Volterra integral equations with rapidly varying kernels and their asymptotic integration

This paper is concerned with a singularly perturbed integral system with rapidly varying kernels. In contrast to integral equations with slowly varying kernels, where regularization is carried out with respect to the spectrum of the “diagonal” kernel, the regularization problem in the case of rapidly varying kernels is solved in terms of the spectrum of a certain extended operator. The goal here is to construct this operator and develop an appropriate algorithm for asymptotic solutions.