  We study the existence and regularity of invariant graphs for bundle maps (or
bundle correspondences with generating bundle maps motivated by ill-posed
differential equations) having some relative partial hyperbolicity on
non-trivial and non-locally-compact bundles. The regularity includes (uniform)
$ C^0 $ continuity, H\"older continuity and smoothness. To illustrate the power
of our results and methods, a number of applications to both well-posed and
ill-posed semilinear differential equations and abstract infinite-dimensional
dynamical systems are given. These applications include 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, the $ C^{k,\alpha} $ theorem and
decoupling theorem, etc., in a general setting.
