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

Given a fractional differential equation of order α ∈ ( 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 ∈ [ 0 , T ] that are stronger than the bounds previously described in the literature.