Аннотация:The paper is devoted to the study of singularly perturbed differential systems in Banach spaces, when the fast subsystem is stable, but is not asymptotically stable. The results are very significant for the theory and applications of singular perturbations problems. For the investigation of such problems the generalized Lyapunov functional method is used. This method allows authors to study some problems of hydrodynamics and gas dynamics and the problem of neuron activity modelling.