A three point boundary value problem for the system of Fredholm integral-differential equations is
considered. Dividing this interval into two parts, two subintervals were obtained. A notation was introduced for the internal compression of this function. The three point boundary value problem is reduced to an equivalent problem. Replacements were made by introducing additional parameters. The initial problem was reduced to a problem with a parameter. The solution to the special Cauchy problem was written using the fundamental matrix, as well as using integral
differential equations and initial conditions. Various transformations were used for the parameter problem. To simplify expressions, special notation has been introduced. The matrix was considered reversible. The limits of expressions at the ends of the interval are found. Limit values are specified by boundary conditions and continuity conditions. A system of equations is obtained from the parameters. The theorem is formulated. The solution to the Cauchy problem was written using a fundamental matrix at fixed values of the parameter. The problems were found to be equivalent. It was shown that the solution to the original problem is unique. The method of counterargument was used. A couple of solutions have been received. Solutions to special Cauchy problems were written using the fundamental matrix. Two different systems of equations for the parameters are obtained. A system of homogeneous equations has been created. It was noted that under the condition of the theorem, the matrix was invertible. It was proved that the system of homogeneous algebraic equations has only zero solutions. The conditions for solving the problem were investigated using the algorithm of the parameterization method.