On the Solvability of a Quasilinear Boundary Value Problem for an Impulsive System

A quasilinear two-point boundary value problem for a system of ordinary Fredholm integro-differential equations with impulse effects at fixed times is studied. Sufficient conditions for the existence of a unique solution to the intermediate special Cauchy problem that arises when applying the Dzhumabaev parametrization method to the original boundary value problem are established. The method of successive approximations is used to prove the existence of a unique solution to this problem. By using the solution to this problem, a relationship between the original boundary value problem and the system of algebraic equations with respect to additional parameters is established.