Convergence of continuous analogues of numerical methods for solving degenerate optimization problems and systems of nonlinear equations

A new approach is proposed for studying the convergence of continuous analogues of the gradient and Newton methods as applied to degenerate nonlinear systems of equations and unconstrained optimization problems in the case when traditional Lyapunov functions are ineffective or inapplicable. The main tool for analyzing degenerate systems is the $p$-factor Lyapunov function, which makes it possible to reduce the original problem to a new one based on constructions of $p$-regularity theory and to construct a method converging to the exact solution in the degenerate case.