Global asymptotic stability for two kinds of higher order recursive sequences

Abstract

In this paper, the method named subsequence analysis is introduced to study the global asymptotic stability of the following two families of difference equations:$$x_n=\frac{x_{n-k}{x}_{n-l}+a}{x_{n-k}+x_{n-l}},n=1,2,\dots ,$$

and$$x_n=\frac{x_{n-k}+x_{n-l}}{1+a{x}_{n-k}{x}_{n-l}},n=1,2,\dots ,$$

where k and l are integers, $1\le k\le l,a>0,x_0,x_{-1},\dots ,x_{-(l-1)}\in (0,\infty ).$ This method is novel and can be used to simplify the proof of similar kinds of stability problems with complicated semi-cycle structure.

Keywords:

• Rational recursive sequence
• subsequence analysis
• global asymptotic stability
• sign chain

Notes

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

Funding

The first author Xun-Yang Wang is supported by the NNSF of China [grant number 11461041], [grant number 51369014]; NSF of Gansu Province [grant number 148RJZA024], as well as the Development Program for HongLiu Outstanding Young Teachers in Lanzhou University of Technology.