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

Invariant manifolds of partially normally hyperbolic invariant manifolds in Banach spaces