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We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized Φ-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.
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Tom
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239--247
Opis fizyczny
Bibliogr. 20 poz.
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- Jan Długosz University Institute of Mathematics and Computer Science 42-200 Częstochowa, Poland [M. Wróbel], m.wrobel@ajd.czest.pl
Bibliografia
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- [5] A. Hernández, Space of bounded Φ-variation in the Schramm sense, Dep. Math., Central Univ. Venezuela, Caracas, September 1995.
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- [12] J. Matkowski, M. Wróbel, Locally defined operators in the space of Whitney differentiable functions, Nonlinear Anal. 68 (2008), 2873–3232.
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- [14] H. Nakano, Modulared Semi-Ordered Spaces, Tokyo, 1950.
- [15] I.P. Natanson, Theory of Functions of a Real Variable, 1974.
- [16] W. Orlicz, A note on modular spaces. I, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 9 (1961), 157–162.
- [17] M. Schramm, Funtions of -bounded variation and Riemann-Stieltjes integration, Trans. Amer. Math. Soc. 267 (1985), 49–63.
- [18] N. Wiener, The quadratic variation of function and its fourier coefficients, Massachusett J. Math. 3 (1924), 72–94.
- [19] M. Wróbel, Locally defined operators and a partial solution of a conjecture, Nonlinear Anal. 72 (2010), 495–506.
- [20] M. Wróbel, Representation theorem for local operators in the space of continuous and monotone functions, J. Math. Anal. Appl. 372 (2010), 45–54.
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Bibliografia
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bwmeta1.element.baztech-article-AGHT-0007-0019