Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients

TL;DR

This paper investigates the existence, uniqueness, and representation of solutions to linear Riemann-Liouville fractional differential equations with variable coefficients in integrable function spaces, including examples of solvable and unsolvable cases.

Contribution

It provides new results on existence and uniqueness, a solving method, and solution representations for these fractional differential equations.

Findings

Existence and uniqueness conditions established

Counterexample showing no solution in certain cases

Explicit solution representation method provided

Abstract

The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of solution for a Cauchy type problem with special initial conditions in the space of integrable functions. Then we provide an example of the problem that has no solution in the space of integrable functions. We give a solving method and a representation of solutions for the Cauchy type problem. Last we give some examples.