共查询到20条相似文献,搜索用时 31 毫秒
1.
N. A. Izobov 《Differential Equations》2008,44(5):618-631
We prove the conditional exponential stability of the zero solution of the nonlinear differential system with L p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X A (t), X A (0) = E, satisfies the condition where P 1 and P 2 are complementary projections of rank k ∈ {1, …, n ? 1} and rank n ? k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R n that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin.
相似文献
$$\dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0,$$
$$\mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0,$$
2.
Let (X, d) be a compact metric and 0 < α < 1. The space Lip α (X) of Hölder functions of order α is the Banach space of all functions ? from X into \(\mathbb{K}\) such that ∥?∥ = max{∥?∥∞, L(?)} < ∞, where is the Hölder seminorm of ?. The closed subspace of functions ? such that is denoted by lip α (X). We determine the form of all bijective linear maps from lip α (X) onto lip α (Y) that preserve the Hölder seminorm.
相似文献
$L(f) = sup\{ \left| {f(x) - f(y)} \right|/d^\alpha (x,y):x,y \in X, x \ne y\} $
$\mathop {\lim }\limits_{d(x,y) \to 0} \left| {f(x) - f(y)} \right|/d^\alpha (x,y) = 0$
3.
Alexander A. Gushchin Uwe Küchler 《Statistical Inference for Stochastic Processes》2011,14(3):273-305
Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equationwhere W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [?r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the formwhere a and b are finite signed measure on [?r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoodsas T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the formwhere \({H\in[1/2,1]}\) and B H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ T = T ?1/(2H) ?(T) with a slowly varying function ?. Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.
相似文献
$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$
$a_\vartheta = a+b_\vartheta-b,$
$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$
$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$
4.
For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) 0}, S, T_u 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval. 相似文献
5.
We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l r,k α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product , and generated by the classical orthogonal Laguerre polynomials L k α (x) (k = 0, 1,...). The polynomials l r,k α (x) are represented as expressions containing the Laguerre polynomials L n α?r (x). An explicit form of the polynomials l r,k+r α (x) is established as an expansion in the powers x r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l r,k α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.
相似文献
$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$
6.
Guang Hua Shi 《数学学报(英文版)》2017,33(3):439-448
In this paper, by the Aubry–Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models d/dt(x/(1-|x|~2~(1/2))+ |x|~(α-1)x=p(t),where p(t) ∈ C~0(R~1) is 1-periodic and α 0. 相似文献
7.
Let X be a Banach space with a weak uniform normal structure and C a non–empty convexweakly compact subset of X. Under some suitable restriction, we prove that every asymptoticallyregular semigroup T = {T(t) : t ∈¸ S} of selfmappings on C satisfying
has a common fixed point, where WCS(X) is the weakly convergent sequence coefficient of X, and\({\left| {{\left\| {T(t)} \right\|}} \right|}\) is the exact Lipschitz constant of T(t). 相似文献
${\mathop {\lim \inf }\limits_{S \mathrel\backepsilon t \to \infty } }{\left| {{\left\| {T(t)} \right\|}} \right|} < {\text{WCS}}(X)$
8.
In the space L 2(?2), we consider the operator . We study the spectrum of H and, for V ∈ C 0 2 (?2), prove the trace formula where c 0 = π ?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ k (i) are the eigenvalues of H.
相似文献
$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$
$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $
9.
Cheng-Kai Liu 《Monatshefte für Mathematik》2011,57(3):297-311
Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f (X 1, . . . , X t ) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such thatfor all \({x_1,\ldots,x_t\in\lambda}\). Then either d = 0 and λ δ(λ) = 0 or λ C = RCe for some idempotent e in the socle of RC and one of the following holds:
相似文献
$d(f(x_{1},\ldots,x_{t}))f(x_{1},\ldots,x_{t})-f(x_{1},\ldots,x_{t})\delta(f(x_{1},\ldots,x_{t}))\in C$
- (1)f (X1, . . . , X t ) is central-valued on eRCe;
- (2)λ(d + δ)(λ) = 0 and f (X1, . . . , X t )2 is central-valued on eRCe;
- (3)char R = 2 and eRCe satisfies st 4(X 1, X 2, X 3, X 4), the standard polynomial identity of degree 4.
10.
Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) = S y (f (h(y))) for a family of linear operators S y : E → F, \({y \in Y}\) , and a function h: Y → X. In this paper, we consider maps having the property: where Z(f) = {f = 0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ∞), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Önal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C *-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such thatThen X is homeomorphic to Y and E is lattice isomorphic (respectively, C *-isomorphic) to F. Some results concerning the continuity of T are also obtained.
相似文献
$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $
$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $
11.
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 where the unbounded operators A s (t), s > 0, in a Hilbert space H have domains D(A s (t)) depending on t, are subordinate to the powers A 1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A 0(t) and the symmetric operators A s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that , where the smoothing operators are defined by .
相似文献
$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$
$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$
$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$
12.
Adel Mahmoud Gomaa 《Czechoslovak Mathematical Journal》2017,67(2):339-365
We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.
相似文献
$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$
(P)
$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$
(Q)
13.
This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation We call soliton a solution of (0.1) of the form u(t,x)=Q c (x?ct?y 0), where c>0, y 0∈? and \(Q_{c}''+Q_{c}^{4}=cQ_{c}\). Since (0.1) is not an integrable model, the general question of the collision of two given solitons \(Q_{c_{1}}(x-c_{1}t)\), \(Q_{c_{2}}(x-c_{2}t)\) with c 1≠c 2 is an open problem.
$\partial_tu+\partial_x(\partial_x^2u+u^4)=0,\quad t,x\in \mathbb{R}.$
(0.1)
We focus on the special case where the two solitons have nearly equal speeds: let U(t) be the solution of (0.1) satisfying for \(\mu_{0}=(c_{2}^{-}-c_{1}^{-})/(c_{1}^{-}+c_{2}^{-})>0\) small. By constructing an approximate solution of (0.1), we prove that, for all time t∈?, with y 1(t)?y 2(t)>2|ln?μ 0|+C, for some C∈?. These estimates mean that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable KdV equation in this regime.
$\lim_{t\to-\infty}\|{U}(t)-Q_{c_1^-}(.-c_1^-t)-Q_{c_2^-}(.-c_2^-t)\|_{H^1}=0,$
$\begin{array}{l}\displaystyle{U}(t)={Q}_{c_1(t)}(x-y_1(t))+{Q}_{c_2(t)}(x-y_2(t))+{w}(t)\\[6pt]\displaystyle\quad\mbox{where }\|w(t)\|_{H^1}\leq|\ln\mu_0|\mu_0^2,\end{array}$
However, unlike in the integrable case, we prove that the collision is not perfectly elastic, in the following sense, for some C>0, 相似文献
$\lim_{t\to+\infty}c_1(t)>c_2^-\biggl(1+\frac{\mu_0^5}{C}\biggr),\quad \lim_{t\to+\infty}c_2(t)
and \({w}(t)\not\to0\) in H 1 as t→+∞.14.
V. G. Zhuravlev 《Journal of Mathematical Sciences》2012,182(4):472-483
For the number n s (α, β; X) of points (x 1 , x 2) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x 1 2 ? ??αx 2 2 ?=?β and such that 0?≤?x 1? ≤?X, x 2? ≥?0, the asymptotic formulais established, and the coefficient c s (α, β) is calculated exactly. Using this, we obtain the following result. Let F m be the Fibonacci numbers, A i ∈ \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A i in the Fibonacci numeral system. Then the number n s (X) of all solutions (A 1 , A 2) of the Diophantine system0?≤?A 1? ≤?X, A 2? ≥?0, satisfies the asymptotic formulaHere τ?=?(?1?+?√5)/2 is the golden ratio, and c s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.
相似文献
$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $
$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $
$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $
15.
Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition operators
For the spectral radius of weighted composition operators with positive weight e φ T α , \({\varphi\in C(X)}\) , acting in the spaces L p (X, μ) the following variational principle holdswhere X is a Hausdorff compact space, \({\alpha:X\mapsto X}\) is a continuous mapping and τ α some convex and lower semicontinuous functional defined on the set \({M^1_\alpha}\) of all Borel probability and α-invariant measures on X. In other words \({\frac{\tau_\alpha}{p}}\) is the Legendre– Fenchel conjugate of ln r(e φ T α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form \({A_{\varphi, {\bf c}}= \sum_{k=0}^n e^{c_k} (e^{\varphi} T_{\alpha})^k}\) , \({{\bf c}=(c_k)\in {\Bbb R}^{n+1}}\) . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r(A φ,c) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ, by fixing c.
相似文献
$\ln r(e^\varphi T_\alpha)=\max_{\nu\in M^1_\alpha} \left\{\int\limits_X\varphi d\nu-\frac{\tau_\alpha(\nu)}{p}\right\},$
16.
In the rectangle G = (0, 1) × (0, T), we consider the family of problems . It is well known that, for α = 0 and α = 1, the corresponding problems with local conditions are solvable, and the solutions are unique and belong to W 2 2,1 (G).
We prove the existence and uniqueness of solutions of the family of problems with nonlocal conditions for each α ∈ (0, 1). For the differences u α ? u 0 and u α ? u 1 (0 < α < 1), we establish a priori estimates and use them to prove that if ? α → ? 0 as α → 0, then u α → u 0 and if ? α → ? 1 as α → 1, then u α → u 1. 相似文献
$$\begin{gathered} \frac{1}{{a(x,t)}}\frac{{\partial u_\alpha }}{{\partial t}} - \frac{{\partial ^2 u_\alpha }}{{\partial x^2 }} = f(x,t), u_\alpha (x,0) = \phi _\alpha (x), u_\alpha (0,t) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ u_0 (1,t) = h(t), \frac{{\partial u_1 (1,t)}}{{\partial x}} = h(t), \frac{{u_\alpha (1,t) - u_\alpha (\alpha ,t)}}{{1 - \alpha }} = h(t), 0 < \alpha < 1, \hfill \\ a_1 \geqslant a(x,t) \geqslant a_0 > 0, h \in W_2^1 (0,T), \phi _\alpha \in W_2^1 (0,T), \phi _\alpha (0) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ \phi _0 (1) = h(0), \phi '_1 (1) = h(0), \frac{{\phi _\alpha (1) - \phi _\alpha (0)}}{{1 - \alpha }} = h(0), 0 < \alpha < 1, f \in L_2 (G) \hfill \\ \end{gathered} $$
17.
For the first-order ordinary delay differential equation where p ∈ L loc(?+; ?+), τ ∈ C(?+; ?+), τ(t) ≤ t for t ∈ ?+, limt→+∞ τ(t) = +∞, and ?+:= [0, ∞), we obtain new criteria for the existence of sign-definite and oscillating solutions, thus generalizing some earlier-known results.
相似文献
$$u'(t) + p(t)u(r(t)) = 0,$$
18.
Igor I. Skrypnik 《Israel Journal of Mathematics》2016,212(1):163-188
We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ [0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces Lp(T,X), periodic Besov spaces Bp,qs(T,X) and periodic Triebel–Lizorkin spaces Fp,qs(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P3) in the above three function spaces. 相似文献
19.
Let {X(t), t∈? N } be a fractional Brownian motion in ? d of index H. If L(0,I) is the local time of X at 0 on the interval I?? N , then there exists a positive finite constant c(=c(N,d,H)) such that where \(\phi(t)=t^{N-dH}(\log\log\frac{1}{t})^{dH/N}\), and m φ (E) is the Hausdorff φ-measure of E. This refines a previous result of Xiao (Probab. Theory Relat. Fields 109: 126–197, 1997) on the relationship between the local time and the Hausdorff measure of zero set for d-dimensional fractional Brownian motion on ? N .
相似文献
$m_\phi\bigl(X^{-1}(0)\cap I\bigr)=cL(0,I),$
20.
In this paper, we consider the existence of positive solutions of second-order periodic boundary value problem , where 0 < ε < 1/2, g: [0, 2π] → ? is continuous, f: [0,∞) → ? is continuous and λ > 0 is a parameter.
相似文献
$$u'' + {\left( {\frac{1}{2} + \varepsilon } \right)^2}u = \lambda g\left( t \right)f\left( u \right),t \in \left[ {0,2\pi } \right],u\left( 0 \right) = u\left( {2\pi } \right),u'\left( 0 \right) = u'\left( {2\pi } \right)$$