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1.
In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential
equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x
γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded.
This project was supported by the Program for New Century Excellent Talents of Ministry of Education of China and the National
Natural Science Foundation of China (Grant No. 10671020 and 10301006). 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.
The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form
\frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} ,
\frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C
1 in
\mathbbRn{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) £ C j(x)eahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and limh? +¥hb-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates
and Lyapunov functional methods. 相似文献
4.
For the equation K(t)u
xx
+ u
tt
− b
2
K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t|
m
, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability
of the boundary value problem u(0, t) = u(1, t), u
x
(0, t) = u
x
(1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1. 相似文献
5.
Vincenzo De Filippis 《Israel Journal of Mathematics》2007,162(1):93-108
Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x
1,..., x
n) a multilinear polynomial over C, I a nonzero right ideal of R.
If [g(f(r
1,..., r
n)), f(r
1,..., r
n)] = 0, for all r
1, ..., r
n ∈ I, then either g(x) = ax, with (a − γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:
Supported by a grant from M.I.U.R. 相似文献
(i) | f(x 1,..., x n) is central valued in eRCe |
(ii) | g(x) = cx + xb, where (c+b+α)e = 0, for α ∈ C, and f (x 1,..., x n)2 is central valued in eRCe |
(iii) | char(R) = 2 and s 4(x 1, x 2, x 3, x 4) is an identity for eRCe. |
6.
WENGPEIXUAN 《高校应用数学学报(英文版)》1997,12(2):155-164
We use the topological degree method to deal with the generalized Sturm-Liouville boundary value problem (BVP) for second order mixed-type functional differential equation x(t)=f(t,xt,xt), 0≤t≤T. Existence principle and theorem for solutions of the BVP are obtained. 相似文献
7.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
8.
We present a fully polynomial time approximation scheme (FPTAS) for minimizing an objective (a
T
x + γ)(b
T
x + δ) under linear constraints A
x ≤ d. Examples of such problems are combinatorial minimum weight product problems such as the following: given a graph G = (V,E) and two edge weights find an s − t path P that minimizes a(P)b(P), the product of its edge weights relative to a and b.
相似文献
9.
On Hilbert''s Integral Inequality 总被引:5,自引:0,他引:5
In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L2[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π2is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫∞0(c(t2x)/(1 + t2)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]]. 相似文献
10.
We present results on total domination in a partitioned graph G = (V, E). Let γ
t
(G) denote the total dominating number of G. For a partition , k ≥ 2, of V, let γ
t
(G; V
i
) be the cardinality of a smallest subset of V such that every vertex of V
i
has a neighbour in it and define the following
We summarize known bounds on γ
t
(G) and for graphs with all degrees at least δ we derive the following bounds for f
t
(G; k) and g
t
(G; k).
相似文献
(i) | For δ ≥ 2 and k ≥ 3 we prove f t (G; k) ≤ 11|V|/7 and this inequality is best possible. |
(ii) | for δ ≥ 3 we prove that f t (G; 2) ≤ (5/4 − 1/372)|V|. That inequality may not be best possible, but we conjecture that f t (G; 2) ≤ 7|V|/6 is. |
(iii) | for δ ≥ 3 we prove f t (G; k) ≤ 3|V|/2 and this inequality is best possible. |
(iv) | for δ ≥ 3 the inequality g t (G; k) ≤ 3|V|/4 holds and is best possible. |
11.
Let G be a multigraph, g and f be integer-valued functions defined on V(G). Then a graph G is called a (g, f)-graph if g(x)≤deg
G(x)≤f(x) for each x∈V(G), and a (g, f)-factor is a spanning (g, f)-subgraph. If the edges of graph G can be decomposed into (g, f)-factors, then we say that G is (g, f)-factorable. In this paper, we obtained some sufficient conditions for a graph to be (g, f)-factorable. One of them is the following: Let m be a positive integer, l be an integer with l=m (mod 4) and 0≤l≤3. If G is an -graph, then G is (g, f)-factorable. Our results imply several previous (g, f)-factorization results.
Revised: June 11, 1998 相似文献
12.
A. V. Zheleznyak 《Vestnik St. Petersburg University: Mathematics》2009,42(4):269-274
In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f
−1(x) = $
\sum\limits_{n = 0}^\infty {b_n x^n }
$
\sum\limits_{n = 0}^\infty {b_n x^n }
b
n
x
n
be such that b
0 > 0 and b
n
≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x
1, x
2, …, x
m
) with positive coefficients in order that the series f
−1(x
1, x
2, …, x
m
) = $
\sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } }
$
\sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } }
satisfies the property b
0, …, 0 > 0, $
bi_1 ,i_2 , \ldots ,i_m
$
bi_1 ,i_2 , \ldots ,i_m
≤ 0, i
12 + i
22 + … + i
m
2 > 0, which is similar to the one-dimensional case. 相似文献
13.
V. G. Krotov 《Ukrainian Mathematical Journal》2010,62(3):441-451
We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t
−a
decreases for a certain a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0 , for which
| f(x) - f(y) | \leqslant [ g(x) + g(y) ]h( d( x,y ) ), x,y ? X | / |
E \left| {f(x) - f(y)} \right| \leqslant \left[ {g(x) + g(y)} \right]\eta \left( {d\left( {x,y} \right)} \right),\,x,y \in {{X} \left/ {E} \right.} 相似文献
14.
The following system considered in this paper:
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