ABSTRACT. We give a new approach for the investigation of existence and construction of an approximate solutions of nonlinear non-autonomous systems of ordinary differential equations under nonlinear integral boundary conditions depending on the derivative. The constructivity of a suggested technique is shown on the example of non-linear integral boundary value problem with two solutions.

Abstract. We show how a suitable interval halving and parametrization technique can help to essentially improve the sufficient convergence condition for the successive approximations dealing with solutions of nonlinear non-autonomous systems of ordinary differential equations under integral boundary conditions.

Sufficient conditions guaranteeing the solvability of non-linear integral boundary value problems for a system of non-linear ordinary differential equations are obtained using a special successive approximation technique. The efficiency of the suggested approach is shown on an example of a non-linear integral boundary value problem possessing at least two solutions.