Upper and lower estimates for the separation of solutions to fractional differential equations

TL;DR

This paper establishes new, tighter upper and lower bounds on how solutions to fractional differential equations with Caputo derivatives diverge based on their initial conditions, enhancing understanding of solution stability.

Contribution

It provides improved quantitative bounds for the difference between solutions of fractional differential equations, advancing previous theoretical results.

Findings

Derived tighter bounds for solution differences

Enhanced understanding of solution dependence on initial conditions

Applicable to fractional differential equations of order α in (0,1]

Abstract

Given a fractional differential equation of order $\alpha \in (0,1]$ with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two solutions $x_1$ and $x_2$, say, of the same differential equation, both of which are assumed to be defined on a common interval $[0,T]$, and provide upper and lower bounds for the difference $x_1(t) - x_2(t)$ for all $t \in [0,T]$ that are stronger than the bounds previously described in the literature.