Uniformly Continuous Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz

Abstract

We show that any uniformly continuous and convex compact valued Nemytskiȋ composition operator acting in the spaces of functions of bounded ϕ-variation in the sense of Riesz is generated by an affine function.

Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz

“…Let clb Y . This considerably extends the main result of the paper [1] where the uniform continuity of the operator H is assumed.…”

“…The first paper concerning composition operators in the space of bounded variation functions was written by J. Miś and J. Matkowski in 1984 [2]; these results shown here have been verified by varying the hypothesis, in other contributions (see for example, [1] [3]- [7]).…”

SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV2α(I)

“…The uniformly continuous composition operators were firstly considered in [12] for the space of differentiable functions and absolutely continuous functions, later in [13] for the space of Hölder function, and in [14] for the space of bounded variation functions. Later, these were used in the main result of the papers [1,2,3,5].…”