In this paper we consider the so called composition operator being a self-mapping of the Banach algebra of the function of two variables with bounded total Φ-variation in the Schramm sense. The main result of the paper characterizes the composition operator mentioned above which has a generating function being Lipschitzian with respect to the second variable. The basic tool used in our considerations is the concept of the left-left regularization.
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Assume that the generator of a Nemytskii composition operator is a function of three variables: the first two real and third in a closed convex subset of a normed space, with values in a real Banach space. We prove that if this operator maps a certain subset of the Banach space of functions of two real variables of bounded Wiener φ-variation into another Banach space of a similar type, and is uniformly continuous, then the one-sided regularizations of the generator are affine in the third variable.
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