On Helly's principle for metric semigroup valued BV mappings to two real variables

Abstract

We introduce a concept of metric space valued mappings of two variables with finite total variation and define a counterpart of the Hardy space. Then we establish the following Helly type selection principle for mappings of two variables: Let X be a metric space and a commutative additive semigroup whose metric is translation invariant. Then an infinite pointwise precompact family of X-valued mappings on the closed rectangle of the plane, which is of uniformly bounded total variation, contains a pointwise convergent sequence whose limit is a mapping with finite total variation.

Maps of several variables of finite total variation. I. Mixed differences and the total variation

“…also [17, Part I, Lemma 6 and (3.5)]). The inequalities in Theorem 2 are also known for metric semigroup-valued maps of two variables [5,16]. However, in the general case Theorem 2 needs a different proof as compared to the cases of maps of one or two variable(s) or M = R.…”

Abstract Superposition Operators on Mappings of Bounded Variation of Two Real Variables. II

“…To estimate the quantities S k , observe that from (8) and the definition of ρ 2 for all (t, s) ∈ I b a we derive ρ(g 1 (t, s), g 2 (t, s)) ≤ ρ(g 1 (a), g 2 (a)) + T W ρ g 1 , g 2 , I b a = ρ 2 (g 1 , g 2 ).…”

Abstract Superposition Operators on Mappings of Bounded Variation of Two Real Variables. I

“…The goal of the present article is an exhausting description for the abstract Lipschitzian Nemytskii superposition operators in the spaces of mappings of bounded variation of several real variables with values in metric semigroups and abstract convex cones and also, as a consequence, a description for the generators of set-valued superposition operators (in this or another context, the mappings of bounded variation with values in metric spaces were studied in [9;14;23;27,Chapter 4;28] (the case of a single variable) and in [22,[29][30][31] (the case of several variables). In this article we only consider the mappings of bounded variation of two variables as introduced in [22,31], since here the principal distinction from the one-dimensional case is most transparent.…”

“…In this article we only consider the mappings of bounded variation of two variables as introduced in [22,31], since here the principal distinction from the one-dimensional case is most transparent. Moreover, our results extend these of [24,25] and [32, § 8.3]; in a short form they were published in [33] and reported in [34].…”

“…, is called the function of total variation of f on I b a and possesses the following properties [25,31]:…”