We present several recent examples of the generic approach to nonlinear problems. In this approach, instead of considering, for instance, the convergence of an algorithm, one investigates an appropriate space of algorithms equipped with some natural complete metric and shows that convergence does indeed occur for most algorithms there. As we shall see, sometimes one can show that the complement...

Ferrando and Lüdkovsky proved that for a non-empty set Ω normed space X, the c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if only X is, respectively, Baire-like. When metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then over field K (of real complex numbers) all functions f:Ω→X such each ε>0 n∈N ω∈Ω:f(ω)n>ε finite ...

The following two theorems give the flavour of what will be proved. THEOREM. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0, 1] to Y coincide if and only if Y is connected and locally connected. THEOREM. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class fun...

Let (Ω,B) be a measurable space, An ⊂ B a sub-σ-field and μn a random probability measure on (Ω,B), n ≥ 1. In various frameworks, one looks for a probability P on B such that μn is a regular conditional distribution for P given An for all n. Conditions for such a P to exist are given. The conditions are quite simple when (Ω,B) is a compact Hausdorff space equipped with the Borel or the Baire σ-...

Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity assumptions. The bangbang property is proved for a new class of “concave” multifunctions, characterized by the existence of suitable linear selections. The proofs rely ...

We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers P not Gδ · Fσ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ P of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. T...

All spaces considered are metric separable and are denoted usually by the letters X, Y , or Z. ω stands for the set of all natural numbers. If a metric separable space is additionally complete, we call it Polish; if it is a continuous image of ω or, equivalently, of a Polish space, it is called Souslin. The main subject of the present paper is the structure of Baire class 1 functions. Recent de...