Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


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Ignatyev M. Y. Asymptotics of Solutions of Some Integral Equations Connected with Differential Systems with a Singularity. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 1, pp. 17-28. DOI: 10.18500/1816-9791-2020-20-1-17-28, EDN: QEZGRI

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Published online: 
02.03.2020
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517.984
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QEZGRI

Asymptotics of Solutions of Some Integral Equations Connected with Differential Systems with a Singularity

Autors: 
Ignatyev M. Yu., Saratov State University
Abstract: 

Our studies concern some aspects of scattering theory of the singular differential systems y′ − x−1Ayq(x)y = ρBy, x > 0 with n × n matrices A, B, q(x), x ∈ (0, ∞), where A, B are constant and ρ is a spectral parameter. We concentrate on investigation of certain Volterra integral equations with respect to tensor-valued functions. The solutions of these integral equations play a central role in construction of the so-called Weyl-type solutions for the original differential system. Actually, the integral equations provide a method for investigation of the analytical and asymptotical properties of the Weyl-type solutions while the classical methods fail because of the presence of the singularity. In the paper, we consider the important special case when q is smooth and q(0) = 0 and obtain the classical-type asymptotical expansions for the solutions of the considered integral equations as ρ → ∞ with o−1) rate remainder estimate. The result allows one to obtain analogous asymptotics for the Weyl-type solutions that play in turn an important role in the inverse scattering theory.

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Received: 
26.06.2019
Accepted: 
01.07.2019
Published: 
02.03.2020