【作者】 王红；

【导师】 王晓敏；

【作者基本信息】 东北大学 ， 概率论与数理统计， 2009， 硕士

【摘要】 20世纪60年代,Hartman和Stampacchia创立了变分不等式理论。变分不等式理论是当今数学技术中的一个非常有力的研究工具,它在运筹学,概率论与数理统计中随机问题,计算机科学,系统科学,工程技术,交通,经济与管理等许多方面有广泛的应用,在二十世纪的最后二十年里,它受到许多学者的特别关注。本文全面系统地介绍了变分不等式理论发展的历史背景、研究现状及许多学者所做的主要工作。受这一领域近年来研究成果的启发,本文做了以下几方面的讨论：(1)应用辅助变分原理技巧求解了一类变分不等式,构建了一类新的计算近似解的迭代算法,证明了该算法产生的序列的收敛性。(2)讨论了一类变分包含解的存在性,我们建立了变分包含问题与其预解方程的等价性,提出了一种三步迭代算法,证明了该迭代算法的收敛性。(3)我们研究了Banach空间中一类增生型映象的集值变分包含解的存在性,得到了迭代序列强收敛于变分包含唯一解的若干等价条件。更多还原

【Abstract】 In the 1960s, Hartman and Stampacchia construct the variational inequality theory. It is well known that Variational inequality theory is powerful tools of the current mathematical technology. Variational inequalities have many important applications in operation research, the random problem of probability and statistics, computer science, system science, engineering technology, transportation, economics and management et. al. In the last 20 years of the twentieth century, they have been paid close attention by many scholars.In the paper, we show the real background of the problems we study and the main works that have been studied by many authors. Based on the results of research in this field in recent years, we discuss the following problems:First, the auxiliary variational principle technique is applied to solve a class of variational inequality. An innovative iterative algorithm to cumpute approximate solution is constructed. The convergence of iterative sequences generated by the algorithm is proved.Second, the existence of solution for a kind of variational inclusion is discussed. We establish the equivalence between the variational inclusion and the general resolvent equation. Give a three-step iterative algorithm for this variational inclusion and prove the convergence of this iterative algorithm.Third, we study the existence of solution for a class of set-valued variational inclusion with accretive type mappings in Banach spaces. Some equivalent conditions as the sequence of iterations strongly converges to the only solution for variation inclusion are obtained.更多还原