + p(t)x'(t) + q(t)f (x(t)) =0, t≧t ₀> 0 with damping where a∈C ¹ ([t ₀,∞)) is a nonnegative function, p, q∈ C([t ₀,∞)) are allowed to change sign on [t ₀,∞), ψ, f∈C(R) with ψ(x) ≠ 0, xf(x)/ψ(x) > 0 for x≠ 0, and ψ, f have continuous derivatives on R{0} with [f(x) / ψ(x)]' ≧ 0 for x≠ 0. This criteria are obtained by using a general class of the parameter functions H(t,s) in the averaging techniques. An essential feature of the proved results is that the assumption of positivity of the function ψ(x) is not required. Consequently, the obtained criteria cover new classes of equations to which known results do not apply.     

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation (r(t)y(x(t))|x^￠(t)|^p-1x^￠(t))^￠+c(t)f(x(t))=0,    t 3 t₀,(r(t)\psi(x(t))\vert x^{\prime}(t)\vert^{p-1}x^{\prime}(t))^{\prime}+c(t)f(x(t))=0,\quad t\ge t_0,     

Oscillation theorems for second-order superlinear neutral differential equations

The purpose of this paper is to investigate the oscillation of the second-order neutral differential equations of the form (E) $$ (r(t)|z'(t)|^{\alpha - 1} z'(t))' + q(t)|x(\sigma (t))|^{\alpha - 1} x(\sigma (t)) = 0, $$ where z(t) = x(t) + p(t)x(τ(t)). The obtained comparison principles essentially simplify the examination of the studied equations. Further, our results extend and improve the results in the literature.     

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.     

一类时标上具状态依赖时滞的中立型泛函微分方程的周期正解

该文利用一个严格集压缩不动点定理,得到了如下形式的一类时标上具状态依赖时滞的中立型泛函微分方程周期正解存在性的充分条件x~Δ(t)=x(t)[r(t)-a(t)x(t)-sum from j=1 to n a_j(t)x(t-Υ_j(t,x(t)))-sum from j=1 to n c_j(t)x~Δ(t-σ_j(t,x(t)))],其中r,a,a_j,c_j∈C(T,R~+)(j=1,2,…,n)是ω-周期函数,Υ_j,σ_j∈C(T×R,T)(j=1,2,…,n)分别是其第一变元的ω-周期函数.     

Eigenvalue Intervals for 2mth Order Sturm—Liouville Boundary Value Problems

Values of λ are determined for which there exist positive solutions of the 2mth order differential equation on a measure chain, (-1)^m x ^?2m (t)=λa(t)f(u(σ(t))), y? [0,1], satisfying α _i+1 u ^?21(0)+0, γ _i+1 u ^?21(σ(1))=0, 0≤i≤m?1 with αi,βi,γi,δi≥0, where a and f are positive valued, and both lim _x-0+ (f(x)/x) and lim _x-0+ (f(x)/x) exist.     

Oscillation criteria of second order half linear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation

$${\left( {r\left( t \right)g\left( {{x^\Delta }\left( t \right)} \right)} \right)^\Delta } + p\left( t \right)f\left( {x\left( {\tau \left( t \right)} \right)} \right) = 0,$$

on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=?, difference equations when T=? but can be applied on different types of time scales such as when T=q^? for q > 1 and also improve most previous results. Finally, we give some examples to illustrate our main results.