On the Solvability of a Nonlocal Boundary Value Problem for Systems of Differential Equations with Involution

We study a nonlocal boundary value problem for a system of functional-differential equations with involution and an additional parameter. The proposed approach is based on a parameterization method developed by D. Dzhumabaev. The original boundary value problem is transformed into an equivalent Cauchy problem posed at the midpoint of the interval, together with a system of algebraic equations for the unknown parameters. By exploiting the symmetry and antisymmetry properties of the solution, we reduce the resulting Cauchy problem to a system of coupled Volterra integral equations of the second kind. This reduction makes it possible to derive explicit solution representations and to establish necessary and sufficient conditions for unique solvability in terms of the invertibility of the matrix generated by the boundary conditions. Finally, a first-order functional-differential equation is analyzed to demonstrate the applicability of the proposed method and to illustrate the obtained solvability conditions.