The Perm Seminar on Functional Differential Equations
(A
survey of the beginning and the development)
Perm State Technical University
29a, Komsomol'sky av., Perm, 614990, RUSSIA
The Rector of
Perm Polytechnic Institute (PPI) Professor M.N.Dedyukin invited Professor
N.V.Azbelev to move with his group to Perm from the Tambov Institute of Chemical
Engineering. At the beginning of 1975 the group came to Perm and founded the
Department of Mathematical Analysis. Soon after the Perm Seminar at the
department started its work. The Seminar was devoted to the functional
differential equation
(1)
the
natural generalization of the classical differential equation
Equation
(1) contains the integro-differential equation
,
the
equation "with delay"
and
other equations containing the derivative of the unknown function.
The
Perm Seminar prolonged traditions of the Tambov Seminar on equations with
deviated argument, but extended the area of its interests by the links between
various functional differential equations considered before without connection
with each other
The
Seminar united post-graduates, doctorants and some other scientists around
actual problems and turned out to be a kind of a research institute without
official status: articles in journals and lectures at conferences amplified the
reports of the PPI and Professor M.N.Dedyukin dreamed to found in course of
time an official research institute on mathematics and mechanics at the PPI and
patronized the Seminar in every way.
The
post graduate students and doctorants got a large section at the student's
hostel with four rooms and a black-board. At the black-board there were going
permanent discussions and the guests of the Seminar rehears their lectures.
From
1976 the Seminar began issue the year book "Boundary Value Problems" and from
1985 issued the other year book "Functional Differential Equations". The
editorial board of the issues contains some famous scientists such as
academicians I.T.Kiguradze, A.A.Pozdeev and the future President of the Russian
Academy of Science Yu.S.Osipov.
As
the guides of the Seminar as well as the advisers of post-graduates and
doctorants are the Professors N.V.Azbelev, V.P.Maksimov and L.F.Rakhmatullina.
These scientists of various character and mentality amplify very well each
other and work successfully on the detailed education of young specialists. At
the center of such a work are the sessions of the "Main Seminar" on Fridays and
the "Post-graduate's Seminar" on Wednesdays. The lecturers at the Seminar are
obliged to pay much attention to the precision and laconic of statements as
well as to the diction and the behavior at the black-board. At the arguing
lectures or a manuscript of an article is often used the following quotation
from Plutarch about Archimedes: "If anybody would try to solve this problem, he
never will find the solution. But after getting acquaintance with the solution
of Archimedes, one will immediately get the impression that he would be able to
solve the problem by himself. So simple and direct were the Archimedes' steps
to the aim.
After
the sessions of the Seminar arise at the hostel's black-board passionate
discussions the role of which it is difficult to overestimate from the point of
view of the education of the young people as well as in solutions of some
problems: the concentration of efforts of mathematicians of various mentality
and various level of the erudition led often to unexpected ideas and original
results and compelled to revise the previous results. The beloved motto at the
Seminar used the words of academician S.P.Novikov: "Mathematics is the
profession, but not an amusement!".
At
the discussions connected with the perennial topic around priority of others,
the following concept is recommended. Any theorem, concept or idea originates
from something that sometimes is difficult to determine and it is more
difficult to believe that getting such clear results has been so circuitous.
And very often, the final results is obtained by a more fortunate investigator
within his own interpretations of the problem and in view of work by his
predecessors rather than by those who got through the most difficult stages.
Sometimes these investigators do not refer and acknowledge their predecessors,
who have prepared the ground for modern developments, and they may consider the
results by the letter trivial. But they forget about the difficult steps
leading to simplicity.
The
investigation of 1975-1985 years have led the Seminar to an elegant theory of
equation (1). Boundary value problems became a key subject of theory. A special
attention was devoted to the equations with deviated argument. These results
are systematized in [1], where the main assertions assume that the operator 
is completely
continuous as one acting from the Banach space 
of absolute
continuous functions 
into the space 
of summable functions
. However equation (1), where 
is not completely
continuous, arises in applications. For instance, in the equations of the
"neutral type" or in the case when the deviated argument depends on the unknown
[2]. Such cases ever
attracted the attention of the Seminar. The discussions around the case when 
is not completely
continuous put the origin to the special direction in the investigations of
nonlinear problems. The direction got the name as "the doctrine on
reducibility": equation (1) was named to be reducible, if there exists a
completely continuous 
such that the
equations 
and 
are equivalent (the
sets of solutions coincide). The compactness of any bounded closed subset of
the whole set of solutions of equations (1) is necessary and sufficient for
reducibility of the equation. The behavior of solutions of non-reducible
equations is sometimes unexpected and paradoxical. The study of these,
so-called black-sheeps, to the habitual equations (1) with completely
continuous 
is up to now confined
by consideration some examples only.
The
logic of the development of our comprehension of functional differential
equations have led the Seminar to the farther generalization of the
differential equation. The idea of such generalization arised as follows. The
theory of equation (1) uses essentially the isomorphism between the space 
of absolutely
continuous functions 
(the functions
permitting the decomposition 
) and the direct product 
of the space 
of summable functions
and the finite-dimensional space 
. The space 
isomorphic to the
direct product 
(
) of an arbitrary Banach space 
and the
finite-dimensional space 
is a far off going
generalization of the space 
of absolutely continuous
functions. It turned out to be that some fundamental theorems on the equations
in the space 
keep in the general
space 
. Thus there arise the elements of the theory of equations in
the space 
(the "abstract
functional differential equations"). The monograph [3] is devoted to the new
broad concept of equations and results of developing the following directions
in the study of classical and new problems.
1.
Stability and asymptotic behavior of solutions
[3, 4]:
2.
Calculus of variations [3, 5].
3.
Singular boundary value problems [3].
4.
Constructive methods [3, 6, 7].
5.
Stochastic functional differential equations
[3, 8, 9].
6.
Reducibility of equations [3, 10].
The members of the Perm
Seminar were awarded 12 Doctor's degrees and more the 50 Candidate's degrees.
Remark. The research described in
this publication was made possible in part by Grant of the Russian Foundation
for Basic Research (Ώ 03-01-00255) and by Grant Ώ UR.04.01.001 of the Program
"Universities of Russia".
1.
N.V.Azbelev, V.P.Maksimov, L.F.Rakhmatullina.
Introduction to the theory of linear functional differential equations. World
Federation Publishers Company, Atlanta, 1995, 172p.
2.
S.A.Gusarenko, E.S.Zhukovskiy, V.P.Maksimov. On
the theory of functional differential equations with locally Volterra
operators. Sov. Math. Dokl., V. 33, 1986, pp.368-371.
3.
N.V.Azbelev, V.P.Maksimov, L.F.Rakhmatullina.
Elements of the contemporary theory of functional differential equations.
Methods and applications. Moscow: Institute of computer-assisted study, 2002,
384 p. (Russian).
4.
N.V.Azbelev, P.M. Simonov. Stability of
differential equations with aftereffect. London and New York, Taylor and
Francis, 2002.
5.
N.V.Azbelev, E.I.Bravyi, S.A.Gusarenko. On the
question of effective sufficient conditions for solvability of variational
problems. Dokl. Ross. Akad. Nauk, V. 381, Ώ 2, 2001, pp.1-4 (Russian).
6.
V.P.Maksimov, A.N.Rumyantsev. Boundary value
problems and problems of pulse control in economic dynamics. Constructive
study. Russ. Math. (Iz. VUZ), V. 37, Ώ 5, 1993, pp.48-62.
7.
A.N.Rumyantsev. Reliable computing experiment
in the study of boundary value problems. Perm: Perm State University, 1999. 174
p. (in Russian).
8.
Ponosov A.V. Fixed point method in the theory
of stochastic differential equations. Sov. Math. Dokl., V. 37, Ώ 2, 1988,
pp.426-429.
9.
A.V.Ponosov. Local operators and stochastic
differential equations. Functional Differential Equations, V. 4, Ώ 1-2, 1997,
pp.173-189.
10. Yu.V.Nepomnyashchikh. On reducibility of functional differential
equations. Nonlinear Analysis and Nonlinear Differential Equations, Editors
V.A.Trenogin and A.F.Filippov, Moscow: FIZMATLIT, 2003, pp.305-316 (Russian).
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