3-Variable Additive {\rho} -Functional Inequalities and Equations

In this paper, we introduce and investigate additive \({\rho}\) -functional inequalities associated with the following additive functional equations $$\begin{array}{lll} \,\,\,\,\,\,\, f(x+y+z) - f(x)-f(y)-f(z) \,\,\,\, = 0 \\ 2f \left(\frac{x+y}{2}+z \right) - f(x)-f(y)-2f(z) = 0 \\ \,\,2f \left(\frac{x+y+z}{2} \right) - f(x)-f(y)-f(z) = 0\end{array}$$ Furthermore, we prove the Hyers–Ulam stability of the additive \({\rho}\) -functional inequalities in complex Banach spaces and prove the Hyers–Ulam stability of additive \({\rho}\) -functional equations associated with the additive \({\rho}\) -functional inequalities in complex Banach spaces.     

Fixed points and additive $${\rho}$$-functional equations

In this paper, we solve the additive \({\rho}\)-functional equations

$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$

where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.

     

An additive ($${\alpha, \beta}$$)-functional equation and linear mappings in Banach spaces

In this paper, we investigate the additive (\({\alpha, \beta}\))-functional equation \({f(x+y) + \bar{\alpha}f({\alpha}z) = \beta^{-1}f(\beta(x+y+z))}\) for all complex numbers \({\alpha}\) with \({|\alpha| = 1}\) and for a fixed nonzero complex number \({\beta}\). Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of this additive (\({\alpha, \beta}\))-functional equation in complex Banach spaces.     

The stability of additive $(\alpha,\beta)$-functional equations

In this paper, we investigate the following $(\alpha,\beta)$-functional equations $$ 2f(x)+2f(z)=f(x-y)+\alpha^{-1}f(\alpha (x+z))+\beta^{-1}f(\beta(y+z)),~~~~~~~~~(0.1) $$ $$ 2f(x)+2f(y)=f(x+y)+\alpha^{-1}f(\alpha(x+z)) +\beta^{-1}f(\beta(y-z)),~~~~~~~~~~~(0.2) $$ where $\alpha,\beta$ are fixed nonzero real numbers with $\alpha^{-1}+\beta^{-1}\neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the $(\alpha,\beta)$-functional equations $(0.1)$ and $(0.2)$ in non-Archimedean Banach spaces.     

玛欣凯维奇函数与泊松核的一个新微分性质

将Stein[On the functions of Littlewood-Paley,Lusin,and Marcinkiewicz,Trans.Amer.Math.Soc.,1958,88:430-466]中的玛欣凯维奇函数的逆向不等式推广到一般情形.主要结果是对于n-维欧几里得空间k-阶球面调和函数空间的任意一基底,得到玛欣凯维奇函数的一般性的逆向不等式,即存在不依赖于函数f正常数C_p,使得||f||_p≤C_pΣ_(j=1)~N=1||μ_j(f)||_p,其中{μ_j(f)}_(j=1)~N是f的由这些球面调和函数生成的玛欣凯维奇函数.此外,对于任意的n-变元的k-阶调和多项式Q(x)以及泊松核P_t(x),有Q(D)P_t(x)=C_n k(tQ(x))/((|x|)~2+t~2~(n+2k+1)/2).     

某类非线性微分方程超越亚纯解的进一步结果

本文研究非线性微分方程f~n+Q_d(z,f)=P_1(z)e~(α_1(z))+p_2(z)e~(α_2(z))超越亚纯解的存在性和形式,其中n≥4是整数,Q_d(z,f)是关于f的次数d≤n-3且系数为有理函数的微分多项式,p_1,p_2是非零的有理函数,α_1,α_2是非常数的多项式.运用Nevanlinna值分布理论,能够得到该方程存在超越亚纯解时p_1,p_2,α_1及α_2所满足的条件.特别地,还考虑了当Q_d(z,f)=a(z)ff'且n=4时方程的超越亚纯解的存在性和形式,其中a(z)是一个非零的有理函数.     

一类积分方程组正解的对称性和单调性

本文讨论积分方程组（？）解的性质,其中G_α是α阶贝塞尔位势核,0≤β〈α（n-α＋β）/n,1/（q＋1）＋1/（r＋1）〉（n-α＋β）/n,1/（r＋1）＋1/（p＋1）〉（n-α＋β）/n.我们用积分形式的移动平面法证明上述积分方程组的正解是径向对称且单调的.     

素变量混合幂丢番图逼近

设λ_1,λ_2,λ_3,λ_4为不全为负的非零实数,λ_1/λ_2是无理数和代数数.■是具有良好间隔的序列,δ>0.本文证明了:对于任意ε>0及v∈■,v≤X,使得不等式|λ_1p_1~2+λ_2p_2~2+λ_3p_3~3+λ_4p_4~3-v|相似文献   

Homomorphisms in quasi-Banach algebras associated with a Pexiderized Cauchy-Jensen functional equation

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）.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

素环上强保持2-Jordan乘积的映射

对于给定的正整数k≥1,环R上的元x,y的k-Jordan乘积定义为{x,y}_k={{x,y}_(k-1),y}_1,其中{x,y}_0=x,{x,y}_1=xy+yx.假设R是包含有单位元与一非平凡幂等元的素环.本文证明了R上的满射f满足{f(x),f(y)}2={x,y}_2对所有x,y∈R成立当且仅当存在λ∈l(R的可扩展中心)且λ~3=1,使得下列之一成立:(1)若R的特征不为2,则f(x)=λx对所有x∈R成立;(2)若R的特征为2,则f(x)=λx+μ(x)对所有x∈R成立,其中μ:R→l是一个映射.作为应用,得到了因子von Neumann代数上保持上述性质映射的结构.     

复平面上解析Banach空间的拟不变子空间

讨论复平面上解析Banach空间具有任意指标的拟不变子空间的存在性问题.首先给出一类复平面上解析Banach空间存在任意指标拟不变子空间的判定定理.作为应用,证明了Fock型空间F~p(C)={f∈Hol(C):1/π∫_C|f(z)|~pe~(-|z|~2)dA(z)+∞,1≤p+∞}与Hilbert空间H={f∈Hol(C):1/π∫_C|f(z)|~2e~(-|z|)dA(z)+∞}具有任意指标的拟不变子空间.     

一阶系统非线性边值问题的奇摄动*

本文应用比较定理研究了一类非线性边界条件的向量非线性奇摄动问题εx='f(t,x,y,e)εy'=g(t,x,y,ε)x(0)=A(ξ₁,ξ₂,x(1)-x(0),y(1)-y(0),ε)y(0)=B(ξ₁,ξ,x(1)-x(0),y(1)-y(0),ε)这里ξ₁,ξ₂为ε的函数。0<ε<<1,在适当的条件下,作出了任意次精度的渐近展式。并得出余项估计。     

On the stability of a mixed functional equation deriving from additive,quadratic and cubic mappings

In this paper, we investigate the general solution and the Hyers–Ulam stability of the following mixed functional equation f(2x + y) + f(2x- y) = 2f(2x) + 2f(x + y) + 2f(x- y)- 4f(x)- f(y)- f(-y)deriving from additive, quadratic and cubic mappings on Banach spaces.     

非线性分数次边值问题改进的ADM解法(英文)

We use the modified Adomian decomposition method（ADM） for solving the nonlinear fractional boundary value problem {D（^α₀） + u（x） = f（x, u（x））, 0 < x < 1, 3 < α≤ 4 u（0） = α₀ , u’’ （0） = α₂ u（1） = β₀ , u’’（1） = β₂} （1） where D（₀^α）+u is Caputo fractional derivative and α₀,α₂,β₀,β₂ is not zero at all,and f:[0,1]×R→ R is continuous.The calculated numerical results show reliability and efficiency of the algorithm given.The numerical procedure is tested on linear and nonlinear problems.     

振荡超奇性Hilbert变换的Sobolev有界性

主要研究R~n上沿曲线Γ(t)=(t~(p_1),t~(p_2),…,t~(p_n))的振荡超奇性Hilbert变换H_(n,α,β)=∫_0~1 f(x-Γ(t))e~(it-β)t~(-1-α),在Sobolev空间上的有界性,其中0p_1P_2…P_n,αβ0.证明了对于0γ(nα)/((n+1))(p_1+α),当|1/p-1/2|(β-(n+1)[α-(β+p_1)γ])/(2β)时,H_(n,α,β)是从L_γ~2(R~n))到L~2(R~n)的有界算子.特别地,当β≥(α-γp_1)/(γ+1/(n+1))等时,H_(n,α,β)是从L_γ~2(R~n)到L~2(R~n)的有界算子·     

奇摄动向量问题的边界层和内层现象

本文考虑非线性向量边值问题:εy″=f(x,y,z,y',ε), y(0)=A₁ y(1)=B₁ εz″=f(x,y,z,z',ε), z(0)=A₂ z(1)=B₂其中ε是正的小参数,0≤x≤1,f,g是R4中的连续函数。在适当的假设下,利用微分不等式理论,我们证明了上述问题的解的存在性,并得到包括边界层和内层在内的解的估计.     

一类源自可加、二次、三次和四次映射的泛函方程的模糊稳定性

在模糊Banach空间中研究了混合泛函方程f(x+ky)+f(x-ky)=k~2f(x+y)+k~2f(x-y)+2(1-k~2)f(x)+(k~2(k~2-1))/12(f(2y)-4f(y))的Hyers-Ulam稳定性,这里k>1是固定的一个整数,f(y)=f(y)+f(-y).     

单位圆到凸区域上的调和拟共形映照

设F（x）=p（x）e^ir（x）为单位圆周到约当凸曲线Γ上的保向同胚映照.本文证明:若ess inf|F’（x）|>0且对于一切的φ∈R有|F（φ+x）+F（φ-x）-2F（φ）|≤M|x|^α,这里α>1,M为正常数,则ω=P[F]（z）为单位圆到凸区域Ω=int（Γ）上为调和拟共形映照.     

$\beta$-齐性赋范空间中近似二次泛函不等式

Using the direct method,we investigate the generalized Hyers-Ulam stability of the following quadratic functional inequality■inβ-homogeneous complex Banach spaces.