  We investigate the existence and regularity of (locally) invariant manifolds
nearby an approximately invariant set satisfying certain (geometric)
hyperbolicity with respect to an abstract `generalized' dynamical system in a
Banach space; such hyperbolicity is between normal hyperbolicity and partial
hyperbolicity which has being studied in the finite-dimension and in some
concrete PDEs. The `generalized' dynamical system is allowed to be non-smooth,
non-Lipschitz, or even `non-mapping', making it applicable to both well-posed
and ill-posed differential equations. As an illustration, we apply our results
to study the dynamics of the whiskered tori.
