The Nemytskii operator on bounded Φ-variation in the mean spaces
DOI:
https://doi.org/10.4067/S0716-09172013000200003Keywords:
(p, α)-variation, Nemytskii operator.Abstract
We introduce the notion of bounded Φ-variation in the sense of LΦ-norm. We obtain a Riesz type result for functions of bounded Φ-variation in the mean. We also show that if the Nemytskii operator act on the bounded Φ-variation in the mean spaces into itself and satisfy some Lipschitz condition there exist two functions g and h belonging to the bounded Φ-variation in the mean space such that f (t,y) = g(t)y + h(t),t ∈ [0, 2π], y ∈ R.References
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[2] Castillo, R., and Trousselot, E., On functions of (p, α)-bounded variation. Real Anal. Exchange, 34, n. 1 , pp. 49-60, (2009).
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[9] Mricz, F. and Siddiqi, A. H., A quatified version of the Dirichlet-Jordan test in L1-norm, Rend. Circ. Mat. Palermo, 45, pp. 19-24, (1996).
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[11] Riesz, F., Untersuchungen ber system intergrierbarer function, Mathematische Annalen, 69, pp. 1449-1497, (1910).
[12] Schramm, M., Functions of Φ-Bounded variation and RiemannStieltjes Integration. Trans of the Amer Math. Soc., 287, 1, pp. 46-63, (1985).
[13] Waterman, D., On Λ-Bounded Variation, Studia Mathematicae, LVII, pp. 33-45, (1976).
[2] Castillo, R., and Trousselot, E., On functions of (p, α)-bounded variation. Real Anal. Exchange, 34, n. 1 , pp. 49-60, (2009).
[3] Jordan, C., Sur la Serie de Fourier, C. R. Math. acad. Sci. Paris, 2, pp. 228-230, (1881).
[4] Maligranda, L. and Orlicz, W., On some properties of Functions of Generalize Variation, Monatshif fr Mathematik (springe Verlag), 104, pp. 53-65, (1987).
[5] Merentes, N., Functions of bounded (Φ, 2) Variation Annals Univ Sci Budapest, XXXIV, pp. 145-154, (1991).
[6] Merentes, N., On Functions of bounded (p, 2)-variation. Collect Math, 43, 2, pp. 115-118, (1992).
[7] Merentes, N. y Rivas, S., El Operador de composicin con algn tipo de variacin acotada, IX Escuela de Matematica, AMV, IVIC, (1996).
[8] Medved’ev, Y., A generalization of certain Theorem of Riesz, Uspekhi. Mat. Nauk, 6, pp. 115-118, (1953).
[9] Mricz, F. and Siddiqi, A. H., A quatified version of the Dirichlet-Jordan test in L1-norm, Rend. Circ. Mat. Palermo, 45, pp. 19-24, (1996).
[10] Neves, M. T., Φ-variacin en el sentido de wiener y Riesz, Trabajo de pasanta (asesorado por S. Rivas) UNA Centro local Aragua, rea de Matemtica, Maracay, (1994).
[11] Riesz, F., Untersuchungen ber system intergrierbarer function, Mathematische Annalen, 69, pp. 1449-1497, (1910).
[12] Schramm, M., Functions of Φ-Bounded variation and RiemannStieltjes Integration. Trans of the Amer Math. Soc., 287, 1, pp. 46-63, (1985).
[13] Waterman, D., On Λ-Bounded Variation, Studia Mathematicae, LVII, pp. 33-45, (1976).
How to Cite
[1]
R. E. Castillo Castillo, N. Merentes, and E. Trousselot, “The Nemytskii operator on bounded Φ-variation in the mean spaces”, Proyecciones (Antofagasta, On line), vol. 32, no. 2, pp. 119-142, 1.
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