Geodesic
Nonoscillatory solutions of discrete fractional order equations with positive and negative terms
Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 461-479
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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} ={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.
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} ={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.
DOI : 10.21136/MB.2022.0157-21
Classification : 26A33, 39A10, 39A13, 39A21
Keywords: fractional difference equation; nonoscillatory; Caputo fractional difference; forcing term 
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Alzabut, Jehad; Grace, Said Rezk; Selvam, A. George Maria; Janagaraj, Rajendran. Nonoscillatory solutions of discrete fractional order equations with positive and negative terms. Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 461-479. doi: 10.21136/MB.2022.0157-21

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