On Smoothness of the Abel Equation Solution in Terms of the Jacoby Series Coefficients

TL;DR

This paper advances the understanding of the Abel equation by establishing conditions for unique solutions expressed via Jacobi series coefficients and analyzing how parameters influence solution smoothness.

Contribution

It improves existing theorems on solution existence and uniqueness, linking parameter values to the smoothness of solutions in the context of Jacobi series.

Findings

Conditions for unique solutions are formulated.

Relationship between parameters and solution smoothness is established.

Independence of certain parameters from solution smoothness is proved.

Abstract

In this paper we continue the investigation of the Abel equation with the right part belonging to a Lebesgue weighted space. We have improved the previously known result - the uniqueness and existence theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find and classify a solution due to an asymptotic of some relation containing the Jacoby coefficients of the right part. The new main results are in the following: The conditions imposed on the parameters, under which the Abel equation has a unique solution represented by the series, are formulated; The relationship between the values of the parameters and the solution smoothness is established. The independence between one of the parameters and the smoothness of the solution is proved.