一类非线性算子方程的求解方法

1.引言首先研究线性方程Au=f (1.1)的求解问题,其中A是H→H的连续线性算子,H是可分Hilbert空间,U,f∈H,||f||=1.利用得到的结论,研究一类非线性算子方程AuBu+Cu=f(1.2)     

三阶线性变系数差分方程的Mikusiski算符解法（Ⅲ）

本文在［３］，［４］工作的基础上，利用变数算符的思想以及Ｍｉｋｕｓｉｎｓｋｉ算符域中移动算符和变系数移动算符级数的有关结果，解决了一般的三阶线性变系数差分方程的求解问题，并且绘出了一些特殊的三阶线性变系数差分方程的更好的解式；此外，还试图为实现更高阶线性变系数差分方程的求解提供思想方法。     

一类微分项中含Hysteresis的抛物型方程

本文通过引入导出算子分离Ｈｙｓｔｅｒｅｓｉｓ与微分,证明了一个微分项中含Ｈｙｓｔｅｒｅｓｉｓ的抛物型方程古典解的存在唯一性。     

n阶变系数线性差分方程的解

本文利用变数算符￣［２］以及给出变数算符和移动算符的乘积关系，并定义变系数移动算符幂级数间的乘积且证明其在Ｍｉｋｕｉｕｓｋｉ收敛意义下是正确的；另外，把一般的ｎ阶变系数线性差分方程转化为一个恰当的算符方程组，从而获得一般ｎ阶变系数线性差分方程的解。     

更一般的常系数线性差分微分方程的解

利用Mikusinski J的算符演算,移动算符级数和微分算符的有关公式,给出更为一般的低阶常系数线性差分微分方程的级数形式解.     

On M-Wright transforms and time-fractional diffusion equations

In this paper, we consider the Mittag-Leffler operator as an analytical solution of time-fractional diffusion equation in the Caputo sense. This solution is presented by an integral representation in terms of the M-Wright functions and the exponential operators. Further, we study the Mittag-Leffler operators associated with the Legendre and Bessel diffusion equations. Finally, we extend the obtained integral representation for the time-fractional diffusion equation of distributed order.     

算子方程AX-XA*=B的正解与实正解

利用算子矩阵分块技巧和算子的广义逆,在A是幂等算子的情况下,给出了算子方程AX-XA＊=B有正解和有实正解的充要条件,并给出了正解和实正解的通式。     

一类非线性算子方程的正算子解的研究

研究了算子方程X^(-1)+A(-1)+A⁺X+X^tA=Q,的正算子解问题,给出了此类非线性算子方程正算子解的范围以及正算子解存在的一些充分必要条件,并用迭代的方法得到了方程的正算子解.     

n阶变系数线性差分微分方程的解

利用Mikusi'nski算符域中变系数算符概念和相应的算符系数移动算符幂级数的概念和结果,获得初值条件下n阶变系数线性差分微分方程的解.     

一类非线性算子方程拟Galerkin逼近意义下的可解性

对非线性算子方程 x+KF x=y,本文构造了一种新的不同于 Galerkin逼近的另一种逼近方程 ,并证明了在此逼近意义下上述方程是逼近可解的     

Convergence rates of regularized solutions of nonlinear ill-posed operator equations involving monotone operators

The convergence rates of regularized solutions of nonlinear ill-posed operator equations involving monotone operators are investigated, and conditions that guarantee convergence rates like and are given, where δ denotes the noise level of the data perturbation. Project supported by the National Natural Science Foundation of China (Grant No. 19671029).     

一类非线性二元算子方程的迭代求解及其应用

运用锥与半序理论与混合单调算子理论,讨论半序Banach空间一类非线性二元算子方程解的存在唯一性,并给出迭代序列收敛于解的误差估计.作为应用,讨论了不具有单调性的算子方程的可解性,所得结果是某些已有结果的本质改进和推广.     

An infinite-dimensional version of the Borsuk-Ulam theorem

We study the solvability of the equation a(x) = f(x) on a sphere in a Banach space, where a is a closed surjective linear operator and f is an odd a-compact map. We also estimate the topological dimension of the solution set and give applications of the corresponding theorem to some problems in differential equations and other fields of mathematics.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 1–5, 2004Original Russian Text Copyright © by B. D. GelmanSupported by the Russian Foundation for Basic Research, Grant No. 02-01-00189.     

在再生核空间中方程AuBu+Cu＝f的求解方法

$ 1 引言 本文研究下面一类非线性算子方程求解问题 AμBμ Cμ=f, (1.1)其中f,μ∈W(Ω),μ(O)=1,||f ||=1,A,B,C∈(W(Ω)→W(Ω)),(W(Ω)→W(Ω))是W(Ω)到W(Ω)的连续线性算子空间,W(Ω)是定义在Ω域上的(Ω是实数域R的有界域)再生核空间。 本文是在再生核空间上,通过将一维非线性算子方程(1.1)转化为二维线性算子方     

Mikusinski算符演算在方程求解中的应用

就 Mikusinski算符演算在方程求解方面的研究进展情况和已获得的重要结果作一综述 ,其内容有常系数线性微分方程、差分方程的 M算符解法 ;变数算符概念及其相关结果 ;变系数线性常微分方程、差分方程、差分微分方程的 M算符解法以及 M算符演算在其他方程求解中的应用 .     

二阶变系数线性差分方程的Mikusinski算符解法

本文利用Mikusinski算符域中变数算符的概念和相应的变系数移动算符幂级数的概念和结果,成功地获得二阶变系数线性差分方程的解。     

一类算子方程的正算子解的研究

对算子方程X+A~*X~(-2)A=Q有正算子解的条件做了进一步的研究,得到了方程有正算子解时A,Q,X的范数、谱半径之间新的关系.并给出了算子方程X+A~*X~(-t)A=Q有正算子解的一些条件.     

A resolvent test for admissibility of Volterra observation operators

Necessary and sufficient conditions are given for finite-time admissibility of a linear system defined by a Volterra integral equation when the underlying semigroup is equivalent to a contraction semigroup, in terms of a pointwise bound on the resolvent of the infinitesimal generator. This generalizes an analogous result known to hold for the standard Cauchy problem. For infinite-time admissibility, however, it is shown by means of an example that the natural generalization of the Weiss resolvent test is no longer valid.     

New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type

We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier-type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener-Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier-type series are used to produce the solution formula of such equations, and a Shannon-type sampling formula is also obtained.     

On some properties of systems of Volterra integral equations of the fourth kind with kernel of convolution type

We consider the system of integral equations of the form Ax +V x = Ψ, where V is the Volterra operator with kernel of convolution type and A is a constant matrix, det A = 0. We prove an existence theorem and establish necessary and sufficient conditions for the kernel of the operator of the system to be trivial.