向量值柯西问题的适定性

设A为复Banach空间X上的稠定闭算子.我们证明了若抽象柯西问题具有C-适定性,则{λ∈C:Reλ≥0）　ρ（A）且存在C>0,使得任给λ∈C,Reλ≥0,都有（λ-A）-1≤C/1+　λ.因而A必为X上某一有界解析半群的无穷小生成元.对于定义在有限区间[0,T]上的抽象柯西问题,我们亦得到了类似的结果.     

Vector subdivision schemes in (Lp(Rs))r(1 ≤ p ≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the form ψ(x) = ∑α∈Zs a(α)ψ(Mx - α), x ∈ Rs,where the vector of functions ψ=(ψ1,…,ψr)T is in (Lp(Rs))r, 1≤p≤∞,a(α),α∈Zs,is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix suchthat lim n→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vectorof compactly supported functions ψ0 ∈ (Lp(Rs))r and use the iteration schemes fn := Qnaψ0,n = 1,2,…,where Qa is the linear operator defined on (Lp(Rs))r given by Qaψ:= ∑α∈Zs a(α)ψ(M·- α),ψ∈ (Lp(Rs))r. This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of somelinear operators determined by the sequence a and the set B restricted to a certain invariant subspace, wherethe set B is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0.     

关于亚纯函数理论中与极点无涉的基本不等式

<正> 关于奈望利纳(Nevanlinna)氏第二基本不等式曾有引入纪(导)数而作之种种不同的推广,在这些推广式中,极点的密指量每见出现且有特殊作用,因之能否将此量消去是一问题.米约(Milloux)氏及熊庆来教授由不同途径各获得与极点无涉之一不等式此二结果形状互异而各有特点.熊庆来教授指出这两个结果尚可推广到更普遍的境地,我们由此方向探研得如下两个定理,为熊、米二氏者之推广.     

Global Smooth Solutions to a System of Dissipative Nonlinear Evolution Equations

The existence and uniqueness are proved for global classical solutions of the following initial-boundary problem for the system of parabolic equations which is proposed by Hsieh as a substitute for the Rayleigh-Benard equation and can lead to Lorenz equations: {ψ_t = -(σ - α)ψ - σθ_x, + αψ_{xx} θ_t = -(1- β)θ + vψ_x + (ψθ)_x + βθ_{xx} ψ(0,t) = ψ(1,t) = 0, θ_x(0,t) = θ_x(1,t) = 0 ψ(x,0) = ψ_0(x), θ(x,0) = θ_0(x)     

Control subspaces of minimal dimension and root vectors

We investigate the following characteristic of a linear operator A in a Banach space X: disc {inf(dim R′:R′?R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (AⁿR: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system \(\dot x = Ax + Bu\) with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.     

用广义多项式逼近在正数轴上的函数

<正> 导言伯恩斯坦曾经证明:设 F(x)是偶的整函数,其泰勒系数不是负数,并且它的性(род,genus)大于零.如果 f(x)在(—∞,∞)上连续,并且适合     

Long time classical solutions of a nonlinear wave equation

The existence and uniqueness of long time classical solutions of the Cauchy problem u_{t t}+μu_t = div(a(u)▽u), where a(u) = 1+u and μ ≥ 0, are studied for the case of two space dimensions. Let the initial data u(0,.) = φ and u_t(0,.) = ψ be supported compactly on R². Then for every T > 0, such a solution exists on [0,T] whenever (φ,ψ) is small enough in H⁴ (R²) x H³(R²). A result on the asymptotic relation between the maximal T and the size of the initial data is given.     

一类拟线性二阶微分方程解的振动与非振动的判定

本文讨论了拟线性微分方程（ｐ（ｘ＇））＇＋ｑ（ｔ）ｐ（ｘ）＝０,ｔ≥ｔ０,ｑ（ｔ）≥０（这里 ｐ（ｕ）＝｜ｕ｜ｐ－１ｕ,ｐ＞０是常数）的解的振动与非振动条件,并改进了文献［２］的结果．     

关于循环环的理想、素理想和极大理想

研究了循环环R=的理想、素理想和极大理想的个数和结构,得到了如下结论:1)理想:(1)若|R|=∞,则R共有无穷多个理想:;(2)若|R|=n,设n的正因数个数为T(n),则R共有T(n)个理想:.2)素理想:(1)若|R|=∞,设a^2=ka(k≥0),①当k=0时,R的素理想只有R;②当k>0时,R的素理想共有无穷多个,它们是:{0}、R及;(2)若|R|=n>1,设a^2=ka,0≤k.3)极大理想:(1)若|R|=∞,则R有无限多个极大理想,它们是;(2)若|R|=n>1,设n的互不相同的素因数个数为ψ(n),则R共有ψ(n)个极大理想:(pa|p是n的素因数).     

有时滞的系統的无条件稳定性

<正> §1.問題的提出及解法 錢学森在[1]中提出了有时滞的系統的无条件稳定性的問題,并叙述了Satche的作图法.对于有时滞的系統的稳定性問題,一般化为超越方程的根的实部的符号的判定問題,这方面有及,Hayes及Bellman等人的工     

一类非线性积分方程非负解的存在性

本文利用凝聚映象的不动点定理得到了如下形式的积分方程：ψ０（ｘ）＝ψ（ｘ）＋ｆ（ψ（ｘ））∫１０Ｒ（ｘ,ｙ）ｘ２－ｙ２ｇ（ψ（ｙ））ｄｙ非负连续解的存在性．     

具非线性边界条件的泛函微分方程边值问题奇摄动(英文)

研究一类具非线性边界条件的泛函微分方程边值问题εx″( t) =f ( t,x( t) ,x( t-τ) ,x′( t) ,ε) ,　 t∈ ( 0 ,1 ) ,x( t) =φ( t,ε) ,　 t∈ [-τ,0 ],　 h( x( 1 ) ,x′( 1 ) ,ε) =A(ε) .我们利用微分不等式理论证明了边值问题解的存在性 ,并给出了解的一致有效渐近展开式     

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.     

Zeros of accretive operators

In the investigation of accretive operators in Banach spaces X, the existence of zeros plays an important role, since it yields surjectivity results as well as fixed point theorems for operators S such that I-S is accretive. Let D?X and T: D→X an operator such that the initial value problems (1) u′(t)=-Tu(t), u(0)=x εD are solvable. Then T has a zero iff (1) has a constant solution for some xεD. Under certain assumptions on D and T it is possible to show that (1) has a unique solution u(t,x) on [0,∞), for every xεD. In this case, define U(t): D→D by U(t)x=u(t,x). If T is accretive it turns out that U(t) is nonexpansive for every t≥0. This fact constitutes the basis for several authors concerned with this subject. They proceed with assumptions on D and X ensuring either that the U(t) must have a common fixed point x_o or that U(_p) has a fixed point x_p for every p≥0. In the first case, U(t)x_o is a constant solution of (1), whence Tx_o=0. In the second case, U(t)x_p is a p-periodic solution of (1). Hence, one has to impose additional conditions on T which imply that a p-periodic solution must be constant, for some p>0. The main purpose of the present paper is to show that, in certain situations, either the operators U(t) are actually strict contractions or T may be approximated by operators T_n such that the corresponding U_n(t) are strict contractions. Thus, we obtain several results in general Banach spaces and a unification of some results in special spaces.     

华林问题中 g(ψ)的估值

<正> 假定 k 是一个整数≥12,(?)(x)=(?)_k(x)=x(x+1)…(x+k-1).记号g((?)_k)表示最小的整数 r 满足条件,使得每个整数 N≥1都能够表示成(?)这里的 x_i 是一个非负整数。Нечаев曾证明有(?)本文的目的是要改善这个不等式为     

Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.     

Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron

We consider gauge-dependent dynamical equations describing homogeneous isotropic cosmologies coupled to a scalar field ψ (scalaron). For flat cosmologies (k = 0), we analyze the gauge-independent equation describing the differential χ(α) ≡ ψ (a) of the map of the metric a to the scalaron field ψ, which is the main mathematical characteristic of a cosmology and locally defines its portrait in the so-called a version. In the more customary ψ version, the similar equation for the differential of the inverse map \(\bar \chi (\psi ) \equiv \chi ^{ - 1} (\alpha )\) is solved in an asymptotic approximation for arbitrary potentials v(ψ). In the flat case, \(\bar \chi (\psi )\) and χ^?1(α) satisfy first-order differential equations depending only on the logarithmic derivative of the potential, v(ψ)/v(ψ). If an analytic solution for one of the χ functions is known, then we can find all characteristics of the cosmological model. In the α version, the full dynamical system is explicitly integrable for k ≠ 0 with any potential v(α) ≡ v[ψ(α)] replacing v(ψ). Until one of the maps, which themselves depend on the potentials, is calculated, no sort of analytic relation between these potentials can be found. Nevertheless, such relations can be found in asymptotic regions or by perturbation theory. If instead of a potential we specify a cosmological portrait, then we can reconstruct the corresponding potential. The main subject here is the mathematical structure of isotropic cosmologies. We also briefly present basic applications to a more rigorous treatment of inflation models in the framework of the α version of the isotropic scalaron cosmology. In particular, we construct an inflationary perturbation expansion for χ. If the conditions for inflation to arise are satisfied, i.e., if v > 0, k = 0, χ² < 6, and χ(α) satisfies a certain boundary condition as α→-∞, then the expansion is invariant under scaling the potential, and its first terms give the standard inflationary parameters with higher-order corrections.     

一个变分不等式的奇异摄动

对变分不等式的奇异摄动问题进行了探索,证明了解的重合集Iε={x∈Ωuε（x)=φ}在Hausdorff距离意义下收敛到ε＝0时解的重合集。     

Banach空间中关于Lipschitz φ-半压缩映象的带误差项的Ishikawa迭代过程

本文在任意Ｂａｎａｃｈ空间中研究了Ｌｉｐｓｃｈｉｔｚ φ－半压缩映象与φ－强拟增生映象的带误差项的Ｉｓｈｉｋａｗａ迭代过程,使用新的分析技巧建立了几个强收敛定理．