Generalized variation of mappings with applications to composition operators and multifunctions

We study (set-valued) mappings of bounded Phi -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 Phi -variation of a metric space valued mapping. We show that the linear span GV(Phi)(I;X) of the set of all mappings of bounded Phi -variation is automatically a Banach algebra provided X is a Banach algebra. If h:Ix X --> Y is a given mapping and the composition operator H is defined by (Hf)(t)=h(t,f(t)), where t is an element ofI and f:I --> X, we show that H:GV(Phi)(I; X)--> GV(Psi)(I;Y) is Lipschitzian if and only if h(t,x)=h(0)(t)+h(1)(t)x, t is an element ofI, x is an element ofX. 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 Phi -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded Phi -variation.