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

DOI: 10.4236/apm.2013.36072   PDF   HTML     3,566 Downloads   6,785 Views  

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.

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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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