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Global attractivity of a higher-order nonlinear difference equation. (English) Zbl 1201.39006

Authors’ abstract: The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation

${x}_{n+1}=\frac{1-{x}_{n}}{A+{x}_{n-k}},\phantom{\rule{1.em}{0ex}}n=0,1,\cdots ,$

where $A\in \left(-\infty ,-1\right),k$ is a positive integer and initial conditions ${x}_{-k},\cdots ,{x}_{0}\in \left(-\infty ,0\right]$. It is shown that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient

##### MSC:
 39A20 Generalized difference equations 39A30 Stability theory (difference equations) 39A22 Growth, boundedness, comparison of solutions (difference equations)
##### References:
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