Oscillation behavior of solutions for a class of a discrete nonlinear fractional-order derivatives

Abstract

Based on the generalized Riccati Transformation technique and some inequality, we study some oscillation behavior of solutions for a class of a discrete nonlinear fractional-order derivative equation
$$
\Delta\left[\gamma(\ell)[\alpha(\ell)+\beta(\ell)\Delta^\mu u(\ell)]^\eta\right]+\phi(\ell)f[G(\ell)]=0, \ell\in N_{\ell_0+1-\mu}
$$
where $G(\ell)=\sum\limits_{j=\ell_0}^{\ell-1+\mu}\left(\ell-j-1\right)^{(-\mu)}u(j)$ and $\Delta^\mu$ is the Riemann-Liouville (R-L) difference operator of the derivative of order $\mu$, $0<\mu\leq 1$ and $\eta$ is a quotient of odd positive integers. Illustrative examples are given to show the validity of the theoretical results.