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.     

Representations of Affine Multifunctions by Affine Selections

The paper deals with affine selections of affine (both convex and concave) multifunctions acting between finite-dimensional real normed spaces. It is proved that each affine multifunction with compact values possesses an exhaustive family of affine selections and, consequently, can be represented by its affine selections. Moreover, a convex multifunction with compact values possesses an exhaustive family of affine selections if and only if it is affine. Thus the existence of an exhaustive family of affine selections is the characteristic feature of affine multifunctions which differs them from other convex multifunctions with compact values. Besides a necessary and sufficient condition for a concave multifunction to be affine on a given convex subset is also proved. Finally it is shown that each affine multifunction with compact values can be represented as the closed convex hull of its exposed affine selections and as the convex hull of its extreme affine selections. These statements extend the Straszewicz theorem and the Krein–Milman theorem to affine multifunctions. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

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.     

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.     

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.     

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.     

Calmness of efficient solution maps in parametric vector optimization

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.     

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.     

Maximal Monotone Multifunctions of Brøndsted–Rockafellar Type

We consider whether the “inequality-splitting” property established in the Brøndsted–Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions form a subclass of Gossez"s maximal monotone multifunctions of type (D); however, in every case where it has been proved that a multifunction is maximal monotone of type (D) then it is also of type (ED). Specifically, the following maximal monotone multifunctions are of type (ED): ? ultramaximal monotone multifunctions, which occur in the study of certain nonlinear elliptic functional equations; ? single-valued linear operators that are maximal monotone of type (D); ? subdifferentials of proper convex lower semicontinuous functions; ? “subdifferentials” of certain saddle-functions. We discuss the negative alignment set of a maximal monotone multifunction of type (ED) with respect to a point not in its graph – a mysterious continuous curve without end-points lying in the interior of the first quadrant of the plane. We deduce new inequality-splitting properties of subdifferentials, almost giving a substantial generalization of the original Brøndsted–Rockafellar theorem. We develop some mathematical infrastructure, some specific to multifunctions, some with possible applications to other areas of nonlinear analysis: ? the formula for the biconjugate of the pointwise maximum of a finite set of convex functions – in a situation where the “obvious” formula for the conjugate fails; ? a new topology on the bidual of a Banach space – in some respects, quite well behaved, but in other respects, quite pathological; ? an existence theorem for bounded linear functionals – unusual in that it does not assume the existence of any a priori bound; ? the 'big convexification" of a multifunction.

     

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.     

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.     

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.     

Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization

This paper investigates vector optimization problems with objective and the constraints are multifunctions. By using a special scalarization function introduced in optimization by Hiriart-Urruty, we establish optimality conditions in terms of Lagrange-Fritz-John and Lagrange-Kuhn-Tucker multipliers. When all the data of the problem are subconvexlike we derive the results by Li, and hence those of Lin and Corley. We also show how the generalized Moreau-Rockafellar type theorem to multifunctions obtained recently by Lin can be derived from the well-known results in scalar optimization. In the last, vector optimization problem in which objective and the constraints are defined by multifunctions and depends on a parameter u, and the resulting value multifunction M(u) are considered. With the help of the generalized Moreau-Rockafellar type theorem we establish the weak subdifferential of M in terms of the weak subdifferential of objective and constraint multifunctions.     

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.     

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.     

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.     

Implicit multifunction theorems in complete metric spaces

In this paper, we establish some new characterizations of metric regularity of implicit multifunctions in complete metric spaces by using lower semicontinuous envelopes of the distance functions to set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the perturbation stability of implicit multifunctions.