Selections for quasi-l.s.c. mappings with uniformly equi-LCn range

Every quasi-lower semi-continuous (q.l.s.c.) mapping admits a lower semi-continuous (l.s.c.) selection preserving all important (from the selection point of view) properties of the former mapping. Special-type extensions of l.s.c. mappings are established on this base.     

Collectionwise normality and selections into Hilbert spaces

A T₁-space X is countably paracompact and collectionwise normal if and only if every l.s.c. mapping from X into a Hilbert space with closed and convex point-images has a continuous selection. This settles a conjecture posed by M. Choban, V. Gutev and S. Nedev [M. Choban, S. Nedev, Continuous selections for mappings with generalized ordered domain, Math. Balkanica (N.S.) 11 (1-2) (1997) 87-95].     

Continuous selections and reflexive Banach spaces

Every l.s.c. mapping from a space into the non-empty closed convex subsets of a reflexive Banach space admits a continuous selection provided it satisfies a ``weak' u.s.c. condition. This result partially generalizes some known selection theorems. Also, it is successful in solving a problem concerning the set of proper lower semi-continuous convex functions on a reflexive Banach space.

     

Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints

We study the quantitative stability of the solution sets, optimal value and M-stationary points of one stage stochastic mathematical programs with complementarity constraints when the underlying probability measure varies in some metric probability space. We show under moderate conditions that the optimal solution set mapping is upper semi-continuous and the optimal value function is Lipschitz continuous with respect to probability measure. We also show that the set of M-stationary points as a mapping is upper semi-continuous with respect to the variation of the probability measure. A particular focus is given to empirical probability measure approximation which is also known as sample average approximation (SAA). It is shown that optimal value and M-stationary points of SAA programs converge to their true counterparts with probability one (w.p.1.) at exponential rate as the sample size increases.     

Paraconvexity of the graphs of lipschitzian functions

The paper deals with the essentiality of the convexity condition in the well-known selection theorems due to E. Michael. It is proved that the convexity of the range of a lower semi-continuous multivalued mapping may be replaced by an essentially weaker assumption: each value of the multivalued mapping is a graph of a Lipschitzian function of n variables in a suitable coordinate system. Bibliography: 3 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 236–252, 1995.     

Convex and alpha-prestarlike subordination chains

In this paper we consider convex subordination chains (c.s.c.) and alpha-prestarlike subordination chains (α-p.s.c.) over (0,1] on the unit disc in the complex plane. We obtain sufficient conditions for a mapping f(z,t) to be an α-p.s.c. over (0,1], which generalize a well-known result of Ruscheweyh [St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l'Université de Montreal, 1982]. We also obtain sufficient conditions for injectivity for nonanalytic mappings on the unit disc, and give certain examples of α-c.s.c. over (0,1] on the unit disc.     

Orlicz空间中支撑映射的上(下)半连续点

该文给出了赋Ｏｒｌｉｃｚ范数的Ｏｒｌｉｃｚ空间Ｌ°Ｍ中的点是支撑映射的上（下）半连续点的充分必要条件；推出了支撑映射为上（下）半连续的判别准则．     

A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces

We introduce a hybrid proximal point algorithm and establish its strong convergence to a common solution of a proximal point of a lower semi-continuous mapping and a fixed point of a demicontractive mapping in the framework of a CAT(0) space. As applications of our new result, we solve variational inequality problems for these mappings on a Hilbert space. Illustrative example is given to validate theoretical result obtained herein.     

A polyhedral study of the semi-continuous knapsack problem

We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Such structure arises in a wide variety of important problems, e.g. blending and portfolio selection. We show how strong inequalities valid for the semi-continuous knapsack polyhedron can be derived and used in a branch-and-cut scheme for problems with semi-continuous variables. To demonstrate the effectiveness of these inequalities, which we call collectively semi-continuous cuts, we present computational results on real instances of the unit commitment problem, as well as on a number of randomly generated instances of linear programming with semi-continuous variables.     

Accretive operators which are always single-valued in normed spaces

Set-valued accretive operators in Banach spaces have been extensively studied for several decades. Our main purpose in this paper is to establish a quite revealing result that says that every set-valued lower semi-continuous accretive mapping defined on a normed space is, indeed, single-valued on the interior of its domain. No reference to the well-known Michael’s Selection Theorem is needed. This result is used to extend known theorems concerning the existence of zeros for such operators, as well as showing existence of solutions for variational inclusions.     

On decomposition of A-continuity

Summary The following two theorems are obtained. (1) A mapping fis A-continuous if and only if it is semi-continuous and C-continuous. (2) A mapping f is A-continuous if and only if it is spr-continuous and LC-continuous.     

Accretive operators which are always single-valued in normed spaces

Set-valued accretive operators in Banach spaces have been extensively studied for several decades. Our main purpose in this paper is to establish a quite revealing result that says that every set-valued lower semi-continuous accretive mapping defined on a normed space is, indeed, single-valued on the interior of its domain. No reference to the well-known Michael’s Selection Theorem is needed. This result is used to extend known theorems concerning the existence of zeros for such operators, as well as showing existence of solutions for variational inclusions.     

Stability Results for Ekeland's ε Variational Principle for Set-Valued Mappings

In this paper, we define the Mosco convergence and Kuratowski-Painleve (P.K.) convergence for set-valued mapping sequence F n . Under some conditions, we derive the following result If a set-valued mapping sequence F n , which are nonempty, compact valued, upper semicontinuous and uniformly bounded below, Mosco (or P.K.) converges to a set-valued mapping F , which is upper semicontinuous, nonempty, compact valued, then Q l >0, u >0, $\varepsilon / \lambda - {\rm ext}\, F := \{ \bar x \in X : (F(x) - \bar y + \varepsilon / \lambda \Vert x - \bar x \Vert e)$     

Boundary value problems for semi-continuous delayed differential inclusions on Riemannian manifolds

The existence of solutions to nonlinear second order boundary value problems for differential inclusions with time lags is investigated. The set-valued mappings generating the differential inclusions are semi-continuous with compact and convex values. It turns out that solutions exist for small time intervals and arbitrary time lags in the state variable. The proof relies on mapping degree methods.     

Selections, extensions and collectionwise normality

We demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well.     

Applications of Michael's selection theorems to fixed point theory

Applying some of Ernest Michael's selection theorems, from recent fixed point theorems on u.s.c. multimaps, we deduce generalizations of the classical Bolzano theorem, several fixed point theorems on multimaps defined on almost convex sets, almost fixed point theorems, coincidence theorems, and collectively fixed point theorems. These results are related mainly to Michael maps, that is, l.s.c. multimaps having nonempty closed convex values.     

抽象凸空间中集值映射的不动点定理及其应用

A new fixed point theorem and the selection property for upper semi-continuous set-valued mappings in abstract convexity space are established. As their applications the existence of Nash equilibrium for n-person non-cooperative generalized games is proved.     

On the continuity of the feasible set in extremally linear problems

The purpose of this paper is to investigate continuity properties of the feasible set in extremally linear problems. Feasible sets of such problems are subsets of Fⁿ (the n-fold cartesian product of a fully ordered group (F,º)) and are described by a finite or an infinite number of inequalities or equalities, which are linear with respect to the operations ? and º. Thereby, the operation ? is induced by the fully-order in F by setting x?y = y iff x ≤ y for x, y in F. In particular, we derive conditions for the upper- and lower-semi-continuity of the feasible-set-mapping Z and investigate the structure of certain parameter sets. Especially, we show that the compactness of the feasible set, resp. the condition that the closure of the set of the strict feasible points is the feasible set, is sufficient for the upper-semi-continuity (u.s.c), resp. the lower-semi-continuity (l.s.c.) of Z, but - unlike to semi-infinite linear optimization - is not necessary for the u.s.c, resp. l.s.c. If we restrict Z on a certain subset of the parameter set, these conditions are also necessary. This paper is a continuation of the author's work in [6] and [7].     

不确定条件下n人非合作博弈均衡点集的通有稳定性

基于经典博弈模型的Nash均衡点集的通有稳定性和具有不确定参数的n人非合作博弈均衡点的概念,探讨了具有不确定参数博弈的均衡点集的通有稳定性.参照Nash均衡点集稳定性的统一模式,构造了不确定博弈的问题空间和解空间,并证明了问题空间是一个完备度量空间,解映射是上半连续的,且解集是紧集（即usco（upper semicontinuous and compact-valued）映射）,得到不确定参数博弈模型的解集通有稳定性的相关结论.     

On the metric structure of non-Kähler complex surfaces

We give a characterization of a locally conformally K?hler (l.c.K.) metric with parallel Lee form on a compact complex surface. Using the Kodaira classification of surfaces, we classify the compact complex surfaces admitting such structures. This gives a classification of Sasakian structures on compact three-manifolds. A weak version of the above mentioned characterization leads to an explicit construction of l.c.K. metrics on all Hopf surfaces. We characterize the locally homogeneous l.c.K. metrics on geometric complex surfaces, and we prove that some Inoue surfaces do not admit any l.c.K. metric. Received: 23 July 1998 / Revised: 2 June 1999