HOW IT IS HAPPENED

(about fundamental stages

in functional-differential equations modern theory development)

N.V.Azbelev

Perm State Technical University

29a, Komsomol’sky av, Perm, 614990, RUSSIA

 

It is given a short survey of the rise and development of the contemporary theory of functional differential equations. The beginning of the theory was originated from the attempts to construct an acceptable concept of equations with deviated argument (EDA). Many actual problems connected with EDA attracted mathematicians and some of them revealed in their works a great skill, but they were forced to prove the assertions of one and the same type in various cases. It was obvious that a concrete theory of EDA was in need, but on the way to such a theory was a durable tradition to use the obsolete definition of the notion of solutions accepted by famous authors such as N.N.Krasovskii, A.D.Myshkis and J.Hale… In the seventies of the past century there was put forth by the Tambov seminar a more contemporary definition based on the ideas of functional analysis. Such a definition gave the possibility to consider EDA as a case of the so-called functional differential equation (FDE), that is a general equation in the space of absolutely continuous functions. Mainly the case of equations with delay and the Chaplygin like theorems on inequalities were in the center of attention. There was discovered the fact that the boundary value problem for EDA turned out to be equivalent to the classical integral equation similarly to the ordinary differential equation. The Tambov seminar introduced a special decomposition of the linear operator acting from the space of absolutely continuous functions in the space of summable functions. On the base of the decomposition the question on the adjoint boundary value problem got the complete solution. In the theory of FDE an important pace belongs to the Green operator and the theorem about the existence in an explicit form of the so-called “elementary Green operator” for a “model boundary value problem”. A connection between Greens operators of boundary value problems for linear FDE and corresponding integral equations was a serious discovery of the seminar.

The Perm seminar developed the theory of FDE. At this work the role of the isomorphism between the space of absolutely continuous functions and the direct product of the space of summable functions and the space of vectors with real components was revealed. Besides it was observed that the replacement of the Lebesgue space by any Banach one leads to a far going generalization of the space of absolutely continuous functions. Thus aroused the theory of equations in such a general space, the theory of the so-called abstract FDE. The Perm seminar thoroughly studied the new generalization of ordinary differential equations. The study opened broad perspectives and allowed the consideration of wide classes of equations from the unique point of view and by unique scheme. Some classes of singular equations and systems of FDE with impulse actions are among such equations. The so-called “neutral equations” are in general up to now a very difficult object for investigation. As for such equations, the seminar developed the studies of “reducibility”. A reducible neutral equation is one of “high quality”: standard methods of the analysis are applicable to it, the set of its solutions is subjected to a clean order. The absence of reducibility means the existence of superfluous solutions, which possibly are not adequate to the original problem…  The results of the Perm seminar makes it possible to offer more perfect approaches to some actual problems, in this number to the variational problems and to the theory of stability.