<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>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
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0022_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><msup><mi>Δ</mi><mi>μ</mi></msup><mi>u</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mi>η</mi></msup><mo stretchy="false">]</mo><mo>+</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mo>,</mo><mi>ℓ</mi><mo>∈</mo><msub><mi>N</mi><mrow><msub><mi>ℓ</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>−</mo><mi>μ</mi></mrow></msub><mo>,</mo></mrow></math><tex-math>
\[\Delta [\gamma (\ell ){[\alpha (\ell ) + \beta (\ell ){\Delta ^\mu }u(\ell )]^\eta }] + \phi (\ell )f[G(\ell )] = 0,\ell \in {N_{{\ell _0} + 1 - \mu }},\] </tex-math></alternatives></inline-formula> where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0022_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>ℓ</mi><mn>0</mn></msub><mo>></mo><mn>0</mn><mo>,</mo><mtext> </mtext><mi>G</mi><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><msub><mi>ℓ</mi><mn>0</mn></msub></mrow><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>μ</mi></mrow></munderover><mrow><msup><mrow><mo stretchy="false">(</mo><mi>ℓ</mi><mo>−</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mo>−</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></msup><mi>u</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></math><tex-math>\[{\ell _0} > 0,\quad G(\ell ) = \sum\limits_{j = {\ell _0}}^{\ell - 1 + \mu } {{{(\ell - j - 1)}^{( - \mu )}}u(j)} \]</tex-math></alternatives></inline-formula> and Δ<sup><italic>μ</italic></sup> is the Riemann-Liouville (R-L) difference operator of the derivative of order <italic>μ</italic>, 0 &lt; <italic>μ</italic> ≤ 1 and <italic>η</italic> is a quotient of odd positive integers. Illustrative examples are given to show the validity of the theoretical results.</p>
</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)Δμu(ℓ)]η]+ϕ(ℓ)f[G(ℓ)]=0,ℓ∈Nℓ0+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 ℓ0>0, G(ℓ)=∑j=ℓ0ℓ−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 Δμ 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.

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

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Based on the generalized Riccati transformation technique and some inequality, we study some oscillation behaviour of solutions for a class of a discrete...

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

Published Online: Jan 01, 2022

Page range: 101 - 118

Received: May 22, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0022

Keywords
oscillation, Riemann-Liouville fractional derivatives, difference equations

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Published Online: Jan 01, 2022

Page range: 101 - 118

Received: May 22, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0022

Keywordsoscillation, Riemann-Liouville fractional derivatives, difference equations

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
oscillation, Riemann-Liouville fractional derivatives, difference equations