On the Cauchy problem for the descriptor equation with perturbation in the right-hand side

We consider the Cauchy problem for a differential equation with a Fredholm operator acting on the derivative whose right-hand side depends on a small parameter. We examine the behavior of solutions as the parameter tends to zero and establish a connection between the sensitivity of solutions to perturbations and the relation between the lengths of some Jordan chains and the orders of the poles of some operators. The conditions for the appearance of boundary layers are determined.