Nemytskii operator on generalized bounded
variation space

RENÉ ERLÍN CASTILLO^a,*, HUMBERTO RAFEIRO^b,
EDUARD TROUSSELOT^c

^a Universidad Nacional de Colombia, Departamento de Matemáticas, Bogotá, Colombia.
^b Pontificia Universidad Javeriana, Departamento de Matemáticas, Bogotá, Colombia.
^c Universidad de Oriente, Departamento de Matemáticas, 6101 Cumaná, Edo. Sucre, Venezuela.

Abstract. In this paper we show that if the Nemytskii operator maps the (φ, α)-bounded variation space into itself and satisfies some Lipschitz condition, then there are two functions g and h belonging to the (φ, α)-bounded variation space such that f(t, y) = g(t)y + h(t) for all t ∈ [a, b], y ∈ ℝ.

Keywords: Riesz p-variation, (φ, α)-bounded variation.
MSC2010: 26A45, 26B30, 26A16, 26A24.

El Operador de Nemytskii en espacios de variación
acotada generalizados

Resumen. En este artículo demostramos que si el operador de Nemytskii lleva el espacio de variación (φ, α)-acotada en sí mismo, y satisface cierta condición de Lipschitz, entonces existen dos funciones g y h perteneciendo al espacio de variación (φ, α)-acotada tal que f(t, y) = g(t)y + h(t) para todo t ∈ [a, b], y ∈ ℝ.

Palabras claves: p-variación de Riesz, variación (φ, α)-acotada.

Texto Completo disponible en PDF

References

[1] Appell J. and Zabrejko P.P., Nonlinear superposition operators, Cambridge University Press, 1990.

[2] Castillo R.E., Rafeiro H. and Trousselot E., "On functions of (φ, α)-bounded variation", (2013), submitted.

[3] Castillo R.E. and Trousselot E., "On Functions of (p, α)-Bounded Variation", Real Anal. Exchange 34 (2008), no. 1, 49-60.

[4] Chakvabarty M.C., "Some result on AC-w-functions", Fundamenta Mathematica LXIV (1969), 219-230.

[5] Chistyakov V.V., "Generalized variation of mappings and applications", Real Anal. Exchange 25 (1999-2000), no. 1, 61-64.

[6] Chistyakov V.V., "On mappings o finite generalized variation and nonlinear operators", Real Anal. Exchange 24th Summer Symp. Conf. Reports (2000), 39-43.

[7] Chistyakov V.V., "Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight", J. Appl. Anal. 6 (2000), no. 2, 173- 186.

[8] Chistyakov V.V., "Generalized variation of mappings with applications to composition operators and multifunctions", Positivity 5 (2001), no. 4, 323-358.

[9] Chistyakov V.V., "Superposition operators in the algebra of functions of two variables with finite total variation", Monatsh. Math. 137 (2002), no. 2, 99-114.

[10] Chistyakov V.V., "Metric semigroups and cones of mappings of inite variation of several variables and multivalued superposition operators", (Russian), Dokl. Akad. Nauk 393 (2003), no. 6, 757-761. English translation: Dokl. Math. Sci. 68, 6/2 (2003).

[11] Cybertowicz Z. and MatuszewskaW., "Functions of bounded generalized variations", Commentat. Math. 20 (1977), 29-52.

[12] Halmos P., Measure Theory, Springer-Verlag, 1974.

[13] Hudjaev S.I. and Vol’pert A.I., Analysis in classes of discontinuous functions and equations of mathematical physics, Springer, 1985.

[14] Jeffery R.L., "Generalized integrals with respect to bounded variation", Canadian Journal of Mathematics 10 (1958), 617-625.

[15] Jordan C., "Sur la série de Fourier", C.R. Acad. Sci. Paris 2 (1881), 228-230.

[16] Merentes N. and Rivas S., "El operador de composición con algún tipo de variación acotada". IX Escuela Venezolana de Matemática A:M:V-IVIC, 1996.

[17] Krasnosel’skii M.A. and Rutickii Ya.B., Convex Functions and Orlicz Spaces, Groningen: P.Noordhoff Ltd, 1961.

[18] Maligranda L. and Orlicz W., "On some properties of Functions of Generalized Variation", Monatshift für Mathematik 104 (1987), 53-65.

[19] Matkowski J., "Functional equations and Nemytskii operators", Funkc. Ekvacioj Ser. Int. 25 (1982), 127-132.

[20] Matkowski J., "Form of Lipschitz operators of substitution in Banach spaces of differentiable functions", Sci. Bull. Lodz Tech. Univ. 17 (1984), 5-10.

[21] Matkowski J., "On Nemytskii operator", Math. Japon. 33 (1988), no. 1, 81-86.

[22] Matkowski J., "Lipschitzian composition operators in some function spaces", Nonlinear Anal. 30 (1997), no. 2, 719-726.

[23] Matkowski J. and Merentes N., "Characterization of globally Lipschitzian composition operators in the Banach space ", Archivum Math. 28 (1992), no. 3-4, 181-186.

[24] Matkowski J. and Miś J., "On a characterization of Lipschitzian operators of substitution in the space Math. Nachr. 117 (1984), 155-159.

[25] Merentes N. and Nikodem K., "On Nemytskii operator and set-valued functions of bounded p-variation", Rad. Mat. 8 (1992), no. 1, 139-145.

[26] Merentes N. and Rivas S., "On characterization of the Lipschitzian composition operator between spaces of functions of bounded p-variation", Czechoslovak Math. J. 45 (1995), no. 4, 627-637.

[27] Riesz F., "Untersuchungen über systeme integrierbarer funktionen", Mathematische Annalen. 69 (1910), 449-497.

[28] Riesz F. and Nagy B., Functional Analysis (translated from the second french edition), Ungar, New York, 1955.

[29] Smajdor A. and SmajdorW., "Jensen equation and Nemytskii operator for set-valued functions", Rad. Mat. 5 (1989), 311-320.

[30] Smajdor W., "Note on Jensen and Pexider functional equations", Demonstratio Math. 32 (1999), no. 2, 363-376.

[31] Zawadzka G., "On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation", Rad. Mat. 6 (1990), 279-293.

*Corresponding author: E-mail: recastillo@unal.edu.co.
Received: 09 November 2013, Accepted: 17 March 2014.
To cite this article: R. E. Castillo, H. Rafeiro, E. Trousselot, Nemytskii operator on generalized bounded
variation space, Rev. Integr. Temas Mat. 32 (2014), no. 1, 71–90.