[For the entire collection see Zbl 0718.00007.]
This paper is a systematic review of the recent achievements of the mathematicians who communicate with the Perm’ seminar on functional- differential equations. The basic concept of this investigation is a non conventional concept of abstract functional-differential equations. This system of views enables a uniform approach to distinct problems for different types of functional-differential equations to be developed. For example, the stability of linear equations is the solutions belong to some functional space. This approach is suitable for linear impulse systems and brought to the new non conventional notion of generalized Green’s operator for a linear boundary value problem. In particular, it is the generalized Green’s operator introduced by the reviewer for the case when kernel and co-kernel of a linear boundary value problem may be infinite-dimensional. Furthermore, the offered conception enables successfully to solve problems of optimal control and to develop some constructive methods of the theory of functional-differential equations. Finally, the abstract functional-differential equation as equation in Banach spaces with special structure brought the Perm’ seminar to the systematic investigation of nonlinear boundary value problems in Banach spaces. In particular, the resonance boundary value problems of which the linear part is a topologically Noetherian operator, i.e. its kernel and co-kernel may be infinite-dimensional, but its kernel and image must be complemented. This case is typical for neutral type functional- differential equations.