Contracting differential equations in weighted Banach spaces

Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite the Riemannian metric. In infinite-dimensional setting, however, such analysis generally restricted to norm-contracting systems. We develop a generalization of geodesic rates Banach spaces using smoothly-weighted semi-inner product structure on tangent spaces. show that negative bijectively weighted imply asymptotic norm-contraction, and apply recent results contractions establish existence fixed points. surjectively verifies non-equilibrium properties, as convergence finite- subspaces, submanifolds, limit cycles, phase-locking phenomena. use Sobolev continuous data dependence nonlinear PDEs, pose method for constructing weak solutions vanishing one-sided Lipschitz approximations. discuss applications control order reduction PDEs.