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Schramm spaces and composition operators

Wróbel Małgorzata

Journal of Applied Mathematics and Computational Mechanics

2023

Vol. 22, nr 2

87--98

We give some properties of Schramm functions; among others, we prove that the family of all continuous piecewise linear functions defined on a real interval I is contained in the space ΦBV (I) of functions of bounded variation in the sense of Schramm. Moreover, we show that the generating function of the corresponding Nemytskij composition operator acting between Banach spaces CΦBV (I) of continuous functions of bounded Schramm variation has to be continuous and additionally we show that a space CΦBV (I) has the Matkowski property.

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.

Uniformly continuous composition operators in the space of functions of two variables of bounded φ-variation in the sense of Wiener

Guerrero J.A., Matkowski J., Merentes N.

Commentationes Mathematicae

2010

Vol. 50, [Z] 1

41-48

Assume that the generator of a Nemytskii composition operator is a function of three variables: the first two real and third in a closed convex subset of a normed space, with values in a real Banach space. We prove that if this operator maps a certain subset of the Banach space of functions of two real variables of bounded Wiener φ-variation into another Banach space of a similar type, and is uniformly continuous, then the one-sided regularizations of the generator are affine in the third variable.