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Stability of solutions for a family of nonlinear difference equations. (English) Zbl 1146.39020

Summary: We consider the family of nonlinear difference equations:
\[ \begin{split} x_{n+1}=\left.\left(\sum^3_{i=1}f_i(x_n,\dots,x_{n-k})+ f_4(x_n,\dots,x_{n-k})f_5(x_n,\dots,x_{n-k})\right)\right/ \\ \left(f_1(x_n,\dots,x_{n-k})f_2(x_n,\dots,x_{n-k})+ \sum^5_{i=3}f_i(x_n,\dots,x_{n-k})\right),\quad n=0,1,\dots,\end{split} \]
where \(f_i\in C((0,+\infty)^{k+1},(0,+\infty))\), for \(i\in \{1,2,4,5\}\), \(f_3\in C([0,+\infty)^{k+1}\), \((0,+\infty))\), \(k\in \{1,2,\dots\}\) and the initial values \(x_{-k},x_{-k+1},\dots,x_0\in (0,+\infty)\). We give sufficient conditions under which the unique equilibrium \(x=1\) of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references.

MSC:

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

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