Oscillation Behaviour of Solutions for a Class of a Discrete Nonlinear Fractional-Order Derivatives

Abstract

Abstract Based on the generalized Riccati transformation technique and some inequality, we study some oscillation behaviour of solutions for a class of a discrete nonlinear fractional-order derivative equation Δ [ γ ( ℓ ) [ α ( ℓ ) + β ( ℓ ) Δ μ u ( ℓ ) ] η ] + ϕ ( ℓ ) f [ G ( ℓ ) ] = 0 , ℓ ∈ N ℓ 0 + 1 − μ , \[\Delta [\gamma (\ell ){[\alpha (\ell ) + \beta (\ell ){\Delta ^\mu }u(\ell )]^\eta }] + \phi (\ell )f[G(\ell )] = 0,\ell \in {N_{{\ell _0} + 1 - \mu }},\] where ℓ 0 > 0 ,   G ( ℓ ) = ∑ j = ℓ 0 ℓ − 1 + μ ( ℓ − j − 1 ) ( − μ ) u ( j ) \[{\ell _0} > 0,\quad G(\ell ) = \sum\limits_{j = {\ell _0}}^{\ell - 1 + \mu } {{{(\ell - j - 1)}^{( - \mu )}}u(j)} \] and Δ μ is the Riemann-Liouville (R-L) difference operator of the derivative of order μ, 0 < μ ≤ 1 and η is a quotient of odd positive integers. Illustrative examples are given to show the validity of the theoretical results.