On superposition operators in spaces \(BV_{\varphi}(0,1)\). (English) Zbl 1348.47047
Summary: For a function \(f : [0, 1] \times \mathbb{R} \to \mathbb{R}\) the superposition operator \(S_f : \mathbb{R}^{[0, 1]} \to \mathbb{R}^{[0, 1]}\) is defined by the formula \(S_f(\varphi)(t) = f(t, \varphi(t))\). We study properties of operators \(S_f\) in Banach spaces \(BV_\varphi(0, 1)\) of all real functions of bounded \(\phi\)-variation on \([0, 1]\) for convex functions \(\phi\). We show that if an operator \(S_f\) maps the space \(B V_\varphi(0, 1)\) into itself, then (1) it maps bounded subsets of \(B V_\varphi(0, 1)\) into bounded sets if additionally \(f\) is locally bounded, (2) \(f = f_{\mathrm{cr}} + f_{\mathrm{dr}}\) where the operator \(S_{f_{\mathrm{cr}}}\) maps the space \(D(0, 1) \cap B V_\varphi(0, 1)\) of all right-continuous functions in \(B V_\varphi(0, 1)\) into itself and the operator \(S_{f_{\mathrm{dr}}}\) maps the space \(B V_\varphi(0, 1)\) into its subset consisting of functions with countable support. Moreover we show that if an operator \(S_f\) maps the space \(D(0, 1) \cap B V_\varphi(0, 1)\) into itself, then \(f\) is locally Lipschitz in the second variable uniformly with respect to the first variable.
MSC:
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
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