Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case

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The paper is devoted to the study of the solution of the Cauchy problem for a first-order differential equation in a Banach space with a small parameter on the right-hand side perturbing the equation. The coefficient of the derivative of the unknown function is a Fredholm operator with index zero and one-dimensional kernel. The case of a regular pair of operator coefficients is considered. An asymptotic expansion of the solution of the problem is constructed by using a method due to Vasil’eva, Vishik, and Lyusternik. In calculating the components of the regular and boundary-layer parts of the expansion, the cascade decomposition of the equations is used. It is proved that this expansion is asymptotic. Conditions for regular degeneration are found. The behavior of the solution as the parameter tends to zero is studied.