集值映射的连续选择与线性算子的齐性右逆

本文对于取值在Ｂａｎａｃｈ空间的几乎下半连续映射引入两个相关的下半连续闭凸集值映射,得到集值映射的连续选择存在性的若干特征,从而将 Ｄｅｕｔｓｃｈ Ｅ,和Ｋｅｎｄｅｒｏｖ Ｐ,Ｄｅ Ｂｌａｓｔ Ｆ．Ｓ．和 Ｍｙｊａｋ Ｊ,Ｐｒｚｅｓｌａｗｓｋｉ Ｋ．和 Ｒｙｂｉｎｓｋｉ Ｌ Ｅ,Ｇｕｔｅｖ Ｖ．等人以及作者自己的关于连续选择存在性的结果作为推论给出．并用这些结论讨论了在开映射定理不成立的。情况下从Ｂａｎａｃｈ空间到赋范空间上线性连续算子的齐性右逆存在问题．     

集值映射空间上的自然映射(英文)

In this paper,we discuss the continuities of some natural mappings on pointcompact continuous set-valued mapping spaces with compact-open topology and obtain the properties of set-valued injective mappings,set-valued diagonal mappings,induced mappings,set-valued evaluation mappings,set-valued topological sum mappings and set-valued topological product mappings.     

On the stable containment of two sets

This paper studies the stability of the set containment problem. Given two non-empty sets in the Euclidean space which are the solution sets of two systems of (possibly infinite) inequalities, the Farkas type results allow to decide whether one of the two sets is contained or not in the other one (which constitutes the so-called containment problem). In those situations where the data (i.e., the constraints) can be affected by some kind of perturbations, the problem consists of determining whether the relative position of the two sets is preserved by sufficiently small perturbations or not. This paper deals with this stability problem as a particular case of the maintaining of the relative position of the images of two set-valued mappings; first for general set-valued mappings and second for solution sets mappings of convex and linear systems. Thus the results in this paper could be useful in the postoptimal analysis of optimization problems with inclusion constraints.      

Generic Existence of Fixed Points for Set-Valued Mappings

We first consider a complete metric space of nonexpansive set-valued mappings acting on a closed convex subset of a Banach space with a nonempty interior, and show that a generic mapping in this space has a fixed point. We then establish analogous results for two complete metric spaces of set-valued mappings with convex graphs.     

完备凸度量空间中(次)相容和集值广义非扩张映象的公共不动点定理

本文在完备凸度量空间中,利用集值和单值映象(次)相容的一些条件,建立了数值广义非扩张映象存在公共不动点的一个充要条件和一个充分条件.我们的结果改进、扩充和发展了文[2～7]中的主要结果.     

凸度量空间中单值和集值映象的公共不动点定理

§ 1.　Introduction　　Anecessaryandsufficientconditionandasufficientconditiontoensurethataset valuedmappingandasingle valuedmappinginacompletemetricspaceandacompleteconvexmetricspacehavingacommonfixedpointarerespectivelygivenin [1 ] .Meanwhilethemainresultsin[2 ]— [5]areimprovedandextendedin [1 ] .Inthispaper ,theexistenceatcommonfixedpointsoftwoset valuedmappingandasing valuedmappingwerestudied ,andthecorrespond ingresultsin [1 ]— [5]wereextendedandimproved .Let(X ,d)beanon emptymetricspa…     

A generic approach to approximate efficiency and applications to vector optimization with set-valued maps

In this paper we focus on approximate minimal points of a set in Hausdorff locally convex spaces. Our aim is to develop a general framework from which it is possible to deduce important properties of these points by applying simple results. For this purpose we introduce a new concept of ε-efficient point based on set-valued mappings and we obtain existence results and properties on the behavior of these approximate efficient points when ε is fixed and by considering that ε tends to zero. Finally, the obtained results are applied to vector optimization problems with set-valued mappings.     

On the Stability of the Motzkin Representation of Closed Convex Sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple $\left( C,D\right) $ formed by a compact convex set C and a closed convex cone D its Minkowski sum C?+?D. The continuity properties of other related mappings are also analyzed.     

DC Optimization Approach to Metric Regularity of Convex Multifunctions with Applications to Infinite Systems

The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the difference of convex (DC) functions. In this way, we establish new formulas for calculating the exact regularity bound of closed and convex multifunctions and apply them to deriving explicit conditions ensuring well-posedness of infinite convex systems described by inequality and equality constraints.     

参数凸二次规划的线性稳定性

本文研究参数凸二次规划的最优解集的稳定性。首先给出参数数学规划的方向线性稳定的定义，然后利用集值映射的微分理论证明线性约束参数凸二次规划是线性稳定的。     

An optimization result for set-valued mappings and a stability property in vector problems with constraints

First, we prove an existence result relative to minimal points of set-valued mappings. Then, conditions about the upper and lower semicontinuity of constraint sets defined through set-valued mappings are given. Finally, a stability result relative to vector problems with abstract constraints is proved.The author thanks the referee for helpful comments on the first version of this paper.     

Exceptional family and solution existence of variational inequality problems with set-valued mappings

In this paper, we introduce a new exceptional family for a variational inequality with a set-valued mapping over a general unbounded closed convex set in a Hilbert space. By means of the exceptional family and topological degree theory of set-valued mappings, an alternative theorem and some solution existence theorems are obtained.     

Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

Minimax theorems for scalar set-valued mappings with nonconvex domains and applications

In this paper, by virtue of the separation theorem of convex sets, we prove a minimax theorem, a cone saddle point theorem and a Ky Fan minimax theorem for a scalar set-valued mapping under nonconvex assumptions of its domains, respectively. As applications, we obtain an existence result for the generalized vector equilibrium problem with a set-valued mapping. Simultaneously, we also obtain some generalized Ky Fan minimax theorems for set-valued mappings, in which the minimization and the maximization of set-valued mappings are taken in the sense of vector optimization.     

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.     

Uniform convexity and the splitting problem for selections

We continue to investigate cases when the Repovš–Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.     

Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces

In this article, we introduce the concept of a family of set-valued mappings generalized Knaster–Kuratowski–Mazurkiewicz (KKM) w.r.t. other family of set-valued mappings. We then prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and 𝒯 and 𝒮 are two families of self set-valued mappings of X such that 𝒮 is generalized KKM w.r.t. 𝒯, under some natural conditions, the set-valued mappings S ∈ 𝒮 have a fixed point. Other common fixed point theorems and minimax inequalities of Ky Fan type are obtained as applications.     

Generalized convex relations with applications to optimization and models of economic dynamics

We examine a notion of generalized convex set-valued mapping, extending the notions of a convex relation and a convex process. Under general conditions, we establish duality results for composite set-valued mappings and for convex programming problems involving convex set-valued mappings. We also present applications to the study of economic dynamical systems, by obtaining the characteristics of optimal paths generated by convex processes, and to optimization problems of a certain class of positively homogeneous increasing functions.     

Implicit mapping theorem for extended metric regularity in metric spaces

In this paper we prove an extension of the contraction mapping principle for set-valued mappings stated by A.L. Dontchev and W.W. Hager dealing with more general assumptions containing modulus instead of pseudo-contractive multifunctions. Using this result we show that the update Graves theorem, in company with the stability of metric regularity under perturbations can be extended to a much broader framework of set-valued mappings acting in abstract spaces.     

The Derived Mappings and the Order of a Set-Valued Mapping Between Topological Spaces

Previous investigations of, in particular, continuous selections have led to the definition of the derived mappings and, here, the order of a set-valued mapping between topological spaces. The relation between the topological spaces and the possible orders of set-valued mappings between them is considered and examples are constructed to show that each ordinal number is the order of some set-valued mapping.