Nonoscillatory solutions of discrete fractional order equations with positive and negative terms

Abstract

summary:This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form \begin{align} \Delta ^{\gamma }u(\kappa )&+\Theta [\kappa +\gamma ,w(\kappa +\gamma )]\\=&\Phi (\kappa +\gamma )+\Upsilon (\kappa +\gamma )w^{\nu }(\kappa +\gamma ) +\Psi [\kappa +\gamma ,w(\kappa +\gamma )],\quad \kappa \in \mathbb {N}_{1-\gamma },\\ u_{0} =&c_{0}, \end{align} where $\mathbb {N}_{1-\gamma }=\{1-\gamma ,2-\gamma ,3-\gamma ,\cdots \}$, $0<\gamma \leq 1$, $\Delta ^{\gamma }$ is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results