Abstract The aim in this work is to investigate oscillation criteria for a class of nonlinear discrete fractional order equations with damping term of the form Δa(t)Δr(t)gΔαx(t)β+p(t)Δr(t)gΔαx(t)β+F(t,G(t))=0,t∈Nt0. $$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta+F(t,G(t))=0, t\in N_{t_0}. \end{array}$$ In the above equation α (0 < α ≤ 1) is the fractional order, G(t)=∑s=t0t−1+αt−s−1(−α)x(s) $\begin{array}{} \displaystyle G(t)=\sum\limits_{s=t_0}^{t-1+\alpha}\left(t-s-1\right)^{(-\alpha)}x(s) \end{array}$ and Δα is the difference operator of the Riemann-Liouville (R-L) derivative of order α. We establish some new sufficient conditions for the oscillation of fractional order difference equations with damping term based on a Riccati transformation technique and some inequalities. We provide numerical examples to illustrate the validity of the theoretical results.

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一类带阻尼项的非线性离散分数阶方程的振荡判据

摘要 这项工作的目的是研究一类非线性离散分数阶方程的振荡判据，其阻尼项的形式为 Δa(t)Δr(t)gΔαx(t)β+p(t)Δr(t)gΔαx( t)β+F(t,G(t))=0,t∈Nt0。$$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right] )\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^ \beta+F(t,G(t))=0, t\in N_{t_0}。\end{array}$$ 上式中 α (0 < α ≤ 1) 是分数阶，G(t)=∑s=t0t−1+αt−s−1(−α)x(s) $ \begin{array}{} \displaystyle G(t)=\sum\limits_{s=t_0}^{t-1+\alpha}\left(ts-1\right)^{(-\alpha)}x (s) \end{array}$ 和 Δα 是 α 阶 Riemann-Liouville (RL) 导数的差分算子。我们基于Riccati变换技术和一些不等式，为带阻尼项的分数阶差分方程的振荡建立了一些新的充分条件。我们提供了数值例子来说明理论结果的有效性。