Singular integral equation equivalent in the space of smooth functions to an ordinary differential equation, method of successive approximations for the construction of its smooth solutions and its nonsmooth solutions

Abstract

We propose a singular integral equation whose definition is extended to a singular point by additional conditions. In the space of smooth functions, this equation becomes equivalent, by the indicated extended definition, to an ordinary differential equation, whereas in the space of piecewise discontinuous functions, it becomes equivalent to an impulsive differential equation. For smooth solutions of the singular equation, we substantiate the method of successive approximations. For the ordinary differential equation, this method turns into a new algorithm for the construction of successive approximations. For the investigated equation, we specify a solution of new type, which is equivalent, for the impulsive differential equation, to a solution with discontinuity of the second kind (a “solution with needle”). We propose an algorithmic formula for the general solution of the initial-value problem for the impulsive differential equation.