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Functions of bounded variations on compact subsets of C

Gimenez J., Merentes N., Vivas M.

Commentationes Mathematicae

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2014

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Vol 54, No. 1

3--19

EN

In this paper we introduce the concept of bounded variation for functions defined on compact subsets of the complex plane C, based on the notion of variation along a curve as defined by Ashton and Doust; We describe in detail the space so generated and show that it can be equipped, in a natural way, with the structure of a Banach algebra. We also present a necessary condition for a composition operator Cφ to act between two such spaces.

2

Integral representation of functions of bounded second Φ-variation in the sense of Schramm

Gimenez J., Merentes N., Rivas S.

Opuscula Mathematica

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2012

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Vol. 32, no. 1

137-151

EN

In this article we introduce the concept of second Φ--variation in the sense of Schramm for normed-space valued functions defined on an interval [a; b] ⊂ R. To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of φ-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function [formula] (where X is a reflexive Banach space) is of bounded second Φ-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm.

3

On strongly midconvex functions

Azocar A., Gimenez J., Nikodem K., Sanchez J. L.

Opuscula Mathematica

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2011

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Vol. 31, no. 1

15-26

EN

In this paper we collect some properties of strongly midconvex functions. First, counterparts of the classical theorems of Bernstein-Doetsch, Ostrowski and Sierpiński are presented. A version of Rod é support theorem for strongly midconvex functions and a Kuhn-type result on the relation between strongly midconvex functions and strongly t-convex functions are obtained. Finally, a connection between strong midconvexity and generalized convexity in the sense of Beckenbach is established

4

Uniformly continuous set-valued composition operators in the space of total phi-bidimensional variation in the sense of Riesz

Aziz W., Gimenez J., Merentes N., Sanchez J. L.

Opuscula Mathematica

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2010

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Vol. 30, no. 3

241-248

EN

In this paper we prove that if a Nemytskij composition operator, generated by a function of three variables in which the third variable is a function one, maps a suitable large subset of the space of functions of bounded total φ-bidimensional variation in the sense of Riesz, into another such space, and is uniformly continuous, then its generator is an affine function in the function variable. This extends some previous results in the one-dimensional setting.