Regularization of an ill-posed Volterra integral equation of the first kind

In the paper we study a nonlinear the first kind Volterra integral equation with a special solution. At the same time, on the base of the developed method of singular perturbations, the questions of regularizability and uniqueness of the solution of the original equation in the introduced space, where a special function of a special type with a small parameter is taken into account, are proved. We note that the equation under study degenerates in many incorrect (conditionally correct) inverse problems of mathematical physics, for example, in the problems of heat conduction, filtration, integral geometry, moisture transfer in soils, hereditary environment [2-4, 10], etc., in which is the relevance of this paper.