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

N. Merentes, and 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 52.4 (2004): 417-430. <http://eudml.org/doc/280638>.

@article{N2004,
abstract = {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].},
author = {N. Merentes, J. L. Sánchez Hernández},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Nemytskij operator; set-valued functions; Jensen equation; composition operator; -variation in the sense of Riesz},
language = {eng},
number = {4},
pages = {417-430},
title = {Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz},
url = {http://eudml.org/doc/280638},
volume = {52},
year = {2004},
}

TY - JOUR
AU - N. Merentes
AU - J. L. Sánchez Hernández
TI - Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2004
VL - 52
IS - 4
SP - 417
EP - 430
AB - 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].
LA - eng
KW - Nemytskij operator; set-valued functions; Jensen equation; composition operator; -variation in the sense of Riesz
UR - http://eudml.org/doc/280638
ER -