Samoilenko had been appointed as a senior research fellow at the Institute of Mathematics of the Academy of Sciences of the Ukraine in Kiev in 1965, and he also taught at the Shevchenko Kiev State University from 1967. In 2000, he reminisced about his years as a young scientist at the Institute (see ):-
In Kiev, at the Institute of Mathematics, great scientists were my teachers ... In many fields of science, they were 'trendsetters' on the scale of the Soviet Union. It is very important for a young scientist to belong to a serious scientific school. Probably, only in this case he has a chance to obtain results at the world level. The atmosphere of a good scientific school itself stimulates a young scientist to carry out his research work at the cutting edge of modern science. And if he suddenly opens a new direction in science, then his name immediately gains recognitionIn 1974 Samoilenko became a professor and headed the Integral and Differential Equations section within the Department of Mechanics and Mathematics at the Kiev State University. Four years later, in 1978, he was elected a Corresponding Member of the Ukrainian Academy of Sciences. In 1987 Samoilenko was appointed head of the Department of Ordinary Differential Equations at the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev. In the following year he became head of the Institute.
Samoilenko worked on both linear and nonlinear ordinary differential equations. In the 1960s he studied nonlinear ordinary differential equations with impulsive action publishing papers such as Systems with pulses at given times (1967). His work on boundary-value problems led to papers Numerical-analytic method for the investigation of systems of ordinary differential equations (2 parts both published in 1966) and many other innovative works. His first major monograph was Method of Accelerated Convergence in Nonlinear Mechanics (Russian) (1969) which he wrote in collaboration with his teachers Nikolai Nikolaevich Bogolyubov and Yurii Alekseevich Mitropolskii. Petryshyn writes in :-
His most original contribution was the numeric-analytic method for the study of periodic solutions of differential equations with periodic right hand side. A monograph on the method of accelerated convergence, written jointly by Samoilenko, N Bogolyubov, and Yu Mitropolskii in 1969, gives an exhaustive analysis of the speed of convergence, error estimates, stability, and applications.Eugene Leimanis writes:-
This monograph contains an account of a basic method in nonlinear mechanics and of some important results obtained by this method. The latter is known as the method of successive changes of variables and its aim is to ensure the convergence of the iteration process in solving systems of nonlinear differential equations.An English translation of this monograph was published in 1976.
With contributions from Mikola Oleksiiovich Perestyuk, Samoilenko put the application of asymptotic methods to solve discontinuous and impulsive systems on a rigorous foundation. Their work continued over a long period and was written up in the important joint monograph Impulsive Differential Equations (Russian) in 1987. An English translation was published in 1995.
Samoilenko was to undertake several joint projects with Mitropolskii who had become the Director of the Institute of Mathematics in Kiev where he worked. In addition to the work mentioned above they worked jointly on the theory of multifrequency oscillation, then later on a system of evolutionary equations with periodic and conditional periodic coefficients. This last work was done in collaboration with D Martyniuk and the three of them published, in 1984, the monograph Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients (Russian) giving an excellent account of their results. In 1987 Samoilenko published the book Elements of the Mathematical Theory of Multifrequency Oscillations. Invariant Tori (Russian). This was translated into English and published in 1991. The authors of  highlight this 1987 publication:-
The beginning of this fruitful creative period was marked by [this] fundamental monograph devoted to the qualitative theory of invariant manifolds of dynamical systems. This monograph served as a foundation for the construction of the general perturbation theory of invariant tori of nonlinear dynamical systems on a torus.Further fundamental monographs continued to be written by Samoilenko and his collaborators. For example, with Mitropolskii and V L Kulik, he wrote Investigation of Dichotomy of Linear Systems of Differential Equations Using Lyapunov Functions (Russian) published in 1990.
Samoilenko has written a series of monographs with N I Ronto. They wrote Numerical-analytic methods for the study of periodic solutions (Russian) in 1976 and followed this with a new work on similar topics entitled Numerical-analytic methods for investigating the solutions of boundary value problems (Russian) in 1986. In 1992 they published Numerical-analytic methods in the theory of boundary value problems for ordinary differential equations. The Preface begins:-
In this monograph we present new promising directions in the development of numerical-analytic methods for studying the solutions of nonlinear boundary value problems in the case of a general form of boundary conditions, problems with controlling parameters, and also boundary value problems for impulse systems. In the tradition of earlier papers, we study, from the same viewpoint, both periodic and non-periodic boundary value problems.With Petryshyn, Samoilenko has written books such as Multifrequency oscillations of nonlinear systems (Ukrainian) (1998) which appeared in an English translation in 2004. In the same year another English book appeared, this being a joint work with A A Boichuk entitled Generalized inverse operators and Fredholm boundary-value problems:-
The book is devoted to the theory of generalized inverses of operators in a Banach space and its applications to linear and weakly nonlinear boundary-value problems for various classes of functional-differential equations, including systems of ordinary differential and difference equations, systems of differential equations with delay, systems with impulse action, and integro-differential systems.A recent book by Samoilenko, written with Yu V Teplinskii, is Elements of the mathematical theory of evolution equations in Banach spaces (Ukranian) (2008). This book is based on lecture notes of courses given by the authors to graduate and postgraduate students at the University of Kiev.
Although we have taken a brief look at Samoilenko's mathematics by looking at a few of the monographs he has written, we must not give the impression that these monographs are his only publications. Nothing could be further from the truth, for MathSciNet lists over 400 publications by Samoilenko. Although some of his publications are single-author, nevertheless, he has nearly 200 co-authors.
As well as having a reputation as an outstanding researcher, Samoilenko is renowned as a fine teacher. The book Differential equations : Examples and problems (Russian) (1984) written with S A Krivosheya and N A Perestyuk contains the following authors' summary:-
We give the solutions of typical problems in a course on ordinary differential equations. The text is structured so as to develop practical skills in students for solving and investigating differential equations describing evolutionary processes in different fields of natural science. Special attention is given to questions that are inadequately discussed (or entirely absent) in existing textbooks, and with which students, as experience shows, are not very familiar. The text is intended for students in mathematical physics departments of universities, technical schools and pedagogical institutes.Another aspect of Samoilenko's contributions is his editorial work for many different journals. He is on the Editorial Board of: Nonlinear Oscillations; the Ukrainian Mathematical Journal; Reports of the Ukrainian Academy of Sciences; the Bulletin of the Ukrainian Academy of Sciences; the Ukrainian Mathematical Bulletin; In the World of Mathematics; the Memoirs on Differential Equations and Mathematical Physics; the Miskolc Mathematical Notes; the Georgian Mathematical Journal; and the International Journal of Dynamical Systems and Differential Equations.
Samoilenko has received, and continues to receive, widespread recognition for his outstanding achievements. The honours he has received include: the Ostrovskii Republican Prize (1968), the Krylov Prize (1981), the State Prize of Ukraine for science and technology (1985), a second award of the State Prize (1996), the Bogolyubov Prize (1998), the Lavrent'ev Prize (2000), the Ostrogradskii Silver Medal (2001), and the Ostrogradski Prize of the Ukrainian Academy of Sciences (2004). He was given the honorary title of "Soros Professor" in 1996 and, two years later, the title of the Honoured Worker of Science and Technology of the Ukraine. He has been elected to the European Academy of Sciences (2002) and the Shevchenko Scientific Society. In 2006 he was elected a Corresponding Member of Accademia Peloritana dei Pericolanti in Messina, Sicily.
Anatoly Mykhailovych Samoilenko is married to Lypa Hryhorivna; they have a son Anatolii who is a geneticist with a doctorate from Göttingen University. Lypa Hryhorivna is also scientist who worked for many years at the Institute of Cybernetics of the Ukrainian Academy of Sciences.
Article by: J J O'Connor and E F Robertson