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

Abstract

We define and study the metric semigroup BV2(I ; M) of mappings of two real variables of bounded total variation in the Vitali-Hardy-Krause sense on a rectangle I with values in a metric semigroup or abstract convex cone M. We give a complete description for the Lipschitzian Nemytskii superposition operators acting from BV2(I ; M) to a similar semigroup BV2(I ; N) and, as a consequence, characterize set-valued superposition operators. We establish a connection between the mappings in BV2(I ; M) with the mappings of bounded iterated variation and study the iterated superposition operators on the mappings of bounded iterated variation. The results of this article develop and generalize the recent results by Matkowski and Mis (1984), Zawadzka (1990), and the author (2002, 2003) to the case of (set-valued) superposition operators on the mappings of two real variables.