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Nemytskii Operator in the Space of Set-Valued Functions of Bounded <i>φ</i>-Variation
Departamento de Fsica y Matemática, Universidad de Los Andes, Trujillo, RBV
- 1 Departamento de Fsica y Matemática, Universidad de Los Andes, Trujillo, RBV
Advances in Pure Mathematics·Volume 03 (2013)·Pages 563–575·Published 29 August 2013·DOI10.4236/apm.2013.36072
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Abstract
In this paper we consider the Nemytskii operator, i.e ., the composition operator defined by ( Nf )( t )= H ( t , f ( t )) , where H is a given set-valued function. It is shown that if the operator N maps the space of functions bounded φ 1 -variation in the sense of Riesz with respect to the weight function α into the space of set-valued functions of bounded φ 2 -variation in the sense of Riesz with respect to the weight, if it is globally Lipschitzian, then it has to be of the form ( Nf )( t )= A ( t ) f ( t )+ B ( t ) , where A ( t ) is a linear continuous set-valued function and B is a set-valued function of bounded φ 2 -variation in the sense of Riesz with respect to the weight.
KeywordsBounded VariationFunction of Bounded Variation in the Sense of RieszVariation SpaceWeight FunctionBanach SpaceAlgebra Space
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