On some results of investigation of singular systems of integrodifferential equations obtained by Yu. Ye. Boyarintsev research group

This paper is dedicated to the 110th anniversary of the dean of the mathematical faculty of Irkutsk State University Vladimir Vladimirovich Vasiliev. We consider statements of problems that arose when developing the theory and numerical methods for solving differential algebraic equations (DAEs) and that were studied and outlined in the works by Yu.Ye. Boyarintsev. The solution of such problems required expansion of the original field of research and incorporated investigations of systems of integral differential equation (IDEs)s as well as systems of Volterra equations with an identically singular matrix at the leading part. Nowadays such systems of integral equations are commonly called integral algebraic equations (IAEs). The results obtained in this area formed the basis for creating efficient numerical methods for IDEs, IAEs, and DAEs. This paper focuses on a general case of linear systems of IDEs with an identically singular matrix multiplying the higher derivative of the desired vector-function. We also study IDEs perturbed by the Fredholm operators and investigate the structure of general solutions to such systems. Our study employs the approach based on the analysis of extended systems and properties of matrix polynomials that in a certain way correspond to the IDEs under scrutiny.