变分不等式与非线性算子方程的逼近

发布时间：2018-07-22 13:46

【摘要】：本学位论文主要研究无限维实Hilbert空间背景下几类新的变分不等式(系统)、不动点、非线性算子方程及它们的公共解问题。本文分别利用投影算子技巧、Wiener-Hopf方程组技巧、辅助原理技巧和强正有界线性算子技巧,改进了Mann迭代方法、混合方法、外逼近方法、外梯度方法和粘性迭代方法,构造了几种新的迭代算法,并证明了迭代序列的收敛性。其结果改进、推广与补充了之前文献中相应的结果。全文共分五章。第一章,介绍了变分不等式和非线性算子方程的研究背景,回顾了文中将要用到的一些基本概念和基本理论,并简述了本文的主要工作与结构安排。第二章,研究了一类新的广义集值非线性隐拟变分不等式系统问题。首先,证明了广义集值非线性隐拟变分不等式系统问题分别等价于相应的不动点问题和Wiener-Hopf方程组问题。其次,利用投影算子技巧和Wiener-Hopf方程组技巧,构造了两种新的迭代算法。最后,证明了广义集值非线性隐拟变分不等式系统问题解的存在性和迭代序列的收敛性。这是首次利用Wiener-Hopf方程组技巧求解变分不等式系统问题。通过比较这两种方法所得到的结果,表明Wiener-Hopf方程组技巧比投影算子技巧更一般。第三章,基于第二章,利用辅助原理技巧,研究了一类新的广义集值强非线性混合隐拟似变分不等式系统问题。首先,证明了相应的辅助变分不等式系统问题解的存在性。然后,利用这个解的存在性结果,构造了一个新的迭代算法。最后,证明了原问题解的存在性和迭代序列的收敛性。这是对Noor等(Korean J.Comput.Appl.Math.1998,1:73-89;J.Comput.Appl.Math.1993,47:285-312)提出的公开问题做的肯定的回答。第四章,为了寻找两族有限极大单调映射的公共零点集、非扩张映射的不动点集和具单调Lipschitz连续映射的变分不等式解集的公共元问题,介绍和研究了一种新的一般混合迭代算法。该算法基于以下列四种著名的方法:Mann迭代方法、混合方法、外逼近方法和外梯度方法。证明了公共元的存在性和迭代序列的强收敛性。本章的结果极大地推广和改进了文献[Wei L.,Tan R.L.Fixed Point Theory and Applications.2014,77(1)]中相应的结果。第五章,基于第四章的内容,首先利用强正线性有界算子技巧,构造了一种新的广义Mann型混合复合外梯度CQ迭代算法。然后利用这个算法,找到了具单调Lipschtiz连续映射的变分不等式的解集,两族有限极大单调映射的零点集,依中间意义渐近κ-严格伪压缩映射的不动点集的公共元。最后,证明了公共元的存在性和迭代序列的强收敛性。
[Abstract]:In this dissertation, we study some new variational inequalities (systems), fixed points, nonlinear operator equations and their common solutions in infinite dimensional real Hilbert spaces. In this paper, we improve Mann iterative method, mixed method, outer approximation method, outer gradient method and viscous iteration method by using projection operator technique, auxiliary principle technique and strongly positive bounded linear operator technique, respectively. Several new iterative algorithms are constructed and the convergence of iterative sequences is proved. The results are improved to extend and supplement the corresponding results in previous literatures. The full text is divided into five chapters. In the first chapter, we introduce the research background of variational inequality and nonlinear operator equation, review some basic concepts and theories that will be used in this paper, and introduce the main work and structure of this paper. In chapter 2, we study a new class of generalized set-valued nonlinear implicit quasi-variational inequality systems. Firstly, it is proved that the generalized set-valued nonlinear implicit quasi-variational inequality problems are equivalent to the corresponding fixed point problems and Wiener-Hopf equations respectively. Secondly, two new iterative algorithms are constructed by using the technique of projection operator and Wiener-Hopf equations. Finally, the existence of solutions and the convergence of iterative sequences for generalized set-valued nonlinear implicit quasi-variational inequality systems are proved. This is the first time that Wiener-Hopf equations are used to solve variational inequality system problems. By comparing the results obtained by the two methods, it is shown that the technique of Wiener-Hopf equations is more general than that of projection operator. In chapter 3, based on the second chapter, a new class of generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequality systems is studied by using the auxiliary principle technique. Firstly, the existence of solutions for the corresponding auxiliary variational inequality systems is proved. Then, a new iterative algorithm is constructed by using the existence of the solution. Finally, the existence of the solution and the convergence of the iterative sequence are proved. This is a yes answer to a public question raised by Noor et al. (Korean J.Comput.Appl.Math.199881: 73-89 J.Comput.appl.Math.1993 47: 285-312). In chapter 4, in order to find the common zero set of two families of finite maximal monotone mappings, the fixed point sets of nonexpansive mappings and the common element problems of solutions of variational inequalities with monotone Lipschitz continuous mappings, A new general hybrid iterative algorithm is introduced and studied. The algorithm is based on the following four famous methods: the: Mann iteration method, the hybrid method, the external approximation method and the outer gradient method. The existence of common elements and the strong convergence of iterative sequences are proved. The results of this chapter greatly extend and improve the corresponding results in [Wei L. Li Tan R. fixed Point Theory and applications s.2014 (1)]. In chapter 5, based on the content of chapter 4, a new generalized Mann type hybrid compound outer gradient CQ iterative algorithm is constructed by using the technique of strongly positive linear bounded operator. Then the solution set of variational inequalities with monotone Lipschtiz continuous mapping, the set of zeros of two families of finite maximal monotone mappings, and the common elements of fixed point sets of asymptotically 魏 -strictly pseudocontractive mappings with intermediate meaning are found. Finally, the existence of common elements and the strong convergence of iterative sequences are proved.
【学位授予单位】：上海师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177.91

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