A two spaces extension of Cauchy-Lipschitz theorem

We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential equations, we consider x˙=f(x,x) for a function f:V×E→E where E is a Banach space and V↪E a normed vector space. This structure allows us to distinguish between the two dependencies of f in x and allows to generalize classical results. We also prove a similar result for partial differential equations.