具无限时滞的积分微分方程的周期解的存在性、唯一性及稳定性

1引言考虑如下的Volterra积分微分方程其中t∈R，x∈Rn；A（t），C（t，s），C（t-S）都是n×n连续函数矩阵；f：R→Rn连续．关于方程（1．1）及（1．2）的周期解的存在性问题，已有不少研究工作[1-4]，例如[1]研究了当n＝1时方程（1．1）的周期解的存在性问题．得到了如下结果：定理A[1]如果下列条件满足：（i）A（t＋T），f（t＋T）=f（t），C（t＋T，s＋T）＝C（t；s）对t，s∈R成立，其中T＞0是常数．（ii）方程（1．1）具有“衰退记忆”．（iii）存在着常数K＞1及μ＞0使得A（t）＋K∫t-∞|C（t，s）|ds＜－μ则方程（1．1）…     

矩阵方程X＋A^TX^—1A=Q存在可稳定化解的充分必要条件

１引言一般的时离散代数Ｒｉｃｃａｔｉ方程具有下面的形式：这里如果方程（１）中的系数矩阵满足：（ｎ＝ｍ）则方程（１）变为当Ｑ＝ＱＴ＞０时,Ｅｎｇｗｅｒｄａ,詹兴致等人研究了方程（２）存在正定解的充分必要条件［１］［２］［３］．本章利用方程（２）与（１）的关系,从另一角度讨论了Ｑ为对称矩阵时,方程（２）存在可稳定化解的充分必要条件．２基本概念与记号首先我们简单回顾一下以前的概念与记号．矩阵束Ｍ—Ｎ,Ｍ,Ｎ为正则的,也就是说ｄｅｔ（λＭ－Ｎ）＝０;如果λ０为ｄｅｔ（λＭ－Ｎ）的ｋ重根,则称λ０为它的ｋ…     

一类半线性热方程整体解的存在性与非存在性

本文研究半线性热方程的初值问题u_t－△u＝u~γ＋cu,（γ＞1）;u（x,0）=（x）非负整体L~P解的存在性与非存在性．首先证明,若C＞0,则不存在非负整体解．而后,对C＜0情形给出了解的整体存在与非存在的充分条件,特别证明了,若P＞（γ一1）或,则当。充分小时存在非负整体L~P解.最后,对系数C和初值（x）得到无穷多个门槛结果．     

奇异半线性热方程初值问题解的存在性与Blow-up问题

本文讨论如下奇异半线性热方程的初值问题其中γ＞1，σ＞O，f（x）连续有界非负但不恒为零．我们讨论了问题（1），（2）非负局部经典解的存在性和非负整体解的不存在问题，得到当0＜σ＜1且时，（1），（2）没有非负整体解．当σ＝1且时，（1），（2）没有非负整体解，当σ＞1时（1），（2）没有非负整体解．     

丢番图方程与实二次域类数的可除性

设ｄ无平方因子，ｈ（ｄ）是二次域的类数。本文证明了：在方程Ｕ￣２－ｄＶ￣２＝４，（Ｕ，Ｖ）＝１有整数解时，丢番图方程４ｘ￣（２ｎ）－ｄｙ￣２＝－１，ｎ＞２无｜ｙ｜＞１的整数解；如果正整数ａ，ｋ，ｎ满足，ｋ＞１，ｎ＞２且而是Ｐｅｌｌ方程ｘ￣２－ｄｙ￣２＝－１的基本解，则ｈ（ｄ）≡０（ｍｏｄｎ）。     

海仑数组的探求

如果三角形的三边长为整数且面积亦为整数，则称之为海仑三角形．海仑三角形的三边长所构成的数组（a，b，c）称之为海会数组．本文对海会数组进行新的探索．假定D＞0，D不是平方数，c是非0整数．设x=u，y=V是不定方程x~2-Dy~2=c的一个解，那么就称u＋v是它的一个解．其中当u≥0，v≥0时，最小的一个叫做基本解．再设x＋y是Pell方程X~2-Dy~2=1的任意一个解，则容易验证（u十v）（X十y）（=ux＋uyD＋（ux＋ut）也是x~2-Dy~2=c的解．设三角形三边长分别为a，从一a十…，C，其中p为奇数（可正可负）．则其面积为由于这个关于C’的M次方程…     

一类具有双重奇性的抛物型方程的整体解和有限熄灭

本文讨论具有双重奇性的抛物型方程ｕｔ＝ ｄｉｖ（｜△ｕ~ａ｜ｐ~－２△ｕ~ａ）,（ｘ,ｔ） ∈ Ｒ~ｎ ×（０,∞）,其中Ｐ＞ １,ａ＞ ０,ｎ≤ ２．证明当１＜ Ｐ＜ｎ（ａ＋１）／（ａｎ＋１）时,存在整体自相似解ｕｇｓ（·,ｔ） ∈ Ｌ~ｑ（Ｒ~ｎ）（ｑ＞ｓ＝~△ｎ［１－ａ（ｐ－１）］／ｐ）,但是ｕｇｓ∈~／Ｌ~ｓ（Ｒ~ｎ）（定理２．１）;同时存在有限熄灭的自相似解ｕｌｓ满足相同的积分条件（定理 ３．１）．     

一类多目标规划的同伦方法

1．引言考虑下述多目标规划问题：其中F（x）一（fi（x），人（x），…，人（x》”，人（x）（j—1，2，…，m）EC’，g；（x）（i—l，2，…，P）EC’，X6R“对于问题（P；），若考虑在最不利的情况下找出一个最有利的方案，依据「l〕，可转化为求下述问题（P。）：其中U（F（x》一max人（x））且有：引理1［‘］问题（P。）的最优解为问题（P；）的弱最优解．显然，问题（P。）等价于下述问题（P。）［‘］：则问题（P。）等价于下列问题（P．）：2．同伦方法的建立由【3j知，相应于（P。）的Kuhn－Tucker方程：其中Y一dia－（－…     

从一道高考题看求解的转化策略

求解超越方程（指数、对数、三角、反三角方程），特别是合参数的方程，一般用等价转化的思想和方法，转化为代数方程求解．下面拟通过一道例题来探讨有关转化策略．例已知关于x的方程lg（ax）=2lg（x—1），（1）求a为何值时方程有解；（2）求出方程的解．（1986年广东省高考题改编）分析（1）即求方程有解的充分条件；（2）即求在（1）中条件下的解．原方程等价于即其中①、②两式成立，则ax＞0必成立．故③式可舍．这样原问题等价于：（1）求a的范围使厂～、T＿“”（。）”IxlM0成立；（2）求出（。）式中方程的解．说明上面通过把原…     

具变系数时滞微分方程解的振动性

研究了时滞微分方程解的振动性，其中P（t）、τ（t）非负连续．我们证明了：如果对充分大的，且，则方程（*）每一解振动．该结论改进和推广了许多已知结果．     

A block monotone domain decomposition algorithm for a semilinear convection–diffusion problem

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed convection–diffusion problem. A block monotone domain decomposition algorithm based on a Schwarz alternating method and on block iterative scheme is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process and converges monotonically to the exact solution of the nonlinear problem. The rate of convergence of the block monotone domain decomposition algorithm is estimated. Numerical experiments are presented.     

On Schwarz's domain decomposition methods for elliptic boundary value problems

Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness . The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results. Received December 28, 1995 / Revised version received November 17, 1998 / Published online September 24, 1999     

Least-squares spectral collocation with the overlapping Schwarz method for the incompressible Navier–Stokes equations

A least-squares spectral collocation scheme is combined with the overlapping Schwarz method. The methods are succesfully applied to the incompressible Navier–Stokes equations. The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The overlapping Schwarz method is used for the iterative solution. For parallel implementation the subproblems are solved in a checkerboard manner. Our approach is successfully applied to the lid-driven cavity flow problem. Only a few Schwarz iterations are necessary in each time step. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.     

Two-level Schwarz method for unilateral variational inequalities

The numerical solution of variational inequalities of obstacletype associated with second-order elliptic operators is considered.Iterative methods based on the domain decomposition approachare proposed for discrete obstacle problems arising from thecontinuous, piecewise linear finite element approximation ofthe differential problem. A new variant of the Schwarz methodology,called the two-level Schwarz method, is developed offering thepossibility of making use of fast linear solvers (e.g., linearmultigrid and fictitious domain methods) for the genuinely nonlinearobstacle problems. Namely, by using particular monotonicityresults, the computational domain can be partitioned into (mesh)subdomains with linear and nonlinear (obstacle-type) subproblems.By taking advantage of this domain decomposition and fast linearsolvers, efficient implementation algorithms for large-scalediscrete obstacle problems can be developed. The last part ofthe paper is devoted to illustrate numerical experiments.     

On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs

Summary The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology. This work was in part supported by a grant from the RGC of HKSAR, China (HKUST6171/99P)     

An additive Schwarz method for a stationary convection-diffusion problem

In this paper, we consider an additive Schwarz method applied to a linear, second order, nonsymmetric, indefinite problem. We discuss the solution of linear system of algebraic equations that arise from the streamline method for the above problem. An alternative linear system, which has the same solution as the system obtained by the streamline method, is derived and the GMRES method is used to solve this system. We show that the rate of convergence does not depend on the mesh size, nor on the number of local problems if the coarse mesh is fine enough.     

A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems

In recent years, competitive domain-decomposed preconditioned iterative techniques of Krylov-Schwarz type have been developed for nonsymmetric linear elliptic systems. Such systems arise when convection-diffusion-reaction problems from computational fluid dynamics or heat and mass transfer are linearized for iterative solution. Through domain decomposition, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow effective solution on parallel machines. A central question is how to choose these small problems and how to arrange the order of their solution. Different specifications of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILU-family of preconditioners, on some nonsymmetric or indefinite elliptic model problems discretized by finite difference methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.     

Overlapping restricted additive Schwarz method applied to the linear complementarity problem with?an?H-matrix

In this paper, a restricted additive Schwarz method is introduced for solving the linear complementarity problem that involves an H ₊-matrix. We show that the sequence generated by the restricted additive Schwarz method converges to the unique solution of the problem without any restriction on the initial point. Moreover, the comparison theorem is given between different versions of the restricted additive Schwarz method by using the weighted max-norm. We also show that the restricted additive Schwarz method is much better than the corresponding additive Schwarz variants in terms of the iteration number and the execution time.     

Fourier analysis of schwarz alternating methods for piecewise hermite bicubic orthogonal spline collocation

The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH ¹-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.This research was supported in part by funds from the National Science Foundation grant CCR-9103451.     

用CROUZEIX－RAVIART元解非自共轭椭圆型问题的重叠型区域分解算法

用ＣＲＯＵＺＥＩＸ－ＲＡＶＩＡＲＴ元解非自共轭椭圆型问题的重叠型区域分解算法顾金生（北京航空航天大学动力系）胡显承（清华大学应用数学系）ＯＶＥＲＬＡＰＰＩＮＧＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＦＯＲＮＯＮＳＥＬＦＡＤＪＯＩＮＴＥＬＬＩ...