A new approach to the question of the existence of bounded solutions of functional differential equations of point type

We develop an approach which we used to deduce conditions of a new type for the existence of periodic solutions of ordinary differential equations and functional differential equations of point type. These conditions are based on the use of asymptotic properties of solutions of differential equations which can be observed on shifts of solutions and stated in terms of averages over the period on a distinguished sphere in the phase space. The development of this approach enables us to obtain conditions for the existence of bounded solutions for the same classes of functional differential equations.