【作者】 陈建领；

【导师】 王俊明；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2011， 硕士

【摘要】 非线性算子的不动点理论作为非线性泛函分析理论的重要组成部分,它与许多近代数学分支有着紧密的联系,并在处理这些分支中的一些问题中发挥着重要作用。在不动点问题研究的众多方向中,将各种非线性算子类型和构造的收敛的迭代序列应用于优化、控制和微分方程等方面成为研究的主流问题。本文研究了非线性算子不动点的迭代逼近问题,全文主要包括以下三方面内容。第一部分,介绍了非线性算子理论的产生背景以及迭代算法的发展情况,为本文的后续研究工作提供了正确方向。第二部分,研究了p-几乎渐近非扩张型映象Ishikawa迭代序列和Mann迭代序列的收敛性。在一致凸Banach空间中映象是一致渐近正则和一致连续的条件下,应用粘性逼近方法得到了当系数序列满足一定条件时,具随机误差的修正的Ishikawa迭代和Mann迭代序列的强收敛性。第三部分,引入一类广义p-渐近非扩张型映象,给出了具混合误差的Ishikawa迭代序列强收敛于广义p-渐近非扩张型映象的某一不动点的充要条件,并在实一致凸Banach空间框架下,采用粘性逼近方法得到了广义p-渐近非扩张型映象的两种迭代算法强收敛定理,所得结论推广和改进了张石生、冯先智、向长合等人的相应结果。更多还原

【Abstract】 The fixed point theory of nonlinear operators, as an important part of nonlinear function analysis, has been closely connected with many branches of modern mathematics and plays an important part for solving problems in these branches. Among many directions of the fixed point researches, it has become the mainstream to realize the application of convergence iterative sequences and various nonlinear operators on optimization, control and derivative equation, etc.This thesis, studying the problem on approximating to the fixed points of nonlinear operators, mainly contains three parts as following:In first part we introduce the background of nonlinear operator theory and the development of iterative methods, both of which lead the later parts of the thesis to a right direction.In second part we study the convergence of modified Ishikawa and Mann iterative sequence with random errors for p-almost asymptotically nonexpansive type of mapping. In uniformly convex Banach spaces, under the condition when Mapping is uniformly asymptotic regular and uniformly continuous, we apply viscosity approximation methods to obtain the convergence of modified Ishikawa and Mann iterative sequences with random errors when the sequence of positive numbers satisfies appropriate conditions.In third part we bring in a generalized p-asymptotically quasi-nonexpansive type of mapping, gives some necessary and sufficient conditions for the modified Ishikawa iterative sequences with mixed errors to converge strongly to a fixed point of generalized p-asymptotically quasi-nonexpansive type of mapping, and with the application of viscosity approximation methods in uniformly convex Banach spaces, the strong convergence theory of two iterative methods of the generalized p-asymptotically quasi-non-expansive type of mapping are obtained. Our results extend and improve the corresponding results of Chang S S, Feng and Xiang.更多还原