共查询到20条相似文献,搜索用时 355 毫秒
1.
Duanxu Dai 《印度理论与应用数学杂志》2018,49(3):571-579
In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: X → Y is said to be a standard ε-isometry provided f(0) = 0 and for all x, y ∈ X. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(xn)}n≥1 is a basic sequence of Y equivalent to {xn}n≥1 whenever {xn}n≥1 is a basic sequence in X and f: X → Y is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.
相似文献
$$\parallel f\left( x \right) - f\left( y \right)\parallel - \parallel x - y\parallel | \leqslant \varepsilon $$
(1)
2.
Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that
$$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$
As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:
This gives an affirmative answer to a question proposed by Vestfrid (2004, 2015). 相似文献
- (i)lim inf t→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ S Y ;
- (ii)\(\tau(f)\equiv sup_{y\epsilon S_{Y}}\) lim inf t→∞dist(ty, f(X))/|t| = 0;
- (iii)there is a surjective linear isometry U: X → Y so that$$\left\| {f(x) - Ux} \right\| \leqslant 2\varepsilon , for all x \in X.$$
3.
We consider integrals of the form , where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? n , and θ ∈ ? k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.
相似文献
$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$
4.
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\} $
$\mathop {\lim }\limits_{d(x,y) \to 0} \left| {f(x) - f(y)} \right|/d^\alpha (x,y) = 0$
5.
We consider some class of non-linear systems of the form where A(·) ∈ ? n × n , A i (·) ∈ ? n × n , b(·) ∈ ? n , whose coefficients are arbitrary uniformly bounded functionals.
$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$
A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation where ρ(·) ∈ ?1, m(·) ∈ ? n , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.
相似文献
$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$
6.
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. $
7.
Hiroshi Suzuki 《Journal of Algebraic Combinatorics》2009,30(3):401-413
Let Γ=(X,R) be a distance-regular graph of diameter d. A parallelogram of length i is a 4-tuple xyzw consisting of vertices of Γ such that ?(x,y)=?(z,w)=1, ?(x,z)=i, and ?(x,w)=?(y,w)=?(y,z)=i?1. A subset Y of X is said to be a completely regular code if the numbers depend only on i=?(x,Y) and j. A subset Y of X is said to be strongly closed if Hamming graphs and dual polar graphs have strongly closed completely regular codes. In this paper, we study parallelogram-free distance-regular graphs having strongly closed completely regular codes. Let Γ be a parallelogram-free distance-regular graph of diameter d≥4 such that every strongly closed subgraph of diameter two is completely regular. We show that Γ has a strongly closed subgraph of diameter d?1 isomorphic to a Hamming graph or a dual polar graph. Moreover if the covering radius of the strongly closed subgraph of diameter two is d?2, Γ itself is isomorphic to a Hamming graph or a dual polar graph. We also give an algebraic characterization of the case when the covering radius is d?2.
相似文献
$\pi_{i,j}=|\Gamma_{j}(x)\cap Y|\quad (i,j\in \{0,1,\ldots,d\})$
$\{x\mid \partial(u,x)\leq \partial(u,v),\partial(v,x)=1\}\subset Y,\mbox{ whenever }u,v\in Y.$
8.
The system where v = C*x is an output, u = S*y is a control, A(·) ∈ R n × n , B(·) ∈ R n × (n–p), C ∈ R n × (n–p), and D ∈ R n × (n–p), is considered. The elements αij(·) and βij(·) of the matrices A(·) and B(·) are arbitrary functionals satisfying the conditions It is assumed that A(·) ∈ Z 1 ∪ Z 3 and A*(·) ∈ Z 1 ∪ Z 3, where Z 1 is the class of matrices in which the first p elements of the kth superdiagonal are sign-definite and the elements above them are sufficiently small. The class Z 3 differs from Z t1 in that the elements between this superdiagonal and the (k + 1)th row are sufficiently small. If k > p, then the elements of the p × p square in the upper left corner of the matrix are sufficiently small as well. By using special quadratic Lyapunov functions, a matrix D for which y(t)–x(t) → 0 exponentially as t → ∞ is first found, and then a matrix S for which the vectors x(t) and y(t) have the same property is constructed.
相似文献
$$\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u,{\kern 1pt} \frac{{dy}}{{dt}} = A\left( \cdot \right)y + B\left( \cdot \right)u + D\left( {C*y - v} \right)$$
$$\mathop {\sup }\limits_{\left( \cdot \right)} |{\alpha _{ij}}\left( \cdot \right)| < \infty \left( {i,j \in 1,n} \right),\mathop {\sup }\limits_{\left( \cdot \right)} |{\beta _{ij}}\left( \cdot \right)| < \infty \left( {i \in 1,n,j \in 1,n - p} \right).$$
9.
A. V. Lasunskii 《Differential Equations》2009,45(3):460-463
For the nonautonomous Lotka-Volterra model where some part φ(x, y) of the prey population is out of reach of the predator, we obtain sufficient conditions for the existence of a positive asymptotically stable equilibrium in the domain of admissible values of the variables x and y. We consider the cases in which φ(x, y) = m, φ(x, y) = mx, and φ(x, y) = my.
相似文献
$\dot x = \alpha (t)(x - M^{ - 1} x^2 - K^{ - 1} (x - \phi (x,y))y),\dot y = \beta (t)y(L^{ - 1} (x - \phi (x,y)) - 1),$
10.
James Hirschorn 《Order》2016,33(1):133-185
A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between two lattices with convex range is a continuous lattice homomorphism. A number of posets are considered; for one of the simplest examples, we prove that every product order embedding σ : ?? → ?? with convex range is of the form and σ(x)(n) = y σ (n) otherwise, for all x ∈ ??, where K σ ? ?, g σ : K σ → ? is a bijection and y σ ∈ ??. The most complex poset examined here is the quotient of the lattice of Baire measurable functions, with codomain of the form ? I for some index set I, modulo equality on a comeager subset of the domain, with its ‘natural’ ordering.
相似文献
$$ \sigma(x)(n)=\left( (x\circ g_{\sigma})+y_{\sigma}\right)(n) ~~~~\text{if}~ n\in K_{\sigma}, $$
(1)
11.
Let J be the Lévy density of a symmetric Lévy process in \(\mathbb {R}^{d}\) with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operatorwhere κ(x, z) is a Borel function on \(\mathbb {R}^{d}\times \mathbb {R}^{d}\) satisfying 0 < κ 0 ≤ κ(x, z) ≤ κ 1, κ(x, z) = κ(x,?z) and |κ(x, z) ? κ(y, z)|≤ κ 2|x ? y| β for some β ∈ (0, 1]. We construct the heat kernel p κ (t, x, y) of \(\mathcal {L}^{\kappa }\), establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel p κ .
相似文献
$$\mathcal{L}^{\kappa}f(x):= \lim_{{\varepsilon} \downarrow 0} {\int}_{\{z \in \mathbb{R}^{d}: |z|>{\varepsilon}\}} (f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$
12.
A. V. Setukha 《Differential Equations》2018,54(9):1236-1255
We consider a bulk charge potential of the form where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(x ? y) is a function with a singularity of the form 1/|x ? y|. We prove that, under certain assumptions on the shape of the surface Σ, the kernel F, and the function g at each point x lying on the midsurface Σ (but not on its boundary), the second derivatives of the function u can be represented as where the function γij(x) does not exceed in absolute value a certain quantity of the order of h2, the surface integral is understood in the sense of Hadamard finite value, and the ni(x), i = 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point x.
相似文献
$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$
$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$
13.
O. Yu. Khachay 《Differential Equations》2008,44(2):282-285
We consider the Cauchy problem for the nonlinear differential equation where ? > 0 is a small parameter, f(x, u) ∈ C ∞ ([0, d] × ?), R 0 > 0, and the following conditions are satisfied: f(x, u) = x ? u p + O(x 2 + |xu| + |u|p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f u 2(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ 0 +∞ f u 2(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].
相似文献
$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$
14.
Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators. 相似文献
15.
The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions. 相似文献
16.
We consider the Schrödinger operatorwhere V is bounded from below and prove a lower bound on the first eigenvalue λ 1 in terms of sublevel estimates: if w V (y) = |{x ∈ [a, b] : V (x) ≤ y}|, thenThe result is sharp up to a universal constant if {x ∈ [a, b] : V(x) ≤ y} is an interval for the value of y solving the minimization problem. An immediate application is as follows: let \({\Omega } \subset \mathbb {R}^{2}\) be a convex domain and let \(u:{\Omega } \rightarrow \mathbb {R}\) be the first eigenfunction of the Laplacian ? Δ on Ω with Dirichlet boundary conditions on ?Ω. We provewhich answers a question of van den Berg in the special case of two dimensions.
相似文献
$$ \text{-} \frac{d^{2}}{d x^{2}} + V {\text{on an interval}}~~[a,b]~{\text{with Dirichlet boundary conditions}},$$
$$\lambda_{1} \geq \frac{1}{250} \min\limits_{y > \min V}{\left( \frac{1}{w_{V}(y)^{2}} + y\right)}.$$
$$\| u \|_{L^{\infty}({\Omega})} \lesssim \frac{1}{\text{inrad}({\Omega})} \left( \frac{\text{inrad}({\Omega})}{\text{diam}({\Omega})} \right)^{1/6} \|u\|_{L^{2}({\Omega})},$$
17.
Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and G x,y \(\underline \triangleleft \) G x . P. Cameron raised the question about the validity of the equality G x,y = 1 in this case. The author proved earlier that, if soc(G) is not a direct power of an exceptional group of Lie type, then G x,y = 1. In the present paper, we prove that, if soc(G) is a direct power of an exceptional group of Lie type distinct from E 6(q), 2 E 6(q), E 7(q), and E 8(q), then G x,y = 1. 相似文献
18.
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)
19.
Let λ > 0 and be the Bessel operator on R+:= (0,∞). We first introduce and obtain an equivalent characterization of CMO(R+, x2λdx). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function b ∈ BMO(R+, x2λdx) is in CMO(R+, x2λdx) if and only if the Riesz transform commutator xxxx is compact on Lp(R+, x2λdx) for all p ∈ (1,∞).
相似文献
$${\Delta _\lambda }: = - \frac{{{d^2}}}{{d{x^2}}} - \frac{{2\lambda }}{x}\frac{d}{{dx}}$$
20.
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