Justification of asymptotic decomposition of a solution for the problem of the motion of weak solutions of polymers near a critical point

We consider the boundary-value problem in a semibounded interval for a third-order integro-differential equation with the small parameter multiplies the product of the integral of unknown function vanishing on the boundary and its highest derivative. Such a problem arises in the description of the motion of weak solutions of polymers near a critical point. Unique solvability for the problem for all values of the parameter in [0,1] is proved in [1]. In this paper the representation of a solution as an asymptotic series in non-negative integer powers of the small parameter is established.