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.

A modification of the parameterization method is proposed to solve a linear two-point boundary value problem for a Fredholm integro-differential equation. The domain of the problem is partitioned and additional parameters are set as the values of the solution at interior points of the partition subintervals. Definition of a regular pair consisting of a partition and chosen interior points is given. The original problem is transformed into a multipoint boundary value problem with parameters. For fixed values of parameters, we get a special Cauchy problem for a system of integro-differential equations on the subintervals. Using the solution to this problem, the boundary condition and continuity conditions of solutions at the interior mesh points of the partition, we construct a system of linear algebraic equations in parameters. It is established that the solvability of the problem under consideration is equivalent to that of the constructed system.

In this paper, a multipoint boundary value problem for systems of integro-differential equations with involution has been studied. To solve the studied problem, the parameterization method is used. Based on the parametrization method, the studied problem is decomposed into two parts, i.e., into the Cauchy problem and a system of linear equations. Necessary and sufficient conditions for the unique solvability of the studied problem are determined.

In this paper, solvability of some inverse problems for a nonlocal analog of a pseudoparabolic equation is studied. The nonlocal analog of a pseudoparabolic equation is formed using transformations that have the involution property. Two types of inverse problems are considered. In the first problem, in addition to the solution, a function in the right-hand side of the equation depending on the spatial variable is determined. In the second problem, a function depending on the time variable is found. The first problem is solved using the Fourier method, and the second problem is solved by reducing to the integral Volterra equation.