A qualitative investigation for some singular functional differential eqation

Abstract

A singular Cauchy problem for functional differential equations of a certain type is considered, solved for the derivative of the unknown function. Solutions are sought in the class of continuously differentiable functions. It is proved that there exists a nonempty set of continuously differentiable solutions having certain asymptotic properties in a sufficiently small neighborhood of the singular point. Construction of the asymptotic behavior of solutions is as important result as proof of the existence of solutions. To study the task, a technique was used that combines elements of the theory of functions and the qualitative theory of differential equations. Moreover, a qualitative analysis was applied not only in constructing a certain nonlinear operator, but also in proving that this operator satisfies the conditions of the fixed-point theorem. This technique, in our opinion, can be used for a wide range of problems of the theory of nonlinear ordinary differential equations.