In this paper, oscillation criteria are obtained for higher-order half-linear delay difference equations involving generalized difference operator of the form Δ b ( p n ( Δ b m − 1 x n ) α ) + q n x n − σ β = 0 , n ≥ n 0 , $${\Delta _b}({p_n}{(\Delta _b^{m - 1}{x_n})^\alpha }) + {q_n}x_{n - \sigma }^\beta = 0,\qquad n \geq {n_0},$$ where ∆ _b is defined by ∆ _b y_n = y _n+1 - by _n , b ∈ ℝ - {0}, p: ℕ → ℝ⁺, α, β are the ratio of odd positive integers with β ≤ α; m, n, n ₀, σ are non-negative integers, q: ℕ → ℝ. The cases of b negative and positive and q_n ≥ 0, which has important role for oscillation of this equation, are considered. Also we provide some examples to illustrate our main results.

In applied sciences, besides the importance of obtaining the analytical solutions of differential equations with constant coefficients, the qualitative analysis of the solutions of such equations is also very important. Due to this importance, in this study, a qualitative analysis of the solutions of a delayed and constant coefficient fractal differential equation with more than one fractional derivative was performed. In the equation under consideration, the derivatives are the Riemann–Liouville fractional derivatives. In the proof of the obtained results, Laplace transform formulas of the Riemann–Liouville fractional derivative and some inequalities are used. We also provide some examples to check the accuracy of our results.

In this paper, we derive new interval oscillation criteria for impulsive conformable fractional differential equations having fixed moments of impulse actions. The results are extended to a more general class of nonlinear impulsive conformable fractional differential equations. Examples are also given to illustrate the relevance of the result.