Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations (Q518543)

The authors obtain the asymptotic approximate solutions of the singularly perturbed system of nonlinear first-order ordinary differential equations of the form: \[ \varepsilon\,\dfrac{dx}{dt} - f(x,t), \quad t \in [0,T)\;x(0) = \alpha, \] on a finite or infinite half-open time interval, when \(\varepsilon\to +0\). A modified Newton's sequence starting from the element \(x_n^0(\varepsilon)\) is constructed. The existence of the limit of the Newton's sequence is proved based on a new variant of the proof of the Kantorovich theorem.