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The main objective of this work is to obtain some new sufficient conditions that are essential for the oscillation of the solutions of forced nonlinear discrete fractional equations of the form ∆ ∆ µ (u(j)) + η(j)Φ(u(j)) = ψ(j), j ∈ N 0 where ∆ µ−1 u(0) = u 0 ; ∆u(j) = u(j+1)−u(j) and ∆ µ is defined as the difference operator of the Riemann-Liouville (R-L) derivative of order µ ∈ (0, 1] and N 0 = {0, 1, 2, • • • }. Numerical examples are presented to show the validity of the theoretical results.

Herein, we examine the oscillatory behavior of all solutions of a fractional order difference equations with damping term of the forms) and ∆ α denotes the Riemann-Liouville difference operator of order 0 < α ≤ 1. We arrive at some new sufficient conditions for the oscillation of solutions of fractional order damped difference equations using generalized riccati type transformation technique under suitable conditions.

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