Strict feasibility of pseudomonotone set-valued variational inequalities

In this article, we use degree theory developed in Kien et al. [B.T. Kien, M.-M. Wong, N.C. Wong, and J.C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res. 193 (2009), pp. 12–22.] to prove a result on the existence of solutions to set-valued variational inequality under a weak coercivity condition, provided that the set-valued mapping is upper semicontinuous with nonempty compact convex values. If the set-valued mapping is pseudomonotone in the sense of Karamardian and upper semicontinuous with nonempty compact convex values, it is shown that the set-valued variational inequality is strictly feasible if and only if its solution set is nonempty and bounded.     

Generalized first class selectors for upper semi-continuous set-valued maps in Banach spaces

In this paper we deal with weakly upper semi-continuous set-valued maps, taking arbitrary non-empty values, from a non-metric domain to a Banach space. We obtain selectors having the point of continuity property relative to the norm topology for a large class of compact spaces as a domain. Exact conditions under which the selector is of the first Borel class are also investigated.This paper was partially supported by DGICYT PB98-03981.     

Fixed-point theorems for set-valued mappings and existence of best approximants

The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlin's extension to normed spaces of Kakutani's Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space.     

Solvability of Langevin differential inclusions with set-valued diffusion terms on Riemannian manifolds

The existence of weak solution is proved for a Langevin type second-order stochastic differential inclusion on a complete Riemannian manifold, having both drift and diffusion terms set-valued. The construction of solution involves integral operators with Riemannian parallel translation and a special sequence of continuous ?-approximations for an upper semicontinuous set-valued mapping with convex bounded closed values, that is proved to converge point-wise to a Borel measurable selection.     

Exceptional family of elements for generalized variational inequalities

This paper introduces a new concept of exceptional family of elements for a finite-dimensional generalized variational inequality problem. Based on the topological degree theory of set-valued mappings, an alternative theorem is obtained which says that the generalized variational inequality has either a solution or an exceptional family of elements. As an application, we present a sufficient condition to ensure the existence of a solution to the variational inequality. The set-valued mapping is assumed to be upper semicontinuous with nonempty compact convex values.     

On the stability of closed-convex-valued mappings and the associated boundaries

This paper deals with the stability properties of those set-valued mappings from locally metrizable spaces to Euclidean spaces for which the images are the convex hull of their boundaries (i.e., they are closed convex sets not containing a halfspace). Examples of this class of mappings are the feasible set and the optimal set of convex optimization problems, and the solution set of convex systems, when the data are subject to perturbations. More in detail, we associate with the given set-valued mapping its corresponding boundary mapping and we analyze the transmission of the stability properties (lower and upper semicontinuity, continuity and closedness) from the given mapping to its boundary and vice versa.     

Stampacchia variational inequality with weak convex mappings

ABSTRACT

In this paper, the concept of weak convex set-valued mapping is introduced and various conditions for a set-valued mapping to be weak convex are given. Then, existence theorems for the Stampacchia variational inequality problem are established, when the involved mapping is weak convex.     

Regularity estimates for convex multifunctions

The main result of the paper contains an exact formula for the rate of regularity of a set-valued mapping with a convex graph. As a consequence we find an exact expression for the rate of regularity of set-valued mappings associated with so-called constraint systems. It turns out that the rate is equal to the upper bound of Robinson-type estimates over all norms in the graph space of the homogenized mapping majorizing the norm of the underlying space. We further introduce a concept of a perfectly regular mapping and find some criteria for perfect regularity.     

锥拟凸集值映射与集值映射的锥半连续性

本文借助锥界集定义了一种新的锥拟凸集值映射,并且利用广义Minkowski泛函讨论了锥半连续及锥外上半连续集值映射的锥拟凸性．     

锥上半连续锥拟凸集值映射多目标优化超有效解集的连通性

本文研究集值映射多目标优化超有效解集的连通性,在目标映射为锥上半连续和锥拟凸的条件下,证明了其超有效解集是连通的.     

Filippov–Pliss lemma and m-dissipative differential inclusions

In the paper we prove a variant of the well known Filippov–Pliss lemma for evolution inclusions given by multivalued perturbations of m-dissipative differential equations in Banach spaces with uniformly convex dual. The perturbations are assumed to be almost upper hemicontinuous with convex weakly compact values and to satisfy one-sided Peron condition. The result is then applied to prove the connectedness of the solution set of evolution inclusions without compactness and afterward the existence of attractor of autonomous evolution inclusion when the perturbations are one-sided Lipschitz with negative constant.     

Locally compact set-valued solutions for the general linear equation

Summary. We give a representation for the set-valued solutions, with closed convex and locally compact values, of a general linear equation as a sum of an additive mapping and a constant set.     

The Hemivariational Inequalities for an Upper Semicontinuous Set-valued Mapping

We establish some existence results for hemivariational inequalities of Stampacchia type involving an upper semicontinuous set-valued mapping on a bounded, closed and convex subset in ? ⁿ . We also derive a sufficient condition for the existence and boundedness of solution, without assuming boundedness of the constraint set.     

On upper semicontinuity of some set-valued iteration semigroups

We study an upper semicontinuity of set-valued iteration semigroups which are the counterparts of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.     

集值映射最优化问题超有效解集的连通性

本文在局部凸空间中对集值映射最优化问题引入超有效解的概念．首先研究了超 有效点的一些重要特性．其后证明了当目标函数为锥类凸的集值映射时，其目标空间里 的超有效点集是连通的；若目标函数为锥凸的集值映射时，其超有效解集也是连通的．     

Equilibrium for perturbations of multifunctions by convex processes

We present a general equilibrium theorem for the sum of an upper hemicontinuous convex-valued multifunction and a closed convex process defined on a noncompact subset of a normed space. The lack of compactness is compensated by inwardness conditions related to the existence of viable solutions of some differential inclusion.     

The Finite Dimensional Approximation Property and the AR-Property in Needle Point Spaces

We introduce the notion of the finite dimensional approximationproperty (the FDAP) and prove that if a subset X of a linearmetric space has the FDAP, then every non-empty convex subsetof X is an AR. As an application we show that every needle point space X containsa dense linear subspace E with the following properties: (i) E contains a non-empty compact convex set with no extremepoints; (ii) all non-empty convex subsets of E are AR.     

Adaptive full-order and reduced-order observers for the Lur’e differential inclusion system

This paper deals with the observer design for the Lur’e differential inclusion system with unknown parameters. The set-valued mapping in the differential inclusion is upper semi-continuous, closed, convex, bounded and monotone. First, under some assumptions an adaptive full-order observer is designed for the system. Then, under the same assumptions, a reduced-order observer is proved to exist. An example is provided to show the validation of the designed observers.     

Semigroups of Jensen set-valued functions

A set-valued function F defined on a convex cone S in a real vector space X into the set n(Y) of all non-empty subsets of a real vector space Y is said to be the Jensen function iff $${1\over 2}\lbrack F(x)+F(x)\rbrack=F\lbrack{1\over 2}(x+y)\rbrack$$ for all x, y ∈ S. This note deals with iteration semigroups of Jensen set-valued functions on S.     

Additive selections of superadditive set-valued functions

Summary A set-valued functionF from a coneC with a cone-basis of a topological vector spaceX into the family of all non-empty compact convex subsets of a locally convex spaceY is called superadditive provided thatF(x) + F(y) F(x + y), for allx, y C. We show that every superadditive set-valued function admits an additive selection.Dedicated to Professor Otto Haupt on his 100th birthday