Adomian decomposition method in the theory of weakly nonlinear boundary value problems

The problem of solvability of nonlinear boundary value problems originates from the classical theory of periodic boundary value problems for systems of ordinary differential equations, developed in the works of A. Poincare, O.M. Lyapunov, I.G. Malkin, Yu.O. Mitropolsky, A.M. Samoilenko, O.A. Boichuk and others. In the classical works of R. Bellman, J. Hale, Y.O. Mitropolsky, A.M. Samoilenko and O.A.~Boichuk, the conditions for solvability of nonlinear boundary value problems for systems of differential equations in critical cases were obtained. To find solutions to nonlinear boundary value problems for systems of differential equations in critical cases, iterative schemes using the method of simple iterations were constructed in the monographs of A.M. Samoilenko and O.A.~Boichuk. In the works of O.A. Boichuk and S.M. Chuiko, iterative schemes based on the Newton--Kantorovich scheme with quadratic convergence were constructed to find solutions to nonlinear boundary value problems, and constructive conditions for convergence were obtained. The technique for constructing approximations to solutions of weakly nonlinear boundary value problems using the Adomian de\-com\-po\-si\-tion method investigated in this paper differs from the authors' previous results in that the boundary condition, the number of components of which, in general, does not coincide with the dimension of the solution. The results obtained can be transferred to weakly nonlinear boundary value problems with a boundary condition using nonlinear bounded vector functions. The article obtains constructive conditions for solvability and a scheme for constructing solutions to a weakly nonlinear boundary value problem for an ordinary differential equation in the critical case using the Adomian decomposition method.