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.