Existence and Uniqueness of Solutions for Riemann-Liouville Fractional High-Order Multi-Point Boundary Value Problems

Shivanian, Elyas; Abdolrazaghi, Fatemeh

doi:10.311581/JAMDA.2604.1031.2.1.5

Existence and Uniqueness of Solutions for Riemann-Liouville Fractional High-Order Multi-Point Boundary Value Problems

Document Type : Research Article

Authors

Elyas Shivanian ¹

Fatemeh Abdolrazaghi ²

¹ Imam Khomeini International University

² Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34148-96818, Iran

10.311581/JAMDA.2604.1031.2.1.5

Abstract

In this paper, we investigate the existence and uniqueness of solutions for a nonlinear fractional differential equation involving the Riemann-Liouville fractional derivative of order $\vartheta \in (m-1, m]$. We construct the Green's function for the corresponding linear boundary value problem and analyze its properties. By employing the Banach contraction mapping principle, we establish sufficient conditions for the existence of a unique solution. A specific emphasis is placed on the structural differences between Riemann-Liouville and Caputo frameworks, particularly regarding the general solutions and the behavior of the Green's function integral bounds.

Keywords

High-order fractional differential equation

Riemann-Liouville fractional derivative

Boundary value problem

Existence and uniqueness

Fixed point theorem