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

• Published in: Lobachevskii journal of mathematics 2021-12, Vol.42 (12), p.3022-3034
• Main Authors: Usmanov, K. I., Nazarova, K. Zh, Yerkisheva, Zh. S.
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Analysis; Boundary value problems; Cauchy problems; Differential equations; Geometry; Integral equations; Kernels; Mathematical analysis; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Parameterization; Probability Theory and Stochastic Processes; Transformations (mathematics)
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080221120374

In 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.