Asymptotic analysis of solutions of ordinary differential equations with distribution coefficients

• Published in: Sbornik. Mathematics Vol. 211; no. 11; pp. 1623 - 1659
• Main Authors: Savchuk, A. M., Shkalikov, A. A.
• Format: Journal Article
• Language: English
• Published: Providence London Mathematical Society, Turpion Ltd and the Russian Academy of Sciences 01.11.2020; IOP Publishing
• Subjects: Asymptotic properties; asymptotics with respect to the spectral parameter; Birkhoff asymptotics; Differential equations; differential equations with distribution coefficients; Linear equations; Mathematical analysis; Mathematical functions; Ordinary differential equations; Representations; spectral asymptotics
• ISSN: 1064-5616; 1468-4802
• DOI: 10.1070/SM9340

Ordinary differential equations of the form on the finite interval are under consideration. Here the functions and are absolutely continuous and positive and the coefficients of the differential expression are subject to the conditions where denotes the th antiderivative of the function in the sense of distributions. Our purpose is to derive analogues of the classical asymptotic Birkhoff-type representations for the fundamental system of solutions of the above equation with respect to the spectral parameter as in certain sectors of the complex plane . We reduce this equation to a system of first-order equations of the form where is a positive function, is a matrix with constant elements, the elements of the matrices and are integrable functions, and as . For systems of this kind, we obtain new results concerning the asymptotic representation of the fundamental solution matrix, which we use to make an asymptotic analysis of the above scalar equations of high order. Bibliography: 44 titles.