On the construction of the approximate solution of a special type integral boundary value problem

We consider the integral boundary value problem (BVP) for a certain class of non-linear system of ordinary differential equations of the form (Formula Presented) where (Formula Presented) and t ∈ [0,T], x ∈ ℝⁿ, f : [0,T] × D → ℝ ⁿ and k : [0-T] × → ℝ ⁿ are continuous vector functions, D ⊂ ℝⁿ is a closed and bounded domain, A, C and d are arbitrary matrices and vector with real components, det C ≠ 0. We give a new approach for studying this problem, namely by using an appropriate parametrization technique the original BVP is reduced to the equivalent parametrized two-point one with linear restrictions without integral term. To study the transformed problem we use a method based upon a special type of successive approximations constructed analytically.