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

Wadie Aziz

Departamento de Fsica y Matemática, Universidad de Los Andes, Trujillo, RBV.

**DOI: **10.4236/apm.2013.36072
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Departamento de Fsica y Matemática, Universidad de Los Andes, Trujillo, RBV.

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.

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

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