ON A REDUCTION OF A NONLINEAR AUTONOMOUS BOUNDARY-VALUE PROBLEM TO A CRITICAL CASE OF THE FIRST ORDER.

We found constructive conditions of solvability and the scheme for constructing solutions of the nonlinear autonomous boundary-value problem in the case of multiple solutions of the equation for generating constants. The convergent iterative scheme was constructed for finding approximations to solutions of the nonlinear autonomous boundary-value problem for the system of ordinary differential equations in the case of multiple solutions of the equation for generating constants. We found the approximations to solutions of the periodic boundary-value problem for autonomous equation of Duffing type as an example of applying the constructed iterative scheme. We applied residuals in the original equation for controlling the accuracy of the found approximations to solutions of the periodic boundary-value problem for the autonomous equation of Duffing type. [ABSTRACT FROM AUTHOR]