Existence and regularity of invariant graphs for cocycles in bundles: partial hyperbolicity case

We study the existence and regularity of the invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relatively partial hyperbolicity in non-trivial bundles without local compactness. The regularity includes (uniformly) $ C^0 $ continuity, Holder continuity and smoothness. A number of applications to both well-posed and ill-posed semi-linear differential equations and the abstract infinite-dimensional dynamical systems are given to illustrate its power, such as the existence and regularity of different types of invariant foliations (laminations) including strong stable laminations and fake invariant foliations, the existence and regularity of holonomies for cocycles, $ C^{k,\alpha} $ section theorem and decoupling theorem, etc, in more general settings.