Method for solving a nonlinear spectral problem for a system of ordinary differential equations with redundant conditions

We consider a nonlinear spectral problem for a system of ordinary differential equations supplemented with nonlocal conditions given by a Stieltjes integral. In addition to the main conditions, the solution is subjected to redundant conditions, which are nonlocal as well. To ensure the nontrivial solvability of the “overdetermined” problem thus obtained, the original problem is replaced by an auxiliary problem, which is consistent under the whole set of conditions. We obtain this auxiliary problem from the original one by allowing a discrepancy of a certain form. We study the resulting problem and provide a numerical solution method.