非线性不适定问题一种双循环的牛顿型迭代格式

本文研究非线性算子方程F(x)=y的解,结合最速下降法,Newton-Landweber迭代格式及正则化思想,在F满足适当的条件下,构造出新的双循环迭代格式.本文对格式的收敛性进行了严格论证,并估计出迭代格式的收敛精度.     

Theory of passively mode-locked lasers with self-frequency shift and saturable absorber response

A general theoretical model for passively mode-locked lasers is presented, in which both the self-frequency shift and either a fast or a slow saturable absorber response are taken into account. An exact soliton-like solution and condition for its existence are obtained under a definite compatible condition. The stability of the solution is analyzed by using a variational method, and a parameter region, in which the solution is linearly stable, is acquired theoretically. To verify the theoretical predictions, a typical example is given for stable pulse propagation over a long distance. The numerical results show that the soliton-like solution is stable under some perturbations within the linearly stable region and an arbitrary Gaussian pulse converges to the exact soliton-like solution after evolution in a distance.     

半参数变量含误差函数关系模型的小波估计

本文研究半参数变量含误差函数关系模型，应用小波估计法和全最小二乘法得出未知参数和未知函数的估计，在一般的条件下，证明了估计的强相合性、一致强相合性，并给出了误差方差估计的强收敛速度。     

关于Banach空间中增生算子方程的迭代法收敛率估计

本文研究Banach空间中增生算子方程的Ishikawa迭代法收敛率估计。本文所得结果在以下方面改进和推广了刘理蔚的结果（Nonlinear Anal.42(2)(2000),271-276):(1)以假设{αn},{βn}在不同区间上独立取值代替刘的假设limn→∞αn=limn→∞βn＝0；（2）以一般的收敛率估计和几何收敛率估计代替刘的收敛率估计||xm=x^*||=O(1/m）。     

一类非线性方程Mann和Ishikawa迭代程序的稳定性

设x是实Banach空间，H：X→X是Lipschitz算子，T:x→x是一致连续的且值域有界，H T是强增生的，则Mann和Ishikawa迭代程序几乎稳定地强收敛到方程Hx Tx＝f的唯一解．     

Convergence Theorems of Mann and Ishikawa Iterative Processes with Errors for Multivalued Φ－strongly Accretive Mapping

Let X be a real Banach space and A : X→ 2^X a bounded uniformly continuous Φ-strongly accretive multivalued mapping. For any f∈ X, Mann and Ishikawa iterative processes with errors converge strongly to the unique solution of Ax(←∈) f. The conclusion in this paper weakens the stronger conditions about errors in Chidume and Moore‘s theorem (J.Math.Anal .Appl.,245(2000),142-160).     

Logistic阻滞增长模型的稳定性与混沌

将Logistic阻滞增长模型的差分形式简化，讨论了它的稳定性，用计算机进行迭代求解。模拟了这一简单差分方程从收敛，分叉，2^n倍周期收敛进入混沌现象的过程。直观地展示了序列{yk}收敛，2倍周期，4倍周期……直至混沌的现象，这对Logistic阻滞增长模型的应用和混沌现象的模拟有很好的参考价值。     

椭圆型方程的广义差分法（二次元）

在文[1]中,李荣华综合了有限元法和差分法的优点,提出了广义差分法,它既保持了差分法的计算简单性,文具有有限元法的精确性,文[2]把广义差分法推广到平面域上二阶椭圆偏微分方程的边值问题,在试探函数空间为分片一次元,检验函数空间为分片常数的情形下得到与线性有限元相同的收敛阶,本文以Poisson方程为模型,取试探函数为分片二次元,检验函数为分片常数,导出了一种二次元的广义差分法,给出最佳收敛阶估计,并做出数值实验。