On the separation of solutions to fractional differential equations of order α ∈ (1,2)

Abstract

Given the Caputo-type fractional differential equation $D^{\alpha }y\left(t\right)=f(t,y(t\left)\right)$ with $\alpha \in (1,2)$, we consider two distinct solutions $y_1,y_2\in C[0,T]$ to this equation subject to different sets of initial conditions. In this framework, we discuss nontrivial upper and lower bounds for the difference $\left|y_1\right(t)-y_2(t\left)\right|$ for $t\in [0,T]$. The main emphasis is on describing how such bounds are related to the differences of the associated initial values.