On the periodicity of a max-type rational difference equation

Wang, Changyou ¹ ; Jing, Xiaotong ² ; Hu, Xiaohong ² ; Li, Rui ³

¹ 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
² College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China
³ College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 9, p. 4648-4661.

Résumé

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.

Détail

DOI : 10.22436/jnsa.010.09.08

Classification : 39A10
Keywords: Max-type, difference equation, positive solution, periodic solution.

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