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On the Unique Solvability of a Boundary Value Problem for Systems of Loaded Integro-Differential Equations with Involution

Authors
  • Usmanov, K. I.1
  • Nazarova, K. Zh.1
  • Yerkisheva, Zh. S.1
  • 1 Khoja Akhmet Yassawi International Kazakh-Turkish University, Turkistan, Kazakhstan , Turkistan (Kazakhstan)
Type
Published Article
Journal
Lobachevskii Journal of Mathematics
Publisher
Pleiades Publishing
Publication Date
Dec 13, 2021
Volume
42
Issue
12
Pages
3022–3034
Identifiers
DOI: 10.1134/S1995080221120374
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

AbstractIn this paper, we consider a boundary value problems for a systems of loaded integro-differential equations with an involutory transformation. The parameterization method is applied to the boundary value problem for a system with continuous kernel. By using the properties of involutory transformation, the problem is transformed to a boundary value problem for systems of loaded integro-differential equations. The latter problem, in turn, is reduced to solving a special Cauchy problem and a system of algebraic equations in parameters introduced. An algorithm for solving the boundary value problem for systems of loaded integro-differential equations is proposed. On the basis of this algorithm, necessary conditions for the unique solvability of the original problem are established. We also consider a boundary value problem for a systems of loaded integro-differential equations with involution in the case of degenerate kernels. By applying the parametrization method and the theory of integral equations, the problem is reduced to solving a system of algebraic equations. Based on the invertibility of the matrix of that system, necessary and sufficient conditions for the unique solvability of the problem under study are established.

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