Abstract superposition operators on mappings of bounded variation of two real variables. II

We define and study the metric semigroup $BV_2(I_a^b;M)$ of mappings of two real variables of bounded total variation in the Vitali–Hardy–Krause sense on a rectangle $I_a^b$ with values in a metric semigroup or abstract convex cone $M$. We give a complete description for the Lipschitzian Nemytskii superposition operators from $BV_2(I_a^b;M)$ to a similar semigroup $BV_2(I_a^b;N)$ and, as a consequence, characterize set-valued superposition operators. We establish a connection between the mappings in $BV_2(I_a^b;M)$ and 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.