International Journal of Pure and Applied Mathematics ————————————————————————– Volume 69 No. 2 2011, 203-235 THEORY OF FUNCTIONAL DIFFERENTIAL EQUATIONS AND APPLICATIONS Nikolay V. Azbelev, Vladimir P. Maksimov 2 § , Pyotr M. Simonov 3 2,3 Perm State University Bukirev Str. 15, 614990, RUSSIA 2 e-mail: maksimov@econ.psu.ru 3 e-mail: simonov@econ.psu.ru Abstract: A survey of the theory of functional differential equations, worked out by participants of the Perm Seminar, Russia, is presented. Examples of new approaches to some classical problems are demonstrated. AMS Subject Classification: 34K10, 34K20, 34K30, 34K35, 34K50 Key Words: functional differential equations, boundary value problems, asymptotic behaviour, variational problems, control problems, computer-assisted study 1. Introduction Over the last 30 years a large group of mathematicians united by the so-called Perm Seminar [4] worked out a unified theory of a wide generalization to the or- dinary differential equation. This theory established a close connection between many problems, previously studied without any connection, and proposed more sophisticated methods to solve those problems. In this paper, we give a brief survey of the general theory and demonstrate some examples of new approaches to a number of classical problems with the aim to illustrate the efficiency of the developed theory. Received: April 11, 2011 c  2011 Academic Publications § Correspondence author