Every l.s.cṁapping from a paracompact space into the non-empty, closed, convex subsets of a (not necessarily convex) $G_{\delta }$-subset of a Banach space admits a single-valued continuous selection provided every such mapping admits a convex-valued usco selection. This leads us to some new partial solutions of a problem raised by E. Michael.

The existence of continuous selections is proved for a class of lower semicontinuous multifunctions whose values are closed convex subsets of a complete metric space equipped with an appropriate notion of convexity. The approach is based on the notion of pseudo-barycenter of an ordered n-tuple of points.

Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if $su{p}_{\left|\right|f\left|\right|\le 1}|f\left(x\right)-{\int }_{K}fd\mu |<\gamma$ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family $\left({\mu }_t\right)$ of regular Borel...

Some sufficient conditions for controllability of nonlinear systems described by differential equation ẋ = f(t,x(t),u(t)) are given.

We discuss here several types of convergence of conditional expectations for unbounded closed convex random sets of the form $E^{{ℬ}_n}{X}_n$ where $\left({ℬ}_n\right)$ is a decreasing sequence of sub-σ-algebras and $\left(X_n\right)$ is a sequence of closed convex random sets in a separable Banach space.

Existence of fixed points of multivalued mappings that satisfy a certain contractive condition was proved by N. Mizoguchi and W. Takahashi. An alternative proof of this theorem was given by Peter Z. Daffer and H. Kaneko. In the present paper, we give a simple proof of that theorem. Also, we define Mann and Ishikawa iterates for a multivalued map $T$ with a fixed point $p$ and prove that these iterates converge to a fixed point $q$ of $T$ under certain conditions. This fixed point $q$ may be different from...

In this work we study countably z-compact spaces and z-Lindelof spaces. Several new properties of them are given. It is proved that every countably z-compact space is pseuodocompact (a space on which every real valued continuous function is bounded). Spaces which are countably z-compact but not countably compact are given. It is proved that a space is countably z-compact iff every countable z-closed set is compact. Characterizations of countably z-compact and z-Lindelof spaces by multifunctions...

In this paper we consider the Darboux type properties for the paratingent. We review some of the standard facts on the multivalued functions and the paratingent. We prove that the paratingent has always the Darboux property but the property D* holds only when the paratingent is a multivalued function.

Let F be a multifunction with values in Lₚ(Ω, X). In this note, we study which regularity properties of F are preserved when we consider the decomposable hull of F.

∗ The first and third author were partially supported by National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under grant MM-701/97.A theorem proved by Fort in 1951 says that an upper or lower semi-continuous set-valued mapping from a Baire space A into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense Gδ -subset of A. In this paper we show that the conclusion of Fort’s theorem holds under the weaker...

We consider the space $D(X,Y)$ of densely continuous forms introduced by Hammer and McCoy and investigated also by Holá . We show some additional properties of $D(X,Y)$ and investigate the subspace $D^{*}\left(X\right)$ of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence ${\tau }_p$. The largest part of the paper is devoted to the study of various cardinal functions for $(D^{*}\left(X\right),{\tau }_p)$, in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, $\pi$-weight, $\pi$-character,...

We prove the existence of solutions of differential inclusions on a half-line. Our results are based on an approximation method combined with a diagonalization method.

This paper is concerned with existence of equilibrium of a set-valued map in a given compact subset of a finite-dimensional space. Previously known conditions ensuring existence of equilibrium imply that the set is either invariant or viable for the differential inclusion generated by the set-valued map. We obtain some equilibrium existence results with conditions which imply neither invariance nor viability of the given set. The problem of existence of strict equilibria is also discussed.

We survey results related to the problem of the existence of equilibria in some classes of infinitely repeated two-person games of incomplete information on one side, first considered by Aumann, Maschler and Stearns. We generalize this setting to a broader one of principal-agent problems. We also discuss topological results needed, presenting them dually (using cohomology in place of homology) and more systematically than in our earlier papers.