Abstract Differential Equations and Caputo Fractional Derivative

TL;DR

This paper investigates the behavior of solutions to abstract Cauchy problems with Caputo fractional derivatives, focusing on how solutions and related operator families depend continuously on the fractional order parameter.

Contribution

It provides new insights into the continuity properties of solutions and Mittag-Leffler operator families for fractional derivatives of order between 0 and 1.

Findings

Solutions depend continuously on the fractional order parameter.

Mittag-Leffler operator families exhibit continuity with respect to lpha.

The study extends understanding of fractional differential equations in abstract settings.

Abstract

In this work I consider the abstract Cauchy problems with Caputo fractional time derivative of order $\alpha\in(0,1]$, and discuss the continuity of the respective solutions regarding the parameter $\alpha$. I also present a study about the continuity of the Mittag-Leffler families of operators (for $\alpha\in(0,1]$), induced by sectorial operators.