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Global asymptotic stability and oscillation of a family of difference equations. (English) Zbl 1055.39017
The authors are concerned with the oscillatory behavior of the positive solutions of the family of difference equations \[ x_{n+1}= \Biggl(\sum^k_{\substack{ i=0\\ i\neq j,j-1}} x_{n-i}+ x_{n-j+1} x_{n-j}+ 1\Biggr)\Biggl/\sum^k_{i=0} x_{n-j},\quad j= 1,\dots,k, \] where \(n\in \{0,1,\dots\}\), \(k\in \{1,2,\dots\}\) and the initial values \(x_{-k}, x_{-k+1},\dots, x_0\) are positive numbers. It is also proved that the unique equilibrium \(\overline x= 1\) is globally asymptotically stable.

MSC:
39A11 Stability of difference equations (MSC2000)
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