Oscillatory behavior of nonlinear Hilfer fractional difference equations

In this paper, we study the oscillation behavior for higher order nonlinear Hilfer fractional difference equations of the type $$\begin{aligned}& \Delta _{a}^{\alpha ,\beta }y(x)+f_{1} \bigl(x,y(x+\alpha ) \bigr) =\omega (x)+f_{2} \bigl(x,y(x+ \alpha ) \bigr),\quad x\in \mathbb{N}_{a+n-\alpha }, \\& \Delta _{a}^{k-(n-\gamma )}y(x) \big|_{x=a+n-\gamma } = y_{k}, \quad k= 0,1,\ldots,n, \end{aligned}$$ Δ a α , β y ( x ) + f 1 ( x , y ( x + α ) ) = ω ( x ) + f 2 ( x , y ( x + α ) ) , x ∈ N a + n − α , Δ a k − ( n − γ ) y ( x ) | x = a + n − γ = y k , k = 0 , 1 , … , n , where $\lceil \alpha \rceil =n$ ⌈ α ⌉ = n , $n\in \mathbb{N}_{0}$ n ∈ N 0 and $0\leq \beta \leq 1$ 0 ≤ β ≤ 1 . We introduce some sufficient conditions for all solutions and give an illustrative example for our results.