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

Abstract

Abstract Given a fractional differential equation of order $$\alpha \in (0,1]$$ 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 $$x_1$$ x 1 and $$x_2$$ 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 $$x_1(t) - x_2(t)$$ x 1 ( t ) - x 2 ( t ) for all $$t \in [0,T]$$ t ∈ [ 0 , T ] that are stronger than the bounds previously described in the literature.