Oscillation Criteria for Nonlinear Fractional Differential Equations

Abstract

Several oscillation criteria are established for nonlinear fractional differential equations of the form ${\left\{a\left(t\right){\left[(r\left(t\right)D_{-}^{\alpha }x\left(t\right){)}^{\prime }\right]}^{\eta }\right\}}^{\prime }-F\left(t, {\int }_t^{\infty }{\left(v-t\right)}^{-\alpha }x\left(v\right)dv\right)=0,$ where $D_{-}^{\alpha }x$ is the Liouville right-side fractional derivative of order $\alpha \in (0, 1)$ of $x$ and $\eta$ is a quotient of two odd positive integers. We also give some examples to illustrate the main results. To the best of our knowledge, the results are initiation for the oscillatory behavior of the equations.