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 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 , $$\begin{aligned}& \Delta _{a}^{\alpha ,\beta }y(x)+f_{1} \bigl(x,y(x+\alpha ) \bigr) =\omega (x)+f_{2} \bigl(x,y(x+ \alpha ) \bigr),\quad x\in \mathbb{N}_{a+n-\alpha }, \\& \Delta _{a}^{k-(n-\gamma )}y(x) \big|_{x=a+n-\gamma } = y_{k}, \quad k= 0,1,\ldots,n, \end{aligned}$$ where ⌈ α ⌉ = n $\lceil \alpha \rceil =n$ , n ∈ N 0 $n\in \mathbb{N}_{0}$ and 0 ≤ β ≤ 1 $0\leq \beta \leq 1$ . We introduce some sufficient conditions for all solutions and give an illustrative example for our results.

In [29], Haider et al. introduced a new definition of a fractional difference operator which is a generalization of Riemann-Liouville and Caputo type difference operator. This operator interpolates the Riemann-Liouville like fractional difference (β = 0) and the Caputo like fractional difference (β = 1). The type-parameter produces more types of stationary states and provides an extra degree of freedom on the initial condition. No one has studied, to the best of our knowledge, the oscillation of equations involving the Hilfer difference operator in the literature.
In [5], Grace et al. initiated the oscillation theory for fractional differential equations of the form where D α a is the Riemann-Liouville differential operator of order α, 0 < α ≤ 1 and the functions f 1 , f 2 , ν are continuous. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo's differential operator.
In [21], Marian et al. gave similar conclusions for the oscillation behavior of the nonlinear fractional difference equations of the form where α denotes the Riemann-Liouville like discrete fractional difference operator of order α, 0 < α ≤ 1. In [22], Marian where α denotes the Riemann-Liouville like discrete fractional difference operator of order α and m ≥ 1.
This paper aims to state some oscillation criteria for a class of higher order nonlinear Hilfer fractional difference equations. Some sufficient conditions will be given for the oscillation of the solution of Hilfer fractional difference equations. The results also contain new conditions for the oscillation of the solutions of the Riemann-Liouville and Caputo difference equations.
In [29], Haider et al. introduced a Hilfer like fractional difference operator.
Definition 3 Assume f : N a → R. Then the fractional difference of order n -1 < α < n and type 0 ≤ β ≤ 1 is defined by
Then the following inequality holds if and only if ξ = χ u-1 : Then the following inequality holds if and only if ξ = χ u-1 :

Lemma 5
The unique solution of the initial value problem (6) is for all x ∈ N a+1 .
Theorem 3 Suppose α ≥ 1 and assume that (11) and (13) valid for μ < ν. If and lim sup x→∞ where K μ,ν (t) is defined as in Theorem 2, then for every sufficiently large T every bounded solution of (6) is oscillatory.