Theorem of Existence and Uniqueness of Solution for Differential Equation of Fractional Order

TL;DR

This paper establishes theorems guaranteeing the existence and uniqueness of solutions for second-order fractional differential equations in Euclidean space, extending results across different fractional derivative definitions.

Contribution

It proves existence and uniqueness theorems for second-order fractional differential equations in Kipriyanov sense, applicable to Riemann-Liouville derivatives via operator reduction.

Findings

Theorems of existence and uniqueness are proved.

Results are valid for fractional derivatives in Kipriyanov and Riemann-Liouville senses.

Applicable in n-dimensional Euclidean space.

Abstract

In this paper we proved a theorems of existence and uniqueness of solutions of differential equation of second order with fractional derivative in the Kipriyanov sense in lower terms. As a domain of definition of the functions we consider the n --- dimensional Euclidean space. By a simple reduction of Kipriyanov operator to the operator of fractional differentiation in the sense of Marchaud these results can be considered valid for the operator of fractional differentiation in the sense of Riemann-Liouville, because of known fact coincidence of these operators on the classes of functions representable by the fractional integral.