Gap in Nonlinear Equivalence for Numerical Methods for PDEs

For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general nonlinear equivalence between stability and convergence. 1. The General Setup The study of linear and nonlinear evolution systems of PDEs, with possibly associated initial and/or boundary problems, as is well known, can be dealt with in the more general framework of semigroups depending on a continuous parameter which represents time. Here, the study of the nonlinear equivalence for numerical methods approximating exact solutions nonlinear evolution PDEs will be dealt with in this more general framework, namely, of numerical methods approximating rather general nonlinear semigroups. Definition 1.1. Given a normed vector space (X, || ||). By a nonlinear semigroup on