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On The Dynamics of a Nonlinear Difference Equation

Year 2019, Volume: 9 Issue: 1, 190 - 201, 28.06.2019

Abstract

In this study we investigate the stability of solutions of difference equation xn+1 = xn-3xn-4 -1. Moreover, we study periodic and eventually periodic solutions of related difference equation.

References

  • [1] C.M. Kent, W. Kosmala, On the Nature of Solutions of the DifferenceEquation xn+1 = xnxn􀀀3 􀀀 1, International Journal of Nonlinear Analysisand Applications 2(2) (2011) 24-43.
  • [2] C.M. Kent, W. Kosmala, M.A. Radin, S. Stevi´c, Solutions of the differ-ence equation xn+1 = xnxn􀀀1 􀀀 1, Abstr. Appl. Anal., 2010, pp. 1-13.doi:10.1155/2010/469683
  • [3] C.M. Kent, W. Kosmala, S. Stevic, Long-term behavior of solutions of thedifference equation xn+1 = xn􀀀1xn􀀀2 􀀀 1, Abstr. Appl. Anal., 2010, pp.1-17. doi:10.1155/2010/152378
  • [4] C.M. Kent, W. Kosmala, S. Stevic, On the difference equation xn+1 =xnxn􀀀2 􀀀 1, Abstr. Appl. Anal., 2011, pp. 1-15. doi:10.1155/2011/815285
  • [5] E. Camouzis, G. Ladas, Dynamics of third order rational di¤erence equa-tions with open problems and conjectures, volume 5 of Advances in DiscreteMathematics and Applications, Chapman & Hall/CRC, Boca Raton, FL,2008.
  • [6] E. Ta¸sdemir, Y. Soykan, On the Periodicies of the Difference Equationxn+1 = xnxn􀀀1 + , Karaelmas Science and Engineering Journal, 6(2)(2016), pp. 329-333.
  • [7] E. Ta¸sdemir, Y. Soykan, Long-Term Behavior of Solutions of the Non-Linear Difference Equation xn+1 = xn􀀀1xn􀀀3􀀀1, Gen. Math. Notes, 38(1)(2017), pp. 13-31.
  • [8] E. Ta¸sdemir, Y. Soykan, Stability of Negative Equilibrium of a Non-LinearDifference Equation, J. Math. Sci. Adv. Appl., 49(1) (2018), pp. 51-57.
  • [9] E. Ta¸sdemir, Y. Soykan, Dynamical Analysis of a Non-Linear DifferenceEquation, J. Comput. Anal. Appl., 26(2) (2019), pp. 288-301.
  • [10] E.A. Grove, G. Ladas, Periodicities in nonlinear di¤erence equations (Vol.4). CRC Press, (2004).
  • [11] E.M. Elsayed, New method to obtain periodic solutions of period two andthree of a rational difference equation, Nonlinear Dyn., 79(1) (2015), pp.241-250.
  • [12] G. Karakostas, Asymptotic 2–periodic difference equations with diagonallyself–invertible responses, J. Di¤erence Equ. Appl., 6(3) (2000), pp. 329-335.
  • [13] H. El-metwallya, E.A. Grove, G. Ladas, H.D. Voulov, On the global attrac-tivity and the periodic character of some difference equations, J. Di¤erenceEqu. Appl., 7(6) (2001), pp. 837-850.
  • [14] ·I. Okumu¸s, Y. Soykan, On the Stability of a Nonlinear Difference Equation,Asian Journal of Mathematics and Computer Research, 17(2) (2017), pp.88-110.
  • [15] ·I. Okumu¸s, Y. Soykan, Some Technique To Show The Boundedness OfRational Difference Equations, Journal of Progressive Research in Mathe-matics, 13(2) (2018), pp. 2246-2258.
  • [16] ·I. Okumu¸s, Y. Soykan, Dynamical Behavior of a System of Three-Dimensional Nonlinear Difference Equations, Adv. Di¤erence Equ.,2018(223) (2018), pp. 1-15.
  • [17] J. Diblik, M. Feckan, M. Pospisil, Nonexistence of periodic solutions andS-asymptotically periodic solutions in fractional difference equations, Appl.Math. Comput., 257 (2015), pp. 230-240.
  • [18] K. Liu, P. Li, F. Han, W. Zhong, Behavior of the Difference Equationsxn+1 = xnxn􀀀1 􀀀 1, J. Comput. Anal. Appl., 22(1) (2017), pp. 1361-1370.
  • [19] M. Göcen, M. Güneysu, The global attractivity of some rational differenceequations, J. Comput. Anal. Appl., 25(7) (2018), pp. 1233-1243.
  • [20] M. Göcen, A. Cebeci, On the Periodic Solutions of Some Systems of HigherOrder Difference Equations, Rocky Mountain J. Math., (2018 to appear).
  • [21] R.P. Agarwal, J. Popenda, Periodic solutions of …rst order linear di¤erenceequations, Mathl. Comput. Modelling, 22(1) (1995), pp. 11-19.
  • [22] R.P. Agarwal, P.J. Wong, Advanced topics in difference equations (Vol.404), Springer Science & Business Media, (2013).
  • [23] S. Elaydi. An Introduction to Difference Equations, Springer-Verlag, NewYork, Inc, 1996.
  • [24] S. Elaydi, R.J. Sacker, Global stability of periodic orbits of non-autonomousdifference equations and population biology, J. Di¤erential Equations,208(1) (2005), pp. 258-273.
  • [25] S. Stevic, Asymptotics of some classes of higher-order differenceequations, Discrete Dyn. Nat. Soc., 2007 (2007), pp. 1-20. doi:http://dx.doi.org/10.1155/2007/56813
  • [26] S. Stevic, J. Diblik, B. Iricanin, Z. Smarda, Z. Solvability of nonlineardifference equations of fourth order, Electron. J. Differential Equations,264 (2014), pp. 1-14.
  • [27] Y. Wang, Y. Luo, Z. Lu, Convergence of solutions of xn+1 = xnxn􀀀1 􀀀 1,Appl. Math. E-Notes, 12 (2012), pp. 153-157.

Lineer Olmayan Bir Fark Denkleminin Dinamikleri Üzerine

Year 2019, Volume: 9 Issue: 1, 190 - 201, 28.06.2019

Abstract

Bu çalışmada, fark denkleminin çözümlerinin kararlılığını araştırıldı. Ayrıca, ilgili fark denkleminin periyodik ve eninde sonunda periyodik çözümlerini de çalışıldı.

References

  • [1] C.M. Kent, W. Kosmala, On the Nature of Solutions of the DifferenceEquation xn+1 = xnxn􀀀3 􀀀 1, International Journal of Nonlinear Analysisand Applications 2(2) (2011) 24-43.
  • [2] C.M. Kent, W. Kosmala, M.A. Radin, S. Stevi´c, Solutions of the differ-ence equation xn+1 = xnxn􀀀1 􀀀 1, Abstr. Appl. Anal., 2010, pp. 1-13.doi:10.1155/2010/469683
  • [3] C.M. Kent, W. Kosmala, S. Stevic, Long-term behavior of solutions of thedifference equation xn+1 = xn􀀀1xn􀀀2 􀀀 1, Abstr. Appl. Anal., 2010, pp.1-17. doi:10.1155/2010/152378
  • [4] C.M. Kent, W. Kosmala, S. Stevic, On the difference equation xn+1 =xnxn􀀀2 􀀀 1, Abstr. Appl. Anal., 2011, pp. 1-15. doi:10.1155/2011/815285
  • [5] E. Camouzis, G. Ladas, Dynamics of third order rational di¤erence equa-tions with open problems and conjectures, volume 5 of Advances in DiscreteMathematics and Applications, Chapman & Hall/CRC, Boca Raton, FL,2008.
  • [6] E. Ta¸sdemir, Y. Soykan, On the Periodicies of the Difference Equationxn+1 = xnxn􀀀1 + , Karaelmas Science and Engineering Journal, 6(2)(2016), pp. 329-333.
  • [7] E. Ta¸sdemir, Y. Soykan, Long-Term Behavior of Solutions of the Non-Linear Difference Equation xn+1 = xn􀀀1xn􀀀3􀀀1, Gen. Math. Notes, 38(1)(2017), pp. 13-31.
  • [8] E. Ta¸sdemir, Y. Soykan, Stability of Negative Equilibrium of a Non-LinearDifference Equation, J. Math. Sci. Adv. Appl., 49(1) (2018), pp. 51-57.
  • [9] E. Ta¸sdemir, Y. Soykan, Dynamical Analysis of a Non-Linear DifferenceEquation, J. Comput. Anal. Appl., 26(2) (2019), pp. 288-301.
  • [10] E.A. Grove, G. Ladas, Periodicities in nonlinear di¤erence equations (Vol.4). CRC Press, (2004).
  • [11] E.M. Elsayed, New method to obtain periodic solutions of period two andthree of a rational difference equation, Nonlinear Dyn., 79(1) (2015), pp.241-250.
  • [12] G. Karakostas, Asymptotic 2–periodic difference equations with diagonallyself–invertible responses, J. Di¤erence Equ. Appl., 6(3) (2000), pp. 329-335.
  • [13] H. El-metwallya, E.A. Grove, G. Ladas, H.D. Voulov, On the global attrac-tivity and the periodic character of some difference equations, J. Di¤erenceEqu. Appl., 7(6) (2001), pp. 837-850.
  • [14] ·I. Okumu¸s, Y. Soykan, On the Stability of a Nonlinear Difference Equation,Asian Journal of Mathematics and Computer Research, 17(2) (2017), pp.88-110.
  • [15] ·I. Okumu¸s, Y. Soykan, Some Technique To Show The Boundedness OfRational Difference Equations, Journal of Progressive Research in Mathe-matics, 13(2) (2018), pp. 2246-2258.
  • [16] ·I. Okumu¸s, Y. Soykan, Dynamical Behavior of a System of Three-Dimensional Nonlinear Difference Equations, Adv. Di¤erence Equ.,2018(223) (2018), pp. 1-15.
  • [17] J. Diblik, M. Feckan, M. Pospisil, Nonexistence of periodic solutions andS-asymptotically periodic solutions in fractional difference equations, Appl.Math. Comput., 257 (2015), pp. 230-240.
  • [18] K. Liu, P. Li, F. Han, W. Zhong, Behavior of the Difference Equationsxn+1 = xnxn􀀀1 􀀀 1, J. Comput. Anal. Appl., 22(1) (2017), pp. 1361-1370.
  • [19] M. Göcen, M. Güneysu, The global attractivity of some rational differenceequations, J. Comput. Anal. Appl., 25(7) (2018), pp. 1233-1243.
  • [20] M. Göcen, A. Cebeci, On the Periodic Solutions of Some Systems of HigherOrder Difference Equations, Rocky Mountain J. Math., (2018 to appear).
  • [21] R.P. Agarwal, J. Popenda, Periodic solutions of …rst order linear di¤erenceequations, Mathl. Comput. Modelling, 22(1) (1995), pp. 11-19.
  • [22] R.P. Agarwal, P.J. Wong, Advanced topics in difference equations (Vol.404), Springer Science & Business Media, (2013).
  • [23] S. Elaydi. An Introduction to Difference Equations, Springer-Verlag, NewYork, Inc, 1996.
  • [24] S. Elaydi, R.J. Sacker, Global stability of periodic orbits of non-autonomousdifference equations and population biology, J. Di¤erential Equations,208(1) (2005), pp. 258-273.
  • [25] S. Stevic, Asymptotics of some classes of higher-order differenceequations, Discrete Dyn. Nat. Soc., 2007 (2007), pp. 1-20. doi:http://dx.doi.org/10.1155/2007/56813
  • [26] S. Stevic, J. Diblik, B. Iricanin, Z. Smarda, Z. Solvability of nonlineardifference equations of fourth order, Electron. J. Differential Equations,264 (2014), pp. 1-14.
  • [27] Y. Wang, Y. Luo, Z. Lu, Convergence of solutions of xn+1 = xnxn􀀀1 􀀀 1,Appl. Math. E-Notes, 12 (2012), pp. 153-157.
There are 27 citations in total.

Details

Primary Language English
Journal Section Mathematics
Authors

Erkan Taşdemir

Publication Date June 28, 2019
Submission Date June 26, 2018
Acceptance Date May 26, 2019
Published in Issue Year 2019 Volume: 9 Issue: 1

Cite

APA Taşdemir, E. (2019). On The Dynamics of a Nonlinear Difference Equation. Adıyaman University Journal of Science, 9(1), 190-201.
AMA Taşdemir E. On The Dynamics of a Nonlinear Difference Equation. ADYU J SCI. June 2019;9(1):190-201.
Chicago Taşdemir, Erkan. “On The Dynamics of a Nonlinear Difference Equation”. Adıyaman University Journal of Science 9, no. 1 (June 2019): 190-201.
EndNote Taşdemir E (June 1, 2019) On The Dynamics of a Nonlinear Difference Equation. Adıyaman University Journal of Science 9 1 190–201.
IEEE E. Taşdemir, “On The Dynamics of a Nonlinear Difference Equation”, ADYU J SCI, vol. 9, no. 1, pp. 190–201, 2019.
ISNAD Taşdemir, Erkan. “On The Dynamics of a Nonlinear Difference Equation”. Adıyaman University Journal of Science 9/1 (June 2019), 190-201.
JAMA Taşdemir E. On The Dynamics of a Nonlinear Difference Equation. ADYU J SCI. 2019;9:190–201.
MLA Taşdemir, Erkan. “On The Dynamics of a Nonlinear Difference Equation”. Adıyaman University Journal of Science, vol. 9, no. 1, 2019, pp. 190-01.
Vancouver Taşdemir E. On The Dynamics of a Nonlinear Difference Equation. ADYU J SCI. 2019;9(1):190-201.

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