Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Schramm

• Autorzy: Ereú T. , Sánchez J. L. , Merentes N. , Wróbel M.
• Języki publikacji: EN

Abstrakty

EN

We show that the one-sided regularizations of the generator of any uniformly continuous set-valued Nemytskij operator, acting between the spaces of functions of bounded variation in the sense of Schramm, is an affine function. Results along these lines extend the study [1].

Słowa kluczowe

EN

functions; functions of bounded; Nemytskij operator; affine function

PL

funkcje; funkcja ograniczona; operator Nemytskija; funkcja afiniczna

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 2011
• Tom: Vol. 16
• Strony: 23--32
• Opis fizyczny: Bibliogr. 26 poz.

Twórcy

autor

Ereú T.

• Universidad Nacional Abierta, Centro Local Lara (Barquisimeto)-Venezuela

autor

Sánchez J. L.

• Universidad Central de Venezuela, Escuela de Matemáticas, Caracas-Venezuela

autor

Merentes N.

• Universidad Central de Venezuela, Escuela de Matemáticas, Caracas-Venezuela

autor

Wróbel M.

• Institute of Mathematics and Computer Science Jan Długosz University in Częstochowa, 42-200 Częstochowa, Poland

Bibliografia

• [1] A. Azócar, J.A. Guerrero, J. Matkowski, N. Merentes. Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener. Opuscula Mathemat., 30,No. 1, 53-60, 2010.
• [2] A. Smajdor, W. Smajdor. Jensen equation and Nemytskii operator for set-valued functions. Radovi Matematicki, 5, 311-320, 1989.
• [3] G. Zawadzka. On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation. Radovi Matematicki, 6, 279-293, 1990.
• [4] H. Rådström. An embedding theorem for space of convex sets. Proc. Amer. Math. Soc., 3, 165-169, 1952.
• [5] H. Nakano. Modulared Semi-Ordered Linear Spaces. Maruzen, Tokyo 1950.
• [6] J. Appell, P.P. Zabrejko. Nonlinear Superposition Operator. Cambridge University Press, New York 1990.
• [7] J.A. Guerrero, J. Matkowski, N. Merentes, J.L. Sánchez. Uniformly continuous set-valued composition operators in the space of functions of the Wiener bounded p-variation, accepted for J. Math. Appl.
• [8] J. Matkowski. Functional equations and Nemytskij operators. Funkc. Ekvacioj Ser. Int., 25, 127-132, 1982.
• [9] J. Matkowski. Lipschitzian composition operators in some function spaces. Nonlinear Anal., 3, 719-726, 1997.
• [10] J. Matkowski. Uniformly continuous superposition operators in the space of differentiable function and absolutely continuous functions. Internat. Ser. Numer. Math., 157, 155-166, 2008.
• [11] J. Matkowski. Uniformly continuous superposition operators in the space of Hölder functions. J. Math. Anal. Appl., 359, 56-61, 2009.
• [12] J. Matkowski. Uniformly continuous superposition operators in the space of bounded variation functions. Math. Nachr., 283, 1060-1064, 2010.
• [13] J. Matkowski, J. Miś. On a charaterization of Lipschitzian operators of substitution in the space BV (a, b). Math. Nachr., 117, 155-159, 1984.
• [14] J. Musielak, W. Orlicz. On generalized variations (I). Studia Math., XVIII, 11-41, 1959.
• [15] K. Nikodem. K-convex and K-concave set-valued functions. Zeszyty Naukowe Politechniki Łódzkiej, Matematyka, 559, Rozprawy Naukowe 114, Łódź 1989.
• [16] L. Maligranda, W. Orlicz. On some properties of functions of generalized variation. Monatsch. Math., 104, 53-65, 1987.
• [17] L.C. Young. Sur une généralisation de la notion de variation de puissance p-ieme bornée au sens de N. Wiener, et sur la convergence des séries deFourier. C. R. Acad. Sci., 204, 7, 470-472, 1937.
• [18] M. Schramm. Functions of φ-bounded variation and Riemann-Stieltjes integration. Trans. Amer. Math. Soc., 287, 49-63, 1985.
• [19] M. Kuczma. An Introduction to the Theory of Functional Equations and Inequalities. Polish Sientific Publishers, Warszawa-Kraków-Katowice 1985.
• [20] N. Merentes. Composition of functions of bounded ϕ-variation. P.U.M.A., Ser. 1, 39-45, 1991.
• [21] N. Wiener. The quadratic variation of function and its Fourier coefficients. Massachusetts J. Math., 3, 72-94, 1924.
• [22] V.V. Chistyakov. Generalized variation of mappings with applications to composition operators and multifunctions. Positivity, 5, 4, 323-358, 2001.
• [23] V.V. Chistyakov. Modular metric spaces, I: Basic concept. Nonlinear Analysis: Theory, Methods & Applications, 72, 1-14, 2010.
• [24] V.V. Chistyakov. Modular metric spaces, II: Application to superposition operators. Nonlinear Analysis: Theory, Methods & Applications, 72, 15-30, 2010.
• [25] W.A.J. Luxemburg. Banach Function Spaces, Ph.D. thesis, Delft Institute of Technology, Assen, The Netherlands 1955.
• [26] W. Orlicz. A note on modular spaces. I. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 9, 157-162, 1961.