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Demonstratio Mathematica

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Volume 47, Issue 4

Issues

On the Nemytskii Operator in the Space of Functions of Bounded (p, 2, α)-Variation with Respect to the Weight Function

Wadie Aziz
Published Online: 2014-12-11 | DOI: https://doi.org/10.2478/dema-2014-0074

Abstract

In this paper, we consider the Nemytskii operator (Hf)(t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p,2,α)-variation (with respect to a weight function α) into the space of functions of bounded (q,2,α)-variation (with respect to α) 1<q<p, then H is of the form (Hf)(t) = A(t)f(t)+B(t). On the other hand, if 1<p<q then H is constant. It generalize several earlier results of this type due to Matkowski-Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.

Keywords: and phrases variation in the sense of De la Vallée Poussin; variation in the sense of Riesz; -convex functions; weight function; composition operator; Jensen equation

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About the article

Received: 2013-09-16

Revised: 2014-02-26

Published Online: 2014-12-11

Published in Print: 2014-12-01


Citation Information: Demonstratio Mathematica, Volume 47, Issue 4, Pages 933–948, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.2478/dema-2014-0074.

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© by Wadie Aziz. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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