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

<p>The system of nonlinear neutral difference equations with delays in the form <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0025_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo> <mtable columnalign="left"><mtr><mtd><mi>Δ</mi><mo stretchy="false">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><msub><mi>τ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mi>g</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>Δ</mi><mo stretchy="false">(</mo><msub><mi>y</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mi>y</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><msub><mi>τ</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mi>f</mi><mi>m</mi></msub><mo stretchy="false">(</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mi>g</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math><tex-math>\[\left\{ \begin{array}{l}
\Delta ({y_i}(n) + {p_i}(n){y_i}(n - {\tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n),\\
\Delta ({y_m}(n) + {p_m}(n){y_m}(n - {\tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n),
\end{array} \right.\]</tex-math></alternatives></inline-formula> for <italic>i</italic> = 1, . . . , <italic>m</italic> − 1, <italic>m</italic> ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (<italic>p<sub>i</sub></italic>(<italic>n</italic>)), <italic>i</italic> = 1,..., <italic>m</italic>, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.</p>
</abstract>

The system of nonlinear neutral difference equations with delays in the form { Δ(yi(n)+pi(n)yi(n−τi))=ai(n)fi(yi+1(n))+gi(n),Δ(ym(n)+pm(n)ym(n−τm))=am(n)fm(y1(n))+gm(n),\[\left\{ \begin{array}{l}
\Delta ({y_i}(n) + {p_i}(n){y_i}(n - {\tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n),\\
\Delta ({y_m}(n) + {p_m}(n){y_m}(n - {\tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n),
\end{array} \right.\] for i = 1, . . . , m − 1, m ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (pi(n)), i = 1,..., m, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.

Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equations

Tatra Mountains Mathematical Publications

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

The system of nonlinear neutral difference equations with delays in the form {...

Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equations

Published Online: Jan 01, 2022

Page range: 149 - 162

Received: Dec 12, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0025

Keywords
Difference equation, delays, neutral type, periodicity, asymptotic behaviour

© 2021 Mathematical Institute, Slovak Academy of Sciences

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

Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equations

Published Online: Jan 01, 2022

Page range: 149 - 162

Received: Dec 12, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0025

KeywordsDifference equation, delays, neutral type, periodicity, asymptotic behaviour

© 2021 Mathematical Institute, Slovak Academy of Sciences

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

Keywords
Difference equation, delays, neutral type, periodicity, asymptotic behaviour