<abstract xmlns="http://www.w3.org/1999/xhtml">We study the asymptotic properties of solutions to nonautonomous difference equations of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2018-0010_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>Δ</mi></mrow><mi>m</mi></msup><msub><mrow><mi>x</mi></mrow><mi>n</mi></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mi>n</mi></msub><mi>f</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>σ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msub><mo>)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mi>n</mi></msub><mo>,</mo><mi> </mi><mi> </mi><mi>f</mi><mo>:</mo><mi>𝕅</mi><mo>×</mo><mi>𝕉</mi><mo>→</mo><mi>𝕉</mi><mo>,</mo><mi> </mi><mi> </mi><mi>σ</mi><mo>:</mo><mi>𝕅</mi><mo>→</mo><mi>𝕅</mi></mrow></math><tex-math>$${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$</tex-math></alternatives></disp-formula>Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.</abstract>

We study the asymptotic properties of solutions to nonautonomous difference equations of the formΔmxn=anf(n,xσ(n))+bn,  f:𝕅×𝕉→𝕉,  σ:𝕅→𝕅$${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.

Approximation of Solutions to Nonautonomous Difference Equations

Tatra Mountains Mathematical Publications

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

{"article-title":"Approximation of Solutions to Nonautonomous Difference Equations"}

We study the asymptotic properties of solutions to nonautonomous difference equations of the formΔmxn=anf(n,xσ(n))+bn,  f:𝕅×𝕉→𝕉,  σ:𝕅→𝕅$${\Delta ^m}{x_n}...

Approximation of Solutions to Nonautonomous Difference Equations

Published Online: Jan 25, 2019

Page range: 109 - 121

Received: Dec 11, 2017

DOI: https://doi.org/10.2478/tmmp-2018-0010

Keywords
nonautonomous equation, difference pair, prescribed asymptotic behavior, asymptotically polynomial solution, bounded solution

© 2018 Janusz Migda et al., published by Sciendo

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

Approximation of Solutions to Nonautonomous Difference Equations

Published Online: Jan 25, 2019

Page range: 109 - 121

Received: Dec 11, 2017

DOI: https://doi.org/10.2478/tmmp-2018-0010

Keywordsnonautonomous equation, difference pair, prescribed asymptotic behavior, asymptotically polynomial solution, bounded solution

© 2018 Janusz Migda et al., published by Sciendo

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

Keywords
nonautonomous equation, difference pair, prescribed asymptotic behavior, asymptotically polynomial solution, bounded solution