Analysis of Higher-Order Fractional Differential Equations with Fractional Boundary Conditions and Stability Insights Involving the Mittag-Leffler Operator.

Abstract

In this paper, we investigate a fractional differential equation of order $2< \mu <3$ with fractional boundary conditions, employing the Mittag-Leffler Law. We adopt a comprehensive approach to analyze the problem's key aspects. Firstly, we establish the existence of solutions by employing the Leray-Schauder alternative fixed point theorem. Secondly, we examine the uniqueness of the solutions using the Banach principle. Thirdly, we explore how solutions depend continuously on their initial data. Finally, the theoretical findings are supported by a practical example that demonstrates the applicability of the obtained results. This study contributes to the understanding of fractional differential equations with fractional boundary conditions, offering insights into their existence, uniqueness, and sensitivity to initial conditions.