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Bulletin of the Australian Mathematical Society Bulletin of the Australian Mathematical Society

Article contents

UNIFORMLY BOUNDED COMPOSITION OPERATORS

Part of: Real functions Commutative Banach algebras and commutative topological algebras Nonlinear operators and their properties Special classes of linear operators

Published online by Cambridge University Press:  30 July 2015

DOROTA GŁAZOWSKA  and
JANUSZ MATKOWSKI
DOROTA GŁAZOWSKA
Affiliation:
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland email D.Glazowska@wmie.uz.zgora.pl
JANUSZ MATKOWSKI
Affiliation:
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland email J.Matkowski@wmie.uz.zgora.pl
Corresponding
E-mail address:
D.Glazowska@wmie.uz.zgora.pl
J.Matkowski@wmie.uz.zgora.pl
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Abstract

We prove that if a uniformly bounded (or equidistantly uniformly bounded) Nemytskij operator maps the space of functions of bounded ${\it\varphi}$ -variation with weight function in the sense of Riesz into another space of that type (with the same weight function) and its generator is continuous with respect to the second variable, then this generator is affine in the function variable (traditionally, in the second variable).

Keywords

composition operator Nemytskij operator bounded operator uniformly bounded operator 𝜙-variation in the sense of Riesz weight function

MSC classification

Primary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
Secondary: 26A45: Functions of bounded variation, generalizations 46J10: Banach algebras of continuous functions, function algebras 47B38: Operators on function spaces (general)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 3 , December 2015 , pp. 463 - 469
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Appell, J. and Zabrejko, P. P., Nonlinear Superposition Operators (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
Armao, F., Głazowska, D., Rivas, S. and Rojas, J., ‘Uniformly bounded composition operators in the Banach space of bounded (p, k)-variation in the sense of Riesz–Popoviciu’, Cent. Eur. J. Math. 11(2) (2013), 357–367.Google Scholar
Aziz, W., Azocar, A., Guerrero, J. and Merentes, N., ‘Uniformly continuous composition operators in the space of functions of 𝜙-variation with weight in the sense of Riesz’, Nonlinear Anal. 74 (2011), 573–576.CrossRefGoogle Scholar
Chistyakov, V. V., ‘Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight’, J. Appl. Anal. 6(2) (2000), 173–186.CrossRefGoogle Scholar
Głazowska, D., Matkowski, J., Merentes, N. and Sánchez Hernández, J. L., ‘Uniformly bounded composition operators in the Banach space of absolutely continuous functions’, Nonlinear Anal. 75(13) (2012), 4995–5001.CrossRefGoogle Scholar
Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities (Polish Scientific Editors and Silesian University, Warszawa, Kraków, Katowice, 1985).Google Scholar
Lichawski, K., Matkowski, J. and Miś, J., ‘Locally defined operators in the space of differentiable functions’, Bull. Polish Acad. Sci. Math. 37 (1989), 315–325.Google Scholar
Matkowski, J., ‘Functional equations and Nemytskij operators’, Funkcial Ekvac. 25 (1982), 127–132.Google Scholar
Matkowski, J., ‘Uniformly bounded composition operators between general Lipschitz function normed spaces’, Topol. Methods Nonlinear Anal. 38(2) (2011), 395–405.Google Scholar
Matkowski, J. and Wróbel, M., ‘Locally defined operators in the space of Whitney differentiable functions’, Nonlinear Anal. 68(10) (2008), 2933–2942.CrossRefGoogle Scholar
Matkowski, J. and Wróbel, M., ‘Representation theorem for locally defined operators in the space of Whitney differentiable functions’, Manuscripta Math. 129(4) (2009), 437–448.CrossRefGoogle Scholar
Wróbel, M., ‘Locally defined operators and a partial solution of a conjecture’, Nonlinear Anal. 72(1) (2010), 495–506.CrossRefGoogle Scholar
Wróbel, M., ‘Representation theorem for local operators in the space of continuous and monotone functions’, J. Math. Anal. Appl. 372(1) (2010), 45–54.CrossRefGoogle Scholar
Wróbel, M., ‘Locally defined operators in Hölder’s spaces’, Nonlinear Anal. 74(1) (2011), 317–323.CrossRefGoogle Scholar

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