Asymptotic Solution of the Cauchy Problem for a First-Order Differential Equation with a Small Parameter in a Banach Space

The Cauchy problem for a first-order differential equation with a small parameter multiplying the derivative in a Banach space is considered. The right-hand side of the equation contains the Fredholm operator perturbed by an additional operator term containing a small parameter. The asymptotic expansion of the solution in powers of the small parameter is constructed by the Vasil'yeva-Vishik-Lyusternik method. To calculate the components of the regular part of the expansion, the cascade decomposition method is used, which consists in the step-by-step splitting of the equation into equations in subspaces of decreasing dimensions. The conditions under which the boundary layer phenomenon occurs in the problem are determined.