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

<p>This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument of the form
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>Δ</mi><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi>x</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mi>n</mi><mo>∈</mo><msub><mi>ℕ</mi><mn>0</mn></msub><mtext> </mtext><mo stretchy="false">[</mo><mo>∇</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi>x</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo stretchy="false">]</mo><mo>,</mo></mrow></math><tex-math>\[\Delta x(n) + p(n)x(\tau (n)) = 0,\,n \in {_0}\quad [\nabla x(n) - q(n)x(\sigma (n)) = 0,\,n \in ],\
</tex-math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></mrow></math><tex-math><?CDATA $\[{(p(n))_{n \ge 0}}\]$?></tex-math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></mrow></math><tex-math><?CDATA $\[{(q(n))_{n \ge 1}}\]$?></tex-math></alternatives></inline-formula> are sequences of nonnegative real numbers and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></mrow></math><tex-math>\[{(\tau (n))_{n \ge 0}},\quad {(\sigma (n))_{n \ge 1}}\]</tex-math></alternatives></inline-formula> are (not necessarily monotone) sequences of integers. The results obtained improve all well-known results existing in the literature and an example, numerically solved in MATLAB, illustrating the significance of these results is provided.</p>
</abstract>

This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument of the form
Δx(n)+p(n)x(τ(n))=0, n∈ℕ0 [∇x(n)−q(n)x(σ(n))=0, n∈ℕ],\[\Delta x(n) + p(n)x(\tau (n)) = 0,\,n \in {_0}\quad [\nabla x(n) - q(n)x(\sigma (n)) = 0,\,n \in ],\
, where (p(n))n≥0 and (q(n))n≥1 are sequences of nonnegative real numbers and (τ(n))n≥0, (σ(n))n≥1\[{(\tau (n))_{n \ge 0}},\quad {(\sigma (n))_{n \ge 1}}\] are (not necessarily monotone) sequences of integers. The results obtained improve all well-known results existing in the literature and an example, numerically solved in MATLAB, illustrating the significance of these results is provided.

Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

School of Pedagogical and Technological Education (ASPETE)

Faculty of Electrical Engineering and Informatics Technical University of Koşsice

Tatra Mountains Mathematical Publications

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

{"article-title":"Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments"}

This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument...

Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

Publié en ligne: 01 janv. 2022

Pages: 81 - 100

Reçu: 23 août 2020

DOI: https://doi.org/10.2478/tmmp-2021-0021

Mots clés
non-monotone argument, retarded argument, advanced argument, oscillation, Grönwall inequality

© 2021 Mathematical Institute, Slovak Academy of Sciences

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

Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

Publié en ligne: 01 janv. 2022

Pages: 81 - 100

Reçu: 23 août 2020

DOI: https://doi.org/10.2478/tmmp-2021-0021

Mots clésnon-monotone argument, retarded argument, advanced argument, oscillation, Grönwall inequality

© 2021 Mathematical Institute, Slovak Academy of Sciences

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

Mots clés
non-monotone argument, retarded argument, advanced argument, oscillation, Grönwall inequality