Regularized asymptotics of solutions of systems of Fredholm integro-differential equations with rapidly varying kernels

We consider a system of singularly perturbed Fredholm integro-differential equations with rapidly varying kernel and develop an algorithm for constructing regularized asymptotic solutions. It is shown that, in the presence of a rapidly decaying factor multiplying the kernel, the original problem is not on the spectrum (i.e., is solvable for any right-hand side). To prove this, we obtain and use a representation of the resolvent (for sufficiently small ? > 0) in the form of a function series that is uniformly convergent in the ordinary sense.