Substantiation of a theorem on limit transition in singularly perturbed integral system with diagonal degeneration of a kernel

We study a problem of limit transition (as the small parameter tends to zero) in integral singularly perturbed system with diagonal degeneration of a kernel. In the proof of the corresponding theorem on the limit transition we essentially use the structure of the main term of asymptotic behavior, the construction of which is performed by use of algorithm of regularization method developed by S.A. Lomov for integro-differential equations. The spectrum of the operator responsible for the regularization is composed of purely imaginary points, therefore the passage to the limit in the classical sense (i. e. in a continuous metric) in general case is impossible. In work we allocate the class of right parts in which a uniform transition in the classical sense will take place.