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

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

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 4, page 461-479
  • ISSN: 0862-7959

Abstract

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This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form Δ γ u ( κ ) + Θ [ κ + γ , w ( κ + γ ) ] = Φ ( κ + γ ) + Υ ( κ + γ ) w ν ( κ + γ ) + Ψ [ κ + γ , w ( κ + γ ) ] , κ ∈ ℕ 1 - γ , u 0 = c 0 , where ℕ 1 - γ = { 1 - γ , 2 - γ , 3 - γ , ⋯ } , 0 < γ ≤ 1 , Δ γ 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.

How to cite

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Alzabut, Jehad, et al. "Nonoscillatory solutions of discrete fractional order equations with positive and negative terms." Mathematica Bohemica 148.4 (2023): 461-479. <http://eudml.org/doc/299406>.

@article{Alzabut2023,
abstract = {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 \}=\lbrace 1-\gamma ,2-\gamma ,3-\gamma ,\cdots \rbrace $, $0<\gamma \le 1$, $\Delta ^\{\gamma \}$ 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.},
author = {Alzabut, Jehad, Grace, Said Rezk, Selvam, A. George Maria, Janagaraj, Rajendran},
journal = {Mathematica Bohemica},
keywords = {fractional difference equation; nonoscillatory; Caputo fractional difference; forcing term},
language = {eng},
number = {4},
pages = {461-479},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonoscillatory solutions of discrete fractional order equations with positive and negative terms},
url = {http://eudml.org/doc/299406},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Alzabut, Jehad
AU - Grace, Said Rezk
AU - Selvam, A. George Maria
AU - Janagaraj, Rajendran
TI - Nonoscillatory solutions of discrete fractional order equations with positive and negative terms
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 4
SP - 461
EP - 479
AB - 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 }=\lbrace 1-\gamma ,2-\gamma ,3-\gamma ,\cdots \rbrace $, $0<\gamma \le 1$, $\Delta ^{\gamma }$ 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.
LA - eng
KW - fractional difference equation; nonoscillatory; Caputo fractional difference; forcing term
UR - http://eudml.org/doc/299406
ER -

References

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