共查询到20条相似文献，搜索用时 390 毫秒
1.
In this paper, we establish the general solution and the generalized Hyers-Ulam-Rassias stability problem for a cubic Jensen-type functional equation,4f（（3x＋y）/4）＋4f（（x＋3y）/4）=6f（（x＋y）/2）＋f（x）＋f（y）,9f（（2x＋y/3）＋9f（（x＋2y）/3）=16f（（x＋y）/2＋f（x）＋f（y）in the spirit of D. H. Hyers, S. M. Ulam, Th. M. Rassias and P. Gaevruta. 相似文献
2.
The connection between the functional inequalities
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$f\left( {\frac{{x + y}}
{2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}}
{2} + \alpha _J \left( {x - y} \right), x,y \in D,$f\left( {\frac{{x + y}}
{2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}}
{2} + \alpha _J \left( {x - y} \right), x,y \in D, 相似文献
3.
In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi- Banach algebras associated with the following Pexiderized Jensen functional equation f（x＋y/2＋z）-g（x-y/2＋z）=h（y）. This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias＇ stability theorem that appeared in his paper： On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 （1978）. 相似文献
4.
For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj （m）, （j = 0, 1,..., m - 1） which satisfy γj （m）γk （m） ＋γk （m）γj （m） = 2ηjk （m）I[m/2], where （ηjk （m）） 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation （m） of the Lorentz Spin group is known. For m≥ 6, we prove that （i） when m = 2n, （n ≥ 3）, （m） is the group generated by the set of matrices {T｜T=1/√ξ（（I＋k） 0 ＋ 0 I-K） （ U 0 0 U）, （ii） when m = 2n ＋ 1 （n≥ 3）, （m） is generated by the set of matrices {T｜T=1/√ξ（I -k^- k I）U,U∈ （m-1）,ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj（m-2）＋a^（m-2） In],K^-=i[m-3∑j=0 a^j γj（m-2）-a^（m-2） In]} 相似文献
5.
This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed. 相似文献
6.
Let D be an increasing sequence of positive integers, and consider the divisor functions：
d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1,
where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）. 相似文献
7.
Abstract Let Ω be the unit ball centered at the origin in
. We study the following problem
By a constructive argument, we prove that for any k = 1, 2, • • •, if ε is small enough, then the above problem has positive a solution uε concentrating at k distinct points which tending to the boundary of Ω as ε goes to 0 +. 相似文献
8.
In this paper,the parameterized Marcinkiewicz integrals with variable kernels defined by μΩ^ρ（f）（x）=（∫0^∞│∫│1-y│≤t Ω（x,x-y）/│x-y│^n-p f（y）dy│^2dt/t1＋2p）^1/2 are investigated.It is proved that if Ω∈ L∞（R^n） × L^r（S^n-1）（r〉（n-n1p＇/n） is an odd function in the second variable y,then the operator μΩ^ρ is bounded from L^p（R^n） to L^p（R^n） for 1 〈 p ≤ max{（n＋1）/2,2}.It is also proved that,if Ω satisfies the L^1-Dini condition,then μΩ^ρ is of type（p,p） for 1 〈 p ≤ 2,of the weak type（1,1） and bounded from H1 to L1. 相似文献
9.
Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive.
Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras. 相似文献
10.
In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation 相似文献
11.
Let u = ( u
n
) be a sequence of real numbers whose generator sequence is Cesàro summable to a finite number. We prove that ( u
n
) is slowly oscillating if the sequence of Cesàro means of ( ω
n
(m−1)( u)) is increasing and the following two conditions are hold:
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$\begin{gathered}
\left( {\lambda - 1} \right)\mathop {\lim \sup }\limits_n \left( {\frac{1}
{{\left[ {\lambda n} \right] - n}}\sum\limits_{k = n + 1}^{\left[ {\lambda n} \right]} {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1}
{q}} = o\left( 1 \right), \lambda \to 1^ + , q > 1, \hfill \\
\left( {1 - \lambda } \right)\mathop {\lim \sup }\limits_n \left( {\frac{1}
{{n - \left[ {\lambda n} \right]}}\sum\limits_{k = \left[ {\lambda n} \right] + 1}^n {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1}
{q}} = o\left( 1 \right), \lambda \to 1^ - , q > 1, \hfill \\
\end{gathered}$\begin{gathered}
\left( {\lambda - 1} \right)\mathop {\lim \sup }\limits_n \left( {\frac{1}
{{\left[ {\lambda n} \right] - n}}\sum\limits_{k = n + 1}^{\left[ {\lambda n} \right]} {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1}
{q}} = o\left( 1 \right), \lambda \to 1^ + , q > 1, \hfill \\
\left( {1 - \lambda } \right)\mathop {\lim \sup }\limits_n \left( {\frac{1}
{{n - \left[ {\lambda n} \right]}}\sum\limits_{k = \left[ {\lambda n} \right] + 1}^n {\left( {\omega _k^{\left( m \right)} \left( u \right)} \right)^q } } \right)^{\frac{1}
{q}} = o\left( 1 \right), \lambda \to 1^ - , q > 1, \hfill \\
\end{gathered} 相似文献
12.
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
with
S. Q. Tang and H. Zhao [4] have considered the problem and obtained the optimal decay property for suitably small data. In
this paper we derive the asymptotic profile using the Gauss kernel G(t, x), which shows the precise behavior of solution as time tends to infinity. In fact, we will show that the asymptotic formula
holds, where D0, β 0 are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing
variables.
(Received: January 8, 2005) 相似文献
14.
In the “lost notebook”, Ramanujan recorded infinite product expansions for
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$\frac{1}
{{\sqrt r }} - \left( {\frac{{1 - \sqrt 5 }}
{2}} \right)\sqrt r and \frac{1}
{{\sqrt r }} - \left( {\frac{{1 + \sqrt 5 }}
{2}} \right)\sqrt r ,$\frac{1}
{{\sqrt r }} - \left( {\frac{{1 - \sqrt 5 }}
{2}} \right)\sqrt r and \frac{1}
{{\sqrt r }} - \left( {\frac{{1 + \sqrt 5 }}
{2}} \right)\sqrt r , 相似文献
15.
Let Θ be a bounded open set in ℝ
n
, n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that
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$
\sup \left\{ {\int_\Omega {\exp \left( {\left( {\frac{{\left| {f(x)} \right|}}
{K}} \right)^{{n \mathord{\left/
{\vphantom {n {(n - 1)}}} \right.
\kern-\nulldelimiterspace} {(n - 1)}}} } \right):f \in W_0^{1,n} (\Omega ),\left\| {\nabla f} \right\|_{L^n } \leqslant 1} } \right\} < \infty
$
\sup \left\{ {\int_\Omega {\exp \left( {\left( {\frac{{\left| {f(x)} \right|}}
{K}} \right)^{{n \mathord{\left/
{\vphantom {n {(n - 1)}}} \right.
\kern-\nulldelimiterspace} {(n - 1)}}} } \right):f \in W_0^{1,n} (\Omega ),\left\| {\nabla f} \right\|_{L^n } \leqslant 1} } \right\} < \infty
相似文献
16.
Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)
$
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)
on the domain $
\Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} }
$
\Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} }
defined by
|
$
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)(x) = {*{20}c}
{\sum\limits_{j = 1}^n {\left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)} ^{\beta _j } (f'(x_1^* (x)))^{\gamma _j } x_1^* (x)x_j } \\
{ + \left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)^{\beta _{n + 1} } (f'(x_1^* (x)))^{\gamma _{n + 1} } \left( {x - \sum\limits_{j = 1}^n {x_1^* (x)x_j } } \right)} \\
$
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)(x) = \begin{array}{*{20}c}
{\sum\limits_{j = 1}^n {\left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)} ^{\beta _j } (f'(x_1^* (x)))^{\gamma _j } x_1^* (x)x_j } \\
{ + \left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)^{\beta _{n + 1} } (f'(x_1^* (x)))^{\gamma _{n + 1} } \left( {x - \sum\limits_{j = 1}^n {x_1^* (x)x_j } } \right)} \\
\end{array}
相似文献
17.
The initial boundary value problem
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$ {*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ $ \begin{array}{*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ \end{array} 相似文献
18.
In this paper, we consider the functional differential equation with impulsive perturbations
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$ \left\{ {{*{20}{c}} {{x^{\prime}}(t) = f\left( {t,{x_t}} \right),} \hfill & {t \geq {t_0},\quad t \ne {t_k},\quad x \in {\mathbb{R}^n},} \hfill \\ {\Delta x(t) = {I_k}\left( {t,x\left( {{t^{-} }} \right)} \right),} \hfill & {t = {t_k},\quad k \in {\mathbb{Z}^{+} }.} \hfill \\ } \right. $ \left\{ {\begin{array}{*{20}{c}} {{x^{\prime}}(t) = f\left( {t,{x_t}} \right),} \hfill & {t \geq {t_0},\quad t \ne {t_k},\quad x \in {\mathbb{R}^n},} \hfill \\ {\Delta x(t) = {I_k}\left( {t,x\left( {{t^{-} }} \right)} \right),} \hfill & {t = {t_k},\quad k \in {\mathbb{Z}^{+} }.} \hfill \\ \end{array} } \right. 相似文献
19.
In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R~+_n:{u(x)=c_n,a∫_?R_+~n(x_n~(1-a_v)(y)/|x-y|~(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n~(1-a_uθ)(x)/|x-y|~(n-a))dx,y∈ ?R_+~n,where n 2, 2-n a 1, κ, θ 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system. 相似文献
20.
If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case. 相似文献
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