Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term

Abstract

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 Δ a ( t ) Δ r ( t ) g Δ α x ( t ) β + p ( t ) Δ r ( t ) g Δ α x ( t ) β + F ( t , G ( t ) ) = 0 , t ∈ N t 0 . $$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta+F(t,G(t))=0, t\in N_{t_0}. \end{array}$$ In the above equation α (0 < α ≤ 1) is the fractional order, G ( t ) = ∑ s = t 0 t − 1 + α t − 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 Δ α 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.