TITLE:
Nemytskii Operator in the Space of Set-Valued Functions of Bounded φ-Variation
AUTHORS:
Wadie Aziz
KEYWORDS:
Bounded Variation; Function of Bounded Variation in the Sense of Riesz; Variation Space; Weight Function; Banach Space; Algebra Space
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.6,
August
29,
2013
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.