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2 Bulletin of the Polish Academy of Sciences. Mathematics

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

Aziz W., Guerrero J., Merentes N.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 1

39-45

We show that any uniformly continuous and convex compact valued Nemytskii composition operator acting in the spaces of functions of bounded φ-variation in the sense of Riesz is generated by an affine function.

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

Merentes N., Hernandez S.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

417-430

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].