共查询到20条相似文献，搜索用时 224 毫秒
1.
We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded
domains in ℝ
n
. We consider integral operators K whose kernels have the form
|
k( x,y ) = c( x,y )\frace - a| x - y || x - y |b , ( x,y ) ? W×W, k\left( {x,y} \right) = c\left( {x,y} \right)\frac{{{e^{ - \alpha \left| {x - y} \right|}}}}{{{{\left| {x - y} \right|}^\beta }}},\,\left( {x,y} \right) \in \Omega \times \Omega, 相似文献
2.
Let X be a normed space and V be a convex subset of X. Let a\colon \mathbb R+ ? \mathbb R+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbb R{f \colon V \to \mathbb{R}} is called α-midconvex if | f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||) for x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V. 相似文献
3.
We give the general and the so-called density function solutions of equation | lllfU(x)fV(y)=fX(\frac1-y1-xy ) fY (1-xy) \fracy1-xy ( (x, y) ? (0,1)2 )\begin{array}{lll}f_{U}(x)f_{V}(y)=f_{X}\left(\frac{1-y}{1-xy} \right) f_{Y} (1-xy) \frac{y}{1-xy} \qquad \left( (x, y) \in (0,1)^2 \right)\end{array} 相似文献
4.
Under some conditions on the functions f and g defined in a real interval I the function
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Q[f,g](x,y):=( \fracfg ) -1 ( \fracf(x) g(y) ) Q^{[f,g]}(x,y):=\left( \frac{f}{g} \right) ^{-1} \left( \frac{f(x)} {g(y)} \right) 相似文献
5.
We consider the semi-infinite optimization problem:
|
f*:=minx ? X {f(x):g(x,y) ￡ 0, "y ? Yx},f^*:=\min_{\mathbf{x}\in\mathbf{X}} \bigl\{f(\mathbf{x}):g(\mathbf{x},\mathbf{y}) \leq 0, \forall\mathbf{y}\in\mathbf {Y}_\mathbf{x}\bigr\}, 相似文献
6.
Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶ f( x + f( x) n y) = f( x) f( y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : | f( x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : | f( x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f( X \ A) = { 0 }) f(X \backslash A) = \{ 0 \}) . 相似文献
7.
Let X, X
1, X
2,… be i.i.d.
\mathbb Rd {\mathbb{R}^d} -valued real random vectors. Assume that E
X = 0 and that X has a nondegenerate distribution. Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms
\mathbb Q[ SN ] \mathbb{Q}\left[ {{S_N}} \right] of the normalized sums S
N
= N
−1/2 ( X
1 + ⋯ + X
N
) and show that, without any additional conditions,
|
DN(a) = supx | \textP{ \mathbbQ[ SN - a ] \leqslant x } - \textP{ \mathbbQ[ G - a ] \leqslant x } - Ea(x) | = O( N - 1 ) \Delta_N^{(a)} = \mathop {{\sup }}\limits_x \left| {{\text{P}}\left\{ {\mathbb{Q}\left[ {{S_N} - a} \right] \leqslant x} \right\} - {\text{P}}\left\{ {\mathbb{Q}\left[ {G - a} \right] \leqslant x} \right\} - {E_a}(x)} \right| = \mathcal{O}\left( {{N^{ - 1}}} \right) 相似文献
8.
We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than
( n - 1 ) \mathord | / |
\vphantom ( n - 1 ) ?{1 - x2} ?{1 - x2} {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} for all x ∈ (−1, 1) such that
| x | ? èk = 0[ n \mathord | / |
\vphantom n 2 2 ] [ cos\frac2k + 12( n - 1 )p, cos\frac2k + 12np ] \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} . 相似文献
9.
We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space
( X, D( Aa ) ? R( Aa ) ) q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}}
as
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{ x ? X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}} 相似文献
10.
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖
X
and ‖.‖
Y
denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖ Tf Tg‖
Y
= ‖ fg‖
X
, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖ fg + α‖
X
= ‖ Tf Tg + α‖
Y
, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c( B) → c( A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c( B) such that for each f ∈ A,
|
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
相似文献
11.
The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form |
f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz 相似文献
12.
We present the solutions of boundary-value and initial boundary-value problems for a nonlinear parabolic equation with Lévy
Laplacian ∆
L
resolved with respect to the derivative
|
\frac?U( t,x )?t = f( U( t,x ),DLU( t,x ) ) \frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {U\left( {t,x} \right),{\Delta_L}U\left( {t,x} \right)} \right) 相似文献
13.
The aim of the paper is to deal with the following composite functional inequalities
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f(f(x)-f(y)) ￡ f(x+y) + f(f(x-y)) -f(x) - f(y), f(f(x)-f(y)) ￡ f(f(x+y)) + f(x-y) -f(x) - f(y), f(f(x)-f(y)) ￡ f(f(x+y)) + f(f(x-y)) -f(f(x)) - f(y),\begin{gathered}f(f(x)-f(y)) \leq f(x+y) + f(f(x-y)) -f(x) - f(y), \hfill \\ f(f(x)-f(y)) \leq f(f(x+y)) + f(x-y) -f(x) - f(y), \hfill \\ f(f(x)-f(y)) \leq f(f(x+y)) + f(f(x-y)) -f(f(x)) - f(y),\end{gathered} 相似文献
14.
Let X be a real inner product space of dimension greater than 2 and f be a real functional defined on X. Applying some ideas from the recent studies made on the alternative-conditional functional equation
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(x, y) = 0 T f(x + y)2 = [f(x) + f(y)]2(x, y) = 0 \Rightarrow f(x + y)^2 = [f(x) + f(y)]^{2} 相似文献
15.
We study the problem of finding the best constant in the generalized Poincaré inequality
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lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1], ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0, 相似文献
16.
Let T
X
be the full transformation semigroup on a set X,
|
TE*(X)={a ? TX:"x,y ? X, (x,y) ? E? (xa,ya) ? E}T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\} 相似文献
17.
Let
C( \mathbb Rm ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions
x:\mathbb Rm ? \mathbb R x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm
|
|| x ||C = || x ||C( \mathbbRm ): = sup{ | x(t) |:t ? \mathbbRm } \left\| x \right\|C = {\left\| x \right\|_{C\left( {{\mathbb{R}^m}} \right)}}: = \sup \left\{ {\left| {x(t)} \right|:t \in {\mathbb{R}^m}} \right\} 相似文献
18.
Let L
p
, 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm
|| f ||p = ( ò - pp | f |p )1 \mathord | / |
\vphantom 1 p p {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} , and let C = L
∞ be the space of continuous 2π-periodic functions with the norm
|| f ||￥ = || f || = maxe ? \mathbbR | f(x) | {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that
P(f) \leqslant M|| f || P(f) \leqslant M\left\| f \right\| for every f ∈ C. By ?k = 0￥ Ak (f) \sum\limits_{k = 0}^\infty {{A_k}} (f) denote the Fourier series of the function f, and let l = { lk }k = 0￥ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty be a sequence of real numbers for which ?k = 0￥ lk Ak(f) \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) is the Fourier series of a certain function f
λ ∈ L
p
. The paper considers questions related to approximating the function f
λ by its Fourier sums S
n
(f
λ) on a point set and in the spaces L
p
and CP. Estimates for || fl - Sn( fl ) ||p {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} and P(f
λ − S
n
(f
λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions
f and f
λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f.
Bibliography: 11 titles. 相似文献
19.
We describe sequences of zeros of functions f≢0 that are analytic in the half-plane ℂ +={ z:Re z> and satisfy the condition where 0≤σ<+∞ and η is a positive function continuously differentiable on [0; +∞) and such that xη′( x)/η( x)→0 as x→+∞.
Drohobych Pedagogic University, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 904–909,
July, 1999. 相似文献
20.
In this paper we study the quenching problem for the non-local diffusion equation
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ut(x,t) = òW J(x - y)u(y,t)dy + ò\mathbbRN\W J(x - y)dy - u(x,t) - lu - p(x,t) {u_t}(x,t) = \int\limits_\Omega {J(x - y)u(y,t)dy + \int\limits_{{\mathbb{R}^N}\backslash \Omega } {J(x - y)dy - u(x,t) - \lambda {u^{ - p}}(x,t)} } 相似文献
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