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Uniformly continuous set-valued composition operators in the space of total \(\varphi\)-bidimensional variation in the sense of Riesz. (English) Zbl 1229.47087

The authors prove that if a Nemytskij composition operator, generated by a function of three variable in which the third variable is a function one, maps suitable large subset of the space of functions of bounded total \(\varphi\)-bidimensional variation in the sense of Riesz, into another such space, and is uniformly continuous, then its generator is an affine function in the function variable. This extends some previous results in the one-dimensional setting.

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
39B52 Functional equations for functions with more general domains and/or ranges
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