Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations

The functional equation $f(x,\varepsilon)=0$ containing a small parameter $\varepsilon$ and admitting regular and singular degeneracy as $\varepsilon\to0$ is considered. By the methods of small parameter, a function $x_n^0(\varepsilon)$ satisfying this equation within a residual error of $O(\varepsilon^{n+1})$ is found. A modified Newton's sequence starting from the element $x_n^0(\varepsilon)$ is constructed. The existence of the limit of Newton's sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton's iterative sequence). The deviation of the limit of Newton's sequence from the initial approximation $x_n^0(\varepsilon)$ has the order of $O(\varepsilon^{n+1})$, which proves the asymptotic character of the approximation $x_n^0(\varepsilon)$. The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.