Singular initial value problem for a system of integro-differential equations unsolved with respect to the derivative

A singular initial value problem for a system of integro-differential equations of the first order unsolved with respect to the derivative of the unknown functions is considered. It is shown that under certain assumptions there is a solution lying in a region, having the vertex coinciding with the initial point, in which the graph of a solution of given singular problem is located. The main result gives asymptotic estimates of a solution and its derivative with respect to a general solution of an auxiliary differential equation. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. The results generalize some previous ones, where the singular initial value problems for integro-differential equations were studied.