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Uniformly bounded Nemytskij operators between the Banach spaces of functions of bounded \(n\)-th variation. (English) Zbl 1321.47124

Summary: The main result says that the generator of any uniformly bounded composition operator acting between Banach algebras of functions of bounded \(n\)-th variation is an affine function with respect to the function variable.

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
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[1] Azócar, A.; Guerrero, J.; Matkowski, J.; Merentes, N., Uniformly continuous set-valued composition operators in the space of continuous functions of bounded variation in the sense of Wiener, Opuscula Math., 30, 53-60 (2010) · Zbl 1216.47090
[2] Chistyakov, V. V., Generalized variation of mapping and applications, Real Anal. Exchange, 25 (1999) · Zbl 0966.54010
[3] Chistyakov, V. V., Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight, J. Appl. Anal., 6, 173-186 (2000) · Zbl 0997.47051
[4] Chistyakov, V. V., Generalized variation of mappings with applications to composition operators and multifunctions, Positivity, 5, 4, 323-358 (2001) · Zbl 1027.47046
[5] Chistyakov, V. V., Mappings of generalized variation and composition operators, J. Math. Sci., 110, 2, 2455-2466 (2002) · Zbl 1033.26013
[6] Chistyakov, V. V., Superposition operators in the algebra of functions of two variables with finite total variation, Monatsh. Math., 137, 99-114 (2002) · Zbl 1033.26019
[7] Chistyakov, V. V., A Banach algebra of functions of several variables of finite total variation and Lipschitzian superposition operators. I, Nonlinear Anal., 62, 559-578 (2005) · Zbl 1083.26011
[8] Guerrero, J. A.; Leiva, H.; Matkowski, J.; Merentes, N., Uniformly continuous composition operators in the space of bounded \(φ\)-variation functions, Nonlinear Anal., 72, 3119-3123 (2010) · Zbl 1225.47078
[9] Kostrzewski, T., Globally Lipschitzian operators of substitution in Banach space \(BC [a, b]\), Sci. Bull. Łódź Techn. Univ. Mat., 602, 17-27 (1993) · Zbl 0814.47078
[10] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities (1985), Polish Scientific Editors and Silesian University: Polish Scientific Editors and Silesian University Warszawa, Kraków, Katowice
[11] Matkowska, A., On characterization of Lipschitzian operators of substitution in the class of Hölder function, Sci. Bull. Łódź Techn. Univ., 17, 81-85 (1984) · Zbl 0599.46032
[12] Matkowski, J., Functional equation and Nemytskij operators, Funkcial. Ekvac., 25, 127-132 (1982) · Zbl 0504.39008
[13] Matkowski, J., Form of Lipschitz operators of substitution in Banach spaces of differentiable functions, Sci. Bull. Łódź Techn. Univ., 17, 5-10 (1984) · Zbl 0599.46031
[14] Matkowski, J., Lipschitzian composition operators in some function spaces, Nonlinear Anal., 30, 2, 719-726 (1997) · Zbl 0894.47052
[15] Matkowski, J., Uniformly continuous superposition operators in the spaces of differentiable functions and absolutely continuous functions, Internat. Ser. Numer. Math., 157, 155-166 (2008) · Zbl 1266.47082
[16] Matkowski, J., Uniformly continuous superposition operators in the space of Hölder functions, J. Math. Anal. Appl., 359, 56-61 (2009) · Zbl 1173.47043
[17] Matkowski, J., Uniformly continuous superposition operators in the spaces of bounded variation functions, Math. Nachr., 283, 7, 1060-1064 (2010) · Zbl 1235.47052
[18] Matkowski, J., Uniformly bounded composition operators between general Lipschitz functions normed spaces, Topol. Methods Nonlinear Anal., 38, 2, 395-406 (2011) · Zbl 1272.47070
[19] Matkowski, J.; Miś, J., On a characterization of Lipschitzian operators of substitution in the space \(BV [a, b]\), Math. Nachr., 117, 155-159 (1984) · Zbl 0566.47033
[20] Matkowski, J.; Wróbel, M., Locally defined operators in the space of Whitney differentiable functions, Nonlinear Anal., 68, 2873-3232 (2008)
[21] Matkowski, J.; Wróbel, M., Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces, Cent. Eur. J. Math., 10, 2, 609-618 (2012) · Zbl 1253.47039
[22] Merentes, N., On a characterization of Lipschitzian operators of substitution in the space of bounded Riesz \(φ\)-variation, Ann. Univ. Sci. Budapest Eötvös Sect. Math., 34 (1991) · Zbl 0808.47050
[23] Popoviciu, M. T., Sur les fonctions convexes dʼune variable reelle, C. R. Acad. Sci. Paris, 190, 1481-1483 (1930) · JFM 56.0241.03
[24] M.T. Popoviciu, Sur quelques proprietes des fonctions dʼune ou de deux variables reelles, DSc thesis, Paris, 1933.; M.T. Popoviciu, Sur quelques proprietes des fonctions dʼune ou de deux variables reelles, DSc thesis, Paris, 1933. · JFM 59.0988.06
[25] Roberts, W.; Varberg, D. E., Connex Functions (1973), Academic Press: Academic Press New York, London
[26] Russell, A. M., A Banach space of functions of generalized variation, Bull. Austral. Math. Soc., 15, 431-438 (1976) · Zbl 0333.46025
[27] Russell, A. M., Functions of bounded kth variation, Proc. Lond. Math. Soc. (3), 26, 547-563 (1973) · Zbl 0254.26017
[28] Wróbel, M., On functions of bounded \(n\)-th variation, Ann. Math. Sil., 15, 79-86 (2001) · Zbl 1085.26004
[29] Wróbel, M., Representation theorem for local operators in the space of continuous and monotone functions, J. Math. Anal. Appl., 372, 45-54 (2010) · Zbl 1208.47052
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