Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions

• Autorzy: Lupa M. , Wróbel M.

Identyfikatory

DOI

10.17512/jamcm.2017.4.04

• Języki publikacji: EN

Abstrakty

EN

We investigate the Nemytskij (composition, superposition) operators acting between Banach spaces of r -times differentiable functions defined on the closed intervals of the real line with the r-derivatives satisfying a generalized Hölder condition. The main result says that if such a Nemytskij operator is uniformly bounded (in a special case uniformly continuous) then its generator is an affine function with respect to the second variable, i.e., the Matkowski representation holds. This extends an earlier result where an operator is assumed to be Lipschitzian.

Słowa kluczowe

EN

Nemytskij operator; generalized Hölder condition; uniformly bounded mapping

PL

funkcja Höldera; operator Niemyckiego; jednostajnie ograniczone odwzorowanie; jednostajnie ciągłe odwzorowan

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2017
• Tom: Vol. 16, nr 4
• Strony: 37--45
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Lupa M.

• Institute of Mathematics, Czestochowa University of Technology Czestochowa, Poland

autor

Wróbel M.

• Institute of Mathematics, Czestochowa University of Technology Czestochowa, Poland

Bibliografia

• [1] Krämer R., Mathé P., Modulus of continuity of Nemytskii operators with application to the problem of option pricing, Journal of Inverse and Ill-posed Problems 2008, 16(5), 435-461.
• [2] Matkowski J., Functional equation and Nemytskij operators, Funkcial Ekv. 1982, 25, 127-132.
• [3] Appell J., Banaś J., Merentes N., Bounded Variation and Around, De Gruyter Studies in Nonlinear Analysis and Applications, Volume 17, Würzburg 2014.
• [4] Appell J., Zabrejko P.P., Nonlinear Superposition Operators, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne-Sydney 1990.
• [5] Knop J., On globally Lipschitzian Nemytskii operator in a special Banach space of functions, Fasciculi Mathematici 1990, 280(21), 79-85.
• [6] Lupa M., Form of Lipschitzian operator of substitution in some class of functions, Zeszyty Naukowe Politechniki Łódzkiej 1990, Matematyka, 21, 87-96.
• [7] Matkowski J., Miś J., On a characterization of Lipschitzian operators of substitution in the space BV[a,b], Math. Nachr. 1984, 117, 155-159.
• [8] Matkowski J., Uniformly continuous superposition operators in the space of Hölder functions, J. Math. Anal. App. 2009, 359, 56-61.
• [9] Matkowski J., Uniformly continuous superposition operators in the spaces of bounded variation functions, Math. Nach. 2010, 283(7), 1060-1064.
• [10] Matkowski J., Uniformly bounded composition operators between general Lipschitz functions normed spaces, Topol. Methods Nonlinear Anal. 2011, 38(2), 395-406.
• [11] Matkowski J., Wróbel M., Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces, Cent. Eur. J. Math. 2012, 10(2), 609-618.
• [12] Wróbel M., Uniformly bounded Nemytskij operators between the Banach spaces of functions of bounded n-th variation, J. Math. Anal. Appl. 2012, 391, 451-456.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).