具热效应的半导体方程组的初边值问题

证明了热了流模型半导体方程组整体光滑解的存在唯一性。如果区域在某一方向充分窄，则我们证明了平衡解的存在唯一性和解的渐近性。     

代数体函数的唯一性问题

讨论了重值和Ｖａｌｉｒｏｎ亏量对代数体函数唯一性问题的影响，证明了两个唯一性定理．     

关于特征值反问题的唯一性

通过构造线性变换的角度证明了一类简单的特征值反问题的唯一性,以利于从本质上去理解唯一性的根源.     

一类多时滞微分方程初值问题解的存在唯一性

研究了一类多时滞微分方程初值问题解的存在唯一性,用Picard方法证明了这类初值问题解的存在唯一性结论,它是常微分方程基本理论中著名的Picard存在唯一性定理的推广.     

一类非线性四阶波动方程的位势井方法

该文讨论非线性波动方程u_{tt}+u_{xxxx}=σ(u_x)_x+f(x,t)的初边值问题.证明了整体弱解的存在性,还证明了整体广义解的存在唯一性和整体古典解的存在唯一性.     

俄勒冈振子极限环的唯一性

本文证明了俄勒冈振子中极限环的唯一性．     

一类奇异拟线性椭圆型方程组的可解性

研究了一类P-Laplacian方程组边值问题正径向整体解的存在性和唯一性.首先,利用隐函数定理证明了该问题的局部解的存在性与唯一性,以及解对初值的连续依赖性.最后,证明了该问题存在唯一的正径向整体解.     

关于两类特殊图的色多项式唯一性

本文用全新的方法证明了两类图的色多项式唯一性,推广了Beatrice Loe-rine关于广义θ-图的色多项式唯一性的结论。     

亚纯映射的唯一性定理

证明了若干个亚纯映射的唯一性定理．     

关于Chevalley群扩群的两点继续研究

构造出Ｄ２ｍ型Ｃｈｅｖａｌｌｅｙ群扩群Ｇ,证明了其唯一性,并讨论了文「１」中构造的Ａｎ型Ｇ的唯一性。     

Toeplitz matrix method and nonlinear integral equation of Hammerstein type

The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, numerical examples, when the kernel takes a logarithmic and Carleman forms, are discussed and the estimate error, in each case, is calculated.     

Trend to Equilibrium for the Coagulation–Fragmentation Equation

For a linear coagulation kernel and a constant fragmentation kernel we prove the existence of equilibrium solutions and examine asymptotic properties for time-dependent solutions which are proved to converge to the equilibria. The rate of the convergence is estimated. It is shown also that all time-dependent solutions with the same density can tend to only one particular steady-state solution. In this sense the equilibrium solution is proved to be unique. Existence, uniqueness and mass conservation of time-dependent solutions has been proved in a previous paper by the authors [10].     

域外奇源分布法场点解的唯一性

应用奇源解的核函数沿距离递减的条件和积分不等式,可证这类奇源在凸域外分布只要满足以它产生的响应来表达的边界条件时,所引起相应的域内场点解是唯一的。文中给出这类奇源的部分例子,如Kelvin的点力,点圆力偶(PRC)等。并给出PRC分布解回转体扭转问题的场点解的唯一性证明作为应用例。     

An identification problem for a conserved phase‐field model with memory

This paper is devoted to recovering a scalar memory kernel in a conserved phase‐field model. For such a problem local in time existence and uniqueness results are proved. The technique used allows to show also the continuous dependence on the kernel of the solution to the direct problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.     

Reconstruction of a convolution operator from the right-hand side on the real half-axis

We study a Volterra convolution integral equation of the first kind on a semi-infinite interval. Under some rather natural constraints on the kernel and the right-hand side of the Volterra integral equation (the kernel has bounded support, while the support of the right-hand side may be unbounded), it is possible to reconstruct the integral operator of the equation (i.e., the solution and the kernel of the integral operator) from the right-hand side of the equation. The uniqueness theorem is proved, the necessary and sufficient conditions for solvability are found, and the explicit formulas for the solution and the kernel are obtained.     

On the Theory of Generalized Toeplitz Kernels

A new proof of the integral representation of the generalized Toeplitz kernels is given. This proof is based on the spectral theory of the corresponding differential operator that acts in the Hilbert space constructed from a kernel of this sort. A theorem on conditions that should be imposed on the kernel to guarantee the self-adjointness of the operator considered (i.e., the uniqueness of the measure in the representation) is proved.     

On solvability of one class of nonlinear integral equations on whole line with a weak singularity at zero

In present work we investigate one class of nonlinear integral equations with singularity at zero and boundary value conditions at ±∞. Above mentioned class of equations has direct applications in string theory and in the case of concrete structure of the kernel it describes the dynamics of the open-closed p-adic string for the scalar tachyon field. We prove the existence of nontrivial solution in a certain weight class of functions.With an additional restriction on the kernel the uniqueness of the obtained solution is proved.     

Convergence of Collocation Method with Delta Functions for Integral Equations of First Kind

Integral equations of first kind with periodic kernels arising in solving partial differential equations by interior source methods are considered. Existence and uniqueness of solution in appropriate spaces of linear analytic functionals is proved. Rate of convergence of collocation method with Dirac’s delta-functions as the trial functions is obtained in case of uniform meshes. In case of an analytic kernel the convergence rate is exponential.     

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.