Solving a nonlinear matrix equation by Newton's method

The study of nonlinear boundary value problems, generally speaking, is not solved with respect to the derivative for ordinary differential equations. This area of research is a traditional focus of the Kyiv school of nonlinear oscillations, initiated in the 1930s by M.M. Krylov and M.M. Bogolyubov. Their work was continued by S. Campbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, and O.A. Boychuk, who focused on differential-algebraic boundary value problems. The linearization of nonlinear systems of differential equations that are not solved with respect to the derivative leads to linear differential-algebraic equations. Solving these linear differential-algebraic equations can be considered an ill-posed problem, as small perturbations in the equations can lead to significant changes in their solutions. The study of linear differential-algebraic equations, using the central canonical form and perfect pairs and triples of matrices, is the subject of monographs by A.M. Samoilenko, M.O. Perestiuk, V.P. Yakovets, O.A. Boychuk, as well as numerous works by foreign authors such as S. Campbell, J.R. Magnus, and V.F. Chistyakov. In the article by O.A. Boychuk and O.A. Pokutnyi, conditions for the solvability of a nonlinear differential-algebraic system are proposed, utilizing a generalized central canonical form. A key assumption in this work is that the linear part of the differential-algebraic system can be reduced to a canonical form introduced by V.F. Chistyakov. Additionally, studies on linear differential equations with an irreversible operator in the derivative in infinite-dimensional spaces were initiated by S.G. Crane, A.G. Rutkas, L.A. Vlasenko, G.V. Demidenko, M.V. Falaleev, and S.P. Zubova, as well as numerous works by foreign authors A. Favini, A. Yagi, and R.E. Showalter. We have studied the problem of determining constructive conditions for existence and methods for constructing solutions to nonlinear differential-algebraic boundary value problems. In particular, we focus on the problem of finding equilibrium positions for these problems. A distinctive feature of such problems is the inability to apply classical Newton’s method. To find the equilibrium positions, the article uses the generalized Newton-Kantorovich method.