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1.
Cao Jiading 《分析论及其应用》1989,5(2):99-109
Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials:
, ifα
n
≡0, then Bn (0, F, x) are Bernstein polynomials.
Let
, we constructe new polynomials in this paper:
Q
n
(k)
(α
n
,f(t))=d
k
/dx
k
B
n+k
(α
n
,F
k
(u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα
n
≡0, k=1, then Qn
(1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifα
n
=0, k=2, then Qn
(2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is:
Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1],
, it is sufficient and necessary that
,
§ 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]:
.
As usual, for the space Lp [a,b](1≤p<∞), we have
and L[a, b]=l1[a, b].
Letα
n
⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials
[3] [4].
The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports. 相似文献
2.
We establish conditions under which, for a Dirichlet series F(z) = Σ
n = 1
∞
d
n
exp(λ
n
z), the inequality ⋎F(x)⋎≤y(x),x≥x
o, implies the relation Σ
n = 1
∞ |d
n
exp(λ
n
z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞).
Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12,
pp. 1610–1616. December, 1997 相似文献
3.
Alessandra Pagano 《Annali dell'Universita di Ferrara》1993,39(1):1-17
We consider a (possibly) vector-valued function u: Ω→R
N, Ω⊂R
n, minimizing the integral
, whereD
iu=∂u/∂x
i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D
1u,…,Dn−1u∈Lq, under suitable assumptions ona
i(x).
Sunto Consideriamo una funzione u: Ω→R N, Ω⊂R n che minimizzi l'integrale , doveD iu=∂u/∂xi, o un funzionale con un comportamento simile; sotto opportune ipotesi sua i(x), dimostriamo la seguente maggiore integrabilità perDu:D 1u,…,Dn−1uεLq.相似文献
4.
Summary The paper investigates the equation(1.1) in the two cases:i) p≡0,ii) p(≠0) is either bounded or satisfies |(p(t,x,y,z,u)|⩽(A0+|y|+|u|+|z| Ψ(t) where A0 is a constant. For the casei) the asymptotic stability (in the large) of the trivial solution x=0 is investigated and for the caseii) a general estimate and two boundedness results are obtained for solutions of(1.1). The results extend those obtained by Harrow[1] for the same equation(1.1).
Entrata in Redazione il 18 novembre 1971. 相似文献
5.
S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298
We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α
k: C
p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008. 相似文献
6.
The spectrum of each symmetric ψ DO of the symbol class S0
1, γ, 0≤γ<1, acting on B3
p,q(w(x)) and F3
p,q(w(x)), is independent of the choice ofs ∈ℝ, 0<p≤∞ (p<∞ in the F-case), 0<q≤∞ and the weight w(x)∈W. 相似文献
7.
G. I. Laptev 《Journal of Mathematical Sciences》2008,150(5):2384-2394
This paper deals with conditions for the existence of solutions of the equations
considered in the whole space ℝn, n ≥ 2. The functions A
i
(x, u, ξ), i = 1,…, n, A
0(x, u), and f(x) can arbitrarily grow as |x| → ∞. These functions satisfy generalized conditions of the monotone operator theory in the arguments u ∈ ℝ and ξ ∈ ℝn. We prove the existence theorem for a solution u ∈ W
loc
1,p
(ℝn) under the condition p > n.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 133–147, 2006. 相似文献
8.
Laurent Veron 《Journal d'Analyse Mathématique》1992,59(1):231-250
We prove the existence and the uniqueness of a solutionu of−Lu+h|u|
α-1u=f in some open domain ℝd, whereL is a strongly elliptic operator,f a nonnegative function, and α>1, under the assumption that ∂G is aC
2 compact hypersurface, lim
x→∂G
(dist(x, ∂G))2α/(α-1)
f(x)=0, and lim
x→∂G
u(x)=∞. 相似文献
9.
Let Ω⊂R
n
(n≥2) be a bounded open set;Q
T
=Ω×[0,T],S
T
=δΩ×[0,T],S
1,S
2 be the partial boundaries of Ω andS
1∪S
2=δΩ,S
1∩S
2=Φ. We denote Γ1.T
=S
1×[0,T], Γ2.T
=S
2×[0,T], and consider the problem
相似文献
10.
L. I. Sazonov 《Mathematical Notes》1999,65(2):202-207
In the exterior domain Ω⊂ℝ2 we consider the two-dimensional Navier-stokes system Δu-▽p=(u,▽)u, div u=0 whose solution possesses a finite Dirichlet integral
and satisfies the condition lim|x|→∞
u(x)=(1, 0). For this solution, we establish the estimate |u(x)−(1, 0)|≤c|x|
−α, where α>1/4. This estimate implies an asymptotic expression for the solution indicating the presence of a track behind the
body.
Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 246–253, February, 1999. 相似文献
11.
On weighted approximation by Bernstein-Durrmeyer operators 总被引:6,自引:0,他引:6
Zhang Zhenqiu 《分析论及其应用》1991,7(2):51-64
In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in Lp[0, 1] (1≤p≤∞), where the weight function w(x)=xα(1−x)β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals.
Supported by Zhejiang Provincial Science Foundation. 相似文献
12.
Jerk Matero 《manuscripta mathematica》1996,91(1):379-391
Assume that Ω is a bounded, strictly convex, smooth domain in ℝN withN≥2. We consider the problem det ((∂
iju(x)))=f(x,u(x)),u(x)→∞ asx→∂Ω, where (∂
iju(x)) denotes the Hessian ofu(x) andf meets some natural regularity and growth conditions. We prove that there exists a unique smooth, strictly convex solution
of this problem. The boundary-blow-up rate ofu(x) is characterized in terms of the distance ofx from ∂Ω.
Partially supported by the Royal Swedish Academy of Sciences, Gustaf Sigurd Magnuson's fund. 相似文献
13.
The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet
problem for the anisotropic parabolic equations with variable nonlinearity $
u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} }
$
u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} }
. Two different cases are studied. In the first case a
i
≡ a
i
(x), p
i
≡ 2, σ
i
≡ σ
i
(x, t), and b
i
(x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there
exists at least one j for which min σ
j
(x, t) > 2 and either b
j
> 0, or b
j
(x, t) ≥ 0 and Σπ
b
j
−ρ(t)(x, t) dx < ∞ with some σ(t) > 0 depending on σ
j
. In the case of the quasilinear equation with the exponents p
i
and σ
i
depending only on x, we show that the solutions may blow up if min σ
i
≥ max p
i
, b
i
≥ 0, and there exists at least one j for which min σ
j
> max p
j
and b
j
> 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine
the absorption (b
i
≤ 0) and reaction terms. 相似文献
14.
In the space of functions B
a3+={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T3/2)=g(x, −t)}, we establish that if the condition aT
3
(2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u
u
−a
2
u
xx
=g(x, t), u(0, t)=u(π, t)=0, u(x, t+T
3
)=u(x, t), ℝ2, is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator.
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997 相似文献
15.
Matteo Dalla Riva Massimo Lanza de Cristoforis 《Complex Analysis and Operator Theory》2011,5(3):811-833
Let Ω
i
and Ω
o
be two bounded open subsets of
\mathbbRn{{\mathbb{R}}^{n}} containing 0. Let G
i
be a (nonlinear) map from
?Wi×\mathbbRn{\partial\Omega^{i}\times {\mathbb{R}}^{n}} to
\mathbbRn{{\mathbb{R}}^{n}} . Let a
o
be a map from ∂Ω
o
to the set
Mn(\mathbbR){M_{n}({\mathbb{R}})} of n × n matrices with real entries. Let g be a function from ∂Ω
o
to
\mathbbRn{{\mathbb{R}}^{n}} . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map from
]1-(2/n),+¥[×Mn(\mathbbR){]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})} to
Mn(\mathbbR){M_{n}({\mathbb{R}})} . Then we consider the problem
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