Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

The singular Cauchy problem for first-order differential and integro-differential equations resolved or unresolved with respect to the derivatives of unknowns is fairly well studied see, e.g., 1–16 , but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular Cauchy problems were widely considered by using various methods see, e.g., 1–13, 16–18 , the method used here is based on a different approach. In particular, we use a combination of the topological method of T. Ważewski see, e.g., 19, 20 and Schauder’s fixed point theorem 21 . Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right neighbourhood of a singular point. Consider the following problem: