共查询到20条相似文献,搜索用时 203 毫秒
2.
Mark C. Veraar 《Journal of Evolution Equations》2010,10(1):85-127
In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:
(SE) {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t), t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right. 相似文献
3.
F. E. Lomovtsev 《Differential Equations》2008,44(6):866-871
We prove the well-posed solvability in the strong sense of the boundary value Problems 相似文献
$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$ $$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$ $$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$ 4.
W. G. Nowak 《Archiv der Mathematik》2002,78(3):241-248
For a convex planar domain D \cal {D} , with smooth boundary of finite nonzero curvature, we consider the number of lattice points in the linearly dilated domain t D t \cal {D} . In particular the lattice point discrepancy PD(t) P_{\cal {D}}(t) (number of lattice points minus area), is investigated in mean-square over short intervals. We establish an asymptotic formula for¶¶ òT - LT + L(PD(t))2dt \int\limits_{T - \Lambda}^{T + \Lambda}(P_{\cal {D}}(t))^2\textrm{d}t ,¶¶ for any L = L(T) \Lambda = \Lambda(T) growing faster than logT. 相似文献
5.
Roland Schnaubelt 《Journal of Evolution Equations》2001,1(1):19-37
We study the long-term behaviour of the parabolic evolution equation $\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]$\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \] If A(t) A(t) converges to a sectorial operator A with s(A)?i \Bbb R = ? \sigma(A)\cap i \Bbb R =\emptyset as t?¥ t\to\infty , then the evolution family solving the homogeneous problem has exponential dichotomy. If also f(t)? f¥ f(t)\to f_\infty , then the solution u converges to the 'stationary solution at infinity', i.e., limt?¥u(t) = -A\sp-1f¥=:u¥, limt?¥u¢(t)=0, limt?¥A(t)u(t)=Au¥. \lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty. . 相似文献
6.
A. Erfanian 《Archiv der Mathematik》2002,78(4):257-262
The aim of this paper is to give a lower bound for h(2, An), where h(2, An) is the maximum number such that Anh(2, An) A_n^{h(2, A_n)} can be generated by 2 elements, where An is the alternating group on n symbols, and n \geqq 5 n \geqq 5 . Kantor and Lubotzky (1990) gave a lower bound¶ \fracn!8 \frac{n!}{8} for sufficiently large n by the probability of generating the symmetric group. I have improved the above lower bound to \fracn!5 \frac{n!}{5} for large n, using a different method. 相似文献
7.
V.G. Pestov 《Geometric And Functional Analysis》2000,10(5):1171-1201
We establish a close link between the amenability property of a unitary representation p \pi of a group G (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system (\Bbb Sp, G) ({\Bbb S}_{\pi}, G) , where \Bbb SH {\Bbb S}_{\cal H} is the unit sphere the Hilbert space of representation. We prove that p \pi is amenable if and only if either p \pi contains a finite-dimensional subrepresentation or the maximal uniform compactification of (\Bbb Sp ({\Bbb S}_{\pi} has a G-fixed point. Equivalently, the latter means that the G-space (\Bbb Sp, G) ({\Bbb S}_{\pi}, G) has the concentration property: every finite cover of the sphere \Bbb Sp {\Bbb S}_{\pi} contains a set A such that for every e > 0 \epsilon > 0 the e \epsilon -neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of p \pi is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \Bbb Sp {\Bbb S}_{\pi} . As another corollary, a locally compact group G is amenable if and only if for every strongly continuous unitary representation of G in an infinite-dimensional Hilbert space H {\cal H} the system (\Bbb SH, G) ({\Bbb S}_{\cal H}, G) has the property of concentration. 相似文献
8.
In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space E $$\left\{ \begin{gathered} u'(t) + Au(t) = f(t,u(t),Gu(t)) t \in J,t \ne t_k , \hfill \\ \Delta _{\left. u \right|_{t = t_k } } = u\left( {t_k^ + } \right) - u\left( {t_k^ - } \right) = I_k \left( {u\left( {t_k } \right)} \right), k = 1,2, \ldots ,m, \hfill \\ u(0) = g(u) + x_0 , \hfill \\ \end{gathered} \right.$$ where A: D(A) ? E → E is a closed linear operator and ?A generates a strongly continuous semigroup T(t) (t ? 0) on E, f ∈ C(J × E × E, E), J = [0, a], 0 < t 1 < t 2 < ... < t m < a, I k ∈ C(E, E), k = 1, 2, ..., m, and g constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that ?A generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered. 相似文献
9.
Adam Bobrowski 《Journal of Evolution Equations》2007,7(3):555-565
Let
be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in
with related Feller processes {X
A
(t), t ≥ 0} and {X
B
(t), t ≥ 0} and let α and β be two non-negative continuous functions on
with α + β = 1. Assume that the closure C of C
0 = αA + βB with
generates a Feller semigroup {T
C
(t), t ≥ 0} in
. It is natural to think of a related Feller process {X
C
(t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point
, with probability α(p) the process behaves like {X
A
(t), t ≥ 0} and with probability β(p) it behaves like {X
B
(t), t ≥ 0}. We provide an approximation of {T
C
(t), t ≥ 0} via a sequence of semigroups acting in
that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki
et al. [17]. 相似文献
10.
Shangquan Bu 《Integral Equations and Operator Theory》2011,71(2):259-274
We study the well-posedness of the fractional differential equations with infinite delay (P
2): Da u(t)=Au(t)+òt-¥a(t-s)Au(s)ds + f(t), (0 £ t £ 2p){D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}, where A is a closed operator in a Banach space ${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)} and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P
2) on Lebesgue-Bochner spaces
Lp(\mathbbT, X){L^p(\mathbb{T}, X)} and periodic Besov spaces
B p,qs(\mathbbT, X){{B} _{p,q}^s(\mathbb{T}, X)} . 相似文献
11.
W. Müller 《Geometric And Functional Analysis》2000,10(5):1118-1170
Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator MP¢|P(t,l), l ? \frak a*P,\Bbb C M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of MP¢|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order £ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles. 相似文献
12.
Litan Yan 《Mathematische Nachrichten》2003,259(1):84-98
Let X = (Xt, ?t) be a continuous local martingale with quadratic variation 〈X〉 and X0 = 0. Define iterated stochastic integrals In(X) = (In(t, X), ?t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I0(t, X) = 1 and I1(t, X) = Xt. Let (??xt(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = supx∈? ??xt(X) and X* = supt≥0 |Xt|. In this paper, we shall establish various ratio inequalities for In(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants cn,p and Cn,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant Cn,p,γ depending only on n, p and γ, where Un is one of 〈In(X)〉1/2∞ and I*n(X), and V one of the three random variables X*, 〈X〉1/2∞ and ??*∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
13.
Iosef Pinelis 《Annals of Combinatorics》2002,6(1):103-106
Let (Ai) i ? I (A_i) _{i \in I} and (Bi) i ? I (B_i) _ {i \in I} be two (possibly infinite) families of finite sets. Let cl(P) denote the closure of the set P : = { (Ai, Bi ): i ? I } P := \{ ({A_i}, {B_i} ): i \in I \} of the pairs with respect to the componentwise union and intersection operations. Then there exists an injective map èi ? I Ai ? èi ? I Bi {\displaystyle \bigcup _ {i \in I}} A_i \rightarrow {\displaystyle \bigcup _ {i \in I }} B_i such that f (Ai) í Bi f (A_i) \subseteq B_i for every i if, and only if, card (A) £ (A) \leq card (B) for every pair (A, B) ? cl (P) (A, B) \in cl (P) . 相似文献
14.
M. Kikuchi 《Archiv der Mathematik》2000,75(4):312-320
Let (W, F, P)(\Omega, \cal F, P) be a complete nonatomic probability space. We shall give a characterization of rearrangement-invariant spaces X over W\Omega with the property that every martingale f = (fn)n \geqq 0f = (f_n)_{n \geqq 0} bounded in X converges with respect to the norm topology of X. Using the results, we shall consider the summability of martingales by Toeplitz matrices. 相似文献
15.
Rumi Shindo 《Mediterranean Journal of Mathematics》2011,8(1):81-95
Let A, B be uniform algebras. Suppose that A
0, B
0 are subgroups of A
−1, B
−1 that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A
0 → B
0 is a surjection with ||T(f)mT(g)n - a||¥ = ||fmgn - a||¥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A0{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)d = (T(f)/T(1))d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A0{f \in A_0}. This result leads to the following assertion: Suppose that S
A
, S
B
are subsets of A, B that contain A
−1, B
−1 respectively. If m, n > 0 and a surjection T : S
A
→ S
B
satisfies ||T(f)mT(g)n - a||¥ = ||fmgn - a||¥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? SA{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)d = (T(f)/T(1))d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? SA{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B. 相似文献
16.
For given 2n×2n matricesS
13,S
24 with rank(S
13,S
24)=2n
we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C
1(x;λ)u-A
T(x)v with
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