A Fite type result for sequential fractional differential equations

TL;DR

This paper establishes a Fite type lower bound for the interval length in solutions of sequential fractional differential equations, aiding the analysis of disconjugacy and fractional interpolation.

Contribution

It introduces a new Fite type result for fractional differential equations, providing bounds that depend only on parameters, extending classical ODE results to fractional cases.

Findings

Derived a positive lower bound for the interval length based on solution zeros.

The bound depends solely on the order of the fractional derivative and the coefficient's supremum.

Results may facilitate the study of disconjugacy and interpolation in 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.