共查询到19条相似文献,搜索用时 281 毫秒
1.
2.
设$K$是自反的并且具有一致Gateaux可微范数的Banach空间$E$的非空有界闭凸子集.设$T:K\rightarrow K$是一致连续的伪压缩映象.假设$K$的每一非空有界闭凸子集对非扩张映象具有不动点性质.设$\{\lambda_n\}$是$(0,\frac{1}{2}]$中序列满足: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$.任给$x_1\in K$,定义迭代序列$\{x_n\}$为:$x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1),n\geq 1.$若$\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$, 则上述迭代产生的$\{x_n\}$强收敛到$T$的不动点. 相似文献
3.
WANG JIAGANG 《数学年刊A辑(中文版)》1981,2(1):13-20
若说\[(\Omega ,\mathcal{F},P)\]为完备概率空间,\[F = {({\mathcal{F}_t})_{t \in [a,b]}}\]为\[\mathcal{F}\]的递增子\[\sigma \]域族,且满足通常
条件,\[b \leqslant \infty \].又\[W = \{ {W_t},0 \leqslant t \leqslant b\} \]为关于F的Wiener过程,\[X = \{ {X_t},0 \leqslant t < b\} \]为
循序讨测过程,且
\[P\{ \int_0^b {X_t^2} dt < \infty \} = 1\],
则可定义X关于W的Ito随机积分
\[{(X \cdot W)_t} = \int_0^t {{X_s}} d{W_s},0 \leqslant t \leqslant b\]
这时若记
\[{Z_t} = \exp \{ \int_0^t {{X_s}} d{W_s} - \frac{1}{2}\int_0^t {{X_s}^2} ds\} \]
它便是一个指数(局部)鞅.本文的目的在于证明当X为循序可测正态过程时,只要X关于W的积分存在,\[{\text{\{ }}{Z_t}0 \leqslant {\text{t < b\} }}\]总是一致可积的。
引理1若\[\{ {Z_t},0 \leqslant t < b\} \]为实可测正态过程且
\[\int_0^{\text{b}} {\left\| {{X_t}} \right\|} d{m_t} < \infty \]
其中\[\left\| {{X_t}} \right\| = {(E|{X_t}{|^2})^{1/2}}\],\[{m_t}\]为[0,b)上右连续递增函数,则X的几乎所有样本函数关于\[{m_t}\]可积,且其轨道积分
\[\tilde I = \int_0^{\text{b}} {{X_t}} d{m_t}\]
为正态分布随机变量.
引理2若\[X = \{ {X_t},0 \leqslant t < b\} \]为可测正态过程,其几乎所有样本函数关于右连续增函数\[{m_t}\]可积,即
\[P(\int_0^b {|{X_t}} |d{m_t} < \infty ) = 1\]
则按轨道积分 \[\tilde I = \int_0^{\text{b}} {{X_t}} d{m_t}\]
是正态分布随机变量.
引理3 若\[\{ {\xi _n},n \geqslant 1\} \]为正态分布随机变量序列,则
\[\sum\limits_{j = 1}^\infty {E{\xi _i}^2} \leqslant {[Eexp( - \frac{1}{2}\sum\limits_{j = 1}^\infty {{\xi _i}^2} )]^{ - 2}}\]
进而若\[\sum\limits_{j = 1}^\infty {E{\xi _i}^2} < 1\],则
\[E[exp(\frac{1}{2}\sum\limits_{j = 1}^\infty {{\xi _i}^2} )] \leqslant {(1 - \sum\limits_{j = 1}^\infty {E{\xi _i}^2} )^{ - \frac{1}{2}}}\]
引理4若\[{m_s}\]为[0, b)上右连续增函数,又\[X = \{ X_t^{(i)},0 \leqslant t < b,1 \leqslant i < \infty \} \]为正态
过程,则当\[P\{ \sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < \infty \} = 1\]时必有
\[\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < \infty \} = 1\]
进而若;\[\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < 1\],必有
\[Eexp(\frac{1}{2}\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_s}} } ) \leqslant {(1 - \sum\limits_{j = 1}^\infty {E\int_0^b {{{({X_t}^{(i)})}^2}d{m_s}} } )^{ - \frac{1}{2}}}\]
定理 若\[W = (W_t^{(1)},...,W_t^{(n)},...)\]为一个具有无限个分量的过程,其分量都是连续
正态独立增量过程且满足
\[\begin{gathered}
E\{ W_t^{(i)} - W_s^{(i)}\} = 0 \hfill \ E\{ (W_t^{(i)} - W_s^{(i)})(W_t^{(j)} - W_s^{(j)})\} = {\delta _{ij}}(m_t^{(i)} - m_s^{(i)}) \hfill \\
\end{gathered} \]
又\[\{ {f_t} = (f_t^{(1)},...,f_t^{(n)},...)\} \]为循序可测正态过程,若
\[P\{ \sum\limits_{i = 1}^\infty {\int_0^b {{{({f_t}^{(i)})}^2}dm_t^{(i)}} } < \infty \} = 1\]
则 \[{Z_t} = \exp \{ \sum\limits_{i = 1}^\infty {\int_0^b {{f_s}^{(i)}dW_s^{(i)} - \frac{1}{2}\int_0^t {{{({f_s}^{(i)})}^2}dm_s^{(i)}} } } \} ,0 \leqslant t < b\]
是一致可积鞅,特别有\[E{Z_0} = 1\]
利用上述结果及正态过程的Hida-Cramer分解,可以象[1]一样方便地讨论正态测
度的等价性问题并求出其Radon-Nikodym导数. 相似文献
4.
1. Let X be the conjugate of a separable Banach space satifying the *-Opial
condition, i. e., if \[\{ {x_n}\} \subset x,{x_n}\mathop \to \limits^{{w^*}} {x_\infty },{x_\infty } \ne y\], then\[\mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - {x_\infty }|| < \mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - y||\]
for rxample \[X = {l_1}\]
Let K be a nonempty weak* closed convex subset of X.
The main results are:
Theorem 1. Suppose T is a ooniinuons mappings of K into itself such that for
every \[x,y \in K\],\[||Tx - Ty|| \le a||x - y|| + b\{ ||x - Tx|| + ||y - Ty||\} + c\{ ||x - Ty|| + ||y - Tx||\} \]
where real numbers \[a,b,c \ge 0\] and \[a + 2b + 2c = 1\]. Suppose also K is bounded.Then T has at least one fixed point in K.
Theorem 2. Let T be a mapping of K into itself, and \[a(x,y),b(x,y),c(x,y)\]be real functions such that for all\[x,y \in K\]
\[||Tx - Ty|| \le a(x,y)||x - y|| + b(x,y)\{ ||x - Tx|| + ||y - Ty||\} + c(x,y)\{ ||x - Ty|| + ||y - Tx||\} \]
and \[a(x{\rm{y}},y){\rm{ + }}2b(x,y){\rm{ + }}2c(x,y) \le 1\]
Suppose there exists \[x \in K\] such that \[O(x) = \{ {T^n}x\} _{n = 1}^\infty \] is bounded and
\[\mathop {\inf }\limits_{y,z \in o(x)} c(y,z) > 0\]
Then T has at least one fixed point z in K and \[{T^n}x\mathop \to \limits^{{w^*}} z\].
2. We denote \[CL(x) = \{ A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} X\} \]
\[K(x) = A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x\} \]
here X is a complete metric space with metric d.
On \[CL(x)\] and \[K(x)\] we introduce the generalized Hausdorff distance \[H(,)\],
The main results are:
Theorem 3. Suppose \[\{ T,S\} \] is a pair of set-valued mappings of X into \[CL(x)\],which satisfies the following condition:
\[H(Tx,Sy) \le hMax\{ d(x,y),D(x,Tx),D(y,Sy),\frac{1}{2}[D(x,Sy) + D(y,Tx)]\} \]
for each \[x,y \in K\], where 0相似文献
5.
HU KE 《数学年刊A辑(中文版)》1981,2(1):21-24
Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\]
\[\begin{gathered}
\frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\
\end{gathered} \]
Milin-Lebedey proved that
\[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \]
where p>l and \[\lambda \]>0.
In this paper, we have proved the following theorems;
Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and
\[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\]
then F(x) is a decreasing function of x on [0, 1].
This theorem is stronger than the result (1).
Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and
\[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \]
then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2,...)In the case p=2 this is contained in the Miiin-Lebedev's result. 相似文献
6.
HU KE 《数学年刊B辑(英文版)》1981,2(1):21-24
Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\]
\[\begin{gathered}
\frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\
\end{gathered} \]
Milin-Lebedey proved that
\[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \]
where p>l and \[\lambda \]>0.
In this paper, we have proved the following theorems;
Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and
\[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\]
then F(x) is a decreasing function of x on [0, 1].
This theorem is stronger than the result (1).
Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and
\[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \]
then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2,...)In the case p=2 this is contained in the Miiin-Lebedev's result. 相似文献
7.
主要研究了B -值双随机Dirichlet级数在不同条件(i) {X_n}服从强大数定律,且0<\mathop{\underline{\lim}}\limits_{n-->\infty}\Big\|\frac{\sum\limits_{i=1}^n EX_i}{n}\Big\|\leq \mathop{\overline{\lim}}\limits_{n\to\infty}\Big\|\frac{\sum\limits_{i=1}^n EX_i}{n}\Big\|<+\infty.(ii) {X_{n}}独立不同分布,且\mathop{\underline{\lim}}\limits_{n-->\infty}E||X_n||>0,\quad \sup\limits_{n\geq 1}E||X_n||^p <+\infty \quad (p >1)等条件下的收敛性,得出了收敛横坐标的简洁公式. 相似文献
8.
在任意实的Banach空间中研究了用具误差的修正的Ishikawa与Mann迭代程序来逼近一致L-Lipschitz的渐近伪压缩映象不动点的强收敛性问题,在去掉条件$$\sum\limits_{n=0}^{\infty}\alpha_{n}^{2}<\infty, \q \sum\limits_{n=0}^{\infty }\gamma_{n}<\infty,\q \sum\limits_{n=0}^{\infty }\alpha_{n}(\beta_{n}+\delta_{n})<\infty,\q \sum\limits_{n=0}^{\infty}\alpha_{n}(k_{n}-1)<\infty$$之下,证明了相关文献的结果仍然成立.所得结果不但改进和推广了最近一些人的最新结果,而且也从根本上改进了定理的证明方法. 相似文献
9.
设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的 相似文献
10.
李云霞 《数学物理学报(A辑)》2006,26(5):675-687
该文主要讨论的是滑线性过程 $X_k=\sum\limits_{i=-\infty}^\infty a_{i+k}\varepsilon_i$,其中 $\{\varepsilon_i; -\infty$\varphi$ -混合或负相伴随机变量序列,$\{a_i;-\inftyp$, 若 $E|\varepsilon_1|^r<\infty$$\lim_{\epsilon\searrow 0}\epsilon^{2(r-p)/(2-p)}\sum\limits_{n=1}^\infty n^{r/p-2}P\{|S_n|\geq \epsilonn^{1/p}\}=\frac{p}{r-p}E|Z|^{2(r-p)/(2-p)},$ 其中 $Z$ 是服从均值为零,方差为 $\tau^2=\sigma^2\cdot(\sum\limits_{i=-\infty}^\infty a_i)^2$的正态分布. 相似文献
11.
设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设K是实Banach空间E中非空闭凸集,{Ti}i=1^N是N个具公共不动点集F的严格伪压缩映像,{αn}包括于[0,1]是实数例,{un}包括于K是序列,且满足下面条件(i)0〈α≤αn≤1;(ii)∑n=1∞(1-αn)=+∞.(iii)∑n=1∞ ‖un‖〈+∞.设x0∈K,{xn}由正式定义xn=αnxn-1+(1-αn)Tnxn+un-1,n≥1,其中Tn=Tnmodn,则下面结论(i)limn→∞‖xn-p‖存在,对所有p∈F;(ii)limn→∞d(xn,F)存在,当d(xn,F)=infp∈F‖xn-p‖;(iii)lim infn→∞‖xn-Tnxn‖=0.文中另一个结果是,如果{xn}包括于[1-2^-n,1],则{xn}收敛,文中结果改进与扩展了Osilike(2004)最近的结果,证明方法也不同。 相似文献
12.
V. Totik 《Analysis Mathematica》1981,7(1):81-84
В РАБОтЕ ДАЕтсь ОтВЕт НА ОДИН ВОпРОс, пОстАВ лЕННыИ В. г. кРОтОВыМ. УстАНОВлЕН О, ЧтО ЕслИ Ф(х) — МОНОтОННО ВО жРАстАУЩАь ФУНкцИь,Ф (0)=0, Ф(2х)≦кФ(х), х[0, ∞), тО $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {S_k - f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\mu _k } \Phi (\lambda _k ) = \infty $$ Дль пРОИжВОльНых НЕО тРИцАтЕльНых ЧИслОВ ых пОслЕДОВАтЕльНОстЕ И {Μk} И {λk}. (жДЕсьS k ОБОжНАЧАЕт ЧАстНУУ с УММУ пОРьДкАk РьДА ФУ РьЕ ФУНкцИИf). УстАНОВлЕН О тАкжЕ, ЧтО ВО МНОгИх слУЧАьх $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {\tilde S_k - \tilde f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\frac{1}{{k\lambda _k }}} \Phi ^{ - 1} \left( {\frac{1}{{k\mu _k }}} \right)< \infty .$$ 相似文献
13.
Nikolay Moshchevitin 《Czechoslovak Mathematical Journal》2012,62(1):127-137
Let Θ = (θ
1,θ
2,θ
3) ∈ ℝ3. Suppose that 1, θ
1, θ
2, θ
3 are linearly independent over ℤ. For Diophantine exponents
$\begin{gathered}
\alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\
\beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\
\end{gathered}$\begin{gathered}
\alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\
\beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\
\end{gathered} 相似文献
14.
设{X_(ni):1≤i≤n,n≥1}为行间NA阵列,g(x)是R~+上指数为α的正则变化函数,r>0,m为正整数,{a_(ni):1≤i≤n,n≥1}为满足条件(?)|a_(ni)|=O((g(n))~1)的实数阵列,本文得到了使sum from n=1 to ∞n~(r-1)Pr(|■multiply from j=1 to m a_(nij) X_(nij)|>ε)<∞,■ε>0成立的条件,推广并改进了Stout及王岳宝和苏淳等的结论。 相似文献
15.
Huang Falun 《数学年刊B辑(英文版)》1989,10(3):332-340
In this paper the author proves a new fundamental lemma of Hardy-Lebesgne class
$\[{H^2}(\sigma )\]$ and by this lemma obtains some fundamental results of exponential stability of $\[{C_0}\]$-semigroup of bounded linear operators in Banach spaces. Specially, if $\[{\omega _s} = \sup \{ {\mathop{\rm Re}\nolimits} \lambda ;\lambda \in \sigma (A) < 0\} \]$ and $\[\sup \{ \left\| {{{(\lambda - A)}^{ - 1}}} \right\|;{\mathop{\rm Re}\nolimits} \lambda \ge \sigma \} < \infty \]$ , where \[\sigma \in ({\omega _s},0)\]) and A is the infinitesimal generator of a $\[{C_0}\]$-semigroup in a Banach space $X$, then $\[(a)\int_0^\infty {{e^{ - \sigma t}}\left| {f({e^{tA}}x)} \right|} dt < \infty \]$, $\[\forall f \in {X^*},x \in X\]$; (b) there exists $\[M > 0\]$ such that $\[\left\| {{e^{tA}}x} \right\| \le N{e^{\sigma t}}\left\| {Ax} \right\|\]$, $\[\forall x \in D(A)\]$; (c) there
exists a Banach space $\[\hat X \supset X\]$ such that $\[\left\| {{e^{tA}}x} \right\|\hat x \le {e^{\sigma t}}\left\| x \right\|\hat x,\forall x \in X.\]$. 相似文献
16.
Let
G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let
W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to
[`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A
p
(G); p > 0; we denote a class of functions f analytic in G and satisfying the condition
|