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 into (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].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-8,
author = {N. Merentes and J. L. S\'anchez Hern\'andez},
title = {Characterization of Globally Lipschitz Nemytski\u\i\ Operators Between Spaces of Set-Valued Functions of Bounded $\phi$-Variation in the Sense of Riesz},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {52},
year = {2004},
pages = {417-430},
zbl = {1097.47050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-8}
}
N. Merentes; J. L. Sánchez Hernández. Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) pp. 417-430. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-8/