On approximate solutions of nonlinear boundary-value problems by the Newton-Kantorovich method (Q2246192)

In this paper, the authors establish necessary and sufficient conditions for the solvability of a nonlinear boundary-value problem in the critical case. They develop a scheme for the construction of solutions of this problem by using the Newton-Kantorovich method. A new iterative scheme is proposed for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples, approximations to the solutions of a periodic boundary-value problem for the Duffing and Liénard equations are considered.