On the Solvability of a Nonlocal Boundary Value Problem for an Integro-Differential Equation With Involution

This scientific paper considers a nonlocal boundary value problem for a certain class of integro-differential equations that include an involutive transformation in their structure. The main focus is on the application of the parameterization method developed and proposed by Professor D. Dzhumabayev, the aim of which is to study the conditions for the existence and uniqueness of solutions for such problems, as well as to determine the spectrum of eigenvalues of the corresponding boundary value problem. As is known from theory, the Cauchy problem for equations involving involutions does not always have a unique solution. To overcome this difficulty, parameters are introduced at the midpoint of the considered interval, and a transformation is performed that ensures the existence of a unique solution to the Cauchy problem. This transformation allows the original nonlocal boundary value problem to be divided into two parts: first, a special Cauchy problem, and second, a system of linear algebraic equations with respect to the introduced parameters. After substituting the solution into the boundary conditions, a system of equations is constructed, the solvability of which depends on the non-degeneracy of the corresponding matrix. In addition, the case of non-uniqueness of the solution is considered, in which the eigenvalues are studied and the paper establishes criteria ensuring the existence of solutions to the initial boundary value problem.