共查询到20条相似文献,搜索用时 31 毫秒
1.
Weiyang Chen & Xiaoli Chen 《数学研究》2014,47(2):208-220
In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha
1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$ 相似文献
2.
We present some comments on the behavior of solutions of the difference equation
where p
i 0, i = 1,..., k, k N, and x
–k
,..., x
–1 R. 相似文献
3.
Let
denote the generalized hypergeometric function
where no denominator parameter can be zero or a negative integer and (a,n) denotes the ascending factorial notation. Ponnusamy and Vuorinen raised the problem of finding conditions on the parameters aj > 0, bj > 0 so that the function
is univalent in . The main aim of this paper is to discuss this problem in detail for the case q = 2. 相似文献
4.
G. Arag��n-Gonz��lez J. L. Arag��n M. A. Rodr��guez-Andrade 《Advances in Applied Clifford Algebras》2011,21(2):259-272
In this work, the equivalence class representatives of integer solutions of the Diophantine equation of the type ${{a_1x_1^2+ .\,.\,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .\,.\,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .\,.\,.\,,p+q,x_{n+1}\neq0)}}${{a_1x_1^2+ .\,.\,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .\,.\,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .\,.\,.\,,p+q,x_{n+1}\neq0)}} are found using simple reflections of orthogonal vectors, manipulated using the Clifford algebra over orthogonal spaces R
p,q
. These solutions are obtained from the application of a useful Lemma: given two different non-zero vectors of the same norm,
we can map one onto the other, or its negative, by means of a simple reflection. With this Lemma, we extend and improve a
previous work [1] concerning generalized Pythagorean numbers, which now can be obtained as a Corollary. We also show that
our technique is promising for solving others Diophantine equations. 相似文献
5.
We study hypersurfaces in Euclidean space
whose position vector x satisfies the condition L
k
x = Ax + b, where L
k
is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed
,
is a constant matrix and
is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form
, with
. This extends a previous classification for hypersurfaces in
satisfying
, where
is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A
53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].
相似文献
6.
Joseph Rosenblatt 《Mathematische Annalen》1977,230(3):245-272
For a mean zero norm one sequence (f
n
)L
2[0, 1], the sequence (f
n
{nx+y}) is an orthonormal sequence inL
2([0, 1]2); so if
, then
converges for a.e. (x, y)[0, 1]2 and has a maximal function inL
2([0, 1]2). But for a mean zerofL
2[0, 1], it is harder to give necessary and sufficient conditions for theL
2-norm convergence or a.e. convergence of
. Ifc
n
0 and
, then this series will not converge inL
2-norm on a denseG
subset of the mean zero functions inL
2[0, 1]. Also, there are mean zerofL[0, 1] such that
never converges and there is a mean zero continuous functionf with
a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c
n
| = 0(n
–) for >1/2, then
converges a.e. and unconditionally inL
2[0, 1]. In addition, for any mean zerof of bounded variation, the series
has its maximal function in allL
p[0, 1] with 1p<. Finally, if (f
n
)L
[0, 1] is a uniformly bounded mean zero sequence, then
is a necessary and sufficient condition for
to converge for a.e.y and a.e. (x
n
)[0, 1]. Moreover, iffL
[0, 1] is mean zero and
, then for a.e. (x
n
)[0, 1],
converges for a.e.y and in allL
p
[0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one. 相似文献
7.
In this paper, sufficient conditions have been obtained for oscillation of all solutions of a class of nonlinear neutral delay
difference equations of the form
$
\Delta \left( {r\left( n \right)\Delta \left( {y\left( n \right) + p\left( n \right)y\left( {n - m} \right)} \right)} \right) + q\left( n \right)G\left( {y\left( {n - k} \right)} \right) = 0
$
\Delta \left( {r\left( n \right)\Delta \left( {y\left( n \right) + p\left( n \right)y\left( {n - m} \right)} \right)} \right) + q\left( n \right)G\left( {y\left( {n - k} \right)} \right) = 0
相似文献
8.
A. K. Tripathy 《Mathematica Slovaca》2008,58(2):221-240
Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral difference equations of the
form
9.
屠规彰 《应用数学学报(英文版)》1984,1(1):26-30
Let(?)=B_ηu:2(q-(?))+(⊿((?)-2q))+(2q_x+(?)_x))η=0,2(r-(?)+(⊿(2(?)-r)+(r_x+2(?)_x))η=0,u=(q,r)~Tbe the Backlund transformation (BT) of the hierarchy of AKNS equations,where η is a parameterand Δ=integral from -∞ to x (qr-(?))dx′.It is shown in this paper the infinitesimal BT B_(η+ε)B_η~(-1) admits thefollowing expansionB_(η+ε)B_η~(-1)u=u+εsum from n=0 to ∞ β_n(JL~(n+1)u)η~n,β_n=1+(-1)~n2~(-n-1),where L is the recurrence operator of the hierarchy and ε is an infinitesimal parameter.Thisexpansion implies the equivalence between the permutabiliy of BTs and the involution in pairs ofconserved densities. 相似文献
10.
《复变函数与椭圆型方程》2012,57(8):727-729
Let T ( f ) and N ( r,c ) denote the usual Nevanlinna characteristic and the counting function for the c -points of a meromorphic function f , respectively. Using a result of Miles and Shea ( Quart. J. Math. Oxford , 24 (2), (1973), 377-383) and two simple estimates for trigonometric functions, we show in connection with a 1929 problem of Nevanlinna for meromorphic functions f of finite order 1 < u < X $$ \limsup\limits_{r\rightarrow \infty } { N(r, 0)+N(r, \infty ) \over T(r, \,f)}\ge {2\sqrt 2 \over \pi} {|\sin \pi \lambda | \over D(\lambda )}\ge (0.9)\, {{|\sin \pi \lambda | \over {D(\lambda )}, }} $$ with D ( u ) = q +|sin ~ u | for $ q\le \lambda \le q + \fraca {1}{2} $ and D ( u ) = q + 1 for $ q + {\fraca {1} {2}} \le \lambda \lt q + 1 $ , where $ q = \lfloor \lambda \rfloor $ . 相似文献
11.
In this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem. 相似文献
12.
《复变函数与椭圆型方程》2012,57(2):95-110
Let $ k \in {\shadN} $ , $ w(x) = (1+x^2)^{1/2} $ , $ V^{\prime} _k = w^{k+1} {\cal D}^{\prime} _{L^1} = \{{ \,f \in {\cal S}^{\prime}{:}\; w^{-k-1}f \in {\cal D}^{\prime} _{L^1}}\} $ . For $ f \in V^{\prime} _k $ , let $ C_{\eta ,k\,}f = C_0(\xi \,f) + z^k C_0(\eta \,f/t^k)$ where $ \xi \in {\cal D} $ , $ 0 \leq \xi (x) \leq 1 $ $ \xi (x) = 1 $ in a neighborhood of the origin, $ \eta = 1 - \xi $ , and $ C_0g(z) = \langle g, \fraca {1}{(2i \pi (\cdot - z))} \rangle $ for $ g \in V^{\,\prime} _0 $ , z = x + iy , y p 0 . Using a decomposition of C 0 in terms of Poisson operators, we prove that $ C_{\eta ,k,y} {:}\; f \,\mapsto\, C_{\eta ,k\,}f(\cdot + iy) $ , y p 0 , is a continuous mapping from $ V^{\,\prime} _k $ into $ w^{k+2} {\cal D}_{L^1}$ , where $ {\cal D}_{L^1} = \{ \varphi \in C^\infty {:}\; D^\alpha \varphi \in L^1\ \forall \alpha \in {\shadN} \} $ . Also, it is shown that for $ f \in V^{\,\prime} _k $ , $ C_{\eta ,k\,}f $ admits the following boundary values in the topology of $ V^{\,\prime} _{k+1} : C^+_{\eta ,k\,}f = \lim _{y \to 0+} C_{\eta ,k\,}f(\cdot + iy) = (1/2) (\,f + i S_{\eta ,k\,}f\,); C^-_{\eta ,k\,}f = \lim _{y \to 0-} C_{\eta ,k\,} f(\cdot + iy)= (1/2) (-f + i S_{\eta ,k\,}f ) $ , where $ S_{\eta ,k} $ is the Hilbert transform of index k introduced in a previous article by the first named author. Additional results are established for distributions in subspaces $ G^{\,\prime} _{\eta ,k} = \{ \,f \in V^{\,\prime} _k {:}S_{\eta ,k\,}f \in V^{\,\prime} _k \} $ , $ k \in {\shadN} $ . Algebraic properties are given too, for products of operators C + , C m , S , for suitable indices and topologies. 相似文献
13.
Elena I. Kaikina Leonardo Guardado-Zavala Hector F. Ruiz-Paredes Jesus A. Mendez Navarro 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(1):63-77
We study nonlinear nonlocal equations on a half-line in the critical case
14.
Pedro Freitas. 《Mathematics of Computation》2005,74(251):1425-1440
We show that integrals of the form
and satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all and in the second case when is even, these integrals are reducible to zeta values. In the case of odd , we combine the known results for Euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight . The proofs are constructive, giving a method for the evaluation of these and other similar integrals, and we present a selection of explicit evaluations in the last section. 15.
王玉玉 《数学年刊A辑(中文版)》2018,39(3):273-286
本文中,通过几何方法证明了σ相关同伦元素在球面稳定同伦群π_mS中是非平凡的,其中m=p~(n+1)q+2p~nq+(s+3)p~2q+(s+3)pq+(s+3)q-8,p≥7是奇素数,n3,0≤sp-3,且q=2(p-1).该σ相关同伦元素在Adams谱序列的E_2-项中由■_s+3■_ng0表示. 相似文献
16.
V. N. Gabushin 《Mathematical Notes》1968,4(2):624-630
In this article we shall concern ourselves with determining exact (least possible) constants in the inequalities of the form $$\parallel f^{(k)} \parallel _{L_q } \leqslant K\parallel f\parallel _{L_p } ^{\tfrac{{l - k - r - 1 + q - 1}}{{l - r - 1 + p - 1}}} \parallel f^{(l)} \parallel _{L_r } ^{\tfrac{{k - q - 1 + p - 1}}{{l - r - 1 + p^{n - 1} }}} $$ for functions defined on the entire (?∞, ∞), absolutely continuous on any interval together with their (l?1)-th derivatives, and having finite $$l = 2,k = 0,k = 1,q = r = \infty ,1 \leqslant p< \infty $$ is considered. 相似文献
17.
This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any
-minihyper, with
, where
, is the disjoint union of
points,
lines,...,
-dimensional subspaces. For q large, we improve on this result by increasing the upper bound on
non-square, to
non-square,
square,
, and (4) for
square, p prime, p<3, to
. In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry
. For the coding-theoretical problem, our results classify the corresponding
codes meeting the Griesmer bound. 相似文献
18.
Abstract. The existence of positive radial solutions to the systems of 相似文献
19.
Let q, h, a, b be integers with q > 0. The classical and the homogeneous Dedekind sums are defined by $$s(h,q) = \sum\limits_{j = 1}^q {\left( {\left( {{j \over q}} \right)} \right)\left( {\left( {{{hj} \over q}} \right)} \right),{\rm{ }}s(a,b,q) = \sum\limits_{j = 1}^q {\left( {\left( {{{aj} \over q}} \right)} \right)\left( {\left( {{{bj} \over q}} \right)} \right),} } $$ respectively, where $((x)) = \left\{ \begin{gathered} x - [x] - \tfrac{1} {2},if x is not an integer; \hfill \\ 0,if x is an integer. \hfill \\ \end{gathered} \right. $ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\sum\limits_{d|n} {\sum\limits_{r = 1}^d {s\left( {{n \over d}a + rq,dq} \right) = \sigma (n)s(a,q),} } $$ $$\sum\limits_{d|n} {\sum\limits_{{r_1} = 1}^d {\sum\limits_{{r_2} = 1}^d s \left( {{n \over d}a + {r_1}q,{n \over d}b + {r_2}q,dq} \right) = n\sigma (n)s(a,b,q),} } $$ where σ(n) =Σ d|n d. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given. 相似文献
20.
A. P. Scheglova 《Journal of Mathematical Sciences》2005,128(5):3306-3333
We consider the boundary-value problem
|