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Techniques on Solving Systems of Nonlinear Difference Equations

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Progress on Difference Equations and Discrete Dynamical Systems (ICDEA 2019)

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Abstract

This paper provides alternative techniques on solving some systems of difference equations. These techniques are analytical and much explanatory in nature as compared to methods used in existing literatures. We applied these methods particularly to the systems studied by Touafek in his paper Touafek (Iran J Math Sci Info 9(2): 303–305, 2014, [33]). We found out that these strategies can be used also in solving other systems that are closely related to our work. Interestingly, some of the systems are found to posses closed-form solutions that consist of intriguing integer sequences, such as those found in nature and polyenoids.

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References

  1. Bacani, J.B., Rabago, J.F.T.: On linear recursive sequences with coefficients in arithmetic-geometric progressions. Appl. Math. Sci. (Ruse) 9(52), 2595–2607 (2015)

    Google Scholar 

  2. Bacani, J.B., Rabago, J.F.T.: An analytical approach in solving a system of nonlinear difference equations. Nat. Res. Counc. Philippines Res. J. 17(3), 37–51 (2018)

    Google Scholar 

  3. Brand, L.: A sequence defined by a difference equation. Am. Math. Mon. 62, 489–492 (1955)

    Article  MathSciNet  Google Scholar 

  4. Cyvin, S.J., Brunvoll, J., Brendsdal, E., Cyvin, B.N., Lloyd, E.K.: Enumeration of polyene hydrocarbons: a complete mathematical solution. J. Chem. Inf. Comput. Sci. 35, 743–751 (1995)

    Article  Google Scholar 

  5. Dunlap, R.A.: The Golden Ratio and Fibonacci Numbers. World Scientific, Singapore (1997)

    Book  Google Scholar 

  6. Elaydi, S.: An Introduction to Difference Equations, 2nd edn. Springer, New York (1999)

    Book  Google Scholar 

  7. Elsayed, E.M., Ibrahim, T.F.: Periodicity and solutions for some systems of nonlinear rational difference equations. Hacet. J. Math. Stat. 44(6), 1361–1390 (2015)

    Google Scholar 

  8. Elsayed, E.M.: Solution for systems of difference equations of rational form of order two. Comput. Appl. Math. 33(3), 751–765 (2014)

    Article  MathSciNet  Google Scholar 

  9. Elsayed, E.M., El-Metwwally, H.: On the solutions of some nonlinear systems of difference equations. Adv. Diff. Equ. 2013, 161 (2013)

    Article  MathSciNet  Google Scholar 

  10. Grove, E.A., Ladas, G.: Advances in Discrete Mathematics and Applications. Chapman & Hall/CRC, Boca Raton (2005)

    Google Scholar 

  11. Halim, Y., Rabago, J.F.T.: On the solutions of a second-order difference equation in terms of generalized Padovan sequences. Math. Slovaca 68(1), 625–638 (2018)

    Article  MathSciNet  Google Scholar 

  12. Horadam, A.F.: Basic properties of a certain generalized sequence of numbers. Fib. Quart. 3, 161–176 (1965)

    MathSciNet  MATH  Google Scholar 

  13. Iričanin, B.D., Liu, W.: On a higher-order difference equation. Disc. Dyn. Nat. Soc. 2010, Article ID 891564, 6 pages (2010)

    Google Scholar 

  14. Jagerman, D.L.: Difference Equations with Applications to Queues. Chapman & Hall/CRC (2000)

    Google Scholar 

  15. Kocic̀, V.L., Ladas, G.: Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands (1993)

    Google Scholar 

  16. Koshy, T.: Fibonacci and Lucas Numbers with Applications. Pure and Applied Mathematics, Wiley-Interscience, New York (2001)

    Book  Google Scholar 

  17. Kulenović, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. Chapman & Hall/CRC, Boca Raton (2002)

    MATH  Google Scholar 

  18. Larcombe, P.J., Bagdasar, O.D., Fennessey, E.J.: Horadam sequences: a survey. Bull. I.C.A. 67, 49–72 (2013)

    Google Scholar 

  19. Lucas, E.: Théorie des Fonctions Numériques Simplement Périodiques. Am. J. Math. 1, 184–240, 289–321 (1878); reprinted as “The Theory of Simply Periodic Numerical Functions”, Santa Clara, CA: The Fibonacci Association (1969)

    Google Scholar 

  20. Mickens, R.E.: Difference Equations: Theory, Applications and Advanced Topics, 3rd edn. Chapman and Hall/CRC (2015)

    Google Scholar 

  21. OEIS Foundation Inc.: The On-Line Encyclopedia of Integer Sequences (2011). http://oeis.org

  22. Proakis, J.G., Manolakis, D.G.: Digital Signal Processing: Principles, Algorithms, and Applications, 4th edn. (2007)

    Google Scholar 

  23. Rabago, J.F.T.: Effective methods on determining the periodicity and form of solutions of some systems of nonlinear difference equations. Int. J. Dyn. Syst. Differ. Equ. 7(2), 112–135 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Rabago, J.F.T.: An intriguing application of telescoping sums. In: Proceedings of the 2016 Asian Mathematical Conference, IOP Conference Series: Journal of Physics: Conference Series, vol. 893, p. 012005 (2017)

    Google Scholar 

  25. Rabago, J.F.T., Halim, Y.: Supplement to the paper of Halim, Touafek and Elsayed: Part I. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 24:2, 121–131 (2017)

    Google Scholar 

  26. Rabago, J.F.T., Halim, Y.: Supplement to the paper of Halim, Touafek and Elsayed: Part II. Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal. 24:5, 333–345 (2017)

    Google Scholar 

  27. Rabago, J.F.T.: On the closed-form solution of a nonlinear difference equation and another proof to Sroysang’s conjecture. Iran. J. Math. Sci. Inform. 13(1), 139–151 (2018)

    MathSciNet  Google Scholar 

  28. Rabago, J.F.T.: On an open question concerning product-type difference equations. Iran. J. Sci. Technol. Trans. Sci. 42, 1499–1503 (2018)

    Article  MathSciNet  Google Scholar 

  29. Sargent, T.J.: Dynamic Macroeconomic Theory. Harvard University Press (1987)

    Google Scholar 

  30. Sedgewick, R., Flajolet, F.: An Introduction to the Analysis of Algorithms. Addison-Wesley (2013)

    Google Scholar 

  31. Sharkovsky, A.N., Maistrenko, Y.L., Yu Romanenko, E.: Difference Equations and Their Applications, Mathematics and Its Applications, Vol. 250. Springer, Netherlands (1993)

    Google Scholar 

  32. Stévic, S.: Representation of solutions of bilinear equations in terms of generalized Fibonacci sequences. Electron. J. Qual. Theory Differ. Equ. 67, 1–15 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Touafek, N.: On some fractional systems of difference equations. Iran. J. Math. Sci. Info. 9(2), 303–305 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Touafek, N.: On a second order rational difference equation. Hacet. J. Math. Stat. 41, 867–874 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Touafek, N., Elsayed, E.M.: On the solutions of systems of rational difference equations. Math. Comput. Model. 55, 1987–1997 (2012)

    Article  MathSciNet  Google Scholar 

  36. Touafek, N., Elsayed, E.M.: on the periodicity of some systems of nonlinear difference equations. Bull. Math. Soc. Sci. Math. Roum. Nouv. Sr. 55(103), 217–224 (2012)

    Google Scholar 

  37. Vajda, S.A.: Fibonacci & Lucas Numbers and The Golden Section: Theory and Applications. Ellis Horwood Ltd., Chishester (1989)

    MATH  Google Scholar 

  38. Vorobév, N.N.: Fibonacci Numbers. Birkhäuser, Basel (2002)

    Book  Google Scholar 

Download references

Acknowledgements

This work was completed in 2016 when JFTR was still at the Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio.

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Authors and Affiliations

  1. University of the Philippines Baguio, 2600, Baguio, Philippines

    JERICO B. BACANI

  2. Department of Complex Systems Science, Graduate School of Informatics, Nagoya University, A4-2 (780) Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan

    Julius Fergy T. Rabago

Authors
  1. JERICO B. BACANI
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  2. Julius Fergy T. Rabago
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Correspondence to JERICO B. BACANI .

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Editors and Affiliations

  1. Department of Mathematics, University College London, London, UK

    Steve Baigent

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

    Martin Bohner

  3. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

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BACANI, J.B., Rabago, J.F.T. (2020). Techniques on Solving Systems of Nonlinear Difference Equations. In: Baigent, S., Bohner, M., Elaydi, S. (eds) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-030-60107-2_7

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