一类系统半变分不等式可解性问题研究

发布时间：2018-09-11 08:47

【摘要】：变分不等式理论是研究非线性边值问题的重要手段之一,经过几十年的发展,其在接触力学、最优控制、流体力学、工程管理以及经济均衡等学科中得到广泛的应用。得益于实际的问题模型对理论及方法不断提出新的要求,关于变分不等式理论的数学研究也越来越丰富而深刻,这其中就包括半变分不等式问题。半变分不等式是变分不等式结合非光滑分析的一类直接推广与发展,同时也是一类特殊的包含问题。从理论角度而言,由于集值映射被直接引入到问题形式中,因而在研究方法上,区别于传统研究变分不等式的方法体系,我们需要借助集值分析的知识,在非线性包含的研究框架下来研究相应的半变分不等式及其系统形式的问题。从应用角度而言,当对自由边值问题使用变分方法时,在能量泛函具有凸性的情况下,通常可以导出变分不等式,而能量泛函如果是非凸非光滑的,就有可能导出半变分不等式,因而,半变分不等式在研究涉及非凸非光滑能量泛函的实际问题中发挥着重要作用。本文研究了一个由一个二阶半变分不等式和一个一阶半变分不等式构成的半变分不等式系统,其可应用于一类具有热粘弹性的材料在非凸非光滑摩擦条件约束下的接触问题分析中。我们基于非线性分析及非光滑分析的框架研究了上述系统半变分不等式问题解的存在性和唯一性。全文由四章组成。第一章中,我们介绍了半变分不等式问题的研究背景和半变分不等式及其系统形式的研究现状。第二章中,我们列举了相关的预备知识,其中包括泛函分析、集值分析、非光滑分析等理论的相关内容以及关于非线性单调算子的一些结论。第三章中,我们在三重发展空间的框架下,定义相关算子,给出我们的问题形式;然后通过在乘积空间上应用伪单调算子方法和满射性定理先研究初值在连续嵌入子空间时的解的存在性;随后通过对算子做先验估计,利用收敛方法论证原始问题的可解性;最后基于一些额外假设如松弛单调等,讨论了使得解的唯一性成立的一种情况。第四章我们对论文工作加以总结并就相关问题的进一步工作提出设想。
[Abstract]:Variational inequality theory is one of the important methods to study nonlinear boundary value problems. After decades of development, it has been widely used in the fields of contact mechanics, optimal control, fluid mechanics, engineering management and economic equilibrium. Due to the new demands on theory and method from the practical problem model, the mathematical research on variational inequality theory is more and more abundant and profound, including the semi-variational inequality problem. The semi-variational inequality is a kind of direct generalization and development of variational inequality combined with nonsmooth analysis, and it is also a special inclusion problem. Theoretically speaking, because set-valued mapping is directly introduced into the form of the problem, so the research method is different from the traditional method system of studying variational inequalities, so we need to use the knowledge of set-valued analysis. The corresponding semi-variational inequalities and their system forms are studied under the framework of nonlinear inclusions. From the application point of view, when the variational method is used for the free boundary value problem, when the energy functional is convexity, the variational inequality can usually be derived, and the energy functional is nonconvex and nonsmooth. It is possible to derive semi-variational inequalities. Therefore, semi-variational inequalities play an important role in the study of practical problems involving nonconvex nonsmooth energy functional. In this paper, a semi-variational inequality system consisting of a second-order semi-variational inequality and a first-order semi-variational inequality is studied. It can be applied to the analysis of contact problems of a class of materials with thermoviscoelasticity under non-convex and non-smooth friction conditions. In this paper, we study the existence and uniqueness of solutions to the problems of semi-variational inequalities for these systems based on the framework of nonlinear analysis and non-smooth analysis. The full text consists of four chapters. In the first chapter, we introduce the research background of semi-variational inequality problems and the research status of semi-variational inequalities and their system forms. In the second chapter, we enumerate the relevant preparatory knowledge, including functional analysis, set-valued analysis, non-smooth analysis and some conclusions on nonlinear monotone operators. In the third chapter, we define the related operators under the framework of triple development space, and give the form of our problem. Then the existence of the solution of initial value in a continuous embedded subspace is studied by using pseudo-monotone operator method and surjectivity theorem in product space, and then the solvability of the original problem is proved by means of a priori estimate of the operator, and the convergence method is used to prove the solvability of the original problem. Finally, based on some additional assumptions such as relaxation monotone, we discuss a case where the uniqueness of the solution holds. In the fourth chapter, we summarize the work of the thesis and put forward some ideas for further work on the related issues.
【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177.91

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