The global attractivity of the rational difference equation \(y_{n}=1+\frac{y_{n-k}}{y_{n-m}}\). (English) Zbl 1109.39004

The authors prove that the solutions of the recursive equation \[ y_n=1+\displaystyle\frac{y_{n-k}}{y_{n-m}},\,\,\,n=0,1,2,\ldots,\eqno(1) \] with \(y_{-s},\,y_{-s+1},\ldots,y_{-1}\in (0,\infty)\), \(k,\,m\in \{1,2,\ldots,\}\), \(s=\text{max}\{k,\,m\}\), \(\text{gcd}(k,m)=1\) and \(k\) odd, converge exponentially to the unique equilibrium solution 2. By using a result due to E. A. Grove and G. Ladas [Periodicities in Nonlinear Difference Equations. Advances in Discrete Mathematics and Applications 4. (Boca Raton,) FL: Chapman & Hall/CRC. (2005; Zbl 1078.39009)], the periodic character of equation (1) is also established.


39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations


Zbl 1078.39009
Full Text: DOI


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