<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2018-0005_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>Δ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mrow><msup><mrow><mi>Δ</mi></mrow><mi>α</mi></msup><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>f</mi><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow><mi>s</mi><mo>=</mo><msub><mrow><mi>t</mi></mrow><mn>0</mn></msub></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>α</mi></mrow></munderover><mrow><msup><mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mo>(</mo><mo>−</mo><mi>α</mi><mo>)</mo></mrow></msup><mi>x</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>∈</mo><msub><mrow><mi>𝕅</mi></mrow><mrow><msub><mrow><mi>t</mi></mrow><mn>0</mn></msub><mo>+</mo><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msub><mo>,</mo></mrow></math><tex-math>$$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, &amp; t \in {_{{t_0} + 1 - \alpha }},$$</tex-math></alternatives></disp-formula>where Δ<italic>α</italic> denotes the Riemann-Liouville fractional difference operator of order <italic>α</italic>, 0 &lt; <italic>α</italic> ≤ 1, ℕ<italic>t</italic>0+1−α={<italic>t</italic>0+1−α<italic>t</italic>0+2−α…}, <italic>t</italic>0 > 0 and <italic>γ</italic> > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.</abstract>

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the formΔ(r(t)g(Δαx(t)))+p(t)f(∑s=t0t−1+α(t−s−1)(−α)x(s))=0,t∈𝕅t0+1−α,$$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

Oscillation Tests for Fractional Difference Equations

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Oscillation Tests for Fractional Difference Equations"}

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the formΔ(r(t)g(Δαx(t)))+p(t)f(∑s=t0t−1+α(t−s−1)(−α)x(s))=0,t∈𝕅t0+1−α,$$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0  0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the...

Oscillation Tests for Fractional Difference Equations

Published Online: Jan 25, 2019

Page range: 53 - 64

Received: Aug 20, 2018

DOI: https://doi.org/10.2478/tmmp-2018-0005

Keywords
oscillation, difference equations, fractional sum

© 2018 George E. Chatzarakis et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Oscillation Tests for Fractional Difference Equations

Published Online: Jan 25, 2019

Page range: 53 - 64

Received: Aug 20, 2018

DOI: https://doi.org/10.2478/tmmp-2018-0005

Keywordsoscillation, difference equations, fractional sum

© 2018 George E. Chatzarakis et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
oscillation, difference equations, fractional sum