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Uniformly bounded set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener

Guerrero J. A., Matkowski J., Merentes N., Wróbel M.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 4

41--51

EN

We show that the one-sided regularizations of the generator of any uniformly bounded set-valued Nemytskij composition operator mapping the space of bounded variation functions in the sense of Wiener into the space of bounded variation functions with closed bounded convex values (in the sense of Wiener) are affine functions.

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On Nemytskii operator in the space of set-valued functions of bounded p-variation in the sense of Riesz with respect to the weight function

Aziz W., Guerrero J. A., Merentes N.

Fasciculi Mathematici

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2013

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Nr 50

17--32

EN

In this paper we consider the Nemytskii operator (Hf) (t) = h(t, f (t)), generated by a given set-valued function h is considered. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded p-variation (with respect to a weight function α) into the space of set-valued functions of bounded q-variation (with respect to α) ) 1 < q < p, then H is of the form (Hϕ)(t) = A(t)ϕ(t) + B(t). On the other hand, if 1 < p < q, then H is constant. It generalizes many earlier results of this type due to Chistyakov, Matkowski, Merentes-Nikodem, Merentes-Rivas, Smajdor-Smajdor and Zawadzka.

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Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener

Azocar A., Guerrero J. A., Matkowski J., Merentes N.

Opuscula Mathematica

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2010

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Vol. 30, no. 1

53-60

EN

We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.

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Uniformly continuous composition operators in the space of functions of two variables of bounded φ-variation in the sense of Wiener

Guerrero J.A., Matkowski J., Merentes N.

Commentationes Mathematicae

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2010

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Vol. 50, [Z] 1

41-48

EN

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.