Russian Mathematics

, Volume 62, Issue 4, pp 18–28 | Cite as

Substantiation of a Theorem on Limit Transition in Singularly Perturbed Integral System With Diagonal Degeneration of a Kernel

  • M. A. Bobodzhanova
  • O. D. Tuichiev
  • D. A. Shaposhnikova
Article
  • 3 Downloads

Abstract

We study a problem of limit transition (as the small parameter tends to zero) in integral singularly perturbed system with diagonal degeneration of a kernel. In the proof of the corresponding theorem on the limit transition we essentially use the structure of the main term of asymptotic behavior, the construction of which is performed by use of algorithm of regularization method developed by S. A. Lomov for integro-differential equations. The spectrum of the operator responsible for the regularization is composed of purely imaginary points, therefore the passage to the limit in the classical sense (i.e., in a continuous metric) in general case is impossible. In work we allocate the class of right parts in which a uniform transition in the classical sense will take place.

Keywords

singularly perturbed operator diagonal degeneration of the kernel limit solution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Imanaliev, M. I. Solution Methods for Nonlinear Inverse Problems and their Applications (Ilim, Frunze, 1977) [in Russian].MATHGoogle Scholar
  2. 2.
    Bobodzhanov, A. A. and Safonov, V. F. “Regularized Asymptotic Solutions of Singularly Perturbed Integral Systems with Diagonal Degeneration of the Kernel”, Differ. Equ. 37 (10), 1399–1411 (2001).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Lomov, S. A. Introduction to the General Theory of Singular Perturbations (Nauka, Moscow, 1981) [in Russian].MATHGoogle Scholar
  4. 4.
    Lomov S. A. and Lomov, I. S. Fundamentals of Mathematical Theory of Boundary Layer (Moscow Univ. Press,Moscow, 2011) [in Russian].MATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  • M. A. Bobodzhanova
    • 1
  • O. D. Tuichiev
    • 2
  • D. A. Shaposhnikova
    • 1
  1. 1.National Research University (MPEI)MoscowRussia
  2. 2.Khujand State University named after B. GafurovKhujandTajikistan

Personalised recommendations