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