Asymptotically Almost Periodic Solutions of Differential Equations

Résumé du livre

Nonlocal problems concerning the conditions of the existence of different classes of solutions play an important role in the qualitative theory of differential equations. The present work belongs to this direction and is dedicated to the study of asymptotically almost periodic motions of dynamical systems and solutions of differential equations. First, the notion of asymptotically almost periodicity of functions was introduced and studied in the works of M. Fréchet (1941). In the present work, the general problem of the asymptotically Poisson stability (in particular, asymptotically almost periodicity) of the motions of dynamical systems and solutions of differential equations is studied. From the point of view of applications, motions of dynamical systems are naturally divided into transitional (nonstabilized) and stabilized. When we try to define a nonstabilized motion exactly, we come to the notion of the asymptotically stability in the sense of Poisson motion. Such motions are of interest for applications and are met, for instance, in systems possessing stable oscillatory regime. This book is the first, where the asymptotically almost periodic motions of dynamical systems and solutions of differential equations are systematically studied. We study this problem in the framework of general nonautonomous dynamical systems.