共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
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
3.
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
4.
Shu-Yu Hsu 《Mathematische Annalen》2006,334(1):153-197
Let a1,a2, . . . ,am ∈ ℝ2, 2≤f ∈ C([0,∞)), gi ∈ C([0,∞)) be such that 0≤gi(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−ai|→−gi(t) as |x−ai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain. 相似文献
5.
Xiao Ping Yuan 《数学学报(英文版)》2001,17(2):253-262
We prove the existence of quasiperiodic solutions and Lagrange stability for a class of differential equations with jumping
nonlinearity
, where a,b > 0, p(t) ∈C(ℝ/2πℤ) and φ : ℝ→ℝ is an unbounded function.
Supported by the National Natural Science Foundation of China 相似文献
6.
TAOYOUSHAN GAOGUOZHU 《高校应用数学学报(英文版)》1998,13(3):271-280
In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied. 相似文献
7.
Jerome A. Goldstein 《Semigroup Forum》1996,52(1):37-47
Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case)T={T
n
/n=0,1,2,...} or (continuous case)T={T(t)/t=≥0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesàro limits (asn→∞) of |〈T
n
f,f〉|2, |〈T
(n)f,f〉|2 are computed (withn∈ℤ+ orn∈ℤ+). Specializing the Hilbert space to beL
2(T,μ) (discrete case) orL
2(ℝ,μ) (continuous case) where μ is a Borel probability measure on the circle group or the line, the Cesàro limit of
(asn→±∞, with,n∈ℤ orn∈ℝ) is obtained and interpreted. Extensions toT
M
, and ℝ
M
are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context.
Partially supported by an NSF grant 相似文献
8.
9.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
10.
T. S. Kopaliani 《Ukrainian Mathematical Journal》2008,60(12):2006-2014
We point out that if the Hardy–Littlewood maximal operator is bounded on the space L
p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L
p
(ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L
p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L
p(t) (ℝ
n
), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ
n
, if and only if p(t) = const.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008. 相似文献
11.
V.Yu. Slyusarchuk 《Ukrainian Mathematical Journal》2010,62(6):970-981
Let E be a finite-dimensional Banach space, let C0(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K:C
0(R ,E) → C
0(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C
0(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation
\fracdx(t)dt + ( Kx )(t)Ax(t) = h(t), t ? \mathbbR \frac{{dx(t)}}{{dt}} + \left( {Kx} \right)(t)Ax(t) = h(t),\quad t \in \mathbb{R} 相似文献
12.
Jean Saint Raymond 《Rendiconti del Circolo Matematico di Palermo》1995,44(1):162-168
Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF:
and byDom(F) its domain, it is the set {uεL
p(T,E):ū(t)=f(u),t)εL
1(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL
p(T, E). 相似文献
13.
Y. Lacroix 《Israel Journal of Mathematics》2002,132(1):253-263
LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ
0
∞
G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ
U
(x)=inf{k⩾1:T
k
xεU}, and defineG
U
(t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U
n
)
n≥1
of neighbourhoods ofx such that {x}=∩
n
U
n
, and for anyG ∈G, there exists a subsequence (n
k
)
k≥1
withG
U
n
k
↑U weakly.
We also construct a uniquely ergodic Toeplitz flowO(x
∞,S, μ), the orbit closure of a Toeplitz sequencex
∞, such that the above conclusion still holds, with moreover the requirement that eachU
n
be a cylinder set.
In memory of Anzelm Iwanik 相似文献
14.
LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC
0-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can
be summarized roughly as follows:
15.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
16.
Let X be a Banach space, A : D(A) X → X the generator of a compact C0- semigroup S(t) : X → X, t ≥ 0, D a locally closed subset in X, and f : (a, b) × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t) = Au(t) + f(t, u(t - q)), t ∈ [to, to + T], with initial condition uto = φ ∈C([-q, 0]; X), is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t, v(-q)); D) = 0 for almost every t ∈ (a, b) and every v ∈ C([-q, 0]; X) with v(0), v(-q)∈ D. 相似文献
17.
Huan-song Zhou Hong-bo Zhu 《应用数学学报(英文版)》2007,23(4):685-696
In this paper,we consider the following ODE problem(P)where f ∈ C((0, ∞)×R,R),f(r,s)goes to p(r)and q(r)uniformly in r>0 as s→0 and s→ ∞,respectively,0≤p(r)≤q(r)∈ L~∞(0,∞).Moreover,for r>0,f(r,s)is nondecreasing in s≥0.Some existenceand non-existence of positive solutions to problem(P)are proved without assuming that p(r)≡0 and q(r)hasa limit at infinity.Based on these results,we get the existence of positive solutions for an elliptic problem. 相似文献
18.
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
19.
We investigate the joint weak convergence (f.d.d. and functional) of the vector-valued process (U
n
(1)
(τ), U
n
(2)
(τ)) for τ ∈ [0, 1], where
and
are normalized partial-sum processes separated by a large lag m, m/n → ∞, and (X
t
, t ∈ ℤ) is a stationary moving-average process with i.i.d. (or martingale-difference) innovations having finite variance. We
consider the cases where (X
t
) is a process with long memory, short memory, or negative memory. We show that, in all these cases, as n → ∞ and m/n → ∞, the bivariate partial-sum process (U
n
(1)
(τ), U
n
(2)
(τ)) tends to a bivariate fractional Brownian motion with independent components. The result is applied to prove the consistency
of certain increment-type statistics in moving-average observations.
This work supported by the joint Lithuania-French research program Gilibert.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 479–500, October–December, 2005. 相似文献
20.
WU Hao & LI Weigu School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2005,48(12):1670-1682
In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case. 相似文献
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