, Volume 38, Issue 1-2, pp 173-180
Date: 30 Dec 2010

Dynamics of the max-type difference equation \(x_{n}=\max\{\frac{ 1}{ x_{n-m}} , \frac{A_{n} }{x_{n-r} }\}\)

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Abstract

In this paper, we study the periodicity, the boundedness and the convergence of the following max-type difference equation $$x_n =\max\biggl\{\frac{ 1}{ x_{n-m}} , \frac{A_n }{x_{n-r} }\biggr \},\quad n =0, 1,2,\ldots,$$ where \(\{A_{n}\}^{+\infty}_{n=0}\) is a periodic sequence with period k and A n ∈(0,1) for every n≥0, m∈{1,2} and r∈{2,3,…} with m<r, the initial values x r ,…,x −1∈(0,+∞). The special case when \(m = 1, r = 2, \{A_{n}\}^{+\infty}_{ n=0}\) is a periodic sequence with period k and A n ∈(0,1) for every n≥0 has been completely investigated by Y. Chen. Here we extend his results to the general case.

Project Supported by NSF of China (10861002) and NSF of Guangxi (2010GXNSFA013106, 2011GXNSFA014781) and SF of Education Department of Guangxi (200911MS212).