On Lipschitzian operators of substitution generated by set-valued functions

• Autorzy: Ludew J. J.
• Języki publikacji: EN

Abstrakty

EN

We consider the Nemytskii operator, i.e., the operator of substitution, defined by (Nφ)(x) := G(x,φ(x)), where G is a given multifunction. It is shown that if N maps a Hölder space Hα into Hβ and N fulfils the Lipschitz condition then G(x,y) = A(x,y) + B(x), where A(x,·) is linear and A(·,y), B ∈ Hβ. Moreover, some conditions are given under which the Nemytskii operator generated by (1) maps Hα into Hβ and is Lipschitzian.

Słowa kluczowe

EN

Nemytskii operator; Holder functions; set-valued functions; Jensen equation

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2007
• Tom: Vol. 27, no. 1
• Strony: 13--24
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Ludew J. J.

• Silesian University of Technology, Institute of Mathematics, ul. Kaszubska 23, 44-100 Gliwice, Poland,

Bibliografia

• [1] J. Appel, P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.
• [2] A. Matkowska, On a characterization of Lipschitzian operators of substitution in the class of Holders functions, Zeszyty Nauk. Politech. Łódz. Mat. 17 (1984), 81-85.
• [3] J. Matkowski, Functional equations and Nemytskij operators, Funkc. Ekvacioj Ser. Int. 25 (1982), 127-132.
• [4] J. Matkowski, Lipschitzian composition operators in some function spaces, Nonlinear Anal., Theory, Methods Appl. 30, Proc. 2nd World Congress of Nonlinear Analysis (1997), 719-726.
• [5] K. Nikodem, K-convex and K-concave set-valued functions, Politechnika Łódzka, Zeszyty Nauk. Politech. Łódz., Mat. 559, Rozprawy Nauk. 114, 1989.
• [6] H. Radstrom, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169.
• [7] W. Rudin, Functional analysis, McGraw-Hill Book Company, 1973.
• [8] A. Smajdor, On regular multivalued cosine families, Ann. Math. Sil. 13 (1999), 271-280.
• [9] A. Smajdor, W. Smajdor, Jensen equation and Nemytskii operator for set-valued functions, Rad. Mat. 5 (1989), 311-320.
• [10] G. Zawadzka, On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation, Rad. Mat. 6 (1990), 279-293.