Lipschitz and Darbo conditions for the superposition operator in some non-ideal spaces of smooth functions (Q1175923)

Let \(X\) be a Banach space of differentiable functions and \(A:X\to X\) be a superposition operator. We consider for \(A\) the conditions \[ (1) \parallel Ax_ 1-Ax_ 2| X\parallel \leq k \parallel x_ 1-x_ 2| X\parallel (x_ 1,x_ 2\in X), \] \[ (2) \alpha (AM| X)\leq K\alpha (M| X) (M\subset X bounded), \] where \(\alpha (M| X)\) denotes the Hausdorff measure of noncompactness. We characterize (1) and (2) in some nonideal spaces \(X\) which occur frequently in applications.