# Thompson’s metric and global stability of difference equations

Thompson’s metric and global stability of difference equations We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation $$y_{n} = \frac{f_{2m+1}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}{f_{2m}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}, \;\;\;\; n = 0,1,2, \ldots,$$ where $${f_{2m+1}^{2m+1}}$$ and $${f_{2m}^{2m+1}}$$ are polynomials in 2m + 1 variables which satisfy certain conditions. Our results include Stević’s (Appl Math Comput 216:179–186, 2010), Zhang et al.’s (Bull Aust Math Soc 81:251–259, 2010) and other related results as special cases. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Thompson’s metric and global stability of difference equations

, Volume 16 (1) – Feb 15, 2011
10 pages

/lp/springer_journal/thompson-s-metric-and-global-stability-of-difference-equations-5xsncABDWh
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0113-0
Publisher site
See Article on Publisher Site

### Abstract

We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation $$y_{n} = \frac{f_{2m+1}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}{f_{2m}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}, \;\;\;\; n = 0,1,2, \ldots,$$ where $${f_{2m+1}^{2m+1}}$$ and $${f_{2m}^{2m+1}}$$ are polynomials in 2m + 1 variables which satisfy certain conditions. Our results include Stević’s (Appl Math Comput 216:179–186, 2010), Zhang et al.’s (Bull Aust Math Soc 81:251–259, 2010) and other related results as special cases.

### Journal

PositivitySpringer Journals

Published: Feb 15, 2011

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