ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

On a New Class of Rational Difference Equation zn+1 = azn+bzn−k+(α +β zn−k)/(A+Bzn−k)

Journal of Applied Nonlinear Dynamics 8(4) (2019) 569--584 | DOI:10.5890/JAND.2019.12.005

Abdul Khaliq$^{1}$, Sk. Sarif Hassan$^{2}$

$^{1}$ Mathematics Department, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore, Pakistan

$^{2}$ Department of Mathematics, Pingla Thana Mahavidyalaya, PaschimMedinipur, 721400, West Bengal, India

Abstract

In this article, we will investigate invariant intervals, periodic character, the character of semi cycles and global asymptotic stability of all positive solutions of a nonlinear rational difference equation of order k+1, zn+1 = azn+bzn−k+(α +β zn−k)/(A+Bzn−k), n = 0,1, ..., where the parameters a, b, α,β and A, B and the initial conditions z−k, z−k+1, z−k+2, ..., z0 are arbitrary positive real numbers. We also have studied the global stability of this equation through numerically solved examples and confirm our theoretical discussion through it.

References

1.  [1] Cinar, C. (2004), On the positive solutions of the difference equation xn+1 = (axn−1)/(1+bxnxn−1) , Appl. Math. Comp., 156, 587-590.
2.  [2] Das, S.E. and Bayram, M. (2010), On a system of rational difference equations, World Applied Sciences Journal, 10(11), 1306-1312.
3.  [3] Din, Q. and Elsayed, E.M. (2014), Stability analysis of a discrete ecological model, Computational Ecology and Software, 4(2), 89-103.
4.  [4] Elabbasy, E.M., El-Metwally, H., and Elsayed, E.M. (2007), On the difference equations xn+1 = (αxn−k)/(β+γ Πki =0 xn−i) , J. Conc. Appl. Math., 5(2), 101-113.
5.  [5] Elabbasy, E.M., El-Metwally, H., and Elsayed, E.M. (2007), Qualitative behavior of higher order difference equation, Soochow Journal of Mathematics, 33(4), 861-873.
6.  [6] Neyrameh, A., Neyrameh, H., Ebrahimi, M., and Roozi, A. (2010), Analytic solution diffusivity equation in rational form, World Applied Sciences Journal, 10(7), 764-768.
7.  [7] Elsayed, E.M. (2013), Behavior and expression of the solutions of some rational difference equations, Journal of Computational Analysis and Applications, 15(1), 73-81.
8.  [8] Elsayed, E.M. (2014), Solution for systems of difference equations of rational form of order two, Computational and Applied Mathematics, 33(3), 751-765.
9.  [9] Wang, C., Su, X., Liu, P., and Li, R. (2017), On the dynamics of five order fuzzy difference equation, Journal of Nonlinear Science and Application, 10, 3303-3319.
10.  [10] Elsayed, E.M. and El-Metwally, H. (2015), Global behavior and periodicity of some difference equations, Journal of Computational Analysis and Applications, 19(2), 298-309.
11.  [11] Elsayed, E.M. (2009), Dynamics of a recursive sequence of higher order, Communications on Applied Nonlinear Analysis, 16(2), 37-50.
12.  [12] Elsayed, E.M. (2009), Qualitative behavior of difference equation of order three, Acta Scientiarum Mathematicarum (Szeged), 75(1-2), 113-129.
13.  [13] Karatas, R., Cinar, C., and Simsek, D. (2006), On positive solutions of the difference equation xn+1 = (xn−5)/(1+xn−2xn−5) , Int. J. Contemp. Math. Sci., 1(10), 495-500.
14.  [14] Alseda, L. and Misiurewicz, M. (2011), A note on a rational difference equation, Journal of Difference Equations and Applications, 17(11), 1711-1713.
15.  [15] Sun, T. and Xi, H. (2007), On convergence of the solutions of the difference equation xn+1 = 1+ xn−1/xn , J. Math. Anal. Appl., 325, 1491-1494.
16.  [16] Elabbasy, E.M., El-Metwally, H., and Elsayed, E.M. (2008), On the Difference Equation xn+1 = (a0xn+a1xn−1+...+akxn−k)/(b0xn+b1xn−1+...+bkxn−k), Mathematica Bohemica, 133(2), 133-147.
17.  [17] El-Metwally, H. (2007), Global behavior of an economic model, Chaos, Solitons and Fractals, 33, 994-1005.
18.  [18] El-Metwally, H. and El-Afifi, M.M. (2008), On the behavior of some extension forms of some population models, Chaos, Solitons and Fractals, 36, 104-114.
19.  [19] El-Metwally, H. and Elsayed, E.M. (2013), Form of solutions and periodicity for systems of difference equations, Journal of Computational Analysis and Applications, 15(5), 852-857.
20.  [20] Chen, H. and Wang, H. (2006), Global attractivity of the difference equation xn+1 = (xn+αxn−1)/(β+xn), Appl. Math. Comp., 181, 1431-1438.
21.  [21] Elsayed, E.M. (2009), Qualitative behavior of difference equation of order two, Mathematical and Computer Modelling, 50, 1130-1141.
22.  [22] Kocic, V.L. and Ladas, G. (1993), Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht.
23.  [23] Kulenovic, M.R.S. and Ladas, G. (2001), Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press.
24.  [24] Kurbanli, A.S. (2010), On the behavior of solutions of the system of rational difference equations, World Applied Sciences Journal, 10(11), 1344-1350.
25.  [25] Memarbashi, R. (2008), Sufficient conditions for the exponential stability of nonautonomous difference equations, Appl. Math. Letter, 21, 232-235.
26.  [26] Saleh, M. and Aloqeili, M. (2006), On the difference equation yn+1 = A+ yn/yn−k with A < 0, Appl. Math. Comp., 176(1), 359-363.
27.  [27] Saleh, M. and Baha, S.A. (2006), Dynamics of a higher order rational difference equation, Applied Mathematics and Computation, 181, 84-102.
28.  [28] Saleh, M. and Aloqeili, M. (2005), On the difference equation xn+1 = A+ xn/xn−k , Appl. Math. Comp., 171, 862-869.
29.  [29] Wang, C., Wang, S., Li, L., and Shi, Q. (2009), Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Advances in Difference Equations, 2009(1), Article ID 214309, 8pages.
30.  [30] Yalçınkaya, I. (2008), On the difference equation xn+1 =α + xn−m/xk n , Discrete Dynamics in Nature and Society, 2008, Article ID 805460, 8 pages, DOI: 10.1155/2008/805460.
31.  [31] Ahmed, A.M. and Youssef, A.M. (2013), A solution form of a class of higher-order rational difference equations, J. Egyp. Math. Soc., 21, 248-253.
32.  [32] Aloqeili, M. (2009), Global stability of a rational symmetric difference equation, Appl. Math. Comp., 215, 950-953.
33.  [33] Aprahamian, M., Souroujon, D., and Tersian, S. (2010), Decreasing and fast solutions for a second-order difference equation related to Fisher-Kolmogorov’s equation, J. Math. Anal. Appl., 363, 97-110.
34.  [34] Wang, C., Wang, S., Wang, Z., Gong, H., and Wang, R. (2010), Asymptotic stability for a class of nonlinear difference equation, Discrete Dynamics in Natural and Society, 2010, Article ID 791610, 10pages.
35.  [35] Wang, Q. and Zhang, Q. (2017), Dynamics of a higher-order rational difference equation, Jr. of App. Anal. and Computation, 7(2), 770-787.
36.  [36] Yal¸cınkaya, I. (2008), On the global asymptotic stability of a second-order system of difference equations, Discrete Dynamics in Nature and Society, 2008, Article ID 860152, 12 pages, DOI: 10.1155/2008/860152.
37.  [37] Zayed, E.M.E. and El-Moneam, M.A. (2005), On the rational recursive sequence xn+1 = (α+β xn+γ xn−1)/(A+Bxn+Cxn−1), Communications on Applied Nonlinear Analysis, 12(4), 15-28.
38.  [38] Zayed, E.M.E. and El-Moneam, M.A. (2005), On the rational recursive sequence xn+1 = (αxn+β xn−1+γ xn−2+δ xn−3)/(Axn+Bxn−1+Cxn−2+Dxn−3), Comm. Appl. Nonlinear Analysis, 12, 15-28.
39.  [39] Zhang, B.G., Tian, C.J. and Wong, P.J. (1999), Global attractivity of difference equations with variable delay, Dyn. Contin. Discrete Impuls. Syst., 6(3), 307-317.
40.  [40] El-Dessoky, M.M. (2016), Dynamics and Behavior of xn+1 = axn +bxn−1+ (α+cxn−2)/(β+dxn−2),J. Comp. Ana. and App., 24(4), 644-655.