Some notes on densely continuous forms

We consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLá, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLY, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLY, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D _p ^*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D _p ^*(X) and on real-valued continuous functions C _p (X) and a generalization of a sufficient condition for the countable cellularity of D _p ^*(X). This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904 and by the Eco-Net (EGIDE) programme of the Laboratoire de Mathématiques de l’Université de Saint-Etienne (LaMUSE), France.     

Equiconnected spaces and Baire classification of separately continuous functions and their analogs

We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.     

On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L ^p (μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L ¹(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.     

Points of joint continuity and large oscillations

For topological spaces X and Y and a metric space Z, we introduce a new class N( X ×Y, Z ) \mathcal{N}\left( {X \times Y,\,Z} \right) of mappings f: X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping f from this class and any countable-type set B in Y, the set C _B (f) of all points x from X such that f is jointly continuous at any point of the set {x} × B is residual in X: We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and f ? N( X ×Y, Z ) f \in \mathcal{N}\left( {X \times Y,\,Z} \right) , then, for any ε > 0, the projection of the set D ^ε (f) of all points p ∈ X × Y at which the oscillation ω _f (p) ≥ ε onto X is a closed set nowhere dense in X.     

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.     

Halpern Iteration in CAT（k） Spaces

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.     

Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces

In this article, we introduce the concept of a family of set-valued mappings generalized Knaster–Kuratowski–Mazurkiewicz (KKM) w.r.t. other family of set-valued mappings. We then prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and 𝒯 and 𝒮 are two families of self set-valued mappings of X such that 𝒮 is generalized KKM w.r.t. 𝒯, under some natural conditions, the set-valued mappings S ∈ 𝒮 have a fixed point. Other common fixed point theorems and minimax inequalities of Ky Fan type are obtained as applications.     

A generic approach to approximate efficiency and applications to vector optimization with set-valued maps

In this paper we focus on approximate minimal points of a set in Hausdorff locally convex spaces. Our aim is to develop a general framework from which it is possible to deduce important properties of these points by applying simple results. For this purpose we introduce a new concept of ε-efficient point based on set-valued mappings and we obtain existence results and properties on the behavior of these approximate efficient points when ε is fixed and by considering that ε tends to zero. Finally, the obtained results are applied to vector optimization problems with set-valued mappings.     

Densely continuous forms, pointwise topology and cardinal functions

We consider the space D(X, Y) of densely continuous forms introduced by Hammer and McCoy [5] and investigated also by Holá [6]. We show some additional properties of D(X, Y) and investigate the subspace D*(X) of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence τ- _p . The largest part of the paper is devoted to the study of various cardinal functions for (D*(X), τ _p ), in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, π-weight, π-character, netweight etc. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank Ľubica Holá for suggestions and comments.     

Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

For metric spaces (X, d _x) and (Y, d _y) we consider the Hausdorff metric topology on the set (CL(X × Y), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology defined on CL(X × Y). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τ _uc on the set F(X, 2^Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y),) of closed subsets of Y metrized by the Hausdorff metric . We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank.ubica Holá for suggestions and comments.     

Weak local homeomorphisms and <Emphasis Type="Italic">B</Emphasis>-favorable spaces

Let X and Y be topological spaces such that an arbitrary mapping f: X → Y for which every preimage f ⁻¹ (G) of a set G open in Y is an F _σ-set in X can be represented in the form of the pointwise limit of continuous mappings f _n : X → Y. We study the problem of subspaces Z of the space Y for which the mappings f: X → Z possess the same property. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1189–1195, September, 2008.     

The Cluster Function of Single-Valued Functions

Given a single-valued function f between topological spaces X and Y, we interpret the cluster set C(f;x) as a multivalued function F=C(f;⋅) associated to f – the cluster function of f. For appropriate metrizable spaces X and Y, we characterize cluster functions C(f;⋅) among arbitrary set-valued functions F and show that every cluster function F=C(f;⋅) admits a selection h of Baire class 2 such that F=C(h;⋅). Mathematics Subject Classifications (2000) Primary: 54C50, 54C60; secondary: 26A21, 54C65.This research was partially supported by DFG Grant RI 1087/2.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Generic existence of a nonempty compact set of fixed points

Let X be a complete metric space, a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset contained in such that for every mapping T in the set {x ε X: Tx = x} is nonempty, it is proved that most mappings (in the Baire category sense) in do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived.     

Linearly ordered compact sets and co-Namioka spaces

We prove that, for an arbitrary Baire space X, a linearly ordered compact set Y, and a separately continuous mapping ƒ: X × Y → R, there exists a G _δ-set A ⊆ X dense in X and such that the function ƒ is jointly continuous at every point of the set A × Y, i.e., any linearly ordered compact set is a co-Namioka space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 1001–1004, July, 2007.     

Joint Continuity and Quasicontinuity of Horizontally Quasicontinuous Mappings

We show that if Xis a topological space, Ysatisfies the second axiom of countability, and Zis a metrizable space, then, for every mapping f: X× Y Zthat is horizontally quasicontinuous and continuous in the second variable, a set of points x Xsuch that fis continuous at every point from {x} × Yis residual in X. We also generalize a result of Martin concerning the quasicontinuity of separately quasicontinuous mappings.     

Lebesgue–Cech Dimensionality and Baire Classification of Vector-Valued Separately Continuous Mappings

For a metrizable space X with finite Lebesgue–Cech dimensionality, a topological space Y, and a topological vector space Z, we consider mappings f: X × Y Z continuous in the first variable and belonging to the Baire class in the second variable for all values of the first variable from a certain set everywhere dense in X. We prove that every mapping of this type belongs to the Baire class + 1.     

Affine Baire functions on Choquet simplices

We construct a metrizable simplex X such that for each n ɛ ℕ there exists a bounded function f on ext X of Baire class n that cannot be extended to a strongly affine function of Baire class n. We show that such an example cannot be constructed via the space of harmonic functions.     

Functions of the first Baire class with values in metrizable spaces

We show that every mapping of the first functional Lebesgue class that acts from a topological space into a separable metrizable space that is linearly connected and locally linearly connected belongs to the first Baire class. We prove that the uniform limit of functions of the first Baire class ƒ _n: X → Y belongs to the first Baire class if X is a topological space and Y is a metric space that is linearly connected and locally linearly connected. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 568–572, April, 2006.