共查询到18条相似文献,搜索用时 296 毫秒
1.
In this paper, the following retarded system has been studied
$\[\dot x(t) = Ax(t) + Bx(t - r),r > 0\]$(1)
where x(t) is an n-vector valued function; A and B are n*n constant matrices, and all the eigenvalues of A are supposed to have negative real parts. The asymptotical stability of equation (1) has been discussed by Halec13 utilizing the following Liapunov functional
$\[V(\phi ) = {\phi ^T}(0)C\phi (0) + \int_{ - r}^0 {{\phi ^T}(\theta )E\varphi (\theta )} d\theta \]$,
where E>0 and the symmetric matrix C>0 is chosen, such that A^TC+CA= — D<0. In this discussion, he remarked that if matrix
$\[H = \left[ {\begin{array}{*{20}{c}}
{D - E}&{ - CB}\{ - {{(CB)}^T}}&E
\end{array}} \right] > 0\]$,
the rate of decay of the solution of equation (1) to zero would be independent of the delay r, that is, would follow the exponential relation as indicated below : $\[||x(t,{t_0},\phi )|| \le K(r){e^{ - \alpha (t - {t_0})}}||\phi ||\]$,where \alpha(\alpha >0) is indepndent of r.
We show that this conclusion is not true, and a new relation between Liapunov functional and it's solution (exponential estimation) has been developed for the general rOtarded functional differential equation
$\[|\dot X(t) = f(t,{X_t})\]$(2)
If there is a functional $\[V(t,\phi ):{R^ + } \times {C_H} \to R\]$ such that
(i)$\[v|\phi (0){|^\eta } \le V(t,\phi ) \le K||\phi ||_\eta ^\eta ,(v,K > 0,\eta > 0)\]$
(ii)$\[\dot V(t,\phi ) \le - {C_1}|\phi (0){|^\eta },({C_1} > 0)\]$
then the solution of equation (2) x(t_0, ф) (t) satisfies
$\[||x({t_0},\phi )(t)|| \le {K_1}(r)||\phi |{|_\eta }{e^{ - {\alpha _1}(r)(t - {t_0})}}\]$
where \alpha _1 depends on r.
The following inverse problem has also been studied: In case the solution x = 0 of equation (1) is asymptotically stable for every value of r> 0, would there exist the matrices C>0 and E>0 such that the corresponding matrix H>0? Counter example is given for this problem. 相似文献
2.
文章对$3\times 3$阶三角矩阵环$$\Gamma = \left(\begin{array}{ccc}T & 0 & 0 \\M & U & 0\\{N \otimes _U M} & N & V \\\end{array}\right)$$上的模作了研究,其中T,U,V均是环, M,N分别是U-T, V-U双模.通过用一个五元组$(A,B,C;f,g)$来描述一个左$\Gamma$-模 (其中$A \in \mod T, B\in {\rm mod} U, C \in {\rm mod} V$, $f:M \otimes _T A \to B \in {\rm mod} U, g:N \otimes _U B \to C \in {\rm mod} V$), 文章分别刻画了$\Gamma$上的一致模、空的模、有限嵌入模,并且确定了${ }_\Gamma (A \oplus B \oplus C)$的根和基座. 相似文献
3.
具有一对零态射的Morita Context 环(Ⅱ) 总被引:1,自引:0,他引:1
设(A,B,V,W,ψ,φ)是一个Morita Context,具有一对零态射ψ=0,φ=0,C= (A V W B)是对应的Morita Context环.本文给出了C与A,B,V,W之间关于环的π-正则性、semiclean性、Mophic性和环的Exchgange性、Potent性、GM性的关系. 相似文献
4.
设$\mathcal {A,\ B}$ 是含单位元的Banach代数, $\mathcal M$ 是一个Banach $\mathcal {A,\ B}$-双模. $\mathcal {T}=\left ( \begin{array}{cc} \mathcal {A} & \mathcal M \\ & \mathcal {B} \\ \end{array} \right )$按照通常矩阵加法和乘法,范数定义为$\|\left( \begin{array}{cc} a & m \\ & b\\ \end{array} \right)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$,构成三角Banach 代数.如果从$\mathcal T$到其$n$次对偶空间$\mathcal T^{n}$上的Lie导子都是标准的,则称$\mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$\mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的. 相似文献
5.
该文结合算子谱论,应用锥不动点定理,建立了四阶边值问题\[\left\{ {\begin{array}{l}u^{(4)} + B(t){u}' - A(t)u = f(t,u),0 < t < 1 ,\\u(0) = u(1) = {u}'(0) = {u}'(1) = 0 \end{array}} \right.\]正解存在性定理,这里$A(t),B(t) \in C[0,1]$,$f(t,u):[0,1]\times[0,\infty ) \to [0,\infty )$连续. 相似文献
6.
Lin Xiaobiao 《数学年刊B辑(英文版)》1982,3(2):147-152
讨论泛函微分方程$\[\dot x = f(t,{x_t})\]$的解的渐近稳定性理论,往往需要假定f的某种全连续性.Burton在他的论文中讨论了f是一般$\[R \times C \to {R^n}\]$的连续泛函的情况.本文的目的是改进Burton的工作.证明方法釆取更简单的直接证法,证明结果不但同样获得有关解的一致渐近稳定性的结论,而且得到一个有趣的不等式,从中能够导出解的收敛于0的估计式.
设f是$\[R \times C \to {R^n}\]$连续泛函.$$是严格上升的连续函数,$$.设u,v,w是单调不减的连续函数u(0)=v(0)=w(0)=0,且对s>0有u(s),v(s),w(s)>0,
又设$\[|\phi {|_\eta } = \eta (|\phi (0)|) + \frac{1}{r}\int_{ - r}^0 {\eta (|\phi (\theta )|)d} \theta \]$,$\[{w_1}(s) = w({\eta ^{ - 1}}(s))\]$,$\[h(s) = \int_0^s {{w_1}(s)ds} \]$,$\[k(s) = v(s) + \frac{{{w_1}(1)}}{2}rs\]$,那么有如下定理:
定理1 设$\[V:R \times C \to R\]$是连续泛函,使得
$\[u(|\phi (0)|) \le V(t,\phi ) \le v(||\phi |{|_\eta })\]$
$\[V(t,\phi ) \le - w(|\phi (0)|)\]$
那么必有另一个连续泛函$\[G:R \times C \to R\]$,使得对$
\[\eta (|\mu |) < 1\]$有
$\[G(t,\phi ) \le - g(G(t,\phi )),V(t,\phi ) \le G(t,\phi )\]$,
其中$\[g:{R^ + } \to {R^ + }\]$定义为$\[g(s) = h(\frac{1}{2}{k^{ - 1}}(s))\]$
定理2 设定理1的条件均满足,设$\[F(y) = \int_1^y {\frac{{dz}}{{g(z)}}} \]$,那么存在s>0使得对于$\[|{\phi _0}| < s\]$有
$\[|x(t;{t_0},{\phi _0})| \le {u^{ - 1}}({F^{ - 1}}(F(G({t_0},{\phi _0})) + {t_0} - t))\]$
且x=0—致渐近稳定
文章最后给出两个实例说明以上定理的应用. 相似文献
7.
设(A,B,V,W,(),[])是一个Morita Context,C=A VW B是对应的Morita Context环.用基本环论方法,给出了C与A,B,V,W之间关于环的诣零性,幂零性,局部幂零性,N—诣零性,P—性等性质的关系. 相似文献
8.
应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性,其中$\phi(s) = |s|^{p-2}s, ~p>1$,非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性. 相似文献
9.
冉启康 《数学物理学报(A辑)》2008,28(2):320-328
设$D$是$R^N$ ($N>1$)中有界开集,$(\Omega, {\cal F}, P)$是一个完备的概率空间.该文研究了下列随机边值问题弱解的存在性问题\[\left\{\begin{array}{ll}-{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &;(x,\omega)\in D\times \Omega,\\u=0, &;(x,\omega)\in \partial D\times \Omega,\end{array}\right.\]其中, div与 $\nabla $ 表示仅对 $x$求微分. 首先,作者引入了弱解的概念; 然后,作者转化随机问题为高维确定性问题;最后,作者证明了该问题弱解的存在性. 相似文献
10.
设$\mathcal{A}$, $\mathcal{B}$是两个因子且$\dim\mathcal{A}>4$.本文证明了双射$\phi:\mathcal{A}\rightarrow\mathcal{B}$ 满足对所有的$A,B,C\in\mathcal A$有$\phi([A,B]\bullet C)=[\phi(A),\phi(B)]\bullet\phi(C)$当且仅当$\phi$是线性*-同构, 共轭线性*- 同构,负的线性*-同构, 负的共轭线性*-同构. 相似文献
11.
Let =(A C X B)be a 2×2 operator matrix acting on the Hilbert space н( )κ.For given A ∈B (H),B ∈B(K)and C ∈B(K,H)the set Ux∈B(H,к)σe(Mx)is determined,where σe(T)denotes the essential spectrum. 相似文献
12.
Thomas Bartsch Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804
We study the radially symmetric Schr?dinger equation
$ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), 相似文献
13.
Let $J$ be an infinite set and let $I={\cal P}_{f}( J)$, i.e., $I$ is the collection of all non empty finite subsets of $J$. Let $\beta I$ denote the collection of all ultrafilters on the set $I$. In this paper, we consider $( \beta I,\uplus ),$ the compact (Hausdorff) right topological semigroup that is the {\it Stone-$\check{C}\!\!$ech} $Compactification$ of the semigroup $\left( I,\cup \right)$ equipped with the discrete topology. It is shown that there is an injective map $A\rightarrow \beta _{A}( I) $ of ${\cal P}( J) $ into ${\cal P}( \beta I) $ such that each $\beta _{A}( I) $ is a closed subsemigroup of $ ( \beta I,\uplus ) $, the set $\beta _{J}( I) $ is a closed ideal of $( \beta I,\uplus ) $and the collection $\{ \beta _{A}( I) \mid A\in {\cal P} ( J) \} $ is a partition of $\beta I$. The algebraic structure of $\beta I$ is explored. In particular, it is shown that {\bf (1)} $\beta _{J}\left( I\right) =\overline{K( \beta I) }$, i.e., $\beta _{J}( I) $is the closure of the smallest ideal of $\beta I$, and {\bf (2)} for each non empty $A\subset J$, the set ${\cal V}_{A}=\tbigcup \{ \beta_{B}( I) \mid B\subset A\} $is a closed subsemigroup of $( \beta I,\uplus ) ,$ $\beta _{A}( I) $ is a proper ideal of ${\cal V}_{A},$ and ${\cal V}_{A}$ is the largest subsemigroup of $( \beta I,\uplus ) $ that has $ \beta _{A}( I) $ as an ideal. 相似文献
14.
In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper. 相似文献
15.
设$R$是环. 称右$R$-模$M$是PS-模,如果$M$具有投射的socle. 称$R$是PS-环,如果$R_R$是PS-模. 称$M$是CESS-模,如果$M$的任意具有基本socle的子模是$M$的某个直和因子的基本子模.本文给出了形式三角矩阵环 $T=\left( \begin{array}{cc} A & 0 \\ 相似文献
16.
Hu Shuan 《数学年刊B辑(英文版)》1987,8(4):428-433
V is an n-dim unitary space.(?)~kV is the k-th tensor product space with the customaryinduced inner product.(?)∈L((?)~kV),W~⊥(?)={((?)x~(?),x(?)|x~(?)=x_1(?)…(?)x_k,x_1,…,x_k o.n}is called the numerical range of (?).Wang Boying proved in[11]that if (?)=A_1(?)…(?)A_k,A_i∈L(V),i=1,…,k,k
17.
令ΛA_1,Λ_2为两个环,M是(A_2-Λ_1)-双模,且N是(Λ_1-Λ_2)-双模.六元组Γ=(Λ_1,Λ_2,N,M,ψ,φ)是一个森田六元组.对于Γ的表示,确定其几乎分裂序列(也称AR-序列)是非常重要的.通过modΛ_1和modΛ_2的右(左)几乎分裂同态、既约同态构造Γ上的相应同态,并进一步确定它的几乎分裂序列. 相似文献
18.
带一类时滞项的生物种群扩散模型的行波解 总被引:1,自引:0,他引:1
本文利用Schauder不动点理论证明了微分积分方程组行波解u(x,t)=U(z),w(x,t)=W(z),z=xγ-ct的存在性.这个方程组描述了一类在植物上繁殖,且靠飞行在空中扩散的生物种群扩散过程.特别当时滞项,中积分核K(t)(反映种群繁殖模式)属于L1(0,∞)时,本文得到极限值W(-∞)(表示最终植物上种群密度)小于M.这个结论较符合生物实际. 相似文献
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