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On the recursive sequence \(x_{n+1}=A+\frac{f(x_n,\dots,x_{n-k+1)}}{x_{n-k}}\). (English) Zbl 1059.39010

Consider the difference equation \[ x_{n+1}=A+\frac{f(x_n,\dots,x_{n-k+1})}{x_{n-k}}\tag{\(*\)} \] where \(k\) is a fixed positive integer, \(A\in (0,\infty)\) and \(f:\mathbb R_{+}^k\to \mathbb R_{+}\) is a continuous function. The authors investigate the existence and uniqueness of stationary solutions \(x^{*}\) such that \(x^{*}=A+f(x^{*},x^{*},\dots,x^{*})/x^*.\) Using the semicycle analysis about a positive equilibrium, they obtain some conditions for which (\(*\)) admits oscillatory solutions. Several theorems are obtained to guarantee the boundedness and the global attractivity of positive solutions of (\(*\)).
For example, Theorem. Assume that \(A>1\) (the case when \(A\geq 1\) has been considered in another theorem) and \(f: \mathbb R_{+}^k\to \mathbb R_{+}\) is a continuous function such that \(f(y_1,y_2,\dots,y_k)\leq c \max \{y_1,y_2,\dots,y_k\}+b,\) for all \(y_1,y_2,\dots,y_k\in \mathbb{R}_{+},\) for some \(c\in (0,A)\) and \(b\geq 0.\) Then every solution of (\(*\)) is bounded.
The paper contains many references that deal with the properties of nonlinear difference equations.
Reviewer: Fozi Dannan (Doha)

MSC:

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