Oscillation tests for fractional difference equations

Year, pages: 2019, 53 - 64

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form \begin{eqnarray*} \Delta\Bl(r(t)g\bl(\Delta^{\alpha}x(t)\br)\Br) +p(t)f \bgl(\sum_{s=t_{0}}^{t-1+\alpha}(t-s-1)^{(-\alpha)}x(s) \bgr) = 0, \qquad t\in \mathbb{N}_{t_{0}+1-\alpha}, \end{eqnarray*} where $\Delta^\alpha$ denotes the Riemann-Liouville fractional difference operator of order $\alpha,$ $ 0 < \alpha \leq 1$, $\mathbb{N}_{t_0+1-\alpha}=\{t_0+1-\alpha,t_0+2-\alpha,\dots\}$, $t_0 > 0$ and $\ \gamma > 0$ is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

Chatzarakis, G., Gokulraj, P., Kalaimani, T. 2019. Oscillation tests for fractional difference equations. In Tatra Mountains Mathematical Publications, vol. 71, no.1, pp. 53-64. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2018-0005

Chatzarakis, G., Gokulraj, P., Kalaimani, T. (2019). Oscillation tests for fractional difference equations. Tatra Mountains Mathematical Publications, 71(1), 53-64. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2018-0005

Publisher: MÚ SAV