Composing infinitely differentiable functions
Given a Banach space X and a smooth function f: [a,b]\(\times X\to X\), we provide necessary and sufficient conditions under which the nonlinear superposition operator \(Fx(s)=f(s,x(s))\) acts between two Roumieu classes \(R_{\mu}(L,X)\) and \(R_{\mu}(L',X)\), or between the projective and inductive limits \(R_{\mu}(0,X)\) and \(R_{\mu}(\infty,X)\), respectively. Moreover, we discuss some boundedness, continuity, and compactness properties of F in such spaces.
scientific article; zbMATH DE number 4171783
scientific article