Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz

Wadie Aziz^1*, Nelson Merentes²

¹Departamento de Física y Matemática, Universidad de Los Andes, Trujillo, Venezuela.
²Escuela de Matemática, Universidad Central de Venezuela, Caracas, Venezuela.

DOI: 10.4236/ijmnta.2015.44017                

Abstract

We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty bounded closed and convex values, are an affine function.

Keywords

j-Variation in the Sense of Riesz, Set-Valued Functions, Left and Right Regularizations, Uniformly Bounded Operator, Composition (Nemytskij) Operator, Jensen Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Aziz, W., Guerrero, J.A. and Merentes, N. (2010) Uniformly Continuous Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz. Bulletin of the Polish Academy of Sciences Mathematics, 58, 39-45. http://dx.doi.org/10.4064/ba58-1-5
[2]	Mis, J. and Matkowski, J. (1984) On a Characterization of Lipschitzian Operators of Substitution in the Space BV[a, b]. Mathematische Nachrichten, 117, 155-159.
[3]	Ereú, T., Merentes, N., Sanchez, J.L. and Wróbel, M. (2012) Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Schramm. Scientific Issues, Jan Dlugosz University in Czestochowa Mathematics, 17, 37-47.
[4]	Glazowska, D., Guerrero, J.A., Matkowski, J. and Merentes, N. (2013) Uniformly Bounded Composition Operators on a Banach Space of Bounded Wiener-Young Variation Functions. Bulletin of the Korean Mathematical Society, 50, 675-685. http://dx.doi.org/10.4134/BKMS.2013.50.2.675
[5]	Merentes, N. (1991) Composition of Functions of Bounded φ-Variation. P.U.M.A., Ser. 1, 39-45.
[6]	Matkowski, J. and Wrobel, M. (2011) Uniformly Bounded Nemytskij Operators Generate by Set-Valued Functions between Generalized Holder Functions Spaces. Discussiones Mathematicae Differential Inclusions, Control and Optimization, 31, 183-198. http://dx.doi.org/10.7151/dmdico.1134
[7]	Matkowski, J. and Wrobel, M. (2012) Uniformly Bounded Set-Valued Nemytskij Operators Acting between Generalized Holder Functions Spaces. Central European Journal of Mathematics, 10, 609-618. http://dx.doi.org/10.2478/s11533-012-0002-1
[8]	Matkowski, J. (2011) Uniformly Bounded Composition Operators between General Lipschitz Functions Normed Space. Nonlinear Analysis, 382, 395-406.
[9]	Smajdor, W. (1999) Note on Jensen and Pexider Functional Equations. Demonstratio Mathematica, 32, 363-376.
[10]	Smajdor, A. and Smajdor, W. (1989) Jensen Equation and Nemytskij Operators for Set-Valued Functions. Radovi Matemati, 5, 311-320.
[11]	Smajdor, W. (1993) Local Set-Valued Solutions of the Jensen and Pexider Functional Equations. Publicationes Mathematicae Debrecen, 43, 255-263.
[12]	Chistyakov, V.V. (2004) Selections of Bounded Variation. Journal of Applied Analysis, 10, 1-82. http://dx.doi.org/10.1515/JAA.2004.1
[13]	Jordan, C. (1881) Sur la série de fourier. Comptes Rendus de l’Académie des Sciences Paris, 2, 228-230.
[14]	Riesz, F. (1910) Untersuchugen über systeme integrierbarer funktionen. Mathematische Annalen, 69, 449-497. http://dx.doi.org/10.1007/BF01457637
[15]	Medvedev, Y.T. (1953) A Generalization of a Theorem of F. Riesz. Uspekhi Matematicheskikh Nauk, 8, 115-118. (In Russian)
[16]	Ciemnoczolowski, J. and Orlicz, W. (1986) Composing Functions of Bounded φ -Variation. Proceedings of the American Mathematical Society, 96, 431-436. http://dx.doi.org/10.2307/2046589
[17]	Luxemburg, W.A. (1955) Banach Function Spaces. Ph.D. Thesis, Technische Hogeschool te Delft, Netherlands.
[18]	Nakano, H. (1950) Modulared Semi-Ordered Spaces. Maruzen Co., Ltd., Tokyo.
[19]	Orlicz, W. (1961) A Note on Modular Spaces I. Bulletin L’Académie Polonaise des Science, Série des Sciences Mathé-matiques, Astronomiques et Physiques, 9, 157-162.
[20]	Mainka, E. (2010) On Uniformly Continuous Nemytskij Operators Generated by Set-Valued Functions. Aequationes Mathematicae, 79, 293-306. http://dx.doi.org/10.1007/s00010-010-0023-4
[21]	Fifer, Z. (1986) Set-Valued Jensen Functional Equation. Revue Roumaine de Mathématiques Pures et Appliquées, 31, 297-302.
[22]	Nikodem, K. (1989) K-Convex and K-Concave Set-Valued Functions. Zeszyty Nauk. Politech. Lódz. Mat. 559, Rozprawy Naukowe 114, Lódz, 1-75.