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Uniformly continuous composition operators in the space of bounded Φ-variation functions in the Schramm sense

Ereu T., Merentes N., Sanchez J. L., Wróbel M.

Opuscula Mathematica

2012

Vol. 32, no. 2

239-247

We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized Φ-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.

On composition operator in the algebra of functions of two variables with bounded total Φ-variation in Schramm sense

Ereu T., Merentes N., Rzepka B., Sanchez J. L.

Journal of Mathematics and Applications

2010

Vol. 33

35-50

In this paper we consider the so called composition operator being a self-mapping of the Banach algebra of the function of two variables with bounded total Φ-variation in the Schramm sense. The main result of the paper characterizes the composition operator mentioned above which has a generating function being Lipschitzian with respect to the second variable. The basic tool used in our considerations is the concept of the left-left regularization.

Some remarks on the algebra of functions of two variables with bounded total Φ-variation in Schramm sense

Ereu T., Merentes N., Sánchez J-L.

Commentationes Mathematicae

2010

Vol. 50, [Z] 1

23-33

This paper is devoted to discuss some generalizations of the bounded total Φ-variation in the sense of Schramm. This concept was defined by W. Schramm for functions of one real variable. In the paper we generalize the concept in question for the case of functions of of two variables defined on certain rectangle in the plane. The main result obtained in the paper asserts that the set of all functions having bounded total Φ-variation in Schramm sense has the structure of a Banach algebra.