Some remarks on local operators in generalized Hölder function spaces

• Autorzy: Wróbel Małgorzata

Identyfikatory

DOI

10.17512/jamcm.2025.1.07

• Języki publikacji: EN

Abstrakty

EN

We discuss the form of local operators, sometimes called operators with memory, acting between spaces of generalized Hölder functions defined on the compact metric spaces and taking values in the special norm spaces. Using the McShane and the Minty extension theorems, we show that in some cases the operators of such a type become Nemytskij composition operators. Moreover, the uniformly bounded as well as the uniformly continuous local operators acting between different Holder function spaces are investigated.

Słowa kluczowe

EN

locally defined operators; Nemytskij composition operator; uniformly bounded operator; uniformly continuous operator; generalized Hölder function

PL

operator lokalnie zdefiniowany; operator kompozycji Nemytskiego; operator jednostajnie ograniczony; operator jednostajnie ciągły; uogólniona funkcja Höldera

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2025
• Tom: Vol. 24, nr 1
• Strony: 78--88
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Wróbel Małgorzata

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

Bibliografia

• [1] Wróbel, M. (2011). Locally defined operators in the H¨ older spaces. Nonlinear Anal., TMA 74, 317-323.
• [2] Wróbel, M. (2020). Locally defined operators in the space of Ck,ω-functions. Math. Slovaca, 70(3), 745-752.
• [3] Brunner, H. (2017). Volterra Integral Equations. An Introduction to the Theory and Applications. Cambridge University Press.
• [4] Runst, T., & Sickel, W. (2001). Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Berlin, New York: De Gruyter.
• [5] Matkowski, J. (2011). Uniformly bounded composition operators between general Lipschitz functions normed spaces. Topol. Methods Nonlinear Anal., 38(2), 395-406.
• [6] Bana´ s, J., & Nalepa, R. (2016). On a measure of noncompactness in the space of functions with tempered increments. J. Math. Anal. Appl., 435(2), 1634-1651.
• [7] McShane, E.J. (1934). Extension of range of functions. Bull. Amer. Math. Soc., 40, 837-842.
• [8] Minty, G.J. (1970). On the extension of Lipschitz, Lipschitz-Hölder continuous and monotone functions. Bull. Amer. Math. Soc., 76(2), 334-339.
• [9] Matkowski, J. (2009). Uniformly continuous superposition operators in the Banach space of Hölder functions. J. Math. Anal. Appl., 359(1), 56-61.
• [10] Appell, J., Bana´s, J., & Merentes, N. (2014). Bounded Variation and Around. Berlin, Boston: De Gruyter.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).