# MR3058477 Reviewed Ereú, Thomás; Sánchez, José L.; Merentes, Nelson; Wróbel, Małgorzata Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Schramm. Pr. Nauk. Akad. Jana Długosza Częst. Mat. 16 (2011), 23–32. ISBN: 978-83-7455-209-7

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## Abstract

In this paper it is established a property of a composition operatorbetween spaces of functions of bounded variation in the sense of Schramm.Let X and Y be two real normed spaces, C a convex cone in X andI a closed bounded interval of the real line. Moreover let cc(Y) bethe family of all non-empty closed convex and compact subsets of Y. Theauthors study the Nemytskij (composition) operator (HF)(t)=h(t,F(t)),where F: I \rightarrow C and h: I\times C \rightarrow cc(Y) is a givenset-valued function. They show that if the Nemytskij operator $H$ isuniformly continuous and maps the space \Phi BV (I;C) of functions(from I to C) of bounded \Phi-variation in the sense of Schramminto the space $BS_{\Psi}(I; cc(Y))$ of set-valued functions (from I tocc(Y)) of bounded \Psi-variation in the sense of Schramm, then theone-sided regularizations h^- and h^+ of h with respect the firstvariable are affine with respect to the second variable. Reviewed by ( L. Di Piazza)
Original language English 1 MATHEMATICAL REVIEWS MR3058477 Published - 2011

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