一类非线性连续分布时滞系统的周期正解

研究一类非线性周期连续时滞传染病模型ny′i(t)=-αi(t)yi(t)+(ci(t)-yi(t))∑j=1βij(t∫)0-TKj(s)yj(t+s)ds,i=1,2,…,n作者主要讨论了该传染病模型的周期正解的全局存在性,运用重合度延拓理论证明了该模型至少存在一个满足容许值的ω-周期正解.     

N（t）=N（t）[a（t）-∑n i=1bi（t）Ni（t-τi（t））]的周期正解

用非线性分析的方法讨论了方程N（t）=N（t）[a（t）-∑n i=1bi(t)Ni(t-τi（t）)]的周期正解的存在性，所获结果包含或推广了已有的几上结论[2，3]。     

一类积分微分方程周期解的存在性和唯一性

本文考虑具连续时滞和离散时滞的非线性积分微分方程x'(t)=A(t,x(t))x(t)+∫-∞tC(t,s)x(s)ds+∑i=1i gi(t,x(t—τi(t)))+b(t)和x’(t)=f(t,x(t))+∫-∞tC(t,s)x(s)ds+∑i=1igi(t,x(t-τi(t)))+b(t)周期解的存在性和唯一性问题,这里t∈R,x∈Rn;A(t,x),C(t,s)为n×n阶连续的函数矩阵; f(t,x),gi(t,x)(i=1,2,…,l),b(t)是n维连续向量.通过利用线性系统指数型二分性理论和泛函分析方法研究上述系统,获得了保证其周期解存在性、唯一性的充分性条件.我们除了实质性的推广和改进了已有的结果外,还得到三个新的定理,这是用已有的方法无法获得的(见文[1-30]).     

具有无穷时滞中立型泛函积分微分方程周期解的存在性

本文讨论具有无穷时滞中立型泛函积分微分方程ddtx(t)-∫t-∞B(t,s)x(s)ds=A(t,x(t))x(t) ∫t-∞C(t,s)x(s)ds ∑i=l1gi(t,x(t-τi(t)))的周期解问题.通过巧妙的构造算子,利用线性系统的指数二分性和Kras-noselskii不动点定理得到了周期解的存在性.我们的结果推广了相关文献的主要结果.     

非线性四阶周期边值问题多个正解的存在性

利用不动点和度理论,证明了四阶周期边值问题u(4)(t)-βu″(t)+αu(t)=λf(t,u(t)),0≤t≤1,u(i)(0)=u(i)(1),i=0,1,2,3,至少存在两个正解,其中β>-2π2,0<α<(1/2β+2π2)2,α/π4+β/π2+1>0,f:[0,1]×[0,+∞)→[0,+∞)是连续函数,λ>0是常数.     

非线性四阶周期边值问题正解的存在性和多重性

研究了四阶两参数常微分方程周期边值问题{u（4）（t）-βu″（t）＋αu（t）=f（t,u（τ（t）））,t∈[0,1],u（i）（0）=u（i）（1）,i=1,2,3正解的存在性、多重性和不存在性.在非线性项f（t,u）变号的情形下,用锥上的不动点指数理论证明了该问题至少n个甚至无穷多个正解的存在性,并且获得了该问...     

一类二阶多点时标边值问题无界解的存在性

借助不动点定理研究边值问题(φp(u△(t)))▽+f(t,u(t))=0,t∈(0,∞)Tu(0)=∑m-2i=1αiu(ηi),φp(u△(∞))=∑m-2i=1βiφp(u△(ηi))多个正解的存在性,得到了正解存在的充分条件.     

一类中立型积分微分方程周期解的存在性和唯一性

考虑具连续时滞和离散时滞的中立型积分微分方程d/dt[x(t) q∑j=1ej(t)x(t-δj(t))]=A(t,x(t))x(t ∫t-∞ C(t,s)x(s)ds 1∑i=1gi(t,x(t-Υi(t))) b(t)和d/dt[x(t) q∑j=1ej(t)x(t-δj(t))]=A(t)x(t) ∫t-∞C(t,s)x(s)ds 1∑j=1gi(t,x(t-Υi(t))) b(t)周期解的存在性和唯一性问题,利用线性系统指数型二分性理论和泛函分析方法,并通过技巧性代换获得了保证中立型系统周期解存在性和唯一性的充分性条件,从而避开了在研究中立型系统时x(t-δ)时滞项的导数x1(t-δ)的出现,推广了相关文献的主要结果.     

一类三阶常微分方程m点边值问题的正解存在性

讨论了一类如下的三阶常微分方程m点边值问题{u'(t)+h(t)f(u)=0,u(0)=u'(0)=0,u(1)=sum from i=1 to(m-2)βiu(ηi)正解的存在性.其中η_i∈(0,1),0<η_1<η_2<…<η_(m-2)<1,β_i∈[0,∞)且sum from i=1 to(m-2)βiηi2<1.通过与一个线性算子相关的第一特征值的讨论,运用不动点指数定理,得到了正解存在的结果.其中允许h(t)在t=0和t=1处奇异.     

一类三阶泛函微分方程周期解的存在唯一性

利用重合度理论研究了一类三阶泛函微分方程x′′′(t)+multiply from i=1 to 2[a_ix~((i))+b_ix~((i))(t-τ_i)]+ g_1(x(t))+g_2(x(t-τ))=p(t)的2π-周期解问题,获得了该方程2π-周期解存在唯一性的若干新结论.     

具$p$-Laplacian 算子的多点边值问题迭代解的存在性

利用单调迭代技巧和推广的Mawhin定理得到下述带有p-Laplacian算子的多点边值问题迭代解的存在性,{(Фp(u'))' f(t,u, Tu)=0, 0(≤)t(≤)1,u(0)=q-1∑i=1γiu(δi),u(1)=m-1∑i=1ηiu(ξi),其中Фp(s)=|s|p-2s,p＞1;0＜δi＜1,γi＞0,1(≤)i(≤)q-1;0＜ξi＜1,ηi(≥)0,1(≤)i(≤)m-1且q-1∑i=1γi＜1,m-1∑i=1ηi(≤)1;Tu(t)=∫t0k(t,s)u(s)ds,k(t,s)∈C(I×I,R ).     

Cocycle Perturbation on Banach Algebras

Let $α$ be a flow on a Banach algebra$\mathcal{B}$, and $t → u_t$ a continuous function from$\boldsymbol{R}$into the group of invertible elements of$\mathcal{B}$such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.     

HOW BIG ARE THE LAG INCREMENTS OF A 2-PARAMETER WIENER PROCESS? (II)

The author investigated how big the lag increments of a 2-parameter Wiener process is in [1]. In this paper the limit inferior results for the lag increments are discussed and the same results as the Wiener process are obtained. For example, if $\[\mathop {\lim }\limits_{T \to \infty } \{ \log T/{a_T} + \log (\log {b_T}/a_T^{1/2} + 1)\} /\log \log T = r,0 \leqslant r \leqslant \infty \] $ then $\[\mathop {\lim }\limits_{\overline {T \to \infty } } \mathop {\sup }\limits_{{a_T} \leqslant t \leqslant T} \mathop {\sup }\limits_{t \leqslant s \leqslant T} \mathop {\sup }\limits_{R \in L_s^*(t)} |W(R)|/d(T,t) = {\alpha _r},a.s.,\] $ $\[\mathop {\lim }\limits_{\overline {T \to \infty } } \mathop {\sup }\limits_{{a_T} \leqslant t \leqslant T} \mathop {\sup }\limits_{R \in {{\tilde L}_T}(t)} |W(R)|/d(T,t) = {\alpha _r},a.s.,\] $ where $\alpha _r=(r/(r+1))^{1/2}$, $L*_s(t)$ and $\tider L_T(t)$ are the sets of rectangles which satisfy some conditions. Moreover, the limit inferior results of another class of lag increments are discussed.     

负相协随机变量非随机和的差的精确大偏差

In this paper,we study precise large deviation for the non-random difference sum from j=1 to n_1(t) X_(1j)-sum from j=1 to n_2(t) X_(2j),where sum from j=1 to n_1(t) X_(1j) is the non-random sum of {X_(1j),j≥1} which is a sequence of negatively associated random variables with common distribution F_1(x),and sum from j=1 to n_2(t) X_(2j) is the non-random sum of {X_(2j),j≥1} which is a sequence of independent and identically distributed random variables,n_1(t) and n_2(t) are two positive integer functions.Under some other mild conditions,we establish the following uniformly asymptotic relation lim t→∞ sup x≥r(n_1(t))~(p+1)|(P(∑~(n_1(t)_(j=1)X_(1j)-∑~(n_2(t)_(j=1)X_(2j)-(μ_1n_1(t)-μ_2n_2(t)x))/(n_1(t)F_1(x))-1|=0.     

The Initial Value Problems and Nonlinear Boundary Value Problems for a Class of the Second order Quasilinear Parabolio Systems

In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below $\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$ are discussed.The boundary value conditions are $\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$ $\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$ Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved.     

一类多点边值问题的可解性

This paper deals with the existence of solutions for the problem
{（Фp（u′））′=f（t,u,u′）,t∈（0,1）,
u′（0）=0,u（1）=∑i=1^n-2aiu（ηi）,
where Фp（s）=｜s｜^p-2s,p〉1.0〈η1〈η2〈…〈ηn-2〈1,ai（i=1,2,…,n-2）are non-negative constants and ∑i=1^n-2ai=1.Some known results are improved under some sign and growth conditions. The proof is based on the Brouwer degree theory.     

Rate Optimality of Wavelet Series Approximations of Fractional Brownian Motion

Consider the fractional Brownian motion process $B_H(t), t\in [0,T]$, with parameter $H\in (0,1)$. Meyer, Sellan and Taqqu have developed several random wavelet representations for $B_H(t)$, of the form $\sum_{k=0}^\infty U_k(t)\epsilon_k$ where $\epsilon_k$ are Gaussian random variables and where the functions $U_k$ are not random. Based on the results of Kühn and Linde, we say that the approximation $\sum_{k=0}^n U_k(t)\epsilon_k$ of $B_H(t)$ is optimal if $$ \displaystyle \left( E \sup_{t\in [0,T]} \left| \sum_{k=n}^\infty U_k(t) \epsilon_k\right|^2 \right)^{1/2} =O \left( n^{-H} (1+\log n)^{1/2} \right), $$ as $n\rightarrow\infty$. We show that the random wavelet representations given in Meyer, Sellan and Taqqu are optimal.     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)&gt;0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N&gt;0,k(s)&gt;0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

具有非线性记忆项的耗散波动方程的整体吸引子

LetΩRn be a bounded domain with a smooth boundary.We consider the longtime dynamics of a class of damped wave equations with a nonlinear memory term utt+αut-△u-∫0t 0μ(t-s)|u(s)| βu(s)ds + g(u)=f.Based on a time-uniform priori estimate method,the existence of the compact global attractor is proved for this model in the phase space H10(Ω)×L2(Ω).