Parametrization for non-linear problems with integral boundary conditions

We consider the integral boundary-value problem for a certain class of non-linear systems of ordinary differential equations of the form where f: [0, T] × D → ℝⁿ is continuous vector function, D ⊂ ℝⁿ is a closed and bounded domain. By using an appropriate parametrization technique, the given prob- lem is reduced to an equivalent parametrized family of two-point boun- dary-value problems with linear boundary conditions without integral terms. To study the transformed problem, we use a method based upon a special type of successive approximations which are constructed ana- lytically. We establish sufficient conditions for the uniform convergence of that sequence and introduce a certain finite-dimensional determining system whose solutions give all the initial values of the solutions of the given boundary-value problem. Based upon properties of the functions of the constructed sequence and of the determining equations, we give efficient conditions for the solvability of the original integral boundary- value problem.