International Scientific Conference

“Differential-Functional Equations and their Application”

Dedicated to the 80th anniversary of

Professor V.I. Fodchuk (1936-1992)

Chernivtsi, Ukraine

September 28-30, 2016

Professor

Vasyl Ivanovych Fodchuk

PERSONAL LIFE AND SCIENTIFIC ACTIVITY

A future scientist-mathematician, Vasyl Fodchuk was born in the Village of Tulova, Sniatyn District on January 30, 1936, in the family of Ivan Grygorovych Fodchuk (born 1908) and Paraska Ivanivna Poraiko (born 1899). During 1947−1951 Vasyl Fodchuk studied at Ustia Elementary School, and during 1951−1953 continued his education at Prut Secondary School, graduating with only good and excellent marks on all subjects.

STUDIES AT THE CHERNIVTSI STATE UNIVERSITY

School teachers insisted, and the parents agreed to continue education of their talented son, followed by his admission to the Faculty of Physics and Mathematics at the Chernivtsi State University in 1953. As part of his undergraduate studies, Vasyl Fodchuk selected specialization at the Department of Differential Equations, led by Professor Vasyl Pavlovych Rubanyk (1.01.1917 – 9.04.1993).

During early years at the university, Vasyl started his scientific research under the direction of Vasyl Rubanyk, Sr. Lecturer at the time. As a second-year university student, Vasyl Fodchuk received a Certificate of Honour from the University Administration and the Student Council, signed by Professor K.M. Leutsky, University President for his student research paper, The Existence of a Quasi-periodical Solution of Linear Differential Equation with Quasi-periodical Coefficients presented at X Scientific Conference on April 12, 1955.

In 1958, Vasyl Ivanovych Fodchuk graduated from the university and was assigned to work in the Village of Verenchanka, Zastavna District in Chernivtsi Region. He was teaching Mathematics at the local Elementary School during the day, and at the same time, instructed literacy evening classes for the parents. Vasyl dreamed of devoting himself to his future family, students and science. All of this has been waiting for him in the capital city.

WORK AT THE INSTITUTE OF MATHEMATICS,

ACADEMY OF SCIENCES OF UKRAINE

A 23-year period of his life in Kyiv started for V.I. Fodchuk in 1959, when he entered a postgraduate program at the Institute of Mathematics, the Academy of Sciences of Ukraine, which took three years to complete.

V.I. Fodchuk’s scientific research started from the exploration of asymptotic methods of complex systems of differential equations, which take into account the delay factor. Such systems of differential equations describe various oscillation processes taking place in many areas of mechanics, physics, technology, biology, ecology and other applied sciences.

New trends in the exploration of non-linear oscillations originated, and have been further developed by the research of M.M. Krylov, N.N. Bogoliubov and Y.O. Mytropolsky. The related research results have been published in many scientific papers, specifically, in fundamental monographies [1 – 5]. The new direction started by M.M. Krylov and N.N. Bogoliubov in mid-30’s in Ukraine was named the non-linear mechanics. Ukrainian academic school of non-linear mechanics has earned a worldwide recognition, and includes the research of Professor V.I. Fodchuk and his students.

At the time of V.I. Fodchuk’s involvement with the Institute of Mathematics, the institution was led by a well-known Ukrainian mathematician, Member of the Academy of Sciences of Ukraine, the Hero of Ukraine Yuri Oleksandrovych Mytropolsky (3.01.1917 – 14.06.2008), who became a scientific advisor for V.I. Fodchuk’s Ph.D. (Candidate of Science) and D.Sc. (Doctorate) theses.

During his postgraduate studies, Vasyl Ivanovych worked persistently and fruitfully, and achieved a number of substantial results. This comes evident from the 4 articles published in popular Ukrainian mathematical journals in 1962, where the scientific results well exceeded the requirements for the Candidate-level thesis publications nowadays. V.I. Fodchuk’s first publication was a 1962 article in the Ukrainian Mathematical Journal, co-authored by Vasyl Pavlovych Rubanyk, the founder and the first Head of the Department of Mathematics and Mechanics at the Chernivtsi State University (1962 – 1972).

The subject of V.I. Fodchuk’s Ph.D. thesis was non-linear differential equations with the argument delay, which also include a small parameter. The thesis looked at the three types of problems: application of Krylov-Bogoliubov asymptotical methods to quasi-linear equations with delay, substantiation of Bogoliubov averaging principal for systems of non-linear differential equations with the argument delay, and generalization of Bogoliubov-Mytropolsky method of integral manifolds for the above-mentioned systems of equations.

V.I. Fodchuk successfully defended his Ph.D. thesis on January 12, 1963 at the meeting of the Joint Scientific Council of the Institute of Mathematics, the Institute of Cybernetics, and the Main Astronomical Observatory of the Academy of Sciences of Ukraine. The thesis was carried out under the scientific guidance of Academician Yuri Oleksandrovych Mytropolsky.

At the Institute of Mathematics, the Academy of Sciences of Ukraine

Left-to-right (lower row): M.V. Azbeliev, V.P. Rubanyk,

O.Y. Romanenko, Y.O. Mytropolsky, A.D. Myshkis, O.M. Sharkovsky;

left-to-right (upper row): V.M. Shevelo, M.I. Skil, V.I. Fodchuk,

A.M. Samoilenko, A.A. Martyniuk

After completing his postgraduate studies, during 1962−1967 V.I. Fodchuk continued to work at the Institute of Mathematics as a Scientific Researcher. In this position, V.I. Fodchuk spent a lot of effort on the organization of national and international scientific events, specifically, international conferences on non-linear oscillations. Being young and actively involved scientist, Vasyl Ivanovych also participated in these events, evident from numerous invitations in his name.

On July 7, 1967 Presidium of the Academy of Sciences of Ukraine confirmed V.I. Fodchuk’s title of Sr. Scientific Researcher in Mathematical Physics.

V.I. Fodchuk carried on his scientific effort, and within 10 years presented his fundamental publication – a Doctorate thesis, The Asymptotical Methods of Non-linear Mechanics in The Theory of Differential-Variant Equations, specialization 01-003 – Differential and Integral Equations. Official opponents of his D.Sc. thesis were well-respected Ukrainian mathematicians, Professors S.M. Shymanov, Y.O. Riabov and K.G. Valeev.

In his D.Sc. thesis, Vasyl Ivanovych carried on the research of differential-variant equations and influence of the delay factor, or after-effect on the behaviour of their solutions, i.e. how the former state of the process affects its behaviour in the present moment. In a real-world example, the current development of a biological population depends on its state in the previous period of time. The delay factor plays significant role in such areas of science as the theory of automatic guidance and control, radio physics, electronics, mechanics and ecology.

Many real-world physical systems with delay are oscillating and non-linear. The influence of these characteristics during mathematical description of the processes which lead to differential equations with delay became the main focus of the V.I. Fodchuk’s Doctorate research. The thesis consists of the introduction, five sections and references, which include 244 publications.

The first section studies an important question of existence and unity of the solution for some differential equations with delay, specifically,

, .

This equation is carved in the tombstone of V.I. Fodchuk in Chernivtsi.

The thesis presents results for new and wide classes of differential equations, which became the classics of the genre. In addition, it includes the proof of the theorem of existence and unity of the global solution of the system of equations which is close to linear. The thesis also included research of a limited solution of the original problem on half-axis, and proved an analog of P. Bol theorem.

The second section is dedicated to the development and substantiation of the averaging method for differential equations with delay and a small parameter. Significant achievement was the theorem of substantiation of the averaging method for differential equations of neutral type, i.e. equations which describe processes dependent on the state and velocity at the previous moments in time. Similar results were obtained for the singularly disrupted system of the neutral type. Such system of equations describes processes with significantly varied component velocities.

The same section presents results of the asymptotic integration of differential equations with distributed parameters. Such equations can be used, for example, to describe the oscillation of a string affected by a small force with delay.

The third section of the thesis is dedicated to the development of algorithms to find approximated solutions in the form of asymptotical series using the powers of a small parameter for differential equations with delay, which can be taken to the standard form. Such equations have found their broad use in the theory non-linear oscillations. The processes described by Vasyl Ivanovych not only presented methods of constructing asymptotical solutions but include formulas for the first, and the improved first approximation, which is important for practical applications. The aspiration for completeness and applicability to practical problems is a signature characteristic of V.I. Fodchuk’s scientific research.

The same section presents the algorithms of constructing periodical solutions for differential equations of delayed and neutral types, as well as asymptotic methodology used to study equations of partial derivatives of neutral type.

Vasyl Ivanovych made a significant contribution into the development of integral manifolds for non-linear differential-functional equations with the argument delay of the specific and generic types, as well as practical applications of this theory. The main focus of his research was around the question of existence stability of integral manifolds of finite dimension.

While looking for solutions to these problems, Vasyl Ivanovych faced a number of principal complexities, which were not possible to overcome using the methods and techniques common for the equations without delay. The developed method of integral manifolds presented in V.I. Fodchuk’s scientific research found its recognition by Ukrainian and foreign mathematicians, and became an important step in the respective theory. Specifically, this method was further developed by the research of his students, I.M. Cherenkov and I.I. Klevchuk.

The fourth section of the D.Sc. thesis offers a method of exploring integral manifolds for generic differential-functional equations which do not have a special form. The obtained results marked a new approach in the methodology of exploring such equations, as they recognized the possibility of studying solution properties in a space with the finite base, as well as constructing integral manifolds with the dimension smaller than the one of the finite space, and which exhibit such properties as stability and limitedness.

The same section explores the question of existence and stability of the integral manifold of toroidal type, and includes the possibilities of generalization and practical application of the obtained research results.

The last section of the thesis presents the theorems of existence and properties of the integral manifold for singularly disrupted systems of differential equations in the Banach space. Such space allows exploring fairly broad categories of differential, difference and other equations.

The main results of V.I. Fodchuk’s thesis have been presented at numerous international and national conferences and summits.

HEAD OF THE DEPARTMENT OF APPLIED MATHEMATICS AND MECHANICS AT THE CHERNIVTSI STATE UNIVERSITY

1972 marked the second big period in personal life and scientific activity of V.I. Fodchuk, spent in the City of Chernivtsi. Over the course of twenty years, he was leading the Department of Applied Mathematics and Mechanics at the Chernivtsi State University, which allowed him to take his scientific, educational and administrative skill and talent to their full potential. In 1972, Vasyl Ivanovych was invited to work at the Chernivtsi State University. The Head of the Department of Applied Mathematics and Mechanics at the time, Professor V.P. Rubanyk created a new Department of Mathematical Problems of Control and Cybernetics, and took over the newly created academic unit. In the meantime, on September 4, 1972 Vasyl Ivanovych Fodchuk was elected as the new Head of the Department of Applied Mathematics and Mechanics.

Numerous awards speak of V.I. Fodchuk as the scientist of high stature, including a reference letter signed by Y.O. Mytropolsky, the Principal of the Institute of Mathematics. Following is a fragment of this reference letter, “Vasyl Ivanovych Fodchuk is the Science Secretary of the Scientific Commission on the Problem of Common Differential Equations at the Department of Mathematics, the Academy of Sciences of Ukraine, and the Science Secretary of the Expert Commission on the Gold Medal of A.M. Liapunov.”

On February 17, 1975 Vasyl Ivanovych Fodchuk was issued a diploma of D.Sc. in Physics and Mathematics, and on May 21, 1976 was awarded a scientific title of Professor at the Department of Applied Mathematics and Mechanics.

Vasyl Fodchuk was the official scientific advisor to eight Ph.D. graduates, namely (in brackets, the year of the thesis defense): V.A. Dombrovsky (1972), A. Holmatov (1975), M.S. Bortei (1980), Y.J. Bihun (1981), I.M. Cherevko (1983), I.I. Klevchuk (1986), M.M. Popov (1987) and I.V. Yakymov (1989).

While planning the topics of scientific research for his students, Vasyl Ivanovych tried to select those that would be relevant, had a potential for further development after the Ph.D. thesis would have been defended, and did not create a topical collision among the scientists in the same field.

Five of V.I. Fodchuk’s students worked at the Department of Applied Mathematics and Mechanics: Sr. Lecturers M.S. Bortei, Y.J. Bihun, I.M. Cherevko, I.I. Klevchuk and I.V. Yakymov. Two of them, Y.J. Bihun and I.M. Cherevko have later been awarded a title of the Doctor of Science, specialized in Differential Equations. One more student, M.M. Popov has been awarded a title of the Doctor of Science, specialized in Mathematical Analysis.

Professor V.I. Fodchuk was paying a close attention to publications in the area of his scientific research. He had a large scientific library, which was donated by his family to the Department of Applied Mathematics and Mechanics after he passed away. Many publications had bookmarks addressed to his followers, including lecturers, postgraduate thesis nominees and undergraduate students. Vasyl Ivanovych conducted an active scientific research with his students, evident from scientific publications with V. Gonchar [79] and I. Verbytska [120].

As a result of his due diligence, the Department of Applied Mathematics and Mechanics has been granted an authorization to run the undergraduate program for future professionals in specialization 0647 – Applied Mathematics.

Professor Vasyl Fodchuk successfully guided the Department of Applied Mathematics and Mechanics in fulfilling the following scientific research topics:

1. The Asymptotical and Numerical-Analytical Methods of Exploring Oscillations and Stability of Non-linear Systems (1986−1990).

2. The Asymptotical and Numerical-Analytical Methods of Exploring Differential-Functional Equations (1991−1992).

Due to continuous effort and scientific recognition of Professor V.I. Fodchuk, a national scientific research program, Asymptotical and Numerical-Analytical Methods of Exploring Non-linear Differential and Differential-Functional Equations developed under his leadership was carried out at the Faculty of Mathematics at the Chernivtsi State University during 1991−1995.

During 1986−1990, another scientific research program was carried out at the faculty under his guidance, The Approximated Methods of Exploring Systems of Regularly and Singularly Disrupted Differential-Functional Equations.

The Department of Applied Mathematics and Mechanics was the organizer of a scientific seminar on continuous basis, where the department researchers had an opportunity not only to present their own achievements, but also the results of the research conducted by scientists from other educational institutions.

By the invitation of Professor Fodchuk, some lecture courses at the department were provided by then Professor, now Corresponding Member of the National Academy of Sciences of Ukraine O.M. Sharkovsky, Professor V.I. Rozhkov and others. In November 1987, a lecture course on The Asymptotical Methods of Exploring Distributed Systems was provided by the former Head of the Department of Applied Mathematics and Mechanics, a colleague and advisor of Vasyl Ivanovych, Professor V.P. Rubanyk.

The scientific results of V.I. Fodchuk have been included in his monography and 135 publications. His scientific achievements were related to a broad spectrum of problems in The Theory of Differential-Functional Equations. He was first to substantiate a method of averaging for differential equations of the neutral type, as well as develop a method of averaging and asymptotical methods of Krylov, Bogoliubov and Mytropolsky for singularly disrupted differential equations of the delayed and neutral types. Vasyl Fodchuk also pioneered a method of integral manifolds for non-linear differential-functional equations, studied the schemes of constructing periodical and limited solutions of the equations with delay, and differential equations with partial derivatives.

Professor V.I. Fodchuk was paying close attention to the use of mathematical methods in applied research, along with the supply and evolution of computing infrastructure at his department. He was always looking after the adequacy of computing capacity for the Auditorium #24, which now bears his name. At them time when personal computers and the respective software started to gain momentum, Vasyl Fodchuk was overseeing the automation of the Computer Lab installed at the Auditorium #19. The following three contractual-economic programs were held at the faculty under Professor Fodchuk’s guidance:

1. The Optimization and Programming Realization of the Evaluation Algorithms in Navigation Systems.

2. The Development of Interpolation Methods for Even Networks and their Applica-tion for Problems of Navigation and Mathematical Physics.

3. The Development of Special Analog Means of Noise Suppression in the Pickup Channel.

The first two programs were carried out jointly with the Institute of Mathematics of the Academy of Sciences of Ukraine, while the third one – jointly with the Department of Optical-Mechanical Memorizing Devices in relation to the relevant problems in the development of magnetic disks for personal computers. He had close scientific ties with his colleagues from Europe and the Middle Asia, who often would send him their monographies, paper collections, and individual articles: D. Bainov, Bulgaria; A. Halanai, Romania; A.M. Zverkin, G.A. Kamensky, S.B. Norkin, L.E. Elsgoltz, Y.O. Riabov, E.O. Grebenikov, O.M. Filipov, V.S. Burd, V.R. Petukhov and many others.

Professor V.I. Fodchuk managed to successfully combine scientific and educational activities. Given the results of his scientific research, Professor Fodchuk prepared such specialized courses as The Additional Chapters of Physics and Mathematics and The Asymptotical Methods in Non-linear Mechanics. He also demonstrated a high level of professionalism while teaching Higher Mathematics courses at the General Technical Faculty of the Chernivtsi State University, provided by the Department of Applied Mathematics and Mechanics at the time, for the total duration of five semesters at the daytime and evening sections of the faculty. Vasyl Ivanovych was very professional and demanding in carrying out these courses: he himself lectured these courses in every semester at least once, and prepared the respective educational handbooks. He was also attending the lectures of junior staff members conducting these courses, leading the discussions on lecture topics, and giving a plethora of interesting advice and useful directions.

As the Department Head, Vasyl Ivanovych attended lectures of other professors, more often those who were just starting their lecturing career. He was taking the notes of his impressions and recommendations in his notebook, which is kept in the Department Office up to date. Discussing the lecture with the professor who was delivering it assisted in the improvement of methodical, scientific level of teaching the respective course, as opposed to putting an accent on critique. Highly qualified mathematician and excellent methodologist, Vasyl Ivanovych had a good sense of various nuances of both the content and the form of conducting lectures by a professor in any relevant educational discipline.

Very memorable for Yaroslav Bihun and Igor Cherevko, the students of Vasyl Ivanovych, became a joint trip to the Mathematical Summer School organized at Skalka Camp by the Uzhgorod State University in 1980. They had an opportunity to listen to lectures of V.I. Fodchuk’s colleagues from the Institute of Mathematics – Professors A.M. Samoilenko and I.O. Lukovsky, as well as other famous mathematicians – A.M. Stiopin and his student, one of the world-leading specialists in Algebra and Dynamic Systems, Rostyslav Grygorchuk, whom Professor Fodchuk knew very well. Y. Bihun and I. Cherevko remembered an episode when they were participating in a very active and loud discussion, thereby disturbing the recovery of one well-known mathematician, Vasyl Ivanovych condemned them for such behaviour, as a strict father, but at the same time, protected them from the undesirable conclusions of the organizing committee. Vasyl Ivanovych was very generous and reasonable, but at the same time, a principal person. He was taking a good care of the department and looking out for the individual staff members. Unforgettable are memories of many of his colleagues about their joint participation in scientific conferences and schools, about his erudition, simplicity, candidness and friendly attitude.

Yaroslav Bihun, an author of this writing, remembers many positive and interesting episodes from the joint trips together with V.I. Fodchuk, to scientific conferences held in Kyiv, Odesa, Kishinev, Tashkent, Dushanbe and Almaty. Vasyl Fodchuk paved the path for his students to participate in the annual scientific schools led by Professor M.V. Azbeliev at the Perm Polytechnic Institute.

LIFE AND WORK IN THE CITY OF CHERNIVTSI

In fall of 1972, Fodchuk family moved to Bukovina, where Vasyl Ivanovych had agreed to take over the Department of Applied Mathematics and Mechanics at the Chernivtsi State University. Here, in the City of Chernivtsi, the Ukrainian scientist began another fruitful stage of his scientific and educational career. Vasyl Fodchuk successfully carried out his scientific research, led a scientific seminar and directed the scientific activity of the staff members and the students at his department. Well deserved, by the decision of the Highest Attesting Commission of the Council of Ministers of May 21, 1976 (Protocol #15), V.I. Fodchuk was awarded a scientific title of Professor at the Department of Applied Mathematics and Mechanics of the Chernivtsi State University.

The Chernivtsi State University has been recognized in Ukraine and internationally for organizing a Community School of Young Mathematicians, as the Department of Applied Mathematics and Mechanics organized a recurring scientific seminar accepting for participation postgraduate students from other higher educational institutions. Brainstorming, achievement discussions, quality consultations, objective opinions – all of these were the reasons why young mathematicians have been heading to the professor from Chernivtsi.

V.I. Fodchuk was part of the Scientific Council of the Academy of Sciences of Ukraine, working on the program developing fundamentally-applied research in the area of mathematical science in the country. He also had a chance to work as part of the Commission at the Ministry of the Highest Educational Institutions of Ukraine, on defining priority directions for the development of science and the related budget distribution. Professor Fodchuk still had a number of thoughts and plans for developing his department and national mathematics in general, but the unexpected illness slowly started cutting into his well-being. Sadly, on May 9, 1992 the heart of Vasyl Ivanovych Fodchuk stopped beating. He was buried at the Central Cemetery of the City of Chernivtsi.

VASYL FODCHUK’S STUDENTS

Dombrovsky , Vasyl Andriyovych – in 1972, under the guidance of Sr. Researcher V.I. Fodchuk, defended his Ph.D. thesis, The Application of Asymptotical Methods to Some Classes of Linear and Quasi-linear Systems and Equations of Partial Derivatives with Delay.

Holmatov , Abdinabi – in 1974, completed his postgraduate studies at the Department of Applied Mathematics and Mechanics of the Chernivtsi State University; in 1975, defended his Ph.D. thesis, The Exploration and Methods of Constructing Periodical and Limited Solutions of Differential-Functional Equations of the Neutral Type.

Bortei , Myroslav Stepanovych – during 1973–1976, was a student of the postgraduate program; in May 1980, defended his Ph.D. thesis, The Exploration and Construction of Quasi-periodical Solutions of Non-linear Differential-Functional Equations; during 1976–1992, worked at the Department of Applied Mathematics and Mechanics of the Chernivtsi State University.

Bihun , Yaroslav Josypovych – during 1977–1980, was part of the postgraduate program; in 1981, defended his Ph.D. thesis,The Development and Substantiation of Asymptotical Methods for Differential-Functional Equations. In 2009, Y.J. Bihun defended his D.Sc. thesis, The Averaging in Multi-frequency Systems of Differential-Functional Equations, scientific advisor – Academician of the National Academy of Sciences of Ukraine Anatoly Mykhailovych Samoilenko. Since 1974, Yaroslav Bihun works at the Department of Applied Mathematics of the Chernivtsi State University: as a Sr. Lecturer, starting 1987; as a Professor, starting 2012; as the Head, during 1992–1996 and again, starting 2002.

Cherevko , Igor Mykhailovych – during 1978–1982, was part of the postgraduate program; in 1983, defended his Ph.D. thesis, The Exploration of Integral Manifolds for Singularly Disrupted Differential-Functional Equations. In 2004, I.M. Cherevko defended his D.Sc. thesis, The Integral Manifolds and Approximating Algorithms of Exploring Differential-Functional Equations, scientific advisor – Academician of the National Academy of Sciences of Ukraine Mykola Oleksiyovych Perestiuk. During 1978–2003, Igor Cherevko worked at the Department of Applied Mathematics of the Chernivtsi State University. Starting September 2003, he is the Head of the Department of Mathematical Modeling at the Chernivtsi State University, and since 2005, a Professor at the same department, and the Dean of the Faculty of Applied Mathematics.

Klevchuk , Ivan Ivanovych – during 1983–1986, was part of the postgraduate program; in 1986, defended his Ph.D. thesis, The Principal of Settling and Dynamic Equivalence of Differential-Functional Equations. During 1979–1994, I.I. Klevchuk worked as a Lecturer and Researcher at worked at the Department of Applied Mathematics and Mechanics of the Chernivtsi State University. Since 1994, Ivan Klevchuk works as a Sr. Lecturer at the Department of Mathematical Modeling of the Chernivtsi State University.

Yakymov , Igor Victorovych – during 1986–1989, was a postgraduate student; in 1989, defended his Ph.D. thesis, The Asymptotical Decomposition of the Solutions of Singularly Disrupted Differential Equations with the Deviating Argument. During 1983–2001, I.V. Yakymov worked at the Department of Applied Mathematics and Mechanics of the Chernivtsi State University, including in the position of a Sr. Lecturer since 1994.

Popov , Mykhailo Mykhailovych – in 1987, defended his Ph.D. thesis, The Geometry of the Inseparable Spaces, in the specialization of Mathematical Analysis, scientific co-advisor – A.M. Plichko. Since 2002, M.M. Popov works at the Department of Mathematical Analysis of the Chernivtsi National University, including in the position of a Professor since 2011.

THE STARS DO NOT DIE…

Vasyl Ivanovych liked the sound of grass under the scythe, the smell of fresh hay, a “symphony orchestra” of birds over Prut River. He always admired gazing across the starlit sky… From early childhood, Vasyl liked to count sparkling spots in the native sky of Tulova Village. His soul flew into the heights of the sky, towards those bright galaxies and stars, to the heavenly Creator, the faith for which he received from his parents, and never lost it. He left us memories of himself as a person, a father, and a scientist. Science was left with the significant contribution in the form of 128 publications, and participation in 53 international summits.

On the occasion of the 60th anniversary of the teacher’s birth, V.I. Fodchuk’s students, under the guidance of Y.J. Bihun, completed editing his unfinished monography, Regularly and Singularly Disrupted Differential-Functional Equations, published by the Institute of Mathematics in Kyiv, in 1996. One of the copies of this edition was gifted to Vasyl’s wife, with the following signing note, “Dear Lilia Oleksandrivna, the edition of this monography became possible, in part, due to your ideas and support. We admire your tireless, sincere and devoted work with Vasyl Ivanovych’s archive, materials from which you generously shared with us. Sincerely and with best wishes, the authors: Yaroslav Bihun, Ivan Klevchuk, Igor Cherevko and Igor Yakymov. December 7, 1996, Chernivtsi.”

In 1996, the Faculty of Mathematics of the Chernivtsi State University held a national scientific conference, Differential-Functional Equations and Their Applications, dedicated to the 60th anniversary of V.I. Fodchuk’s birth. The conference included 217 reports by 204 scientists from 12 countries. A group picture of the conference participants taken at the grounds of the Residence of Archbishops of Bukovina, shows some famous mathematicians who closely knew Vasyl Ivanovych, namely: A.D. Myshkis from Moscow, Y.S. Baris from Olstyn (Poland), M.O. Perestiuk, O.A. Boichuk, V.I. Tkachenko, A.Y. Luchka from Kyiv, V.O. Plotnikov, M.Y. Tykhonenko from Odesa, V.Y. Sliusarchuk from Rivne, S.P. Lavreniuk from Lviv, V.V. Merynets from Uzhgorod and others. A memorial requiem was held at the Church of the Three Prelates, in loving memory of Vasyl Fodchuk, which was also attended by the conference participants.

Building #1 of Yuri Fedkovych Chernivtsi National University has Professor Vasyl Fodchuk Auditorium, which stands display some of V.I. Fodchuk’s publications and materials on the educational and scientific activity of the scientist. Several years ago, on February 3, 2011 the Chernivtsi National University held a scientific seminar dedicated to the 75th anniversary of Professor Fodchuk’s birth. People keep a thankful memory of the effort of their famous countryman.

MAIN DATES IN V.I. FODCHUK’S PERSONAL LIFE AND CAREER

January 30, 1936 – born in the Village of Tulova, Sniatyn District.

1943–1947 – student of Tulova Beginner School.

June 3, 1947 – awarded a Honorary Certificate by the Office of the Principal and Educational Council of Tulova Beginner School.

1947 – began studies at Ustia Elementary School.

June 15, 1948 – awarded a Honorary Certificate by the Office of the Principal of Ustia Secondary School, for the excellent results in studies and behaviour.

1951 – graduated from Ustia Elementary School.

September 1, 1951 – July 2, 1953 – studied at Prut Secondary School.

September 1, 1953 – June 30, 1958 – student of the Faculty of Physics and Mathematics at the Chernivtsi State University, specializing at the Department of Differential Equations.

August 29, 1958 – September 10, 1959 – teacher of Mathematics at Verenchanka Secondary School, Kitsman District.

July 1, 1959 – July 1, 1962 – postgraduate student at the Institute of Mathematics, the Academy of Sciences of Ukraine, Kyiv.

January 12, 1963 – defended his Ph.D. thesis, Some Problems of Differential Equations with Delay, scientific advisor – Academician Y.O. Mytropolsky.

May 31, 1963 – married Lilia Oleksandrivna Arkhipova.

July 1, 1962 – November 20, 1972 – worked as a Sr. Researcher at the Institute of Mathematics, the Academy of Sciences of Ukraine.

September 5, 1965 – awarded a scientific title of a Sr. Researcher specialized in Mathematical Physics.

May 1972 – defended his D.Sc. thesis, The Asymptotical Methods of Non-linear Mechanics in The Theory of Differential-Variant Equations, scientific consultant – Academician Y.O. Mytropolsky.

November 20, 1972 – May 9, 1992 – Head of the Department of Mathematics and Mechanics at the Chernivtsi State University.

May 21, 1976 – awarded a scientific title of a Professor at the Department of Mathematics and Mechanics of the Chernivtsi State University.

May 9, 1992 – passed away; buried at Chernivtsi Central Cemetery.