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

Abstract

We define and study the metric semigroup BV 2 I b a ; M of mappings of two real variables of bounded total variation in the Vitali-Hardy-Krause sense on a rectangle I b a 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 b a ; M to a similar semigroup BV 2 I b a ; N and, as a consequence, characterize set-valued superposition operators. We establish a connection between the mappings in BV 2 I b a ; 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 Miś (1984), Zawadzka (1990), and the author (2002, 2003) to the case of (set-valued) superposition operators on the mappings of two real variables.

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.…”

“…In what follows we will be interested to what extent the three properties above of the Jordan variation carry over to maps of several real variables so that the Helly theorem still holds. These properties are also of independent importance in the study of (multi-valued) nonlinear superposition operators acting on BV maps of several variables (in a paper under preparation) as it was exposed in more particular cases in [16] and [17]. Basing on the results and technique of this part, two extensions of the Helly-type pointwise selection principle to metric semigroup-valued maps of several variables will be established in part II of this paper (we refer to part II in this issue for more details).…”

Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

“…These are relations between mixed differences of all orders and properties of the total variation (2.3). For real valued functions of n variables the main properties of mixed differences of all orders were elaborated in [1,11,17,22,28,32,38] and for metric semigroup valued maps of two variables-in [5,13,16,35]. For our purposes we need their variants in the multiindex notation, as presented in [17] with M = R, for maps of n variables with values in a metric semigroup.…”

“…also [17, Part I, Lemma 6 and (3.5)]). The inequalities in Theorem B are also known for metric semigroup valued maps of two variables [5,16]. However, in the general case Theorem B needs a different proof as compared to the cases of maps of one or two variable(s) or M = R. These two theorems are extensions of two more properties of the Jordan variation for maps of one variable; in this case (3.2) is actually the equality known as the additivity of Jordan's variation (e.g., [37, Theorem 83 2 ]).…”