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Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

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Merentes N., Hernández J.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

tom 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 Nemytskiĭ 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 $RV_{φ₁}([a,b];K)$ into $RW_{φ₂}([a,b];CC(Y))$ (both are spaces of functions of bounded φ-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 φ₂-variation in the sense of Riesz. This generalizes results of G. Zawadzka [12], A. Smajdor and W. Smajdor [11], N. Merentes and K. Nikodem [5], and N. Merentes and S. Rivas [7].

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Hernández J.

1 Merentes N.

1 2004

Wyniki wyszukiwania

Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz