A Fite type result for sequential fractional differential equations

Abstract

Given the solution $f$ of the sequential fractional differential equation ${}_a{D}_t^{\alpha }{(}_a{D}_t^{\alpha }f)+P(t)f=0$, $t\in [b,c]$, where $-\infty <a<b<c<+\infty$, $\alpha \in (1/2,1)$ and $P:[a,+\infty )\to [0,P_{\infty }]$, $P_{\infty }<+\infty$, is continuous, assume that there exist $t_1,t_2\in [b,c]$ such that $f(t_1)={(}_a{D}_t^{\alpha }f)(t_2)=0$. Then, we establish here a positive lower bound for $c-a$ which depends solely on $\alpha ,P_{\infty }$. Such a result might be useful in discussing disconjugate fractional differential equations and fractional interpolation, similarly to the case of (integer order) ordinary differential equations.