摘要
本文主要研究两类广义方程.首先，我们考虑如下广义方程问题：(p1)0∈f(x)+F(x),
其中，X,Y是Banach空间.f:Ω(?)X→Y是单值函数,Ω是X中的开子集,F:X(?)2Y是具有闭图像的集值映射.其次我们考虑如下形式的广义方程：(p2) O∈T(x),
其中,X是Banach空间,T:x(?)2x是图像为局部闭的集值映射.本文主要内容包含两部分：
在第一部分中,特别是在第三章,我们引进并研究了当f是光滑函数,即f是Fr(?)chet可微的,并且f可用经典的线性化表示时.解决广义方程(P1)的高斯牛顿方法.我们给出了高斯-牛顿法的半局部和局部收敛性.更进一步,我们研究了当f是非光滑函数,即f不具有Fr(?)chet导数,并且f不具有经典的线性化表示时,解决广义方程(P1)的高斯-牛顿方法.同样给出了高斯-牛顿法的半局部和局部收敛性.
在第二部分中,特别是在第四章,我们通过选取有界的下极限非零的常数序列{入κ},引进了解决广义方程(P2)的高斯型临近点算法.我们建立了该高斯型临近点算法的收敛准则,它保证了在一般的条件下：由该算法生成的序列的存在性及其收敛性.更精确的说.我们分析了当T是度量正则时,该高斯型临近点方法的半局部和局部收敛性.并且,我们研究了该算法的稳定性.
In this work, we deal with two types of generalized equations. Firstly, we consider the following generalized equation problem (P1)0∈f(x)+F(x), where X and Y are Banach spaces,f:Ω(?)X→Y is single-valued function,Ω is an open subset of X and F:X(?)2Y is set-valued mapping with closed graph, and secondly we consider the generalized equation of the following form (P2)0∈T(x), where X is a Banach space and T:X(?)2x is a set-valued mapping with locally closed graph. This work consists two parts and the main works we have done in this dissertation that are organized as follows.
In the first part, particulaly in Chapter3, We introduce and study the Gauss-Newton method for solving the generalized equation (P1) when f is smooth function, that is, when f is Frechet differentiable and it can be expressed as a classical linearization. We provide here semilocal and local convergence of the Gauss-Newton method. Furthermore, we introduce and study the Gauss-Newton method for solving the generalized equation (P1) when f is nonsmooth function, that is, when f doesn't possess a Frechet derivative and the classical linearization of f is no longer available. We also present here semilocal and local convergence of the Gauss-Newton method.
In the second part, specifically in Chapter4, we introduce the Gauss-type proximal point algorithm for solving the generalized equation (P2) by choosing a sequence of bounded constants{λk} which are away from zero. We establish the convergence criteria of the Gauss-type proximal point algorithm, which guarantees the existence and the convergence of any sequence generated by the algorithm under mild conditions. More precisely, semilocal and local convergence of the Gauss-type proximal point method are analyzed when T is metrically regular. We also study the stability properties of this algorithm.
引文
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