Cengiz Çinar, Stevo Stevi\'c, Ibrahim Yalçinkaya
A Note on Global Asymptotic Stability of a Family of Rational Equations
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 59, 41-49(2005)

MSC:

39A10 Difference equations, See also {33Dxx}

39A11 Stability of difference equations

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.), See also {30E15}

Abstract: In this note we prove that all positive solutions of the
difference equations
\[
x_{n+1}=\frac{1+x_n\sum_{i=1}^k
x_{n-i}}{x_{n}+x_{n-1}+x_{n}\sum_{i=2}^k x_{n-i}},\quad n=0,1,...,
\]
where $k\in {\bf N},$ converge to the positive equilibrium $\bar
x=1.$ The result generalizes the main theorem in the paper: Li
Xianyi and Zhu Deming, Global asymptotic stability in a rational
equation, {\it J.\ Differ.\ Equations Appl.} {\bf 9} (9), (2003),
833-839. We present a very short proof of the theorem. Also, we
find the asymptotics of some of the positive solutions.
Keywords: rational difference equation, global asymptotic stability, equilibrium point, positive solution, asymptotics