集值变分不等式问题的例外簇

本文首先在Banach空间中证明了零调集值映射的一个Leray-Schauder型不动点定理,然后在Hilbert空间中定义了零调集值映射的变分不等式的例外簇,利用本文给出的不动点定理给出了无界集上的变分不等式问题存在解的一个充分条件．此条件弱于许多已知的关于变分不等式问题的解的存在性条件,并由此得到Hilbert空间中几个变分不等式约解的存在性定理．     

弱向量变分不等式的解集及其连通性

建立Hausdorff拓扑向量空间的非空凸子集到其值域为连续线性映射空间L(X, Y) 内的C -单调映射的弱向量变分不等式和它的纯量型变分不等式问题解的存在性, 讨论该弱向量变分不等式与之相联系的纯量型变分不等式解集的关系, 利用映射的C -弱次连续和C -单调性及其集值映射的不动点定理,通过纯量型变分不等式解集所诱导的集值映射所具有的特性给出弱向量变分不等式解集的连通性.     

Banach空间中变分不等式的例外簇与严格可行性问题

本文在Banach空间中对变分不等式的例外簇,严格可行性及解的存在性三者之间的关系进行了研究,将补问题中的相应结果进行了推广.首先通过定义一类新的例外簇,在映射为拟单调的情况下,证明了变分不等式解的存在性与例外簇之间的关系,即若变分不等式不存在例外簇,则解一定存在.其次主要证明了变分不等式的严格可行性与例外簇之间的关系,即若变分不等式存在严格可行性,则例外簇不存在.     

变分不等式问题的解的存在性

对一般凸集约束下的变分不等式问题提出了一个新的例外簇概念.基于此概念,给出了变分不等式问题解存在的一个充分条件,此条件弱于许多已知的关于变分不等式问题的解的存在性条件.对于伪单调变分不等式问题,它是解存在的充要条件.对于P₀非线性互补问题,利用例外簇的概念,给出了其解存在的充分条件.     

弱向量变分不等式解集的连通性

研究拓扑向量空间到连续线性映射空间映射的弱向量变分不等式和与之相关 的纯量型变分不等式解集的关系, 引入弱和强一致连续概念,利用纯量型变分不等式 解集所表征的集值映射的特性给出弱向量变分不等式解集连通的一个充分条件。     

广义投影下F隐变分不等式解的存在性

首先引进了广义(F,g)-投影算子,提出了一个新的例外簇概念,并运用Fan-KKM定理以及例外簇,分别在无单调性与渐近h-g伪单调假设下,研究了自反Banach空间中一类F隐变分不等式的可解性与解集特征问题,得到了新的解的存在性定理.     

一类积集上的广义向量变分不等式

研究一类积集上具某种权向量的广义向量变分不等式组及其广义向量变分不等式的有关问题,刻画它们之间解的相互关系.在映射的次连续性和关于某向量广义单调性的条件下,利用集值映射的不动点定理,对所讨论的几种类型的广义向量变分不等式给出解的存在性.     

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。     

一类混合似变分不等式问题解的存在唯一性

本文中,我们首先给出了一类混合似变分不等式问题．接着,在Banach空间中研究了它的解的存在性和唯一性．最后,讨论了混合似变分不等式问题的扰动问题,并证明了扰动问题的解的存在唯一性定理．     

向量变分不等式的严格可行性问题

本篇文章首先定义了向量变分不等式的严格可行点概念,其次在假设了映射是强(D)-伪单调的情况下,证明了向量变分不等式解集非空有界与其严格可行点存在的等价性问题,推广了在数量变分不等式上得到的相应结果．     

Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces

By employing the notion of exceptional family of elements, we establish some existence results for generalized variational inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability for the pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is strictly feasible.     

Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications

In this paper, we propose a new notion of ‘exceptional family of elements’ for convex optimization problems. By employing the notion of ‘exceptional family of elements’, we establish some existence results for convex optimization problem in reflexive Banach spaces. We show that the nonexistence of an exceptional family of elements is a sufficient and necessary condition for the solvability of the optimization problem. Furthermore, we establish several equivalent conditions for the solvability of convex optimization problems. As applications, the notion of ‘exceptional family of elements’ for convex optimization problems is applied to the constrained optimization problem and convex quadratic programming problem and some existence results for solutions of these problems are obtained.     

Existence results for solutions of mixed tensor variational inequalities

By employing the notion of exceptional family of elements, we establish existence results for the mixed tensor variational inequalities. We show that the nonexistence of an exceptional family of elements is a sufficient condition for the solvability of mixed tensor variational inequality. For positive semidefinite mixed tensor variational inequalities, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the nonemptiness of the solution sets. We derive several sufficient conditions of the nonemptiness and compactness of the solution sets for the mixed tensor variational inequalities with some special structured tensors. Finally, we show that the mixed tensor variational inequalities can be defined as a class of convex optimization problems.

     

Variational inequalities on weakly compact sets

In this paper, we derive an existence result for generalized variational inequalities associated with multivalued mappings on weakly compact sets under a continuity assumption which is much weaker than the regular complete continuity. As an application, we prove the existence of exceptional families of elements for such mappings on closed convex cones in reflexive Banach spaces when the corresponding complementarity problems have no solutions.     

Banach空间中k-次增生型变分包含解的存在与收敛性

在实自反Banach空间中,引入并研究一类k-次增生型变分包含问题,证明了这类变分包含解的存在与唯一性,并在去掉α_n→0,β_n→0(n→∝)以及序列{x_n)和{_η(g(x_n))}有界限制的条件下,建立了k-次增生型变分包含和变分不等式解的具有混合误差的多步迭代序列的强收敛性定理,给出了收敛率的估计式,从而改进和推广了前人的研究结果.     

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.     

Non-zero solutions for a class of generalized variational inequalities in reflexive Banach spaces

In this paper, we study the existence of nonzero solutions for a class of generalized variational inequalities involving set-contractive mappings by using the fixed point index approach in reflexive Banach spaces. Under some suitable assumptions, we show some new existence theorems of nonzero solutions for this class of generalized variational inequalities in reflexive Banach spaces.     

A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance

In this paper, using a hybrid extragradient method, we introduce a new iterative process for approximating a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. For this purpose, we introduce Bregman–Lipschitz-type condition for a pseudomonotone bifunction. It seems that these results for pseudomonotone bifunctions are first in reflexive Banach spaces. This paper concludes with certain applications, where we utilize our results to study the determination of a common point of the solution set of a variational inequality problem and the fixed point set of a finite family of multi-valued relatively nonexpansive mappings. A numerical example to support our main theorem will be exhibited.     

Exceptional Families and Existence Theorems for Variational Inequality Problems

This paper introduces the concept of exceptional family for nonlinear variational inequality problems. Among other things, we show that the nonexistence of an exceptional family is a sufficient condition for the existence of a solution to variational inequalities. This sufficient condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. From the results in this paper, we believe that the concept of exceptional families of variational inequalities provides a new powerful tool for the study of the existence theory for variational inequalities.