Strict Feasibility for Generalized Mixed Variational Inequality in Reflexive Banach Spaces

The purpose of this paper is to investigate the nonemptiness and boundedness of solution set for a generalized mixed variational inequality with strict feasibility in reflexive Banach spaces. A concept of strict feasibility for the generalized mixed variational inequality is introduced, which recovers the existing concepts of strict feasibility for variational inequalities and complementarity problems. By using the equivalence characterization of nonemptiness and boundedness of the solution set for the generalized mixed variational inequality due to Zhong and Huang (J. Optim. Theory Appl. 147:454–472, 2010), it is proved that the generalized mixed variational inequality problem has a nonempty bounded solution set is equivalent to its strict feasibility.     

反Banach空间内涉及广义混合似臣分不等式问题的双水平广义混合平衡问题

在自反Banach空间内,引入和研究了一类新的涉及广义混合似变分不等式问题的双水平广义混合平衡问题(BGMEP).首先,为了计算BGMEP的近似解,引入了一类辅助广义混合平衡问题(AGMEP).由使用一极小极大不等式,在没有任何强制条件的相当温和假设下,证明了AGMEP解的存在性和唯一性.利用辅助原理技巧,建议和分析了一类计算BGMEP的近似解的新迭代算法.在没有任何强制条件的相当温和假设下,证明了由算法生成的迭代序列的强收敛性.这些结果是新的并且推广了这一领域内某些最近结果.     

Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).     

Lower Semicontinuity of the Solution Map to a Parametric Generalized Variational Inequality in Reflexive Banach Spaces

This paper is concerned with the study of solution stability of a parametric generalized variational inequality in reflexive Banach spaces. Under the requirements that the operator of a unperturbed problem is of class (S)_?+? and operators under consideration are pseudo-monotone and demicontinuous, we show that the solution map of a parametric generalized variational inequality is lower semicontinuous. The obtained results are proved without conditions related to the degree theory and the metric projection.     

Quasi-Strict Feasibility of Generalized Mixed Variational Inequalities in Reflexive Banach Spaces

In this paper, quasi-strict feasibility of a generalized mixed variational inequality as a new notation is introduced, which is weaker than its strict feasibility and recovers the existing concept of strict feasibility for a generalized variational inequality. By using the equivalent characterization of the nonemptiness and boundedness of the solution set for the generalized mixed variational inequality, it is proved that quasi-strict feasibility is a sufficient condition for the generalized mixed variational inequality with a f-pseudomonotone and upper hemicontinuous mapping to have a nonempty and bounded solution set in reflexive Banach spaces. Our results generalize and extend some known results in Zhong and Huang (J Optim Theory Appl 152(3):696–709, 2012).     

自反Banach空间上的半强制单调变分问题

仅用变分法的经典定理，研究了自反Ｂａｎａｃｈ空间上的半强制单调变分问题．获得了自反Ｂａｎａｃｈ空间上半线性方程解的存在性     

Existence of Solutions of Generalized Vector Variational Inequalities in Reflexive Banach Spaces

The purpose of this paper is to study the solvability for a class of generalized vector variational inequalities in reflexive Banach spaces. Utilizing the KKM-Fan lemma and the Nadler’s result, we prove the solvability results for this class of generalized vector variational inequalities for monotone vector multifuctions. On the other hand, we first introduce the concepts of complete semicontinuity and strong semicontinuity for vector multifunctions. Then we prove the solvability for this class of generalized vector variational inequalities without monotonicity assumption by using these concepts and by applying the Brouwer fixed point theorem. The results in this paper are extension and improvement of the corresponding results in Huang and Fang (2006).     

自反Banach空间内一类新的广义混合平衡问题组的辅助原理和逼近可解性

在自反Banach空间内引入和研究了一类新的涉及广义混合似变分不等式问题的广义混合平衡问题组(SGMEP)．首先,为了求解 SGMEP,引入了一类辅助广义混合平衡问题组(SAGMEP)．在没有任何强制条件的相当温和假设下, 对SAGMEP证明了解的存在性和唯一性．其次, 利用辅助原理技巧,对求解SGMEP建议和分析了一类新的迭代算法．最后,在没有任何强制条件的相当温和假设下,证明了由算法生成的迭代序列的强收敛性．这些结果改进、统一和推广了这一领域内某些最近结果．     

Solvability of Variational Inequality Problems

This paper presents the new concept of exceptional family of elements for the variational inequality problem with a continuous function over a general unbounded closed convex set. We establish a characterization theorem that can be used to derive several new existence and compactness conditions on the solution set. Our findings generalize well-known results for various types of variational inequality problems. For a pseudomonotone variational inequality problem, our new existence conditions are both sufficient and necessary.     

Banach空间中广义混合平衡问题,变分不等式问题和不动点问题的混杂投影方法

在Banach空间中,一个新的混杂投影迭代程序被引入来逼近广义混合平衡问题解集,变分不等式问题解集和一个相对弱非扩张映射的不动点集的公共元．所得结果改进和推广了最近一些文献的相应结果．     

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     

自反Banach空间中的相补问题

在自反Banach空间中证明了相补问题解的存在性定理，改进了[1]中的一个主要结果．     

Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces

In this paper, we introduce a localized version of generalized normal maps as well as generalized natural mappings. By using these concepts, we study continuity properties of the solution map of parametric variational inequalities in reflexive Banach spaces. This localization permits us to deal with variational conditions posed on sets that may not be convex and to establish existence and continuity of solutions. We also establish homeomorhisms between the solution set of variational inequalities and the solution set of generalized normal maps. Using these homeomorphisms and the degree theory, we show that the solution map of parametric variational inequalities is lower semicontinuous. Our results extend some results of Robinson (Set-Valued Anal 12:259–274, 2004). The authors wish to express their sincere appreciation to Professor Stephen M. Robinson, Department of Industrial and Systems Engineering, University of Wisconsin-Madison, for his valuable comments and suggestions. This research was partially supported by a grant from National Science Council of Taiwan, ROC.     

Strict Feasibility of Variational Inequalities in Reflexive Banach Spaces

Strict feasibility is proved to be an equivalent characterization of （dual） variational inequalities having a nonempty bounded solution set, provided the mappings involved are stably properly quasimonotone. This generalizes an earlier result from finite-dimensional Euclidean spaces to infinitedimensional reflexive Banach spaces. Moreover, the monotonicity-type assumptions are also mildly relaxed.     

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.     

Slice-Continuous Sets in Reflexive Banach Spaces: Some Complements

In this paper we study a class of closed convex sets introduced recently by Ernst et al. (J Funct Anal 223:179–203, 2005) and called by these authors slice-continuous sets. This class, which plays an important role in the strong separation of convex sets, coincides in ℝ ⁿ with the well known class of continuous sets defined by Gale and Klee in the 1960s. In this article we achieve, in the setting of reflexive Banach spaces, two new characterizations of slice-continuous sets, similar to those provided for continuous sets in ℝ ⁿ by Gale and Klee. Thus, we prove that a slice-continuous set is precisely a closed and convex set which does not possess neither boundary rays, nor flat asymptotes of any dimension. Moreover, a slice-continuous set may also be characterized as being a closed and convex set of non-void interior for which the support function is continuous except at the origin. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

On the Index of Solvability for Variational Inequalities in Banach Spaces

We define the index of solvability, a topological characteristic, whose difference from zero provides the existence of a solution for variational inequalities of Stampacchia’s type with S ⁺-type and pseudo-monotone multimaps on reflexive separable Banach spaces. Some applications to a minimization problem and to a problem of economical dynamics are presented. The work is supported by the Russian FBR Grants 05-01-00100 and 07-01-00137 and by the NATO Grant ICS.NR.CLG 981757.     

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.     

Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, a key assumption similar to that of Li and Chen is introduced by virtue of a gap function for a class of parametric set-valued weak vector variational inequalities in Banach spaces. By using this key assumption, sufficient and necessary conditions of the continuity and Hausdorff continuity of the solution set mapping for such parametric set-valued weak vector variational inequalities are given in Banach spaces when the image space is infinite dimensional. The results presented in this paper generalize and improve some main results of Li and Chen.     

Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, we study the solution stability of parametric weak Vector Variational Inequalities with set-valued and single-valued mappings, respectively. We obtain the lower semicontinuity of the solution mapping for the parametric set-valued weak Vector Variational Inequality with strictly C-pseudomapping in reflexive Banach spaces. Moreover, under some requirements that the mapping satisfies the degree conditions, we establish the lower semicontinuity of the solution mapping for a parametric single-valued weak Vector Variational Inequality in reflexive Banach spaces, by using the degree-theoretic approach. The results presented in this paper improve and extend some known results due to Kien and Yao (Set-Valued Anal. 16:399–412, 2008) and Wong (J. Glob. Optim. 46:435–446, 2010).