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1.
Antonio Tineo 《Annali di Matematica Pura ed Applicata》2003,182(2):113-128
In this paper, we prove a result of Ambrosetti–Prodi type for the problem x′=f(t,x)+λx, where f(t,x) is T-periodic in t, f(t,0)≡0 and f(t,x) has “cubic nonlinearities”.
Received: February 4, 2000?Published online: April 14, 2003
RID="*"
ID="*"This paper was partially supported by CDCHT, Universidad de los Andes. 相似文献
2.
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
3.
D. Bothe 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1999,114(1):779-808
We study the differential equation x"+g(x¢)+m(x) sgn x¢+f(x)=j(t)x''+g(x')+\mu(x)\,{\rm sgn}\, x'+f(x)=\varphi(t) with T-periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T-periodic solution.¶In the more difficult case g = 0, we concentrate on the model equation x"+m(x) sgn x¢+x=j(t)x''+\mu(x)\,{\rm sgn}\,x'+x=\varphi(t) and obtain sufficient conditions for the existence of a T-periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on 7 we prove boundedness of all solutions.¶Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the "linearized" differential equation is an impulsive Hill equation. 相似文献
4.
E. G. Kir’yatskii 《Russian Mathematics (Iz VUZ)》2010,54(2):84-86
Let a set B have the following properties: if z ∈ B, then z ± 2π ∈ B and the intersection of B with the vertical strip 0 ≤ Re x ≤ π is a closed and bounded set. In this paper we study the approximation of a continuous on B and 2π-periodic function f(z) by trigonometric polynomials T
n
(z). We establish the necessary and sufficient conditions for the function f(z) to be entire and specify a formula for calculating its order. In addition, we describe some metric properties of periodic
sets in a plane. 相似文献
5.
Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C
d
n of the n-cycle C
n for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of C
n
d for n≡0 or (mod d+1), and lower and upper bounds for other cases.
Received: May 13, 1996 Revised: December 8, 1997 相似文献
6.
W?odzimierz B?k 《Potential Analysis》2010,32(1):17-27
A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C
n
:n = 1,2,..}, allows for constructing measures nxC, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M
f(x)=?g ? Gf(g)nCx(g). f(x)=\sum_{\gamma\in\Gamma}f(\gamma)\nu^{\mathbf{C}}_x(\gamma). 相似文献
7.
Let T:x↦2x (mod 1) be the doubling map of the circle ?=ℝ/ℤ. We construct a trigonometric polynomial f:?→ℝ with the following property: ∫f
dμ≥0 for every T-invariant probability measure μ, so that f is cohomologous to a non-negative Lipschitz function, yet f is not cohomologous to any non-negative C
1 function.
Oblatum 28-VI-2001 & 4-X-2001?Published online: 18 January 2002 相似文献
8.
Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s
pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this
Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x0, δ) denote the set of points x in M such that the distance from x0 to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that
dimE(x0,d) 3 dimM - \fracC(T)|logd| \dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\vert\!\log \delta \vert}
if τ = 1 and
dimE(x0,d) 3 dimM - \fracC(T)log|logd|\dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\log \vert \log \delta \vert}
if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient
generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case. 相似文献
9.
Sergiy Maksymenko 《Annals of Global Analysis and Geometry》2006,29(3):241-285
Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ
1 or the circle S
1, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h
−1 for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ
f
the set of critical points of f, D(M,Σ
f
) the subgroup of D(M) preserving Σ
f
, and S(f), S (f,Σ
f
), O(f), and O(f,Σ
f
) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ
f
). In fact S(f) = S(f,Σ
f
).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ
f
). It is proved that except for few cases the connected components of S(f) and O(f,Σ
f
) are contractible, π
k
O(f) = π
k
M for k ≥ 3, π2 O(f) = 0, and π1 O(f) is an extension of π1 D(M) ⊕ Z
k
(for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ
f
) are tame Fréchet manifolds.
Communicated by Peter Michor Vienna
Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45. 相似文献
10.
Suppose that f(x) = (f
1(x),...,f
r
(x))
T
, x∈R
d
is a vector-valued function satisfying the refinement equation f(x) = ∑Λ
c
κ
f(2x−κ) with finite set Λ of Z
d
and some r×r matricex c
κ. The requirements for f to have accuracy p are given in terms of the symbol function m(ξ).
Supported by NSFC 相似文献
11.
Pierluigi Benevieri Massimo Furi Maria Patrizia Pera 《Journal of Differential Equations》2007,233(2):404-416
We prove a global bifurcation result for T-periodic solutions of the T-periodic delay differential equation x′(t)=λf(t,x(t),x(t−1)) depending on a real parameter λ?0. The approach is based on the fixed point index theory for maps on ANRs. 相似文献
12.
Michael T. Lacey 《Journal d'Analyse Mathématique》1995,67(1):199-206
LetT
1 andT
2 be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ2, letT
t
=T
1
u
T
2
v
. LetS
1 denote the unit circle in ℝ2 and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA
t
f(x) = ∫
s
1
f(T
ts
x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions. 相似文献
13.
Carl E Langenhop 《Journal of Differential Equations》1985,58(3):391-403
Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝0∞ [dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation. 相似文献
14.
Yu. A. Kilizhekov 《Mathematical Notes》1996,60(4):378-382
LetW
n
2
M be the class of functionsf: Δ
n
→ ℝ (when Δ
n
is ann-simplex) with bounded second derivative (whose absolute value does not exceedM>0) along any direction at an arbitrary point of the simplex Δ
n
. LetP
1,n
(f;x) be the linear polynomial interpolatingf at the vertices of the simplex. We prove that there exists a functiong ∈ W
n
2
M such that for anyf ∈W
n
2
M and anyx ∈ Δ
n
one has |f
(x)−P
1,
n
(f;x)|≤g(x).
Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 504–510, October, 1996.
I thank Yu. N. Subbotin for posing the problem and for his attention to my work. 相似文献
15.
V. D. Zalizko 《Ukrainian Mathematical Journal》2007,59(1):28-44
The Jackson inequality
relates the value of the best uniform approximation E
n
(f) of a continuous 2π-periodic function f: ℝ → ℝ by trigonometric polynomials of degree ≤ n − 1 to its third modulus of continuity ω
3(f, t). In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity
on [−π, π) only at every point of a fixed finite set consisting of an even number of points are approximated by polynomials
coconvex to them.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 29–43, January, 2007. 相似文献
16.
LetT(t) be the translation group onY=C
0(ℝ×K)=C
0(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY
⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y
⊙=L
1(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT
0(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT
0(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ
⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C
0(Ω;X) andL
p
(μ;X).
This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful
to the CWI and the Dutch National Science Foundation NWO for financial support. 相似文献
17.
Jipu Ma 《Frontiers of Mathematics in China》2008,3(2):305-316
Let E and F be Banach spaces, f: U ⊂ E → F be a map of C
r
(r ⩾ 1), x
0 ∈ U, and ft (x
0) denote the FréLechet differential of f at x
0. Suppose that f′(x
0) is double split, Rank(f′(x
0)) = ∞, dimN(f′(x
0)) > 0 and codimR(f′(x
0)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x
0) near x
0. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators,
i.e., x
0 is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the
conclusion of the theorem to be held is given.
相似文献
18.
Zaihong Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(4):592-608
In this paper, we study the existence of periodic solutions of Rayleigh equation
19.
Zbigniew Grande 《Central European Journal of Mathematics》2011,9(4):772-777
A sequence (f
n
)
n
of functions f
n
: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f
n+1(x) ≤ f
n
(x) (f
n+1(x) ≥ f
n
(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions. 相似文献
20.
Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx
1<x
2 <...<x
k
such thatg(x
1)≦g(x
2)≦...≦g(x
k
). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it
will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1)
k
.
Dedicated to Paul Erdős on his seventieth birthday
Research supported in part by the National Science Foundation under ISP-80-11451. 相似文献
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