Oscillation Behavior for a Class of Differential Equation with Fractional-Order Derivatives

Abstract

By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation $[a\left(t\right)$ ${\left(p\left(t\right)+q\left(t\right)\left(D_{-}^{\alpha }x\right)\left(t\right){)}^{\gamma }\right]}^{\mathrm{\prime }}$ − $b(t)f\left({\int }_t^{\mathrm{\infty }}\mathrm{‍}(s-t{)}^{-\alpha }x(s)ds\right)$ = $0$, for $t⩾t_0>0$, where $D_{-}^{\alpha }x$ is the Liouville right-sided fractional derivative of order $\alpha \in (\mathrm{0,1})$ of $x$ and $\gamma$ is a quotient of odd positive integers. The results in this paper extend and improve the results given in the literatures (Chen, 2012).