Nebojša Đurić, MA, Senior Assistant for the narrow scientific field of Mathematical Analysis and Applications, employed at the Faculty of Architecture, Civil Engineering and Geodesy at the University of Banja Luka, challenged the result of the famous Russian mathematician Vyacheslav Yurko.
Namely, Đurić and Sergey Buterin, a Russian mathematician from the Saratov State University, published the scientific paper „On non-uniqueness of recovering Sturm-Liouville operators with delay“ in the prestigous scientific journal „Communication in Nonlinear Science and Numerical Simulation“ with the impact factor of 4,11. It is a paper in which the result of Professor Yurko, one of the world's leading scientists, specialized for the inverse spectral theory, was refuted.
Yurko is a professor at Saratov State University and up to date, he has published nine monographs and more than 500 research articles, over 170 scientific papers of which are dedicated to inverse spectral problems for different classes of operators, including functional-differential operators with delay.
Seven months ago, Nebojša Đurić and Sergey Buterin published the paper „On an open question in recovering Sturm-Liouville-type operators with delay“ in the scientific journal „Applied Mathematics Letters“ with the impact factor of 3,85. It is a paper in which they solved a four-decade-old mathematical problem, which caused great interest in the global mathematics community.
,,In this paper, we gave a negative answer to the question: “Is there a unique solution to the inverse problem of Sturm-Liouville operators with constant delay and Dirichlet/Neumann boundary conditions?“. Even then, there were doubts about Yurko's result concerning the uniqueness theorem of the inverse problem of the Sturm-Liouville operators with constant delay and Robin boundary conditions“, Đurić says.
„When we talk about the inverse problem of the Sturm-Liouville operators with delay, there are two problems. One refers to Dirichlet/Neumann boundary conditions and the other to Robin boundary conditions. If the uniqueness theorem is not valid for one problem, then it is expected that it is not valid for the other problem either“, Đurić explains.
„However, it turned out that the problem with Robin boundary conditions was much more difficult than we could have imagined. At times I thought it would take us a few years to solve the problem“, Đurić says.
Nonetheless, after six months of working on this problem, Đurić and Buterin managed to construct a counterexample and challenge the uniqueness theorem argued for by the famous Professor Yurko.
Many scientists who have been in this field for many years believed in the theorem of uniqueness. Recently, our mathematician Nebojša Đurić began to argue in favour of the idea that in the general case, the solution does not have to be unique. With this idea, he addressed Professor Yurko, who rejected the claim and stated that it was wrong. Shortly afterwards, Đurić's idea was accepted by Sergey Buterin. Their collaboration proved to be successful, and their results changed the approach to inverse spectral theory for differential operators with delay.
It is important to emphasize that many processes in nature often have non-local behavior, so operators with delay and other types of non-local operators often have applications in natural sciences and engineering.
Papers also available at ResearchGate
The papers of Nebojša Đurić and Sergey Buterin are also available in the well-known ResearchGate database, and can be accessed at the following links:
1. The paper ,,On non-uniqueness of recovering Sturm-Liouville operators with delay“
2. The paper „https://www.researchgate.net/publication/345597892_On_an_open_question_in_recovering_Sturm-Liouville-type_operators_with_delay“