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.

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Mathematics Subject Classification (2010): Primary 93B05; Secondary 93C25.

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In this note we give a condition on a function h : R → R, both necessary and sufficient, under which the nonlinear composition operator Hf = h • f maps the space of all regular functions f : [a, b] → R into itself. Moreover, we show that in this case the operator H is automatically bounded in the supremum norm.