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1.
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation
where
(i) r,c ∈ C([t
0, ∞), ℝ := (− ∞, ∞)) and r(t) > 0 on [t
0, ∞) for some t
0 ⩾ 0;
(ii) Φ(u) = |u|p−2
u for some fixed number p > 1.
We also generalize some results of Hille-Wintner, Leighton and Willet. 相似文献
2.
Michael Bildhauer 《manuscripta mathematica》2003,110(3):325-342
Suppose that f: ℝ
nN
→ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|2) if N>1. If f satisfies an ellipticity condition of the form
then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem
provided that the given boundary data u
0
W
1
1
(ω;ℝ
N
) are additionally assumed to be of class L
∞(ω;ℝ
N
). Moreover, if μ<3, then the boundedness of u
0 yields local C
1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem
In our paper we show that the restriction u
0L
∞(ω;ℝ
N
) is superfluous in the two dimensional case n=2, hence we may prescribe boundary values from the energy class W
1
1
(ω;ℝ
N
) and still obtain the above results.
Received: 12 February 2002 / Revised version: 7 October 2002 Published online: 14 February 2003
Mathematics Subject Classification (2000): 49N60, 49N15, 49M29, 35J 相似文献
3.
This paper studies the existence of solutions to the singular boundary value problem
, where g: (0, 1) × (0, ∞) → ℝ and h: (0, 1) × [0, ∞) → [0, ∞) are continuous. So our nonlinearity may be singular at t = 0, 1 and u = 0 and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and
lower solutions.
The research is supported by NNSF of China(10301033). 相似文献
4.
Let B
0,B
1, ⋯ ,B
n
be independent standard Brownian motions, starting at 0. We investigate the tail of the capture time
where 0<b
i
≤ 1, 1 ≤i≤n. In particular, we have ?τ3=∞ and ?τ5<∞. Various generalizations are also studied.
Received: 10 January 2000 / Revised version: 12 January 2001 /?Published online: 14 June 2001 相似文献
5.
Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications
also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement
equation
where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.
The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in
terms of conditions on the pair (a, F).
To have Lp solutions from F ∈ Lp(ℝ), we construct by the cascade algorithm a sequence of functions φ0 ∈ Lp(ℝ) from a compactly supported initial function ℝ as
A necessary and sufficient condition for the sequence {φn} to converge in Lp(ℝ)(1 ≤ p ≤ ∞) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property
of the p-norm joint spectral radius (1 ≤ p ≤ ∞) is presented.
Finally, the general theory is applied to some examples and multiple refinable functions.
Acknowledgements and Notes. Research supported in part by Research Grants Council and City University of Hong Kong under Grants #9040281, 9030562, 7000741. 相似文献
6.
V. E. Slyusarchuk 《Mathematical Notes》2000,68(3):386-391
Necessary and sufficient conditions for the Lipschitzian invertibility of the difference map
, wheref: ℝ → ℝ is a continuous function, in the spacesl
p
(ℤ, ℝ), where 1≤p≤∞, of two-sided numerical sequences are obtained.
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 448–454, September, 2000. 相似文献
7.
We analyze polynomials P
n
that are biorthogonal to exponentials
, in the sense that
Here α>−1. We show that the zero distribution of P
n
as n→∞ is closely related to that of the associated exponent polynomial
More precisely, we show that the zero counting measures of {P
n
(−4nx)}
n=1∞ converge weakly if and only if the zero counting measures of {Q
n
}
n=1∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial
under appropriate assumptions on {Δ
n,j
}.
相似文献
8.
In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second
order neutral delay differential equation (NDDE)
are obtained, where q, h ∈ C([0, ∞), ℝ) such that q(t) ≥ 0, r ∈ C
(1) ([0, ∞), (0, ∞)), p ∈ C ([0, ∞), ℝ), G ∈ C (ℝ, ℝ) and τ ∈ ℝ+. Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.
相似文献
9.
Dominique Barbolosi 《Monatshefte für Mathematik》1999,28(4):189-200
For any irrational , let denote the regular continued fraction expansion of x and define f, for all z > 0 by and by J. GALAMBOS proved that (μ the Gauss measure)
In this paper, we first point out that for all , ( has no limit for for almost all , proving more precisely that: For all , one has for almost all
相似文献
10.
In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation
has a positive and bounded solution, where q, h, f ∈ C ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, r ∈ C
(1) ([0, ∞), (0, ∞)), p ∈ C
(2) [0, ∞), ℝ), G ∈ C(ℝ, ℝ) and τ ∈ ℝ+. In our work r(t) ≡ 1 is admissible and neither we assume G is non-decreasing, xG(x) > 0 for x ≠ 0, nor we take G is Lipschitzian. Hence the results of this paper improve many recent results.
相似文献
11.
László Hatvani 《Periodica Mathematica Hungarica》2008,56(1):71-82
The first part of this review paper is devoted to the simple (undamped, unforced) pendulum with a varying coefficient. If
the coefficient is a step function, then small oscillations are described by the equation
Using a probability approach, we assume that (a
k
)
k=1∞ is given, and {t
k
}
k=1∞ is chosen at random so that t
k
− t
k−1 are independent random variables. The first problem is to guarantee that all solutions tend to zero, as t → ∞, provided that a
k
↗ ∞ (k → ∞). In the problem of swinging the coefficient a
2 takes only two different values alternating each others, and t
k
− t
k−1 are identically distributed. One has to find the distributions and their critical expected values such that the amplitudes
of the oscillations tend to ∞ in some (probabilistic) sense. In the second part we deal with the damped forced pendulum equation
In 1999 J. Hubbard discovered that some motions of this simple physical model are chaotic. Recently, using also the computer
(the method of interval arithmetic), we gave a proof for Hubbard’s assertion. Here we show some tools of the proof.
Supported by the Hungarian NFSR (OTKA T49516) and by the Analysis and Stochastics Research Group of the Hungarian Academy
of Sciences. 相似文献
12.
We consider the singular Cauchy problem
, where x: (0, τ) → ℝ, g: (0, τ) → (0, + ∞), h: (0, τ) → (0, + ∞), g(t) ≤ t, and h(t) ≤ t, t ∈ (0, τ), for linear, perturbed linear, and nonlinear equations. In each case, we prove that there exists a nonempty set
of continuously differentiable solutions x: (0, ρ] → (ρ is sufficiently small) with required asymptotic properties.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1344–1358, October, 2005. 相似文献
13.
L. V. Rozovsky 《Journal of Mathematical Sciences》2009,159(3):341-349
Let Sn = X1 + · · · + X
n
, n ≥ 1, and S
0 = 0, where X
1, X
2, . . . are independent, identically distributed random variables such that the distribution of S
n/B
n converges weakly to a nondeoenerate distribution F
α
as n → ∞ for some positive B
n
. We study asymptotic behavior of sums of the form
where
a function d(t) is continuous at [0,1] and has power decay at zero,
Bibliography: 13 titles.
Translated from Zapiski Nauchnylch Serninarov POMI, Vol. 361, 2008, pp. 109–122. 相似文献
14.
J. Schoißengeier 《Monatshefte für Mathematik》2000,131(3):227-234
For a real number x let be the fractional part of x and for any set M let c
M
be the characteristic function of M. For and a positive integer N let
be the discrepancy of the sequence modulo 1. In this paper we prove that
(Received 2 May 2000; in revised form 19 June 2000) 相似文献
15.
J. Schoi?engeier 《Monatshefte für Mathematik》2000,74(4):227-234
For a real number x let be the fractional part of x and for any set M let c
M
be the characteristic function of M. For and a positive integer N let
be the discrepancy of the sequence modulo 1. In this paper we prove that
相似文献
16.
Let (S, o) be a semigroup. We determine all solutions of the functional equation
under the assumption thatg : ℝ → ℝ is continuous andf : ℝ →S. 相似文献
17.
18.
Linghai ZHANG 《数学年刊B辑(英文版)》2008,29(2):179-198
Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0. 相似文献
19.
Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅)
H
. Let us consider three linear bounded operators,
We define the functions
where a
i
∈H and α
i
∈ℝ. In this paper, we discuss the closure and the convexity of the sets Φ
H
⊂ℝ2 and F
H
⊂ℝ3 defined by
Our work can be considered as an extension of Polyak’s results concerning the finite-dimensional case. 相似文献
20.
John S. Caughman 《Graphs and Combinatorics》1998,14(4):321-343
Let Y=(X,{R
i
}0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A
0, A
1,…, A
D
of the associate matrices, and Q-polynomial with respect to the ordering E
0, E
1,…,E
D
of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv).
(i)
(ii) D is even, and
(iii) θ*
0>θ0, and
(iv) θ*
0>θ0, D is odd, and
Received: February 13, 1996 / Revised: October 16, 1996 相似文献