Branching equation for a first-order differential equation in a Banach space with quadratic perturbations of a small parameter

This paper is devoted to the study of the behavior as $\varepsilon\to0$ of solutions of the Cauchy problem for a first-order differential equation in a Banach space with quadratic operator pencils with the derivative of the unknown function. The branching equation is obtained and analyzed by using the Newton diagram. The conditions of the appearing of a boundary layer near the initial point are identified and the structure of boundary-layer functions is determined.