﻿ 求解非线性算子方程的两步组合方法的收敛性分析 Convergence Analysis of the Two-Step Combined Method for Solving Nonlinear Operator Equations

Vol.06 No.01(2017), Article ID:19641,14 pages
10.12677/AAM.2017.61011

Convergence Analysis of the Two-Step Combined Method for Solving Nonlinear Operator Equations

Weidi Wu, Weiping Shen, Lihua Xu

Department of Mathematics, Zhejiang Normal University, Jinhua Zhejiang

Received: Jan. 3rd, 2017; accepted: Jan. 21st, 2017; published: Jan. 24th, 2017

ABSTRACT

In this paper, we consider the convergence of the two-step combined method for solving nonlinear operator equations. A semi-local convergence of the method is presented under some continuity conditions. Moreover, we establish the uniqueness result of the solutions. Finally, a numerical example is provided to demonstrate our theoretical results.

Keywords:Semi-Local Convergence, Two-Step Combined Method, Divided Differences

1. 引言

(1)

(4)

(5)

2. 定义与引理

(7)

(2)可逆且存在非负数使得

(3)

3. 半局部收敛性定理

(2)存在非负常数，使得对，下列条件成立：

(8)

(9)

(10)

(3)且存在非负数使得

(11)

(4) 闭球。假设实序列由如下定义：

(12)

(13)

(14)

(i)为非负，递减序列且收敛于满足

(ii)算法(6)是适定的。所得序列收敛于解，且满足不等式

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

4. 数值例子

(32)

，则将代入问题(32)，我们即可得

(33)

(34)

，则通过计算(13)，(14)，(15)可得：

Convergence Analysis of the Two-Step Combined Method for Solving Nonlinear Operator Equations[J]. 应用数学进展, 2017, 06(01): 90-103. http://dx.doi.org/10.12677/AAM.2017.61011

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