Characterization of globally Lipschitz Nemytskii operators between spaces of set-valued functions of bounded [phi]- variation in the sense of Riesz

• Autorzy: Merentes N. , Hernandez S.
• Języki publikacji: EN

Abstrakty

EN

Let (X, || . ||) and [Y, || . ||] be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskii operators, i.e. the composition operators defined by [Nu)(t) = H(t,u[t)), where H is a given set-valued function. It is shown that if the operator N maps the space RV[phi]1 ([a, b]; K) into RW[phi]2([a, b]; CC[Y)) (both are spaces of functions of bounded [phi]- variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u[t)) = A(t]u(t)+B(t), where A(t) is a linear continuous set-valued function and B is a set-valued function of bounded [phi]2-variation in the sense of Riesz. This generalizes results of G. Zawadzka [12], A. Smajdor and W. Smajdor [II], N. Merentes and K. Nikodem [5], and N. Merentes and S. Rivas [7].

Słowa kluczowe

EN

Nemytskii operator; set-valued functions; Jensen equation; composition operator; phi-variation in the sense of Riesz

PL

równanie Jensena

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 4
• Strony: 417--430
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Merentes N.

• Departamento de Matemática, Universidad Central de Venezuela, Caracas 1020A, Venezuela

autor

Hernandez S.

• School of Mathematics, Georgia Tech, Atlanta, GA 30332-0160, U.S.A.

Bibliografia

• [1] H. Herda, Modular spaces of generalized variation, Studia Math. 30 (1968), 21-42.
• [2] L. Maligranda and W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math. 104 (1987), 53-65.
• [3] J. Matkowski, Functional equations and Nemytskiĭ operators, Funkcial. Ekvac. 25 (1982), 127-132.
• [4] J. Matkowski and J. Miś, On a characterization of Lipschitzian operators of substitution in the space BV [a; b], Math. Nachr. 117 (1984), 155-159.
• [5] N. Merentes and K. Nikodem, On Nemytskii operator and set-valued functions of bounded p-variation, Rad. Mat. 8 (1996), 139-145.
• [6] N. Merentes, K. Nikodem and S. Rivas, Continuity of the superposition of set-valued functions, J. Appl. Anal. 3 (1997), 43-48.
• [7] N. Merentes and S. Rivas, On Nemytskiĭ operator in the space of set-valued functions of bounded p-variation in the sense of Riesz, Publ. Math. Debrecen 47 (1995), 15-27.
• [8] K. Nikodem, K-convex and K-concave set-valued functions, Politech. Łódź., Zeszyty Nauk. 559, Rozprawy Naukowe Z 114, Łódź, 1989.
• [9] H. Radström, An embedding theorem for space of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169.
• [10] F. Riesz, Untersuchugen über Systeme integrierbarer Funktionen, Math. Ann. 69 (1910), 449-497.
• [11] A. Smajdor and W. Smajdor, Jensen equation and Nemytskij operator for set-valued functions, Rad. Mat. 5 (1989), 311-319.
• [12] G. Zawadzka, On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation, ibid. 6 (1990), 179-193.