Affine and eclipsing multifunctions

In this paper, we want to compare two classes of multifunctions which can be used as approximating multifunctions in differentiability theory: affine and eclipsing multifunctions. We show how the notion of eclipsing multifunctions is an extension of affine multifunctions, and what kinds of difficulties arise in this extension.     

Metric Regularity of the Sum of Multifunctions and Applications

The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported.     

On multifunctions

In this paper, we introduce and study γ-continuous multifunctions as a generalization of quasi-continuous multifunctions due to Popa in 1985 and precontinuous multifunctions due to Popa in 1988. Some characterizations and several properties concerning upper (lower) γ-continuous multifunctions are obtained. The relationships between upper (lower) γ-continuous multifunctions and some known concepts are also discussed.     

Uniform convergence on spaces of multifunctions

We study spaces of multifunctions with closed values, multifunctions with closed graphs, USCO multifunctions, minimal USCO multifunctions and the space of densely continuous forms as metric spaces, equipped with the topology of uniform convergence. We give conditions under which these metric spaces are complete.      

On upper and lower δ-precontinuous multifunctions

In this paper we introduce and study δ-precontinuous multifunctions as a generalization of precontinuous multifunctions due to Popa [Problemy Mat. 10 (1988) 9]. Some characterizations and several properties concerning upper (lower) δ-precontinuous multifunctions are obtained. The relationships between upper (lower) δ-precontinuous multifunctions and some known concepts are also discussed.     

Jensen measures and analytic multifunctions

This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowski's theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions. The author was partially supported by an NSF Grant.     

An application of continuous fuzzy multifunctions

In this paper, the upper and lower δ-continuous multifunctions in fuzzy setting have been presented as a strong form and an application of fuzzy continuous multifunctions. Certain characterizations and several properties of these fuzzy multifunctions along with their mutual relationships are obtained. Attempts are also made to correlate this new class with the corresponding known types of fuzzy multifunctions. Also, applicability of the above new concepts to superstrings and space time could be probably possible in the near future.     

伪凸集值映射的误差界

考虑了伪凸集值映射的误差界.证明了对于伪凸集值映射,局部误差界成立意味着整体误差界成立.通过相依导数,给出了伪凸集值映射存在误差界的一些等价叙述.     

THE NORMALITY OF ALGEBROID MULTIFUNCTIONS AND THEIR COEFFICIENT FUNCTIONS

In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent to their coefficient functions in some conditions.Furthermore, we obtain some new normality criteria for algebroid multifunctions families based on these results. We also provide some examples to expound that some restricted conditions of our main results are necessary.     

Evolutors and Invariance

This paper concerns invariance of multifunctions under evolutors and its characterization through tangency of evolutors to multifunctions. The theory is illustrated by its applications to identity evolutors and to evolutors generated by semilinear perturbed equations.     

Some topological problems in multifunctions theory

The problem of upper semicontinuity of graph-closed multifunctions is considered. Also, several recent results on extension of multifunctions are presented.     

Convergence of an iterative scheme for multifunctions

Recently, Reich and Zaslavski have studied a new inexact iterative scheme for fixed points of contractive and nonexpansive multifunctions. In this paper, we generalize some of their results to Suzuki-type multifunctions.     

Quasi-interiorly e-tangent cones to multifunctions

Tbe notion of quasi-interiorly e-tangent cones to multifunctions is considered. General properties as the sum of multifunctions, the composition of amultifunction with a mapping,... and the relationships with the e-tangent cones are studied.     

Evolutors and Invariance

This paper concerns invariance of multifunctions under evolutors and its characterization through tangency of evolutors to multifunctions. The theory is illustrated by its applications to identity evolutors and to evolutors generated by semilinear perturbed equations.     

Convexity of the optimal multifunctions and its consequences in vector optimization

Studies on cone-convexity of optimal multifunctions in vector optimization are given. Under the convexity assumptions we present conditions guaranteeing a continuous behaviour of the optimal multifunctions.     

A random fixed point theorem for multifunctions

A Class of multifunctions is introduced and a random fixed point theorem for pairs of measurable multifunctions belonging to this class is proved. The result is then used to study the existence of solutions for a class of random operator equations     

超凸空间上一类非扩张集值映象的性质

研究超凸空间上具有非空闭球交值的非扩张集值映象的不动点及近似不动点的存在性，以及其单值非扩张选择的唯一性问题.     

On an existence theorem for generalized quasi-variational inequalities

We obtain some characteristic properties of a subclass of multifunctions introduced by B. Ricceri and give a new proof for the result of P. Cubiotti on the existence of solutions to generalized quasi-variational inequalities involving multifunctions from the class.     

Lagrangian Duality for Minimization of Nonconvex Multifunctions

Two alternative type theorems for nearly convexlike or * quasiconvex multifunctions are presented. They are used to derive Lagrangian conditions and duality results for vector optimization problems when the objectives and the constraints are nearly convexlike or * quasiconvex multifunctions.     

Fixed points for better admissible multifunctions on proximity spaces

We set out a rigorous presentation of Park?s classes of admissible multifunctions and we obtain a fixed point theorem for better admissible multifunctions defined on a proximity space via the Samuel-Smirnov compactification.