Analysis and qualitative behaviour of a tenth-order rational difference equation
DOI :
https://doi.org/10.5269/bspm.64858Résumé
In this article, we examine the qualitative behavior of the solutions
of the following di¤erence equation
zn+1 = aZn-4 +bZn-4/cZn-4-dzn4; n = 0,1,....
where the initial conditions Z_9; Z_8; Z_7; Z_6; Z_5; Z_4; Z_3; Z_2; Z_1;Z0 are arbitrary non-zero real numbers and a, b, c, d are positive constants.
Références
[1] R. Abo-Zeid and C. Cinar, Global Behavior of The Difference Equation xn+1 =Axn-1/(B – Cxnxn-2), Boletim da Sociedade Paranaense de Matematica, 2013, 31(1), 43.49.
[2] R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Advanced Studies in Contemporary Mathematics, 2008, 17(2), 181.201.
[3] M. B. Almatra., E. M. Elsayed, E. and F. Alzahrani, Qualitative behavior of two rational difference equations.Fundamental Journal of Mathematics and Applications,2018, 1(2), 194- 204.
[4] M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp.,176(2)(2006), 768-774.
[5] F. Belhannache, Asymptotic stability of a higher order rational difference equation, Electronic Journal of Mathematical Analysis and Applications, 2019, 7(2),1.8.
[6] C. Cinar, On The Positive Solutions of The Difference Equation xn+1 = axnô€€€1 1+bxnxnô€€€1 ; Applied Mathematics and Computation, 2004, 156, 587.590.
[7] C. Cinar, On the difference equation xn+1= xnô€€€1 -1+xnxnô€€€1 , Applied Mathematicsand Computation, 158(2004), 813-816.
[8] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Qualitative behavior of higherorder difference equation, Soochow Journal of Mathematics, 33(4)(2007), 861-873.
[9] E. M. Elabbasy and E. M. Elsayed, On the solutions of a class of difference equations of higher order, International Journal of Mathematics and Statistics, 6(A09)(2010),57-68.
[10] E. M. Elabbasy and E. M. Elsayed, Global Attractivity and Periodic Nature of a Difference Equation, World Applied Sciences Journal, 2011, 12(1), 39.47.
[11] M. M. El-Dessoky and M. El-Moneam, On The Higher Order Difference Equation xn+1 = Axn+ Bxn-1 + Cxn-k + (xn-k)/(Dxn-s + Exn-t), Journal of Computational Analysis and Applications, 2018, 25(2), 342.354.
[12] H. El-Metwally and M. M. El-A.., On the behavior of some extension forms of some population models, Chaos, Solitons and Fractals, 36(2008), 104-114.
[13] H. El-Metwally and E. M. Elsayed, Solution and Behavior of a Third Rational Difference Equation, Utilitas Mathematica, 2012, 88, 27.42.
[14] M. A. El-Moneam, and E. M. E. Zayed, Dynamics of the rational difference equation. Information Sciences Letters, (2014), 3(2), 45-53.
[15] M. Elsayed, Dynamics of a Recursive Sequence of Higher Order, Communications on Applied Nonlinear Analysis, 2009, 16(2), 37.50.
[16] E. M. Elsayed, A. Alghamdi, Dynamics and Global Stability of Higher OrderNonlinear Difference Equation, Journal of Computational Analysis and Applications, 2016, 21(3), 493.50.
[17] E. M. Elsayed, Qualitative behavior of s rational recursive sequence, Indagationes Mathematicae, New Series, 19(2),(2008), 189-201.
[18] E. M. Elsayed and F. Alzahrani, Periodicity and solutions of some rationaldifference equations systems, Journal of Applied Analysis and Computation, 2019, 9(6), 2358.2380.
[19] E. M. Elsayed, On the Global attractivity and the solution of recursive sequence, Studia Scientiarum Mathematicarum Hungarica, 47(3),(2010), 401-418.
[20] E. M. Elsayed, F. Alzahrani, I. Abbas and N. H. Alotaibi, Dynamical Behavior and Solution of Nonlinear Difference Equation Via Fibonacci Sequence, Journal of Applied Analysis and Computation, 2020, 10(1), 282.296.
[21] E. M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta Applicandae Mathematicae, 110(2)(2010), 589-604.
[22] E. M. Elsayed, F. Alzahrani and H. S. Alayachi, Formulas and Properties of some Class of Nonlinear Difference Equation, Journal of Computational Analysis and Applications, 2018, 4(1), 141.155.
[23] E.M. Elsayed, Dynamics of Recursive Sequence of Order Two, Kyungpook Math. J. 50(2010), 483-497.
[24] M.Ghazel, E. M. Elsayed, A. E. Matouk and A. M. Mousallam, Investigating dynamical behaviors of the difference dquation xn+1 = Cxn-5/(A + Bxn-2xn-5). Journal of Nonlinear Sciences and Applications,(2017), 10, 4662-4679.
[25] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall / CRC Press, 2005.
[26] M. Gumus, Global Dynamics of Solutions of A New Class of Rational Difference Equations, Konuralp Journal of Mathematics, 2019, 7(2), 380.387.
[27] S.S. Hassan, and A. Khaliq, Dynamics of a rational difference equation xn+1 = axn+ (a + bxn-k)/(A + Bxn-k). International Journal of Advances in Mathematics, 2018(1), 159-179.
[28] T. F. Ibrahim, Generalized partial ToDD.s difference equation in n-dimensional space, Journal of Computational Analysis and Applications, 2019, 26(5), 910.926.
[29] T. F. Ibrahim, On The Third Order Rational Difference Equation xn+1 = (xnxn-2)/(xn-1(a+bxnxn-2)), International Journal of Contemporary Mathematical Sciences, 2009, 4(27), 1321.1334.
[30] R. Karatas, Global Behavior of a Higher Order Difference Equation, International Journal of Contemporary Mathematical Sciences, 2017, 12(3), 133.138.
[31] A. Khaliq, F. Alzahrani and E. M. Elsayed, Global Attractivity of a Rational Difference Equation of Order Ten, Journal of Nonlinear Sciences and Applications, 2016, 9, 4465.4477.
[32] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
[33] Y.Kostrov, and Z. Kudlak, On a second-order rational difference equation with a quadratic term. International Journal of Difference Equations, 11(2), (2016),179-202.
[34] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001.
[35] I. Okumus and Y. Soykan, On the Solutions of Four Second-Order Nonlinear Difference Equations, Universal Journal of Mathematics and Applications, 2019, 2(3), 116.125.
[36] M. Saleh and S. Abu-Baha, Dynamics of a higher order rational difference equation, Appl. Math. Comp., 181(2006), 84-102.
[37] M. Saleh and M. Aloqeili, On The Difference Equation yn+1 = A + yn/yn-k , Applied Mathematics and Computation, 2006, 176(1), 359.363.
[38] S. Sadiq and M. Kalim, Global attractivity of a rational difference equation of order twenty, International Journal of Advanced and Applied Sciences, 2018, 5(2), 1.7.
[39] D. T. Tollu and ·I. Yalçinkaya, Global behavior of a three-dimensional system of difference equations of order three, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 2019, 68(1), 1.16
[40] Y. Yazlik and M. Kara, On a solvable system of difference equations of higherorder with period two coefficients, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 2019, 68(2), 1675.1693.
[2] R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Advanced Studies in Contemporary Mathematics, 2008, 17(2), 181.201.
[3] M. B. Almatra., E. M. Elsayed, E. and F. Alzahrani, Qualitative behavior of two rational difference equations.Fundamental Journal of Mathematics and Applications,2018, 1(2), 194- 204.
[4] M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp.,176(2)(2006), 768-774.
[5] F. Belhannache, Asymptotic stability of a higher order rational difference equation, Electronic Journal of Mathematical Analysis and Applications, 2019, 7(2),1.8.
[6] C. Cinar, On The Positive Solutions of The Difference Equation xn+1 = axnô€€€1 1+bxnxnô€€€1 ; Applied Mathematics and Computation, 2004, 156, 587.590.
[7] C. Cinar, On the difference equation xn+1= xnô€€€1 -1+xnxnô€€€1 , Applied Mathematicsand Computation, 158(2004), 813-816.
[8] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Qualitative behavior of higherorder difference equation, Soochow Journal of Mathematics, 33(4)(2007), 861-873.
[9] E. M. Elabbasy and E. M. Elsayed, On the solutions of a class of difference equations of higher order, International Journal of Mathematics and Statistics, 6(A09)(2010),57-68.
[10] E. M. Elabbasy and E. M. Elsayed, Global Attractivity and Periodic Nature of a Difference Equation, World Applied Sciences Journal, 2011, 12(1), 39.47.
[11] M. M. El-Dessoky and M. El-Moneam, On The Higher Order Difference Equation xn+1 = Axn+ Bxn-1 + Cxn-k + (xn-k)/(Dxn-s + Exn-t), Journal of Computational Analysis and Applications, 2018, 25(2), 342.354.
[12] H. El-Metwally and M. M. El-A.., On the behavior of some extension forms of some population models, Chaos, Solitons and Fractals, 36(2008), 104-114.
[13] H. El-Metwally and E. M. Elsayed, Solution and Behavior of a Third Rational Difference Equation, Utilitas Mathematica, 2012, 88, 27.42.
[14] M. A. El-Moneam, and E. M. E. Zayed, Dynamics of the rational difference equation. Information Sciences Letters, (2014), 3(2), 45-53.
[15] M. Elsayed, Dynamics of a Recursive Sequence of Higher Order, Communications on Applied Nonlinear Analysis, 2009, 16(2), 37.50.
[16] E. M. Elsayed, A. Alghamdi, Dynamics and Global Stability of Higher OrderNonlinear Difference Equation, Journal of Computational Analysis and Applications, 2016, 21(3), 493.50.
[17] E. M. Elsayed, Qualitative behavior of s rational recursive sequence, Indagationes Mathematicae, New Series, 19(2),(2008), 189-201.
[18] E. M. Elsayed and F. Alzahrani, Periodicity and solutions of some rationaldifference equations systems, Journal of Applied Analysis and Computation, 2019, 9(6), 2358.2380.
[19] E. M. Elsayed, On the Global attractivity and the solution of recursive sequence, Studia Scientiarum Mathematicarum Hungarica, 47(3),(2010), 401-418.
[20] E. M. Elsayed, F. Alzahrani, I. Abbas and N. H. Alotaibi, Dynamical Behavior and Solution of Nonlinear Difference Equation Via Fibonacci Sequence, Journal of Applied Analysis and Computation, 2020, 10(1), 282.296.
[21] E. M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta Applicandae Mathematicae, 110(2)(2010), 589-604.
[22] E. M. Elsayed, F. Alzahrani and H. S. Alayachi, Formulas and Properties of some Class of Nonlinear Difference Equation, Journal of Computational Analysis and Applications, 2018, 4(1), 141.155.
[23] E.M. Elsayed, Dynamics of Recursive Sequence of Order Two, Kyungpook Math. J. 50(2010), 483-497.
[24] M.Ghazel, E. M. Elsayed, A. E. Matouk and A. M. Mousallam, Investigating dynamical behaviors of the difference dquation xn+1 = Cxn-5/(A + Bxn-2xn-5). Journal of Nonlinear Sciences and Applications,(2017), 10, 4662-4679.
[25] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall / CRC Press, 2005.
[26] M. Gumus, Global Dynamics of Solutions of A New Class of Rational Difference Equations, Konuralp Journal of Mathematics, 2019, 7(2), 380.387.
[27] S.S. Hassan, and A. Khaliq, Dynamics of a rational difference equation xn+1 = axn+ (a + bxn-k)/(A + Bxn-k). International Journal of Advances in Mathematics, 2018(1), 159-179.
[28] T. F. Ibrahim, Generalized partial ToDD.s difference equation in n-dimensional space, Journal of Computational Analysis and Applications, 2019, 26(5), 910.926.
[29] T. F. Ibrahim, On The Third Order Rational Difference Equation xn+1 = (xnxn-2)/(xn-1(a+bxnxn-2)), International Journal of Contemporary Mathematical Sciences, 2009, 4(27), 1321.1334.
[30] R. Karatas, Global Behavior of a Higher Order Difference Equation, International Journal of Contemporary Mathematical Sciences, 2017, 12(3), 133.138.
[31] A. Khaliq, F. Alzahrani and E. M. Elsayed, Global Attractivity of a Rational Difference Equation of Order Ten, Journal of Nonlinear Sciences and Applications, 2016, 9, 4465.4477.
[32] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
[33] Y.Kostrov, and Z. Kudlak, On a second-order rational difference equation with a quadratic term. International Journal of Difference Equations, 11(2), (2016),179-202.
[34] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001.
[35] I. Okumus and Y. Soykan, On the Solutions of Four Second-Order Nonlinear Difference Equations, Universal Journal of Mathematics and Applications, 2019, 2(3), 116.125.
[36] M. Saleh and S. Abu-Baha, Dynamics of a higher order rational difference equation, Appl. Math. Comp., 181(2006), 84-102.
[37] M. Saleh and M. Aloqeili, On The Difference Equation yn+1 = A + yn/yn-k , Applied Mathematics and Computation, 2006, 176(1), 359.363.
[38] S. Sadiq and M. Kalim, Global attractivity of a rational difference equation of order twenty, International Journal of Advanced and Applied Sciences, 2018, 5(2), 1.7.
[39] D. T. Tollu and ·I. Yalçinkaya, Global behavior of a three-dimensional system of difference equations of order three, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 2019, 68(1), 1.16
[40] Y. Yazlik and M. Kara, On a solvable system of difference equations of higherorder with period two coefficients, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 2019, 68(2), 1675.1693.
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Publié
2024-05-03
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Research Articles
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