一类抽象算子方程组的迭代解及其应用

在Banach空间中, 利用半序方法讨论了一类抽象算子方程组解的存在唯一性, 推广和统一了以前的一些结果. 然后应用到 Banach 空间非线性积分方程组, 得到了方程组的唯一解, 构造了收敛于方程组唯一解的迭代序列并给出了相应的误差估计.     

二元非混合单调算子方程组的迭代求解

利用锥与半序理论和混合单调算子理论,讨论Banach空间中非单调二元非线性算子方程组解的存在性与唯一性,并给出了收敛于方程组解的迭代序列和误差估计。改进和推广了混合单调算子方程和一元算子方程的某些相应结果。     

一类高阶非线性中立型差分方程组非振动解的存在性

研究了一类高阶非线性中立型差分方程组非振动解的存在性.利用Banach空间的压缩映象原理,获得了该方程组存在非振动解的充分条件.     

Banach空间一阶脉冲微分积分方程组初值问题的解

研究了Banach空间非线性一阶脉冲微分积分方程组初值问题解的存在性，改进并推广了近期的一些结果．     

Banach空间非线性脉冲Volterra积分方程组的整体解

研究Banach空间中定义在无穷区间R+上具有无穷多个脉冲点的非线性脉冲Volterra积分方程组解的存在性。给出了若干极值解的存在定理,改进了定义在有限区间上具有有限个脉冲点情形时该类方程的相应结果,并利用该结果讨论了一个无穷维积分方程组。     

Banach空间中二元算子方程组解的存在唯一性及其迭代解法

利用锥理论与半序方法对Banach空间中几类二元算子方程组解的存在唯一性进行探讨,给出它们的迭代求解法,得到了一些新结果.     

一类非线性耗散方程组的不变子空间及其精确解

利用不变子空间方法研究一类在孤立子理论中具有广泛应用的非线性耗散方程组,确定出非线性耗散方程组在它所容许的不变子空间W_3~1×W_2~2中的完全分类,构造了方程组的精确解或者将方程组约化为有限维动力系统.文中的结果进一步推广了不变子空间理论在非线性偏微分方程中的应用.     

弱拓扑下的非线性随机积分和微分方程组的解*

在本文中,我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理.利用这一定理,我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则.作为应用,我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理.也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果.我们的定理改进和推广了Szep,Mitchell－Smith,Cramer－Lakshmikantham,Lakshmikantham-Leela和丁的相应结果.     

非线性非单调二元算子方程组的解及其应用

利用非线性泛函分析中的锥理论和单调迭代的方法,研究了一类非线性非单调二元算子方程组的解的存在性,并给出了收敛于解的迭代序列,然后作为应用,得到了B anach空间中的一类非线性V olterra型积分方程组的解,改进了最近的许多结果.     

二元算子方程组的迭代求解方法

利用锥理论和单调迭代方法,本文在Banach空间中对三类二元算子方程组的求解进行了探讨,利用较简捷的条件得出方程组的唯一解和迭代逼近式及误差估计式并推广到了n元算子方程组的情形,得到相应结果．     

非线性二元算子方程组的迭代求解方法

Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are improved and generalized.     

Numerical solutions of coupled systems of nonlinear elliptic equations

This article deals with numerical solutions of a general class of coupled nonlinear elliptic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear elliptic equations. This monotone convergence leads to existence‐uniqueness theorems for solutions to problems with reaction functions of quasi‐monotone nondecreasing, quasi‐monotone nonincreasing and mixed quasi‐monotone types. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating, is proposed. An application to a reaction‐diffusion model in chemical engineering is given. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 621–640, 2012     

Banach空间中非线性积-微分方程组唯一解的存在性

利用新的比较结果和半序方法,研究TBanach空间中二阶积-微分方程组初值问题解的存在唯一性及逼近解的迭代序列和误差估计．     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

Global Existence, Uniqueness and Pathwise Property of Solutions to a Stochastic R¨ossler-Lorentz System

The authors integrate two well-known systems, the R¨ossler and Lorentz systems, to introduce a new chaotic system, called the Lorentz-R¨ossler system. Then, taking into account the effect of environmental noise, the authors incorporate white noise in both R¨ossler and Lorentz systems to have a corresponding stochastic system. By deriving the uniform a priori estimates for an approximate system and then taking them to the limit, the authors prove the global existence, uniqueness and the pathwise property of solutions to the Lorentz-R¨ossler system. Moreover, the authors carried out a number of numerical experiments, and the numerical results demonstrate their theoretic analysis and show some new qualitative properties of solutions which reveal that the Lorentz-R¨ossler system could be used to design more complex and more secure nonlinear hop-frequence time series.     

Matrix correction of a dual pair of improper linear programming problems

A matrix is sought that solves a given dual pair of systems of linear algebraic equations. Necessary and sufficient conditions for the existence of solutions to this problem are obtained, and the form of the solutions is found. The form of the solution with the minimal Euclidean norm is indicated. Conditions for this solution to be a rank one matrix are examined. On the basis of these results, an analysis is performed for the following two problems: modifying the coefficient matrix for a dual pair of linear programs (which can be improper) to ensure the existence of given solutions for these programs, and modifying the coefficient matrix for a dual pair of improper linear programs to minimize its Euclidean norm. Necessary and sufficient conditions for the solvability of the first problem are given, and the form of its solutions is described. For the second problem, a method for the reduction to a nonlinear constrained minimization problem is indicated, necessary conditions for the existence of solutions are found, and the form of solutions is described. Numerical results are presented.     

Global existence and regularity of solutions to a system of nonlinear Maxwell equations

We consider the model that has been suggested by Greenberg et al. (Physica D 134 (1999) 362-383) for the ferroelectric behavior of materials. In this model, the usual (linear) Maxwell's equations are supplemented with a constitutive relation in which the electric displacement equals a constant times the electric field plus an internal polarization variable which evolves according to an internal set of nonlinear Maxwell's equations. For such model we provide rigorous proofs of global existence, uniqueness, and regularity of solutions. We also provide some preliminary results on the long-time behavior of solutions. The main difficulties in this study are due to the loss of compactness in the system of Maxwell's equations. These results generalize those of Greenberg et al., where only solutions with TM (transverse magnetic) symmetry were considered.     

Banach空间半直线上非线性二阶微分方程组正解的存在性

In this paper, the cone theory and MSnch fixed point theorem combined with the monotone iterative technique are used to investigate the positive solutions for a class of systems of nonlinear singular differential equations with multi-point boundary value conditions on the half line in a Banach space. The conditions for the existence of positive solutions are formulated. In addition, an explicit iterative approximation of the solution is also derived.