Lipschitz and Darbo conditions for the superposition operator in ideal spaces (Q1099398)

We give necessary and sufficient conditions for the superposition operator \(Fx(s)=f(s,w(s))\) to satisfy a Lipschitz condition \(\| Fx_ 1-Fx_ 2\| \leq k\| x_ 1-x_ 2\|\) or a Darbo condition \(\alpha\) (FN)\(\leq k\alpha (N)\) in ideal spaces of measurable functions, where \(\alpha\) is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.