Top

Published in:

2018 | OriginalPaper | Chapter

Approximative Solutions to Autonomous Difference Equations of Neutral Type

Author : Janusz Migda

Published in: Differential and Difference Equations with Applications

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Asymptotic properties of solutions to difference equations of the form

$$ \varDelta ^m(x_n-u_nx_{n-k})=a_nf(x_{\sigma (n)})+b_n $$

Using a new version of the Krasnoselski fixed point theorem and the iterated remainder operator, we establish sufficient conditions under which a given solution of the equation

$$ \varDelta ^m(x_n-u_nx_{n-k})=b_n $$

is an approximative solution to the above equation. Our approach, based on the iterated remainder operator, allows us to control the degree of approximation. We use $\mathrm {o}(n^s)$, for a given nonpositive real s, as a measure of approximation.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Collocation Method to Solve Second Order Cauchy Integro-Differential Equations

next chapter Asymptotic Properties of Nonoscillatory Solutions of Third-Order Delay Difference Equations

Chatzarakis, G.E., Diblik, J., Miliaras, G.N., Stavroulakis, I.P.: Classification of neutral difference equations of any order with respect to the asymptotic behavior of their solutions. Appl. Math. Comput. 228, 77–90 (2014)MathSciNetMATH

Dzurina, J.: Asymptotic behavior of solutions of neutral nonlinear differential equations. Arch. Math. 38(4), 319–325 (2002)MathSciNetMATH

Guo, Z., Liu, M.: Existence of non-oscillatory solutions for a higher-order nonlinear neutral difference equation. Electron. J. Differ. Equ. 146, 1–7 (2010)

Hasanbulli, M., Rogovchenko, Y.V.: Asymptotic behavior of nonoscillatory solutions to n-th order nonlinear neutral differential equations. Nonlinear Anal. 69, 1208–1218 (2008)MathSciNetCrossRef

Jankowski, R., Schmeidel, E.: Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences. Discret. Contin. Dyn. Syst. (B) 19(8), 2691–2696 (2014)MathSciNetCrossRef

Liu, M., Guo, Z.: Solvability of a higher-order nonlinear neutral delay difference equation. Adv. Differ. Equ. Art. ID 767620, 14 pp (2010)

Liu, Z., Xu, Y., Kang, S.M.: Global solvability for a second order nonlinear neutral delay difference equation. Comput. Math. Appl. 57(4), 587–595 (2009)MathSciNetCrossRef

Liu, Z., Jia, M., Kang, S.M., Kwun, Y.C.: Bounded positive solutions for a third order discrete equation. Abstr. Appl. Anal. Art. ID 237036, 12 pp (2012)

Liu, Z., Sun, W., Ume, J.S., Kang, S.M.: Positive solutions of a second-order nonlinear neutral delay difference equation. Abstr. Appl. Anal. Art. ID 172939, 30 pp (2012)

10.

Migda, J.: Iterated remainder operator, tests for multiple convergence of series and solutions of difference equations. Adv. Differ. Equ. 2014(189), 1–18 (2014)MathSciNet

11.

Migda, J.: Approximative solutions of difference equations. Electron. J. Qual. Theory Differ. Equ. 13, 1–26 (2014)

12.

Migda, J.: Approximative solutions to difference equations of neutral type. Appl. Math. Comput. 268, 763–774 (2015)MathSciNet

13.

Migda, J.: Asymptotically polynomial solutions to difference equations of neutral type. Appl. Math. Comput. 279, 16–27 (2016)MathSciNetCrossRef

14.

Migda, M., Migda, J.: On a class of first order nonlinear difference equations of neutral type. Math. Comput. Model. 40, 297–306 (2004)MathSciNetCrossRef

15.

Migda, M., Migda, J.: Oscillatory and asymptotic properties of solutions of even order neutral difference equations. J. Differ. Equ. Appl. 15(11–12), 1077–1084 (2009)MathSciNetCrossRef

16.

Zhou, Y., Zhang, B.G.: Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients. Comput. Math. Appl. 45(6–9), 991–1000 (2003)MathSciNetCrossRef

Title: Approximative Solutions to Autonomous Difference Equations of Neutral Type
Author: Janusz Migda
Publisher: Springer International Publishing
Book: Differential and Difference Equations with Applications
Print ISBN: 978-3-319-75646-2

Electronic ISBN: 978-3-319-75647-9

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-75647-9_26

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner