Lp stability and linearization

A theorem by Hadamard gives a two-part condition under which a map from one Banach space to another is a homeomorphism. The theorem, while often very useful, is incomplete in the sense that it does not explicitly specify the family of maps for which the condition is met. Recently, under a typically weak additional assumption on the map, it was shown that Hadamard´s condition is met if and only if the map is a homeomorphism with a Lipschitz continuous inverse. Here, an application is given concerning the relation between the L_p stability (with 1 ≤ p < ∞) of a nonlinear system and the stability of related linear systems. We also give a result that directs attention to a fundamental limitation concerning what can be proved about linearization and stability for a related familiar family of feedback systems.