Singular perturbation for an abstract non-densely defined Cauchy problem

This work is devoted to the study of a class of singularly perturbed non-densely defined abstract Cauchy problems. We extend the Tikhonov theorem for ordinary differential equations to the case of abstract Cauchy problems. Roughly speaking we prove that the solutions rapidly evolve and stay in some neighbourhood of the slow manifold. As a consequence we conclude that the solutions of the problem converge on each compact time interval, as the singular parameter goes to zero, towards the solutions of the so-called reduced problem. These results are applied to an example of age-structured model as well as to a class of functional differential equations.