Existence results for systems of strong implicit vector variational inequalities

The purpose of this paper is to introduce and study systems of strong implicit vector variational inequalities. Under suitable conditions, some existence results for systems of strong implicit vector variational inequalities are established by the Kakutani--Fan--Glicksberg fixed point theorem. This revised version was published online in June 2006 with corrections to the Cover Date.     

APPROXIMATION OF COMMON SOLUTIONS OF VARIATIONAL INEQUALITIES VIA STRICT PSEUDOCONTRACTIONS

In this paper, a convex feasibility problem is considered. We construct an iterative method to approximate a common element of the solution set of classical variational inequalities and of the fixed point set of a strict pseudocontraction. Strong convergence theorems for the common element are established in the framework of Hilbert spaces.     

广义强向量拟均衡问题组解的存在性

本文研究了一类集值广义强向量拟均衡问题组解的存在性问题.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Kakutani-Fan-Glicksberg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,建立了该类集值广义强向量拟均衡问题组解的存在性定理.所得结果推广了该领域的相关结果.     

关于一个非线性变分不等式系统的逼近可解性

本文研究了一个包含两个不同松弛强制映像的非线性变分不等式系统的逼近可解性. 利用数值和逼近的方法, 分析了一个新的迭代序列,改进和推广了近年来的一些最新结果.     

一般多值向量变分不等式问题

本文引入一般多值向量变分不等式问题（GMVVI）,这推广和统一了现有的向量变分不等式,文内还引入了弱C－伪单调映射和半连续映的概念,在弱C－伪单调性和半连续性的假设下,给出了（GMVVI）的广义线性化引理和解的存在定理,本文的结果即使对一般向量变分不等式问题和广义向量变分不等式问题也是全新的。     

AN ITERATIVE METHOD FOR SPLIT VARIATIONAL INCLUSION PROBLEM AND FIXED POINT PROBLEM FOR A FAMILY OF GENERALIZED ASYMPTOTICALLY NONEXPANSIVE SEMIGROUP

In this paper, strong convergence of an iterative sequence is proved, which computes an approximate solution of the set of solutions of split variational inclusion problem, the set of fixed points of a nonexpansive mapping and the set of common fixed points of a family of generalized asymptotically nonexpansive semigroup. Results obtained in this paper extend and unify the previously known results in the previous literatures.     

一类拟双线性型变分不等式解的存在性

本文引入和讨论了一类拟双线性型变分不等式解的存在性问题，其结果改进了文［４－６］中的相应结果．     

向量似变分不等式解的存在性及解集的稳定性

本文首先得到一类广义向量似变分不等式问题的解的存在性定理,然后利用usco映射的性质,讨论广义向量似变分不等式的解集的通有稳定性,得到大多数(在Baire分类意义下)广义向量似变分不等式问题的解集是稳定的;另外还引入广义向量似变分不等式解集的本质连通区的概念,并证明了满足一定连续性、凸性条件的广义向量似变分不等式的解集至少存在一个本质连通区.     

BOUNDARY ELEMENT APPROXIMATION OF THE SEMI-DISCRETE PARABOLIC VARIATIONAL INEQUALITIES OF THE SECOND KIND

The boundary element approximation of the parabolic variational inequalities of the second kind is discussed. First, the parabolic variational inequalities of the second kind can be reduced to an elliptic variational inequality by using semidiscretization and implicit method in time; then the existence and uniqueness for the solution of nonlinear non-differentiable mixed variational inequality is discussed. Its corresponding mixed boundary variational inequality and the existence and uniqueness of its solution are yielded. This provides the theoretical basis for using boundary element method to solve the mixed vuriational inequality.     

关于一类变分不等式的迭代算法

本文通过对「１」中相应结构的讨论与改进,利用变分不等式求解技巧,讨论了Ｈｉｌｂｅｒｔ空间中一类广义变分不等式的解的迭代算法及其收敛性,指出了文「１」中主要结果的证明的一些不足之处,本文所得结果推广并改进了「１」的相应结论。     

System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

A new class of locally finite continuous topological spaces （for short, locally FC-spaces） and a class of system of generalized vector quasi-equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces. These theorems improve, unify and generalize many known results in the literatures.     

解变分不等式的一种二次投影迭代算法

本文研究了一类新的求解伪单调变分不等式的二次投影迭代算法.利用Armijo型线性搜寻程序,建立了一类新的超平面,他们严格分离当前迭代点与变分不等式的解集.运用超平面的这种分离性质,在较弱的条件下证明了该算法生成的无穷序列是全局收敛的.数值实验证明该算法是有效的.     

A COMPARISON OF DIFFERENT CONTRACTION METHODS FOR MONOTONE VARIATIONAL INEQUALITIES

It is interesting to compare the efficiency of two methods when their computational loads in each iteration are equal. In this paper, two classes of contraction methods for monotone variational inequalities are studied in a unified framework. The methods of both classes can be viewed as prediction-correction methods, which generate the same test vector in the prediction step and adopt the same step-size rule in the correction step. The only difference is that they use different search directions. The computational loads of each iteration of the different classes are equal. Our analysis explains theoretically why one class of the contraction methods usually outperforms the other class. It is demonstrated that many known methods belong to these two classes of methods. Finally, the presented numerical results demonstrate the validity of our analysis.     

一般化凸空间上的截口定理和变分不等式定理

摘要：本文利用Brouwer不动点定理或已知的KKM型定理，得到一般化凸空间上的截口定理．讨论了变分不等式解的存在性问题。对文中的相应结论进行了一般化和改进。     

On Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, we study the solvability for a class of vector variational inequalities in reflexive Banach spaces. By using Brouwer fixed point theorem, we prove the solvability for this class of vector variational inequalities without monotonicity assumption. The solvability results for this class of vector variational inequalities with monotone mappings are also presented by using the KKM-Fan lemma This paper is dedicated to Professor Franco Giannessi for his 68th birthday     

非扩张半群的变分不等式的不动点解的粘性逼近方法

本文研究了非扩张半群的变分不等式的不动点解的迭代算法.利用变分不等式与不动点问题的解的关系,结合粘性逼近方法,建立了非扩张半群的不动点的两步迭代格式,证明了该方法所得到的迭代序列在一定条件下的强收敛性,并收敛于某变分不等式的唯一解.     

广义非线性变分包含的带误差的近似点算法

引入和研究了一类新的广义非线性变分包含．在Hilbert空间中利用与极大η-单调映象相联系的预解算子的性质，对新的广义非线性变分包含建立了一个新的寻求近似解的带误差的近似点算法，并证明了求近似解序列强收敛于精确解．其所得结果是近期相关结果的改进和推广．     

广义混合隐拟h变分不等式的解的存在性与算法

本文在Hilbert空间中,引入了一类广义混合隐拟h变分不等式.运用变分原理,给出了广义混合隐拟h变分不等式逼近解的迭代算法,证明了这类变分不等式解的存在性定理,同时,得到迭代序列的收敛性.并改进和推广了[6~8]一些已知结果.     

拓扑空间上的KKM定理,不动点定理,广义变分不等式

引进没有任何凸结构的拓扑空间上的广义R-KKM映射的定义并利用古典的KKM原理得到一般拓扑空间上的KKM型定理以及若干个变形结果,然后作为应用给出了非紧的拓扑空间上不动点定理和广义变分不等式解的存在性定理.     

关于一类变分不等式的LQP算法

本文研究一类ξ-单调的变分不等式问题.利用KKT条件将原问题转换为非线性互补问题(nonlinear complementarity problem,NCP)的方法,获得了基于logarithmic-quadratic proximal(LQP)的算法及其改进形式,推广了LQP算法的适用范围.