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Approximative Solutions to Autonomous Difference Equations of Neutral Type

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Differential and Difference Equations with Applications (ICDDEA 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 230))

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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.

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References

  1. 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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. 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)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. 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)

    Google Scholar 

  9. 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)

    Google Scholar 

  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  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul.Umultowska 87, 61-614, Poznań, Poland

    Janusz Migda

Authors
  1. Janusz Migda
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Correspondence to Janusz Migda .

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Editors and Affiliations

  1. Departamento de Ciências Exactas e Engenharia, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain

    Tomás Caraballo

  3. Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei, China

    Peter Kloeden

  4. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee, USA

    John R. Graef

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Migda, J. (2018). Approximative Solutions to Autonomous Difference Equations of Neutral Type. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J. (eds) Differential and Difference Equations with Applications. ICDDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-319-75647-9_26

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