PARAMETRIZATION FOR NON-LINEAR PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS

We consider the integral boundary-value problem for a cer- tain class of non-linear systems of ordinary differential equations of the form x 0 (t) = f (t, x(t)) , t 2 (0, T), Ax(0) + Z T 0 P(s)x(s)ds + Cx(T) = d, where f : (0, T) × D ! R n is continuous vector function, DR n 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.

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