Persistence of a normally hyperbolic manifold for a system of non densely defined Cauchy problems

We consider a system of non densely defined Cauchy problems and we investigate the persistence of normally hyperbolic manifolds. The notion of exponential dichotomy is used to characterize the normal hyperbolicity and a generalized Lyapunov-Perron approach is used in order to prove our main result. The result presented in this article extend the previous results on the center manifold by allowing a nonlinear dynamic in the unperturbed central part of the system. We consider two examples to illustrate our results. The first example is a parabolic equation coupled with an ODE that can be considered as an interaction between an antimicrobial and bacteria while the second one is a Ross-Macdonald epidemic model with age of infection. In both examples we were able to reduce the infinite dimensional system to an ordinary differential equation. (C) 2019 Elsevier Inc. All rights reserved.