The Rate of Convergence of a Certain Mixed Monotone Difference Equation
DOI:
https://doi.org/10.5644/SJM.20.01.11Keywords:
difference equations, rate of convergence, stability, stable manifoldAbstract
This paper investigates the rate of convergence of a certain mixed monotone rational second-order difference equation with quadratic terms. More precisely we give the precise rate of convergence for all attractors of the difference equation $x_{n+1}=\frac{Ax_{n}^{2}+Ex_{n-1}}{x_{n}^{2}+f}$, where all parameters are positive and initial conditions are non-negative.
The mentioned methods are illustrated in several characteristic examples.
2020 Mathematics Subject Classification. 39A10, 39A20, 65L20.
Downloads
References
R. P. Agarwall and M. Pituk, Asymptotic expansions for higher-order scalar difference equations, Adv. Difference Equ. 2007, Art. ID 67492, 12 pp.
L. J. S. Allen, An Introduction to Mathematical Biology, Prentice Hall, (2006).
S. Bodine and D. A. Lutz, Asymptotic solutions and error estimates for linear systems of difference and differential equations, J. Math. Anal. Appl., 290 (2004), 343–362.
S. Bodine and D. A. Lutz, Exponentially asymptotically constant systems of difference equations with an application to hyperbolic equilibria, J. Difference Equ. Appl., 15 (2009), 821–832.
A. Brett and M. R. S. Kulenović, Basins of Attraction of Equilibrium Points of Monotone Difference Equations, Sarajevo J. Math., 5(2009), 211–233.
H. Caswell, 2001, Matrix Population Models: Construction, Analysis and lnterpretation. 2nd ed. Sinauer Assoc. Inc., Sunderland, Mass. (2001)
M. Garić-Demirović, M. R. S. Kulenović, and M. Nurkanović, Basins of Attraction of Equilibrium Points of Second Order Difference Equations, Appl. Math. Letters, 25(2012), 2110–2115.
S. Elaydi, Asymptotics for linear difference equations I. Basic theory J. Difference Equ. Appl., 5 (1999), pp. 563–589.
S. Elaydi, An Introduction to Difference Equations, 3rd ed., Undergraduate Text in Mathematics, Springer, New York, 2005.
W. Jamieson and O. Merino, Asymptotic behavior results for solutions to some nonlinear difference equations. J. Math. Anal. Appl. 430 (2015), no. 2, 614–632.
S. Jašarević Hrustić, M. R. S. Kulenović and M. Nurkanović, Local Dynamics and Global Stability of Certain Second-Order Rational Difference Equation with Quadratic Terms, Discrete Dyn. Nat. Soc., Volume 2016, Article ID 3716042, 14 pages.
S. Kalabušić and M. R. S. Kulenović, Rate of Convergence of Solutions of Rational Difference equation of Second Order, Adv. Difference Equ., (2002), 121–139.
U. Krause, A theorem of Poincare type for non-autonomous nonlinear difference equations S. Elaydi, I. Gyori, G. Ladas (Eds.), Advances in Difference Equations – Proceedings of the Second International Conference on Difference Equations, Veszprem, 1995, Gordon and Breach Publ., Amsterdam (1997), pp. 107–117.
M. R. S. Kulenović, M. Nurkanović and Z. Nurkanović, Global Stability of Certain Mix Monotone Difference Equation via Center Manifold Theory and Theory of Monotone maps, Sarajevo J. Math., Vol. 15 (28), No.2, (2019), 129-154.
M. R. S. Kulenović and Z. Nurkanović, The Rate of Convergence of Solution of a ThreeDimmensional Linear Fractional System Difference Equations, Zbornik radova PMF Tuzla Svezak matematika, 2 (2005), 1-6.
M. R. S. Kulenović and A.-A. Yakubu, Compensatory versus Overcompensatory Dynamics in Density-dependent Leslie Models, J. Difference Equations Appl. 10 (2004), 1251–1266.
L. Lorentzen, Asymptotic behavior of solutions of three-term Poincare difference equations, Rocky Mountain J. Math., 1 (1997), pp. 187–229.
H. Matsunaga and S. Murakami, Asymptotic behavior of solutions of functional difference equations, J. Math. Anal. Appl., 305 (2005), pp. 391–410.
R. Obaya and M. Pituk, A variant of the Krein–Rutman theorem for Poincare difference equations, J. Difference Equ. Appl., 18 (2012), pp. 1751–1762.
M. Pituk, Asymptotic behavior of a Poincare difference equation J. Difference Equ. Appl., 3 (1997), pp. 33–53.
M. Pituk, More on Poincare’s theorems for difference equations, J. Difference Equ. Appl. 8 (3) (2002), 201–216.
U. Ufuktepe and S. Kapc¸ak, Applications of Discrete Dynamical Systems with Mathematica, Conference: RIMS, Vol. 1909, 2014.
G. G. Thomson, ”A proposal for a treshold stock size and maximum fishing mortality rate”, in Risk Evaluation and Biological Reference Points for Fisheries Management, Canad. Spec. Publ. Fish. Aquat. Sci, S.J. Smith, J.J. Hunt, and D. Rivard, Eds., vol.120, pp. 303-320, NCR Research Press, Ottawa Canada, 1993.
