Mathematica Bohemica, first online, pp. 1-19

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

Jehad Alzabut, Said Rezk Grace, A. George Maria Selvam, Rajendran Janagaraj

Received October 13, 2021.   Published online August 29, 2022.

Abstract:  This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form $\Delta^{\gamma}u(\kappa)+\Theta[\kappa+\gamma,w(\kappa+\gamma)] =\Phi(\kappa+\gamma)+\Upsilon(\kappa+\gamma)w^{\nu}(\kappa+\gamma) +\Psi[\kappa+\gamma,w(\kappa+\gamma)]$, $\kappa\in\mathbb{N}_{1-\gamma}$, $u_0 =c_0$, where $\mathbb{N}_{1-\gamma}=\{1-\gamma,2-\gamma,3-\gamma,\cdots\}$, $0<\gamma\leq1$, $\Delta^{\gamma}$ 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.
Keywords:  fractional difference equation; nonoscillatory; Caputo fractional difference; forcing term
Classification MSC:  26A33, 39A10, 39A13, 39A21

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Affiliations:   Jehad Alzabut (corresponding author), Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia and Department of Industrial Engineering, OSTİM Technical University, Ankara 06374, Turkey e-mail:,; Said Rezk Grace, Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza 12221, Egypt, e-mail:; A. George Maria Selvam, Department of Mathematics, Sacred Heart College, Tirupattur-635601, Tamil Nadu, India, e-mail:; Rajendran Janagaraj, Department of Mathematics, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore-641021, Tamil Nadu, India, e-mail:

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