On the Solvability of a Boundary Value Problem for Delay Integro‐Differential Equation

The article addresses a two‐point boundary value problem for an integro‐differential equation with a delay argument. The problem is studied by means of the Dzhumabaev parametrization method. Within this framework, additional parameters and new functions are introduced, reducing the original boundary value problem to an equivalent problem for a system of integro‐differential equations with delay involving these additional parameters. On the basis of the parametrization method, sufficient conditions for solvability are established in terms of the coefficient matrices, the initial vector functions, the delay, and the boundary conditions. Furthermore, an algorithm is proposed for constructing the solution of the two‐point boundary value problem for the integro‐differential equation with delay.