几类变分不等式问题解的存在性和误差界

发布时间：2018-04-03 06:24

本文选题：广义集值向量似变分不等式　切入点：广义强向量拟均衡问题组　出处：《渤海大学》2017年硕士论文

【摘要】：变分不等式属均衡优化问题,在国民经济的诸多领域有重要应用.向量变分不等式、向量均衡问题是变分不等式的推广形式,其解的存在性研究是变分不等领域的研究热点问题,受到许多学者的特别关注.而误差界是研究变分不等式解映射稳定性和灵敏度分析的前提,间隙函数是获得变分不等式解的误差界的一个有效的办法.因此对几类变分不等式问题解的存在性和误差界的研究具有重要的理论意义和应用价值.本论文主要研究了广义集值向量似变分不等式、广义强向量拟均衡问题组解的存在性以及广义混合拟变分不等式的间隙函数和全局误差界.本论文所取得的主要结果概括如下:第二章主要考虑Banach空间中两类广义集值向量似变分不等式问题.利用KKM定理,在集合的紧性和凸性条件下,并利用集值映射的下半连续性和映射的仿射性条件得到了两类广义集值向量似变分不等式解的存在性结果.第三章利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义和Glicksberg-Fan-Kakutani不动点定理,在不要求锥C的对偶锥C*具有弱*紧基的情况下,建立了集值广义强向量拟均衡问题组解的存在性定理.第四章在Hilbert空间中引入了一类新的广义混合拟变分不等式利用引入的正则间隙函数和D-间隙函数,在所涉及映射的强混合单调性和Lipschitz连续性等条件下,得到了广义混合拟变分不等式的全局误差界结果.
[Abstract]:Variational inequalities are equilibrium optimization problems and have important applications in many fields of national economy.Vector variational inequalities and vector equilibrium problems are the generalization of variational inequalities. The existence of their solutions is a hot issue in the field of variational inequality, which has been paid special attention to by many scholars.The error bound is the premise to study the mapping stability and sensitivity analysis of variational inequality solution. The gap function is an effective method to obtain the error bound of variational inequality solution.Therefore, the study on the existence of solutions and error bounds for some variational inequality problems has important theoretical significance and application value.In this paper, we study the existence of solutions for generalized set-valued vector-like variational inequalities, generalized strong vector quasi-equilibrium problems, and the gap functions and global error bounds of generalized mixed quasi-variational inequalities.The main results obtained in this paper are summarized as follows: in chapter 2, we consider two classes of generalized set-valued vector variational-like inequalities in Banach spaces.By using KKM theorem, under the conditions of compactness and convexity of sets, and by using the lower semicontinuity of set-valued mappings and affine conditions of mappings, the existence of solutions of two classes of generalized set-valued vector variational-like inequalities is obtained.In chapter 3, by using the definition of natural quasi C convexity of set-valued mappings and the lower Ca-continuity of set-valued mappings and the Glicksberg-Fan-Kakutani fixed point theorem, the dual cone C * of cone C is not required to have a weak * compact basis.The existence theorems of solutions for set-valued generalized strong vector quasi equilibrium problems are established.In chapter 4, we introduce a new class of generalized mixed quasi-variational inequalities in Hilbert spaces under the conditions of strong mixed monotonicity and Lipschitz continuity of the mappings, using the regular gap function and the D-gap function.The global error bounds for generalized mixed quasi-variational inequalities are obtained.
【学位授予单位】：渤海大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O178

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