系统半变分不等式问题的适定性研究

发布时间：2018-08-02 16:07

【摘要】：半变分不等式代表着一类与Clake次微分算子有关的非线性包含问题,在非线性分析和非光滑分析理论框架下,半变分不等式已成为了一种强有力的数学模型,并被广泛的应用于单边接触,非凸半渗透,多层结构分层等力学问题以及石工工程,非线性摩擦接触等工程问题中。鉴于半变分不等式在各类实际问题中的广泛应用价值,上世纪90年代,半变分不等式问题便受到海内外各个领域的学者和专家的高度关注,并且获得了大量与半变分不等式理论相关的论文、专著等学术成果。适定性概念在最优化问题、变分不等式、均衡问题以及其相关问题领域的研究中起着至关重要的作用。它对相关问题的可解性、解的唯一性、稳定性以及算法研究等都有着重要的影响。系统半变分不等式和系统可分半变分不等式是变分不等式、半变分不等式问题的两类重要推广形式,它们在工程、力学、经济等领域都有着重要应用价值。在本篇文章中,我们通过提出相应包含问题的适定性概念,对系统半变分不等式问题以及系统可分半变分不等式问题的适定性进行了研究,同时给出了适定性的度量性质以及与适定性相关的一些等价结果。本文具体工作如下:首先,在非线性分析理论、单调算子理论以及非紧性理论的框架下,本文通过定义近似序列,提出了系统半变分不等问题的适定性概念。在一定的假设条件下,对给定集合????与????的性质以及这两个集合之间的相互关系进行了研究。基于集合????与????的性质,本文在第二章对系统半变分不等式问题适定性的度量性质进行了描述。由于系统半变分不等式问题可以看作一类涉及Clake次微分算子的系统包含问题,本文在第三章定义了系统包含问题的适定性,证明到了系统包含问题与相对应的系统半变分不等式问题适定性是相互等价的。最后,本文在第四章研究了系统可分半变分不等式问题及其适定性。而且,在不同的单调性假设条件下,我们得到了系统可分半变分不等式问题的强适定性和弱适定性分别与其解的存在唯一性的等价性结果。
[Abstract]:Semi-variational inequalities represent a class of nonlinear inclusion problems related to Clake subdifferential operators. In the framework of nonlinear analysis and non-smooth analysis theory, semi-variational inequalities have become a powerful mathematical model. It is widely used in mechanical problems such as unilateral contact, non-convex semi-permeation, multi-layer structure delamination and other engineering problems, such as stone engineering, nonlinear friction contact and so on. In view of the extensive application value of semi-variational inequalities in various practical problems, in the 1990s, scholars and experts in various fields at home and abroad paid close attention to the problems of semi-variational inequalities. A large number of papers and monographs related to the theory of semi-variational inequalities have been obtained. The concept of fitness plays an important role in the study of optimization problems, variational inequalities, equilibrium problems and their related problems. It has an important influence on the solvability, uniqueness, stability and algorithm research of the related problems. The semi-variational inequalities of systems and divisible semi-variational inequalities of systems are two important generalizations of variational inequalities and semi-variational inequalities. They have important application value in engineering, mechanics, economy and so on. In this paper, by introducing the concept of fitness for the corresponding inclusion problem, we study the fitness of the semi-variational inequality problem and the divisible semi-variational inequality problem of the system. At the same time, the metric properties of fitness and some equivalent results related to fitness are given. The main work of this paper is as follows: firstly, under the framework of nonlinear analysis theory, monotone operator theory and non-compactness theory, the concept of fitness for semi-variational inequality problems of systems is proposed by defining approximate sequences. Under certain assumptions, for a given set? With? And the relationship between the two sets is studied. Based on the set? With? In the second chapter, we describe the metric properties of the fitness of the semi-variational inequality problems. Since the semi-variational inequality problem of the system can be regarded as a class of inclusion problems involving Clake subdifferential operators, in Chapter 3, we define the fitness of the inclusion problem of the system. It is proved that the adequacy of the inclusion problem and the corresponding semi-variational inequality problem is equivalent to each other. Finally, in chapter 4, we study the divisible semi-variational inequality problem and its fitness. Furthermore, under different monotonicity assumptions, we obtain the equivalence results between the existence and uniqueness of strong and weak proper definiteness and solution for divisible semi-variational inequality problems.
【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O178

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