A Memorial Tribute to Professor Constantin Corduneanu, The Outstanding Mathematician

This paper contains the biographical sketch and reviews scientific contributions of Professor Constantin Corduneanu, an outstanding researcher in stability and control theory, and oscillations.

A nal year course on topological groups (following Pontrjagin's book -the English edition) prompted him to write his thesis, required for obtaining the Diploma of Licentiate in Mathematics (something between a Bachelor's and a Master's degree), on "the group of automorphisms of a topological group." He de ned a topology on the group of automorphisms in the case of a bounded topological group (i.e., all Markov's seminorms are bounded on this topological group); his rst results published were part of his thesis in 1950. In 1951, when preparing his thesis for the degree of Licentiate, he discovered an error in a paper published in Portugaliae Mathematica due to the University of California Los Angeles Professor Szetsen Hu. The error could not be repaired under accepted hypotheses. In 1953, Corduneanu changed his eld of research to di erential and related equations. Corduneanu's research activities began sixty-eight years ago.
During the period 1948-1956, due to the changes brought by the new regime in Romania, the Romanian connections with the Western World were abolished. They did not get publications coming from the Western World, the student exchanges that were common until the WW2 were suspended, and a period of isolation began. Fortunately, under the new circumstances, for mathematicians, the ourishing of the domain had continued because the material received from the former Soviet Union, regarding mathematical research as well as teaching, were of the highest quality. Corduneanu used in his training as mathematician books or other publications authored by such professors as I. G. Petrovski, I. V. Smirnov, A. N. Kolmogorov, S. L. Sobolev, V. V. Stepanov and V. V. Niemytskii, A. N. Tychono and others of the same caliber. The rst mathematical book he studied in entirety was Pontrjagin's famous book on Topological Groups (1949)(1950). Corduneanu even had access to publications authored by Western mathematicians, which were translated to Russian. He could read books on di erential equations by G. Sasone, Lamberto Cesari, E. A. Coddington and N. Levinson, E. Kamke, and others.
In 1957, he organized, with the help of his colleagues at AICU, a seminar on "Qualitative Theory of Di erential Equations." In 1961, he participated at the Congress of the International Union of Mechanical Sciences, organized by Iurii Mitroploskii in Kiev. In that meeting, he met for the rst time several well-known mathematicians from various countries; Solomon Lefschetz, Jack Hale, and L. Cesari, all from the United States of America (the US), and V. V. Niemytskii from the U.S.S.R.
In 1977, Corduneanu decided to expatriate from Romania, and reside in the US. He went to Italy and taught some courses at the International Center for Theoretical Physics (UNESCO) in Trieste, but the Romanian authorities only allowed him to travel to Italy. In January 1978, Corduneanu moved to the US and had a teaching position at the University of Rhode Island. He was a visiting professor there in 1967-1968 and 1973-1974 academic years; hence familiar with the place and colleagues. In the academic year 1978-1979, he was a visiting professor at the University of Tennessee at Knoxville. Corduneanu obtained a tenured position as a Professor in the fall of 1979 at the University of Texas at Arlington (UTA). V. Lakshmikantham was the Chair of the mathematics department at the UTA. He brought Corduneanu to strengthen the mathematics doctoral program. At UTA, Corduneanu made signi cant contributions to the rise of the department's doctoral program, which had been created in 1974. By 1987, the American Mathematical Society ranked UTA's Department of Mathematics 89th out of 620 mathematics doctoral-granting institutions in the US.
I had the opportunity to meet Corduneanu for the rst time in my life at the International Conference on Theory and Applications of Di erential Equations held in Columbus, Ohio, in March 1988. Reza Aftabizadeh from Ohio University, a former student of Corduneanu, organized this conference, and I was his assistant. I had already taken two semesters of di erential equations with Aftabizadeh, and he used Corduneanu's book, Principles of Di erential and Integral Equations for the rst semester. In that meeting, I asked Corduneanu to be my doctoral advisor and after I told him that I had taken all the required courses like Real Analysis, Di erential Equations, etc., he agreed. I transferred to the UTA in 1988. I did not know anybody there. Corduneanu and his wife, Ms. Alice, deceased in 2005, gave me a room in their house until I found an apartment after two weeks. Every year the Corduneanus invited me to their house for Thanksgiving and Christmas and treated me as if I were their son. They did not have children. I never forget their kindness and generosity.
Upon my arrival, Corduneanu gave me the manuscript of his book Integral Equations and Applications and asked me to read it. My Ph.D. dissertation was based on the materials of this book, which was published by the Cambridge University Press in 1991. Corduneanu organized a research seminar during the period September 1990 to May 1994, in the department. All of his students and faculty members attended the seminar, and students presented their research work. Visitors occasionally participated and presented their research results, including V. Barbu, Y. Hamaya, M. Kwapisz, I. Gyori, and Cz. Olech. Corduneanu was well-liked and respected by all faculty members in the department. They called him affectionately C. Corduneanu always assisted everybody, including students, faculty members, and mathematicians he met for the rst time. He was intellectually generous to all. Late Professor Bernfeld, whose specialty was also di erential equations, always said that he was so amazed by Corduneanu's vastness of knowledge in di erential equations. He mentioned that anytime he asked Corduneanu a question, Corduneanu sent him to a speci c paper in a particular journal that would provide the answer to his question.
Corduneanu taught numerous courses, and his classes were always fully attended, including many engineering students. He was very popular and respected among students, and they all wanted to take his classes. Doctoral students wanted to have Corduneanu as a member of their oral comprehensive examinations committee or as a member of their dissertation defense committee because he genuinely wished that students would succeed in their academic endeavors. Corduneanu did his utmost to help the students answering their questions, guiding them, and advising them on what books or papers to read.
In May 1990, Corduneanu, along with faculty members from the engineering department, organized the Integral Methods in Science & Engineering conference at the UTA. In May 1996, Corduneanu, along with several faculty members, including faculty from engineering, organized the Volterra Centennial Symposium at the UTA. About 100 mathematicians and engineers from 15 countries attended the conference and presented their results. In January 2000, Corduneanu and I organized a special session on Integral Equations and Applications for the American Mathematical Society at the Joint Mathematics Meetings in Washington, D. C. We invited 25 mathematicians from various countries and the US to present talks. Corduneanu retired in September 1996, after 47 years in higher education in Romania and the US, holding the title of Emeritus Professor of Mathematics. Corduneanu was very active after his retirement. He published three research monographs and attended numerous conferences and meetings in the US and around the world.
The Russian School in uenced most of Corduneanu's research in di erential equations and related elds, given the abundance of publications in Russian largely available to him during the years of his formation as a researcher, as well as the long tradition of excellence established by Liapunov, Chetayev, and Persidskii. He made constant use of the literature in Russian concerning di erential equations and their applications.
In the next few paragraphs, Corduneanu's encounters and connections with his Russian colleagues are presented more or less in chronological order, sometimes just casual encounters with I. G. Petrovski, N. K. Bary, B. Gnedenko, A. P. Norden, and N. E mov, or  Corduneanu had mathematical interactions with V. V. Niemytskii, whom he met several times in Moscow and Kiev, starting in 1961. They had been in correspondence. Niemytskii was the editor of Referativnyi Zhurnal and knew about Corduneanu's research from that journal. They had fruitful (for him) discussion, and when Niemytskii asked him about his current study, he answered in Russian "Ja vrashchayu vokrug nepodvizhnoi tochki." That was the method he was using for obtaining global existence of solutions and studying the existence of almost periodic or just bounded solutions to nonlinear di erential equations (ordinary differential equations, sometimes partial di erential equations). Corduneanu was applying Banach, Schauder and Tychono xed point theorems to obtain those results. Niemytskii visited AICU for at least one week and gave two or three lectures, participated in a seminar meeting, and interacted with some of the faculty there. Corduneanu had more opportunities to talk to him because he accompanied Niemytskii in a two-day excursion in the Carpathian Mountains. Moreover, before he had met him, he translated to Romanian Niemytskii's book Topological Methods in the Theory of Integral Equations.
In the early 1960s, Corduneanu decided to shift his interest from ordinary di erential equations or delay equations to integral equations. This shift was mainly due to his encounter with Krasnoselskii after reading his book. Professor V. V. Rumiantsev had known about Corduneanu's work on comparison method and partial stability, and guided Laszlo Hatvani, from the University of Szeged, to write his dissertation at Moscow University, based on that research (1975). Rumiantsev visited the UTA on the occasion of an International Conference on Di erential Equations organized by the school.
Professor V. A. Pliss from Sankt-Petersburg studied the boundedness problem of solutions of ordinary di erential equations and included Corduneanu's results in a book he published in 1964. The book was later translated into English. Corduneanu met Professor V. M. Aleksejev at the International Congress of Mathematicians in Moscow in August 1966. Before, they had met in publications. Aleksejev cited Corduneanu's work regarding the comparison method in his papers. Corduneanu used Aleksejev's book on control theory when teaching his graduate students at the UTA.
Professor N. V. Azbelev had founded a school dealing with functional equations, in Perm, Russia. Azbelev visited the UTA in 1996. Corduneanu was in communication with some of Azbelev's former students and collaborators who were spread around the world in various countries. Azbelev and Corduneanu had somewhat di erent approaches to the study of functional di erential equations, but complementary to each other.
Professor N. N. Krasovskii and Corduneanu met in Athens (Greece) in 1966, and in Moscow in 1992. Corduneanu was aware of Krasovkii's work and had read and used his results since 1956. In 1956, Professors Krasovskii and Germaidze published a paper on the stability of general ordinary di erential equations, with respect to perturbations bounded in the mean. The underlying assumption on the ordinary di erential equation system was its uniform asymptotic stability. In January 1957, at a session of the Romanian Academy, Corduneanu presented a similar paper, without having seen before the article of Krasovskii-Germaidze. His underlying assumption was the exponential asymptotic stability of the zero solution of the nonlinear system (unperturbed). The way he had measured the perturbation was the same that the Krasovskii-Germaidze paper was using. However, at that time, he did not know that the integral norm he was using is equivalent to the supremum norm used by Krasovski-Germaidze. One year later, Corduneanu found the equivalence of the norms in papers by Massera and Scha er. The result established by Krasovskii-Germaidze was better, because of the weaker assumption on the unperturbed system. In 1960, Corduneanu used his comparison method that he developed and proved more results on the preservation of stability under perturbations, including nonlinear perturbations and the result of Krasovskii-Germaidze.

Corduneanu's Research Work Global Problems in the Theory of Ordinary Di erential Equations
These types of problems kept Corduneanu's attention at the beginning of his career. His doctoral thesis, which he defended in 1956 at the University of Iaşi, contained problems of that type. Professors Miron Nicolescu, then president of the Romanian Academy, Grigore Moisil, and Nicolae Teodorescu from Bucharest, a former student of J. Hadamard at Sorbonne were members of his thesis defense committee. Corduneanu continued research work in this eld for several years, studying global existence, stability problems, oscillation theory, with particular regard to the almost periodic behavior of solutions to various classes of nonlinear equations.

Qualitative Theory of Di erential Equations, with Special Regard to Stability Theory
His work in this category was mainly directed to ordinary di erential equations and equations with causal operators. Corduneanu published his seminal paper in Russian in 1960 titled "Method Di erential Neravenstv v teorii Ustoichivosti." In that paper, he made one of the rst steps in applying the so-called Comparison Method and proving in a single theorem all basic results on Liapunov stability, based on using the Chaplyguine -Wazewski approach to di erential inequalities, and the Liapunov's function in general form simultaneously. This method had been widely applied by the School of Academician V. M. Matrosov, Russia; and in Ukraine by Academician A. A. Martynyuk and his followers. The result Corduneanu published in 1960, was included in several monographs and treatises, by authors like V. Lakshmikantham and S. Leela, W. Hahn, T. Yoshizawa, A. Halanay, G. Sansone and R. Conti and others.

Theory of Integral Equations
In this domain, Corduneanu contributed to generalizing the method due to Massera and Scha er, from di erential equations to integral equations. His book Integral Equations and Applications published by Cambridge University Press in 1991 contains the basic results he had obtained until 1987. This book became one of the most often quoted references in the literature. In this book, Corduneanu illustrated that integral equations constitute a very useful and successful tool in contemporary research, unifying many particular results available for other classes of functional equations (di erential, integrodi erential, delayed argument). Also, his book Integral Equations and Stability of Feedback Systems published by Academic Press in 1973 contains qualitative results with applications to the stability of systems of automatic control.

Equations with Causal Operators
Corduneanu aimed to present, as much as possible, a uni ed theory of equations with causal operators (according to Volterra -Tonelli -Tychono ), that can cover the classical types of ordinary di erential equations, equations with delay, integrodi erential equations with Volterra type integral, and some discrete evolution Corduneanu presented over 140 papers at various meetings and conferences on mathematical topics, held in Romania, Hungary, Czechoslovakia, Bulgaria, Soviet Union, Russia, Ukraine, Germany, Belgium, Italy, The US, The Netherlands, England, Scotland, Japan, Canada, France, Morocco, Greece, Poland, China, Portugal, and Chile. He was invited to present his research work at over 31 national and international conferences. He was an invited lecturer at 53 Colloquium and Exchange Programs in various countries outside of the US. From 1968 till 2017, he was an invited lecturer at 36 universities in the US. Corduneanu was the founding editor of the journal Libertas Mathematica, a publication of the American Romanian Academy of Arts and Sciences. He published the rst volume in 1981 and continued this task until 2011. From 1990 to 1994, Li and I assisted him in the publication of this journal. We were doing most of the preparation of the journal, copying pages, pagination, collating, and organizing; we submitted the last stage of the work to a publishing shop for binding the journal. After long hours of work on Fridays, Saturdays, and Sundays, Corduneanu would take us to Cici's Pizza, a pizza joint close to campus that had a bu et, and we had lunch or dinner there.
Corduneanu attended more than 100 national and international conferences, had short visits and gave talks about his research work in over 60 universities or institutes, in over 20 countries including Russia, Ukraine, Germany, England, France, Italy, China, Japan, Hungary, Poland, Portugal, and Chile. Authors around the world have quoted Corduneanu's work in over 110 books, monographs, and textbooks.

Editorial Activities
Editor: 1981-2011 Libertas Mathematica, the Mathematical Journal of the American Romanian Academy of Arts and Sciences.

Invited Lectures (Colloquium Programs, Exchange Programs)
. Czechoslovakia: The Mathematical Institutes of the Academies of Sciences and the Universities in Prague, Brunno, and Bratislava (1962, 1966, 1971). . Belgium: The University of Louvain (1971Louvain ( , 1976. . United Kingdom: The Universities of Warwick, Durham, andSussex (1971, 1973); The University of Wales (1989); The University of Dundee (1992); University of Strathclyde (1994). . Canada: The University of Montreal (1973); McGill University (1987); Montreal Polytechnic (1989); University of Victoria (1993); University of Waterloo (1994). . Italy: The Universities in Milano, Florence, Perugia, Naples, and Politecnico in Torino (1965Torino ( -1993; Instituto di Alta Mathematica in Rome (1971). . Morocco: The University of Marrakech (1994,1995  In 2017 Corduneanu fell ill. He was in and out of nursing facilities and hospitals frequently. However, he continued his schedule of traveling to conferences and meetings. He visited Romania several times. In August 2018, he attended a conference that was held in honor of his 90th birthday, at Ural State University in Ekaterinburg, Russia. On December 10, 2018, I received a phone call from a Corduneanu's friend who was taking care of him at his home in Arlington, Texas, that he was very ill and about to pass away. He was taken to the Intensive Care Unit at a hospital in Arlington. On December 14, I ew to Arlington and stayed there until December 20, visiting Corduneanu every day at the hospital. On December 27, I received a text message from Corduneanu's friend that he had passed away the night before, December 26, at 10:30 pm. I was so saddened, and my heart ached. I have known Constantin Corduneanu for 30 years. He was an exemplary mathematician and, more importantly, a decent, kind, generous, and honorable man. Corduneanu did not have any children and was preceded in death by his wife, Alice, in 2005. Corduneanu's body was taken to Iaşi, and he was buried there, next to his wife, Alice.