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.

NONLINEAR NEUTRAL CAPUTO-FRACTIONAL DIFFERENCE EQUATIONS WITH APPLICATIONS TO LOTKA-VOLTERRA NEUTRAL MODEL

In this paper, we consider a nonlinear neutral fractional difference equations. By applying Krasnoselskii's fixed point theorem, sufficient conditions for the existence of solutions are established, also the uniqueness of solutions is given. As an application of the main theorems, we provide the existence and uniqueness of the discrete fractional Lotka-Volterra model of neutral type. Our main results extend and generalize the results that are obtained in <cite>Azabut</cite>.

Existence of local solutions for fractional difference equations with left focal boundary conditions

For 1 < ν ≤ 2 a real number and T ≥ 3 a natural number, conditions are given for the existence of solutions of the νth order Atıcı-Eloe fractional difference equation, Δ ν y(t) + f(t + ν − 1, y(t + ν − 1)) = 0, t ∈ {0, 1, …, T}, and satisfying the left focal boundary conditions Δy(ν − 2) = y(ν + T) = 0.