Generalized variation of mappings with applications to composition operators and multifunctions

Abstract

We study (set-valued) mappings of bounded Φ-variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the Φ-variation of a metric space valued mapping. We show that the linear span GVΦ(I;X) of the set of all mappings of bounded Φ-variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X → Y is a given mapping and the composition operator ℋ is defined by (ℋf)(t)=h(t,f(t)), where t∈I and f:I → X, we show that ℋ:GVΦ(I;X)→ GVΨ(I;Y) is Lipschitzian if and only if h(t,x)=h0(t)+h1(t)x, t∈I, x∈X. This result is further extended to multivalued composition operators ℋ with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded Φ-variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded Φ-variation.