Boundary Value Problem for a System of Fredholm Integro-Differential Equations with Maxima

Abstract

A Dzhumabayev’s parameterization method is proposed to solve a ”linear” two-point
boundary value problem for a Fredholm integro-differential equation with maxima. The system
of differential equations is changed with the system of integral equations. Then by the method of
contracting mapping is studied the integral equation in the space BD [0, T], Rⁿ. As a practical way of solving the original problem, it is transformed into a multipoint boundary value problem
with parameters. Introduction of additional parameters yields a special Cauchy problem for a
system of integro-differential equations with parameters on the subintervals. Using the solution
to this problem, the boundary condition and continuity conditions of solutions at the interior
points of the partition, we construct a system of linear algebraic equations in parameters. We
give the algorithms of how to calculate the solutions of multipoint boundary value problem.