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Uniformly continuous superposition operators in the space of functions of bounded n-dimensional Φ-variation

Bracamonte M., Ereú J., Giménez J., Merentes N.

Demonstratio Mathematica

2014

Vol. 47, nr 1

56--68

We prove that if a superposition operator maps a subset of the space of all metric-vector-space-valued-functions of bounded n-dimensional Φ-variation into another such space, and is uniformly continuous, then the generating function of the operator is an affine function in the functional variable.

On bi-dimensional second μ-variation

Ereú J., Giménez J., Merentes N.

Demonstratio Mathematica

2014

Vol. 47, nr 4

910--932

In this paper, we present a generalization of the notion of bounded slope variation for functions defined on a rectangle Iba in R2. Given a strictly increasing function μ, defined in a closed real interval, we introduce the class BVμ,2 (Iba), of functions of bounded second μ-variation on Iba ; and show that this class can be equipped with a norm with respect to which it is a Banach space. We also deal with the important case of factorizable functions in BVμ,2 (Iba) and finally we exhibit a relation between this class and the one of double Riemann–Stieltjes integrals of functions of bi-dimensional bounded variation.

Superposition Operators in the Space of Functions of Waterman-Shiba Bounded Variation

Giménez J., Merentes N., Rodriguez L.

Commentationes Mathematicae

2014

Vol 54, No. 1

79--93

In this paper we present a necessary condition for an autonomous superposition operator to act in the space of functions of Waterman-Shiba bounded variation. We also show that if a (general) superposition operator applies such space into itself and it is uniformly bounded, then its generating function satisfies a weak Matkowski condition.

The Nemitskij operator on Lipk-type and BVk-type spaces

Giménez J., López L., Merentes N.

Demonstratio Mathematica

2013

Vol. 46, nr 3

543--558

In this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of R. We obtain a result concerning the integrability of products of the form (…) and a generalized version of the chain rule for functions a.e differentiable, in the sense of Lebesgue. As an application, we get a generalization of a theorem due to V. I. Burenkov for the case of functions of bounded Riesz-p-variation.