Mappings of Bounded Φ-Variation with Arbitrary Function Φ

Abstract

We develop the general theory of mappings of bounded Φ-variation in the sense of L. C. Young that are defined on a subset of the real line and take values in metric or normed spaces. We single out the characterizing properties for these mappings, prove the structural theorem for them, and study their continuity properties. We obtain the existence of a geodesic path of bounded Φ-variation between two points of a compact set with certain regularity of its modulus of continuity. The classical Helly selection principle from the theory of functions of bounded variation is generalized for mappings of bounded Φ-variation. Under natural restrictions on the function Φ, we show that the space of all normed space-valued mappings under consideration can be endowed with a metric. Finally, we consider the problem of existence of selections of a continuous set-valued mapping Fof bounded Φ-variation with respect to the Hausdorff distance. We show that if Φ′(0) is finite> 0, then Fhas a continuous selection of bounded Φ-variations if Φ′(0) e ∞, then Fis a constant mappings and if Φ′(0) e 0, then, under additional assumptions on Φ, we give examples of mappings Fwith no continuous selection and with no selection of bounded Φ-variation.