Reduction principles in the theory of selections

In the present article many theorems about existence of continuous selections for a set-valued mapping into the metrizable spaces are extended to mappings into non-metrizable spaces.     

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.     

Selections for paraconvex-valued mappings on non-paracompact domains

We prove that Michael?s paraconvex-valued selection theorem for paracompact spaces remains true for C^′(E)-valued mappings defined on collectionwise normal spaces. Some possible generalisations are also given.     

Covering dimension d by a normal base

S.D. Iliadis introduced the concept of the dimension-like functions of type dim using the notion of a normal base. Since he considered this notion only from the point of view of the existence of universal spaces in different classes of spaces, in the present paper the basic theorems of dimension theory are obtained for these functions. Their relationship with the relative dimensions of A. Chigogidze and the uniform dimensions of M. Charalambous are shown.     

Coupled fixed points in partially ordered metric spaces and application

In this paper, we prove some coupled fixed point theorems for mappings having a mixed monotone property in partially ordered metric spaces. The main results of this paper are generalizations of the main results of Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379-1393]. As an application, we discuss the existence and uniqueness for a solution of a nonlinear integral equation.     

On coincidence point and fixed point theorems for nonlinear multivalued maps

Several characterizations of MT-functions are first given in this paper. Applying the characterizations of MT-functions, we establish some existence theorems for coincidence point and fixed point in complete metric spaces. From these results, we can obtain new generalizations of Berinde-Berinde?s fixed point theorem and Mizoguchi-Takahashi?s fixed point theorem for nonlinear multivalued contractive maps. Our results generalize and improve some main results in the literature.     

Three fixed point theorems for generalized contractions with constants in complete metric spaces

We prove three fixed point theorems for generalized contractions with constants in complete metric spaces, which are generalizations of very recent fixed point theorems due to Suzuki. We also raise one problem concerning the constants.     

DAS WALLMAN-VERFAHREN UND INVERSE LIMITES

Abstract

The relationship between Wallman's construction of a compact T₁-space [9] and Flachsmeyer's inverse limit spaces of inverse systems of decomposition spaces [2] is investigated. There are connections between Wallman spaces and inverse limits, which were initiated by Alexandroff in 1928. Some old theorems using inverse limits have shorter proofs now. On the other hand we obtain a new method to treat Wallman compactifications in terms of inverse limit spaces. A suitable notion in this context is the “prime-filter space”, having an interesting maximality property. This space seems to be proper to examine prime ideals in C(X).     

Closed images of spaces having g-functions

This paper deals with the study of closed images or quasi-perfect images of Nagata spaces, contraconvergent spaces, weak contraconvergent spaces, ks-spaces, γ-spaces and wγ-spaces and, of metrization theorems involving these spaces. We prove that the closed images of contraconvergent (weak contraconvergent) spaces are contraconvergent (weak contraconvergent) and that quasi-perfect images of γ- (wγ-)spaces are γ (wγ).     

Pointfree forms of Dowker’s and Michael’s insertion theorems

In this paper we prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space they are equivalent to the insertion statements of the classical theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. In addition, a study of perfect normality for frames is made.     

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.     

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.     

Dissipated compacta

The dissipated spaces form a class of compacta which contains both the scattered compacta and the compact LOTSes (linearly ordered topological spaces), and a number of theorems true for these latter two classes are true more generally for the dissipated spaces. For example, every regular Borel measure on a dissipated space is separable.The standard Fedor?uk S-space (constructed under ?) is dissipated. A dissipated compact L-space exists iff there is a Suslin line.A product of two compact LOTSes is usually not dissipated, but it may satisfy a weakening of that property. In fact, the degree of dissipation of a space can be used to distinguish topologically a product of n LOTSes from a product of m LOTSes.     

COHOMOLOGICAL DIMENSION OF UNIFORM SPACES

Abstract

In this paper we obtain classification and extension theorems for uniform spaces, using the ?ech cohomology theory based on the finite uniform coverings, and study the associated cohomological dimension theory. In particular, we extend results for the cohomological dimension theory on compact Hausdorff spaces or compact metric spaces to those for our cohomological dimension theory on uniform 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].     

Totally Bounded Quasi-Uniformities and Pseudocompactness of Bispaces

 We characterize pairwise Tychonoff bispaces that admit only totally bounded quasi-uniformities in terms of a suitable notion of bitopological pseudocompactness. We also show that a pairwise Tychonoff bispace has a unique (up to equivalence) bicompactification if and only if it admits a unique totally bounded quasi-unifomity. These results extend classical theorems of R. Doss for uniform spaces to the quasi-uniform (bitopological) setting, and are applied to the study of T ₀ topological spaces that admit a unique quasi-uniformity and a unique bicompactification, respectively. Finally, we discuss the problem of extending the classical Glicksberg theorem on product of pseudocompact spaces to bispaces and a partial solution is obtained. Supported by the Spanish Ministry of Science and Technology, grant BFM2000-1111. Supported by a grant from Generalitat Valenciana. Received November 7, 2001; in revised form August 14, 2002     

Small codimension subvarieties in homogeneous spaces

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in X² where X is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of X. Finally we prove transplanting theorems à la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.     

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.     

Fixed point results for set-valued contractions by altering distances in complete metric spaces

Nadler’s contraction principle has led to fixed point theory of set-valued contraction in non-linear analysis. Inspired by the results of Nadler, the fixed point theory of set-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi–Takahashi, Feng–Liu and many others. In the present paper, the concept of generalized contractions for set-valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions. Our first result extends and generalizes the Nadler, Feng–Liu and Klim–Wardowski theorems and the second result is different from the Reich and Mizoguchi–Takahashi results. As a consequence, we derive some results related to fixed point of set-valued maps satisfying certain conditions of integral type.     

PROXIMAL LIMIT SPACES

Abstract

The proximal limit spaces are introduced which fill the gap arising from the existence of proximity spaces, uniform spaces, and uniform limit spaces. It is shown that the proximal limit spaces can be considered as a bireflective subcategory of the topological category of uniform limit spaces. A limit space is induced by a proximal limit space if and only if it is a S₁-limit space.