一些随机重合点定理

本文引入了一种满足更一般的收缩不等式的多重函数类,并证明了属于该类的可测多重函数对的一些随机重合点定理。     

Random fixed point theorems for composites of acyclic multifunctions

In this paper we obtain random versions of Kakutani–Fan type fixed point theorems for a class V _c ⁺ of multifunctions which contains Kakutani factorizable maps and composites of acyclic maps. As applications, we derive some random approximation theorems.     

Upper and lower pre-strong na continuous multifunctions

The aim of this paper is to introduce a new class of multifunctions namely pre-strong na continuous multifunctions and to obtain some characterizations and properties of it.     

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.     

On the transitivity of multifunctions and density of orbits in generalized topological spaces

The connection between transitivity and existence of a dense orbit for multifunctions $\phi \: X\multimap X$ in generalized topological spaces is studied. Moreover strongly transitive multifunctions and functions in generalized topological spaces are investigated.     

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.     

Toward A geometric theory of minimax equalities

We present new ideas and concepts in minimax equalities. Two important classes of multifunctions will be singled out, the Weak Passy-Prisman multifunctions and multifunctions possessing the finite simplex property. To each class of multifunctions corresponds a class of functions. We obtain necessary and sufficient conditions for a multifunction to have the finite intersection property, and necessary and sufficient conditions for a function to be a minimax function. All our results specialize to sharp improvements of known theorems, Sion, Tuy, Passy-Prisman, Flåm-Greco. One feature of our approach is that no topology is required on the space of the maximization variable. In a previous paper [6] we presented a “method of reconstruction of polytopes” from a given family of subsets, this in turn lead to a “principle of reconstruction of convex sets” Theorem 3, which plays a major role in this paper. Our intersection theorems bear no obvious relationship to other results of the same kind, like K.K.M. or other more elementary approaches based on connectedness. We conclude our work with a remark on the role of upper and lower semicontinuous regularization in mimmax equalities     

Integral representation of random set-valued measures

Abstract

We consider random set-valued measures with values in a separable Banach space. We prove two integral representation theorems using measurable multifunctions and set-valued integrals. The first theorem is valid for all separable Banach spaces, while the second holds for reflexive separable Banach spaces.     

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.     

Some extensions of fan's best approximation theorem

A well-known Ky Fan's best approximation theorem which has been of great importance in nonlinear analysis, game theory, and minimax theorems is extended to a class of factorizable multifunctions.     

Variational Analysis of Paraconvex Multifunctions

Journal of Optimization Theory and Applications - Our aim in this article is to study the class of so-called $$\rho -$$ paraconvex multifunctions from a Banach space X into the subsets of another...     

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.     

Komlós Theorem for Unbounded Random Sets

We present the Komlós theorem for multivalued functions whose values are closed (possibly unbounded) convex subsets of a separable Banach space. Komlós theorem can be seen as a generalization of the SLLN for it deals with a sequence of integrable multivalued functions that do not have to be identically distributed nor independent. The Artstein–Hart SLLN for random sets with values in Euclidean spaces is derived from the main result. Finally, since the main theorem concerns multifunctions whose values are allowed to be unbounded, we can restate it in terms of normal integrands (random lower semicontinuous functions).     

On the existence of solutions for random differential inclusions in a Banach space

We prove two existence theorems for random differential inclusions defined in a separable Banach space. One is about differential inclusions defined on all of the Banach space X and the other for differential inclusion defined on a closed convex subset K. Both theorems are proved through the use of analogous deterministic results, which we also include, and techniques from the theory of measurable multifunctions.     

Nonconvex differential calculus for infinite-dimensional multifunctions

The paper is concerned with generalized differentiation of set-valued mappings between Banach spaces. Our basic object is the so-called coderivative of multifunctions that was introduced earlier by the first author and has had a number of useful applications to nonlinear analysis, optimization, and control. This coderivative is a nonconvex-valued mapping which is related to sequential limits of Fréchet-like graphical normals but is not dual to any tangentially generated derivative of multifunctions. Using a variational approach, we develop a full calculus for the coderivative in the framework of Asplund spaces. The latter class is sufficiently broad and convenient for many important applications. Some useful calculus results are also obtained in general Banach spaces.This research was partially supported by the National Science Foundation under grants DMS-9206989 and DMS-9404128, by the USA-Israel grant 94-00237, and by the NATO contract CRG-950360.     

An Extension of the Birkhoff Integrability for Multifunctions

A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.     

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.     

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.     

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.     

On generalized cluster sets of functions and multifunctions

This paper introduces and studies generalized cluster sets (g-cluster sets) of functions and multifunctions on GTS, which unifies the existing notions of cluster sets, θ-cluster sets, δ-cluster sets, S-cluster sets, s-cluster sets, p-cluster sets and many more. Several properties of the functions and multifunctions as well as their range and domain spaces are observed via degeneracies of their g-cluster sets. Characterizations of g-cluster sets through filterbases and grills on a typical class of GTS’s are also obtained. Moreover, μ-compactness of a GTS is characterized through g-cluster sets of multifunctions.