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1.
For any , let Pk denote the natural projections on ℓ1. Let |||||| be an equivalent norm of ℓ1 that satisfies all of the following four conditions: - (1) There are α>4 and a positive (decreasing) sequence (αn) in (0,1) such that for any normalized block basis {fn} of (ℓ1,||||||) and xℓ1 with Pk−1(x)=x and |||x|||<αk,
- (2) There are two strictly decreasing sequences {βk} and {γk} with such that for any normalized block basis {fn} of (ℓ1,||||||) and x with (I−Pk)(x)=x,
- (3) For any , I−Pk=1.
- (4) The unit ball of (ℓ1,||||||) is σ(ℓ1,c0)-closed.
In this article, we prove that the space ( ℓ1,||||||) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author.
Keywords: Renorming; Fixed point property 相似文献
2.
We consider a Cauchy problem for a semilinear heat equation with p> pS where pS is the Sobolev exponent. If u( x, t)=( T− t) −1/(p−1)φ(( T− t) −1/2x) for xRN and t[0, T), where φ is a regular positive solution of then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡( p−1) −1/(p−1) for all p>1. Let pL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for pS< p< pL. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p> pL. 相似文献
3.
Let I be a finite or infinite interval and dμ a measure on I. Assume that the weight function w( x)>0, w′( x) exists, and the function w′( x)/ w( x) is non-increasing on I. Denote by ℓ k's the fundamental polynomials of Lagrange interpolation on a set of nodes x1< x2<< xn in I. The weighted Lebesgue function type sum for 1≤ i< j≤ n and s≥1 is defined by In this paper the exact lower bounds of Sn( x) on a “big set” of I and are obtained. Some applications are also given. 相似文献
4.
With the notation
, we prove the following result. Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies||p||L2(K)An1/2 and ||p′||L2(K)Bn3/2. ThenwithWe also prove that and for every
, where
denotes the collection of all trigonometric polynomials of the form 相似文献
5.
Let
be a probability space and let Pn be the empirical measure based on i.i.d. sample ( X1,…, Xn) from P. Let
be a class of measurable real valued functions on
For
define Ff( t):= P{ ft} and Fn,f( t):= Pn{ ft}. Given γ(0,1], define n,γ(δ):=1/( n1−γ/2δ γ). We show that if the L2( Pn)-entropy of the class
grows as −α for some α(0,2), then, for all
and all δ(0,Δ n), Δ n=O( n1/2), and where
and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define Then for all
uniformly in
and with probability 1 (for
the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory. 相似文献
6.
In the paper sufficient conditions are given under which the differential equation y(n)= f( t, y,…, y(n−2)) g( y(n−1)) has a singular solution y :[ T,τ)→ R, τ<∞ fulfilling 相似文献
7.
In this paper, the authors study the asymptotic behavior of solutions of second-order neutral type difference equations of the form Δ2(yn+pyn−k)+f(n,yn−ℓ)=0,n | , n ε, and Δ2(yn+pyn−k)+f(n,yn−ℓ,Δyn−ℓ)=0,n
,n ε using some difference inequalities. We establish conditions under which all nonoscillatory solutions are asymptotic to an + b as n → ∞ with a and b ε . 相似文献
8.
The
n ×
n generalized Pascal matrix
P(
t) whose elements are related to the hypergeometric function
2F1(
a,
b;
c;
x) is presented and the Cholesky decomposition of
P(
t) is obtained. As a result, it is shown that
is the solution of the Gauss's hypergeometric differential equation,
x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0
. where
a and
b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of
P(
t) is given.
相似文献
9.
In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear
nth-order differential equations with delays of the form
x(n)(t)+f(t,x(n−1)(t))+g(t,x(t−τ(t)))=e(t).
相似文献
10.
In Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {
αn}
∞n=0be a sequence in the open unit disk in the complex plane, let
(
/|
αk|=−1 when
αk=0), and let
We consider the following “moment” problem: Given a positive-definite Hermitian inner product ·, · on
×
, find a non-decreasing function
μon [−
π,
π] (or a positive Borel measure
μon [−
π,
π)) such that
In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that
If this series diverges the solution is always unique.
相似文献
11.
Let
ζ be the Riemann zeta function and
δ(
x)=1/(2
x-1). For all
x>0 we have
(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),
with the best possible constant factors
This improves a recently published result of Cerone et al., J. Inequalities Pure Appl. Math. 5(2) (43) (2004), who showed that the double-inequality holds with and .
相似文献
12.
For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.
相似文献
13.
Let {
pk(
x;
q)} be any system of the
q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the
q-difference equation
Dq(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {
pk(1)(
x;
q)} be associated polynomials of the polynomials {
pk(
x;
q)}. Explicit forms of the coefficients
bn,k and
cn,k in the expansions
are given in terms of basic hypergeometric functions. Here
k(
x) equals
xk if σ
+(0)=0, or (
x;
q)
k if σ
+(1)=0, where σ
+(
x)σ(
x)+(
q−1)
xτ(
x). The most important representatives of those two classes are the families of little
q-Jacobi and big
q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous
q-difference equation satisfied by
pn−1(1)(
x;
q) in a special form, recurrence relations (in
k) for
bn,k and
cn,k are obtained in terms of σ and τ.
相似文献
14.
Consider Robin problem involving the
p(
x)-Laplacian on a smooth bounded domain
Ω as follows
Applying the sub-supersolution method and the variational method, under appropriate assumptions on
f, we prove that there exists
λ*>0 such that the problem has at least two positive solutions if
λ(0,
λ*), has at least one positive solution if
λ=
λ*<+∞ and has no positive solution if
λ>
λ*. To prove the results, we prove a norm on
W1,p(x)(
Ω) without the part of ||
Lp(x)(Ω) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.
相似文献
15.
The authors consider the semilinear SchrSdinger equation
-△Au+Vλ(x)u= Q(x)|u|γ-2u in R^N,
where 1 〈 γ 〈 2* and γ≠ 2, Vλ= V^+ -λV^-. Exploiting the relation between the Nehari manifold and fibrering maps, the existence of nontrivial solutions for the problem is discussed.
相似文献
16.
Let
p be a nonnegative locally bounded function on
,
N3, and 0<
γ<1. Assuming that the oscillation sup
|x|=rp(
x)−inf
|x|=rp(
x) tends to zero as
r→∞ at a specified rate, it is shown that the equation
Δu=
p(
x)
uγ admits a positive solution in
satisfying lim
|x|→∞u(
x)=∞ if and only if
相似文献
18.
We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium
N*=1/(
a+∑
i=0mbi) of the following differential equation with piecewise constant arguments:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, ∑
i=0mbi>0,
bi0,
i=0,1,2,…,
m, and
a+∑
i=0mbi>0. These new conditions depend on
a,
b0 and ∑
i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case
m=0 and
r(
t)≡
r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, 1−
ax−
g(
x,
x,…,
x)=0 has a unique solution
x*>0 and
g(
x0,
x1,…,
xm)
C1[(0,+∞)×(0,+∞)××(0,+∞)].
相似文献
19.
We consider the integral of a function
and its approximation by a quadrature rule of the form
i.e., by a rule which uses the values of both
y and its derivative at nodes of the quadrature rule. We examine the cases when the integrand is either a smooth function or an ω dependent function of the form
y(
x)=
f1(
x) sin(ω
x)+
f2(
x) cos(ω
x) with smoothly varying
f1 and
f2. In the latter case, the weights
wk and α
k are ω dependent. We establish some general properties of the weights and present some numerical illustrations.
相似文献
20.
In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)
ut=ρu−Δφ(u)−(1+iγ)Δu−νΔ2u−(1+iμ)|u|2σu+αλ1(|u|2u)+β(λ2)|u|2
is studied. The existence of global attractor for this equation with periodic boundary condition is established and upper bounds of Hausdorff and fractal dimensions of attractor are obtained.
相似文献