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Nemytskii Operator in the Space of Set-Valued Functions of Bounded φ-Variation

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DOI: 10.4236/apm.2013.36072    3,406 Downloads   6,514 Views  
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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 φ1-variation in the sense of Riesz with respect to the weight function αinto the space of set-valued functions of bounded φ2-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 φ2-variation in the sense of Riesz with respect to the weight.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

W. Aziz, "Nemytskii Operator in the Space of Set-Valued Functions of Bounded φ-Variation," Advances in Pure Mathematics, Vol. 3 No. 6, 2013, pp. 563-575. doi: 10.4236/apm.2013.36072.

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