An Approach to Solving A Nonlinear Boundary Value Problem for A Fredholm Integro-differential Equation

A nonlinear boundary value problem for a Fredholm integro-differential equation is consid-ered. The interval where the problem is considered is partitioned and the values of a solution to the problem at the left endpoints of the subintervals are introduced as additional parameters. The introduction of addi-tional parameters gives initial values at the left endpoints of subintervals for new unknown functions. The  considered integro-differential equation is reduced to a special Cauchy problem with parameters for a sys-tem of integro-differential equations. If this problem is solvable, then its solution can be represented using the introduced parameters and known values of the integro-differential equation. By substituting these repre-sentations into the boundary condition and the continuity conditions of the solution at the interior partition points, a system of nonlinear algebraic equations in the introduced parameters is constructed. The solvabil-ity of the boundary value problem is reduced to that of the system of algebraic equations. The conditions for the existence of a solution to the auxiliary system of algebraic equations are established