Continuous selections for set-valued mappings with finite-dimensional convex values

We show selection theorems for set-valued mappings with finite-dimensional convex values, one of which is a partial extension of a selection theorem due to S. Barov (2008) [1].     

Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.     

On C‐Suslin spaces

We prove a closed graph theorem for Baire locally convex spaces (for Baire linear topological spaces) in the domain and weakly C‐Suslin locally convex spaces (respectively, for C‐Suslin linear topological spaces) in the range which improves some classic closed graph theorems and other, more recent, related results.     

Some representations of midconvex set-valued functions

Summary In this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.     

On closedness assumptions in selection theorems

We begin by a short survey of various attempts in selection theory to avoid the closedness assumption for values of multivalued mappings. We collect special cases when Michael's Gδ-problem admits an affirmative solution and we prove some unified theorems of such type. We also show that in general this problem has a negative solution. In comparison with a recent result of Filippov, we work directly in the Hilbert cube rather than in the space of all probabilistic measures endowed with different topologies.     

Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space is an absolute retract if and only if X is an absolute retract of weight ?ω₁. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω₁ is homeomorphic to Iω₁. An analogous result is also proved for the cone over Iω₁. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.     

Mappings between inverse limits of continua with multivalued bonding functions

We investigate the limit mappings between inverse limits of continua with upper semi-continuous bonding functions. Results are obtained when the coordinate mappings are surjective, one-to-one or homeomorphisms. We construct examples showing the hypothesis of the theorems are essential. Further, we construct an example showing that, unlike for the inverse limits with single valued maps, properties of being monotone, confluent or weakly confluent mappings between factor spaces are not preserved in the inverse limit map.     

Fixed point theorems for partially outward mappings

We generalize some classical theorems related to dimension. We extend Brouwer's fixed point theorem to a class of mappings whose images are not necessarily a subset of the domain. These results also generalize theorems of B.R. Halpern and G.M. Bergman. As applications, we prove some theorems for maps that pull absolute retracts outward into attached sphere collars. We note relationships to the relative Nielsen theory and show that certain of our applications can also be obtained using results of H. Schirmer.     

On the topological entropy of continuous and almost continuous functions

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.     

Fixed points,coincidence points and maximal elements with applications to generalized equilibrium problems and minimax theory

In this paper, using a generalization of the Fan–Browder fixed point theorem, we obtain a new fixed point theorem for multivalued maps in generalized convex spaces from which we derive several coincidence theorems and existence theorems for maximal elements. Applications of these results to generalized equilibrium problems and minimax theory will be given in the last sections of the paper.     

Extending paired multifunctions

As a rule, the classical Michael-type selection theorems for the existence of single-valued selections are analogues and, in certain respects, generalisations of ordinary extension theorems. In contrast to this, the theorems for the existence of multi-selections deal with natural generalisations of cover properties of topological spaces. This paper continues the study of the latter problem, and its main purpose is to furnish a mapping characterisation of a cover-extension property—the so-called Katětov spaces.     

Multivalued generalized nonlinear contractive maps and fixed points

We introduce some notions of generalized nonlinear contractive maps and prove some fixed point results for such maps. Consequently, several known fixed point results are either improved or generalized including the corresponding recent fixed point results of Ciric [L.B. Ciric, Multivalued nonlinear contraction mappings, Nonlinear Anal. 71 (2009) 2716-2723], Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132-139], Feng and Liu [Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112] and Mizoguchi and Takahashi [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188].     

Generalized selection theorems without convexity

In this paper, we extend new selection theorems for almost lower semicontinuous multifunctions T on a paracompact topological space X to general nonconvex settings. On the basis of the Kim-Lee theorem and the Horvath selection theorem, we first show that any a.l.s.c. C-valued multifunction admits a continuous selection under a mild condition of a one-point extension property. Finally, we apply a fundamental selection theorem, due to Ben-El-Mechaiekh and Oudadess, to modify our selection theorems by adjusting a closed subset Z of X with its covering dimension dim_XZ≤0. The results derived here generalize and unify various earlier ones from classic continuous selection theory.     

Bead spaces and fixed point theorems

The notion of a bead metric space defined here (see Definition 6) is a nice generalization of that of the uniformly convex normed space. In turn, the idea of a central point for a mapping when combined with the “single central point” property of the bead spaces enables us to obtain strong and elegant extensions of the Browder-Göhde-Kirk fixed point theorem for nonexpansive mappings (see Theorems 14-17). Their proofs are based on a very simple reasoning. We also prove two theorems on continuous selections for metric and Hilbert spaces. They are followed by fixed point theorems of Schauder type. In the final part we obtain a result on nonempty intersection.     

Selections and topological convexity

A simple natural proof of van de Vel's selection theorem for topological convex structures is given. The technique developed to achieve this proof allows to give also a direct simple proof of the classical Michael's selection theorem in Fréchet spaces, and the Horvath's selection theorem in metric l.c.-spaces.     

Farkas-type Results for Max-functions and Applications

We present some Farkas-type results for inequality systems involving finitely many convex constraints as well as convex max-functions. Therefore we use the dual of a minmax optimization problem. The main theorem and its consequences allows us to establish, as particular instances, some set containment characterizations and to rediscover two famous theorems of the alternative.     

Fixed point theorems in fuzzy metric spaces using property E.A.

We prove two common fixed point theorems for a pair of weakly compatible maps in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani, by using E.A. property.     

Positional dimension-like functions of the type Ind

In Iliadis (2005) [4] positional dimension-like functions of the type ind are given. All these functions are studied only with respect to the property of universality. In a later paper by the present authors, and in two papers by V.V. Tkachuk (1981, 1982) (see [7] and [8]), these dimension-like functions have been studied with respect to the other standard properties of dimension theory. In R. Koga, Subspace-dimension with respect to total spaces, Master Thesis, Osaka Kyoiku University, 1998 (see also K.P. Hart, Jun-iti Nagata, J.E. Vaughan, Encyclopedia of General Topology, Elsevier Science Publishers, B.V., Amsterdam, 2004) a positional dimension-like function of the type Ind is given. Here we define new positional dimension-like functions of the type Ind, and present for all these functions, theorems concerning subspace theorems, partition theorems, sum theorems, and product theorems. Finally, we give some open questions concerning these functions.     

Existence theorems for usual and approximate solutions of optimal control problems

The theory of measurable set-valued mappings allows us to study some problems of optimal control in the framework of minimization of convex functionals and thus to obtain existence theorems. When the functionals are nonconvex, we obtain the existence theorems for control problems which are weakly perturbed from the initial one. In this regard, we specify some theorems of nonconvex optimization.     

没有开纤维的广义拟均衡问题(英文)

In this paper, we first prove some new selection and fixed point theorems in generalized convex spaces. Then, we establish some existence theorems of quasi-equilibrium and generalized quasi-equilibrium without the conditions of open fibers, by applying our selection and fixed point theorems.