On the periodicity of a max-type rational difference equation
-
2422
Downloads
-
4940
Views
Authors
Changyou Wang
- School of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, P. R. China.
- College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.
Xiaotong Jing
- College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.
Xiaohong Hu
- College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.
Rui Li
- College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China.
Abstract
This paper shows that every well-defined solution of the following max-type difference equation
\[{x_{n + 1}} = \max \{ \frac{A}{{{x_n}}},\,\frac{A}{{{x_{n - 1}}}},\,{x_{n - 2}}\} ,\quad n \in {N_0},\]
where \(A \in R\) and the initial conditions \({x_{ - 2}},\,{x_{ - 1}},\,{x_0}\) are arbitrary non-zero real numbers is eventually periodic with period three by using new iteration method for the more general nonlinear difference equations and inequality skills as well as the mathematical induction. Our main results considerably improve results appearing in the literature.
Share and Cite
ISRP Style
Changyou Wang, Xiaotong Jing, Xiaohong Hu, Rui Li, On the periodicity of a max-type rational difference equation, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4648--4661
AMA Style
Wang Changyou, Jing Xiaotong, Hu Xiaohong, Li Rui, On the periodicity of a max-type rational difference equation. J. Nonlinear Sci. Appl. (2017); 10(9):4648--4661
Chicago/Turabian Style
Wang, Changyou, Jing, Xiaotong, Hu, Xiaohong, Li, Rui. "On the periodicity of a max-type rational difference equation." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4648--4661
Keywords
- Max-type
- difference equation
- positive solution
- periodic solution.
MSC
References
-
[1]
M. M. El-Dessoky , On the periodicity of solutions of max-type difference equation , Math. Methods Appl. Sci., 38 (2015), 3295–3307.
-
[2]
E. M. Elsayed, B. Iričanin, S. Stević, On the max-type equation \(x_{n+1} = max\{\frac{A_n}{x_n} , x_{n-1}\}\) , Ars Combin., 95 (2010), 187–192.
-
[3]
E. M. Elsayed, S. Stević, On the max-type equation \(x_{n+1} = max\{\frac{A}{x_n} , x_{n-2}\}\), Nonlinear Anal., 71 (2009), 910–922.
-
[4]
A. Gelişken, C. Çinar , On the global attractivity of a max-type difference equation , Discrete Dyn. Nat. Soc., 2009 (2009), 5 pages.
-
[5]
T. F. Ibrahim, N. Touafek, Max-type system of difference equations with positive two-periodic sequences, Math. Methods Appl. Sci., 37 (2014), 2541–2553.
-
[6]
B. D. Iričanin, E. M. Elsayed , On the max-type difference equation \(x_{n+1} = max\{\frac{A}{x_n} , x_{n-3}\}\) , Discrete Dyn. Nat. Soc., 2010 (2010), 13 pages.
-
[7]
W. T. Jamieson, O. Merino , Asymptotic behavior results for solutions to some nonlinear difference equations, J. Math. Anal. Appl., 430 (2015), 614–632.
-
[8]
D. Jana, E. M. Elsayed, Interplay between strong Allee effect, harvesting and hydra effect of a single population discrete-time system, Int. J. Biomath., 9 (2016), 25 pages.
-
[9]
L.-J. Kong , Homoclinic solutions for a second order difference equation with p-Laplacian , Appl. Math. Comput., 247 (2014), 1113–1121.
-
[10]
W.-T. Li, Y.-H. Zhang, Y.-H. Su, Global attractivity in a class of higher-order nonlinear difference equation , Acta Math. Sci. Ser. B Engl. Ed., 25 (2005), 59–66.
-
[11]
W.-P. Liu, S. Stević, Global attractivity of a family of nonautonomous max-type difference equations, Appl. Math. Comput., 218 (2012), 6297–6303.
-
[12]
J. E. Macías-Díaz, A positive finite-difference model in the computational simulation of complex biological film models, J. Difference Equ. Appl., 20 (2014), 548–569.
-
[13]
J. Migda , Approximative solutions to difference equations of neutral type, Appl. Math. Comput., 268 (2015), 763–774.
-
[14]
D. P. Mishev, W. T. Patula, H. D. Voulov , A reciprocal difference equation with maximum, Comput. Math. Appl., 43 (2002), 1021–1026.
-
[15]
A. D. Myškis, Some problems in the theory of differential equations with deviating argument, (Russian) Uspehi Mat. Nauk, 32 (1977), 173–202.
-
[16]
I. Niven, H. S. Zuckerman, H. L. Montgomery , An introduction to the theory of numbers, Fifth edition, John Wiley & Sons, Inc., New York (1991)
-
[17]
W. T. Patula, H. D. Voulov, On a max type recurrence relation with periodic coefficients, J. Differ. Equ. Appl., 10 (2004), 329–338.
-
[18]
E. C. Pielou, Population and community ecology: principles and methods, Gordon and Breach Science Publishers, Inc., New York (1974)
-
[19]
E. P. Popov, Automatic regulation and control, (Russian) Nauka, Moscow (1966)
-
[20]
B. Qin, T.-X. Sun, H.-J. Xi , Dynamics of the max-type difference equation \(x_{n+1} = max\{\frac{A}{x_n} , x_{n-k}\}\) , J. Comput. Anal. Appl., 14 (2012), 856–861.
-
[21]
M. Shojaei, R. Saadati, H. Adibi , Stability and periodic character of a rational third order difference equation , Chaos Solitons Fractals, 39 (2009), 1203–1209.
-
[22]
S. Stević , On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl., 376 (2011), 317–328.
-
[23]
S. Stević, On a symmetric system of max-type difference equations, Appl. Math. Comput., 219 (2013), 8407–8412.
-
[24]
T.-X. Sun, Q.-L. He, X. Wu, H.-J. Xi , Global behavior of the max-type difference equation \(x_n = max \{\frac{1}{x_{n-m}} , \frac{A_n}{x_{n-r}}\}\) , Appl. Math. Comput., 248 (2014), 687–692.
-
[25]
Q. Xiao, Q.-H. Shi, Eventually periodic solutions of a max-type equation, Math. Comput. Modelling, 57 (2013), 992–996.